tag:blogger.com,1999:blog-36248392726905055932024-02-20T06:42:43.926-08:00SonantometryMusic as math discipline. Discuss please the ideas concerning tonal functions of higher order, which are developing here.Mykhaylo Khramovhttp://www.blogger.com/profile/00483331671036933638noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-3624839272690505593.post-23052125093144281382007-06-05T02:29:00.000-07:002007-07-25T08:02:33.257-07:00<P align=center><FONT face="Times New Roman"></P> <P align=center><FONT face="Times New Roman" size=5><B>Sonantometry. </B></FONT><FONT face=Verdana size=2> </FONT></P> <P align=center><FONT face="Times New Roman" size=3><B>© 2007 Mykhaylo Khramov.</B> </FONT><FONT face=Verdana size=3> </FONT></P> <P align=center><FONT face="Times New Roman" size=3><B> The article is writing during the </B></FONT><A href="http://geocities.com/commator/" target=_blank><FONT face="Times New Roman" color=#800080 size=3><B><U>project COMMATOR.</U></B></FONT></A><FONT face="Times New Roman" size=3><B> </B> </FONT><FONT face=Verdana size=3> </FONT></P> <P align=center><FONT face="Times New Roman" size=3><B>The last modifications are introduced on Jule, 25, 2007.</B> </FONT><FONT face=Verdana size=3> </FONT></P> <P align=center><FONT face="Times New Roman" size=3><B>The English translation of </B></FONT><A href="http://docs.google.com/Doc?id=dhj4pnk_1f3d2z2" target=_blank><FONT face="Times New Roman" color=#800080 size=3><B><U>the original text</U></B></FONT></A><FONT face="Times New Roman" size=3><B> has performed </B></FONT><A href="http://geocities.com/kmy180753/" target=_blank><FONT face="Times New Roman" color=#800080 size=3><B><U>Mykhaylo Khramov.</U></B></FONT></A><FONT face="Times New Roman" size=3> </FONT><FONT face=Verdana size=3> </FONT></P> <P align=center><FONT face="Times New Roman" size=3>In offered notes you'll find developing of </FONT><A href="http://www.sibeliusmusic.com/cgi-bin/show_score.pl?scoreid=53047" target=_blank><FONT face="Times New Roman" color=#000080 size=3><U>ideas concerning tonal functions of higher order</U></FONT></A><FONT face="Times New Roman" size=3>. Is talking on </FONT><A href="http://www.sibeliusmusic.com/cgi-bin/show_score.pl?scoreid=96007" target=_blank><FONT face="Times New Roman" color=#000080 size=3><U>practice of application these tonal functions as sonants</U></FONT></A><FONT face="Times New Roman" size=3> to the analysis and synthesis of music. </FONT><FONT face=Verdana size=3> </FONT></P> <P><FONT face="Times New Roman" size=3>Sonantometry, or the </FONT><A href="http://docs.google.com/Doc?id=dhj4pnk_16fkptr2" target=_blank><FONT face="Times New Roman" color=#000080 size=3><U>algebra of tonal functions</U></FONT></A><FONT face="Times New Roman" size=3>, updates and expands possibilities of the functional analysis and synthesis of music. Example of my own experience in application of a subject of article is available in the file: <A title="R. Wagner's Tristan opening. 7-limit 53EDO approximation." href="http://commons.wikimedia.org/wiki/Image:Wagner_Tristan_opening_by_Khramov53e7L.jpg" target=_blank>R. Wagner's Tristan opening. 7-limit 53EDO approximation.</A> Also <A title="MIDI - model of this fragment sounding." href="http://commons.wikimedia.org/wiki/Image:Wagner_Tristan_opening_by_Khramov53e7L.mid" target=_blank>the </A><A title="MIDI - model of this fragment sounding." href="http://commons.wikimedia.org/wiki/Image:Wagner_Tristan_opening_by_Khramov53e7L.mid" target=_blank>M</A><A title="MIDI - model of this fragment sounding." href="http://commons.wikimedia.org/wiki/Image:Wagner_Tristan_opening_by_Khramov53e7L.mid" target=_blank>IDI - model of this fragment sounding</A> is given. </FONT><FONT face=Verdana size=3> </FONT></P> <P><FONT face="Times New Roman" size=3>The following table can explain the essence of sonantometry: </FONT><FONT face=Verdana size=3> </FONT></P></FONT> <DIV style="MARGIN: 1ex"><FONT size=3> <CENTER> <TABLE cellSpacing=0 width=0 border=2> <TBODY> <TR vAlign=top> <TD width="37%" colSpan=4 height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>5-limit Just Intonation Diatonic Scale </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="25%" colSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>Tonal Function </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="37%" colSpan=4> <P align=center><FONT face="Times New Roman"><FONT size=2>Sonant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="10%" colSpan=2 height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>Grade </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="26%" colSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>Frequency Ratio of Grade and 1st Grade </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%" rowSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>Name </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%" rowSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>Symbol </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="18%" colSpan=3> <P align=center><FONT face="Times New Roman"><FONT size=2>Formulas by Order of Affinity to Tonant </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="18%" rowSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>Name </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>No. </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Symbol </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Fraction </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Factorization </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>0 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>2 </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>8 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VIII ~ I </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman"><FONT size=2>2/1 ~ 1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>2<SUP>1 </SUP>~ 1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Tonic </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:Dd </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="18%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Tonant and Dedant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>7 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VII<</FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman"><FONT size=2>15/8~15 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>1</SUP>×3<SUP>1</SUP>×2<SUP>-3 </SUP>~ 5<SUP>1</SUP>×3<SUP>1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Leading</FONT><FONT size=3> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:MD or :DM </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="18%"> <P align=center><FONT face="Times New Roman"><FONT size=2>MeDent or DeMent </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>6 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VI<</FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5/3 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>1</SUP>×3<SUP>-1</SUP>×2<SUP>0 </SUP>~ 5<SUP>1</SUP>×3<SUP>-1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Submediant </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:Md or :dM </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="18%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Medant or daMent </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>5 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>V </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3/2~3 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3<SUP>1</SUP>×2<SUP>-1 </SUP>~ 3<SUP>1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Dominant </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>D </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:D </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="18%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Dominant or Dent </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>4 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>IV </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman"><FONT size=2>4/3~1/3 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3<SUP>-1</SUP>×2<SUP>2 </SUP>~ 3<SUP>-1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Subdominant </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>S </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:d </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="18%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Subdominant or dant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>3 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>III<</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5/4~5 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>1</SUP>×2<SUP>-2 </SUP>~ 5<SUP>1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Mediant </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>M </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:M </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="18%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Mediant or Ment </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>2 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>II </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman"><FONT size=2>9/8~9 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3<SUP>2</SUP>×2<SUP>-3 </SUP>~ 3<SUP>2</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Double Dominant </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>DD or 2D </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:2D or :DD </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="18%"> <P align=center><FONT face="Times New Roman"><FONT size=2>TwoDent or DeDent </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>I ~ VIII </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman"><FONT size=2>1/1=1 ~ 2/1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>2<SUP>0</SUP>=1 ~ 2<SUP>1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Tonic </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:dD </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="18%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Tonant and daDent </FONT><FONT size=2> </FONT></FONT></P></TD></TR></TBODY></TABLE></CENTER> <P><FONT face="Times New Roman"> <P><FONT size=3></P> <P><FONT face="Times New Roman" size=3>This table shows relations among well-known 5-limit just intonation major diatonic scale, also well-known tonal functions and those ones that are named here sonants and underlies discipline named here sonantometry. </FONT><FONT face=Verdana size=3> </FONT></P> <P><FONT face="Times New Roman" size=3>The comparison of grades VIII and I indicates matching by them of the same function T, or tonic. In other words these grades are functionally equivalent, and it is possible to write VIII ~ I and I ~ VIII. The frequency ratios of grades point thus, that 2 ~ 1 and all degrees of a prime 2 in factorizations of frequency ratios of all grades it is possible to replace by 1 and not to show. It is clear, that to tonic corresponds number 1, just to this number corresponds the first entered here sonant :T (to pronounce [Te]) named Tonant. The sonant formula :T has the zero order of affinity to Tonant, as this order is equal to a sum of absolute values of exponents of odd prime bases in a factorization and for :T is equal 0. </FONT><FONT face=Verdana size=2> </FONT></P> <P><FONT face="Times New Roman" size=3>A grades V and IV are matching the functions D and S, or dominant and subdominant. It is easy to notice, that to the name 'dominant' corresponds number 3<SUP>1</SUP>, and the prefix 'sub' indicates only change of a sign of exponent of this number. In sonantometry it is fixed by that anyone sonant :S assumes availability of own subsonant :s. Therefore the function dominant corresponds to sonant :D (to pronounce [De]), with the name of Dominant or Dent. Its subsonant :d (to pronounce [da]), with the name of subdominant or dant corresponds to subdominant function. The formula :S in sonantometry designates anyone possible sonant with its subsonant :s, and the subdominant function corresponds to subsonant :d of the sonant :D. From a factorization it is clear, that Dent and dant has the order of affinity to Tonant equal to 1. </FONT><FONT face=Verdana size=2> </FONT></P> <P><FONT face="Times New Roman" size=3>In 5-limit just intonation major diatonic scale there is one more sonant of the order 1. It corresponds to number 5<SUP>1</SUP> and mediant function that matches grade III<. The sign "<" indicates flat pitch bend of a Pythagorean scale grade, and will be described further, together with the sharp pitch bend sign ">". Sonant :M (to pronounce [Me]), with the name Mediant or Ment, and its subsonant :m (to pronounce [ma]), with the name submediant or mant are last of possible sonants of the order 1 in 5-limit just intonation major diatonic scale. In considered diatonic scale there is no mant and submediant function is compared to other sonant. However harmonic version of the 5-limit just intonation major diatonic scale substitutes the grade VI< by the grade VIb> (8/5 ~ 5-1) and the correspondence of submediant to mant is restored. </FONT><FONT face=Verdana size=2> </FONT></P> <P><FONT face="Times New Roman" size=3>The existence of the function with the name double dominant helps to understand the essence of complex sonants. These are all sonants of the order 2 and above. Similarly to a double dominant, or dominant from dominant, anyone sonant of the order 2 exists as prime sonant of the order 1 from adopted instead of Tonant other sonant, which prime to Tonant by affinity. It is necessary to notice, that subsonant of the own sonant is equal to Tonant, and the change of the places of prime sonants in the formulas of the complex ones does not cause other sonant. </FONT><FONT face=Verdana size=2> </FONT></P> <P><FONT face="Times New Roman" size=3>In given diatonic scale exist complex sonants :2D or :DD, :MD or :DM, :Dd or :dD, :Md or :dM. Their names and correspondence to grades are reflected in a table. It is clear, that diatonic scale envelops not all possible sonants of 5-limit and order of 2 just intonation. The full set of these sonants is reflected in a table below. </FONT><FONT face=Verdana size=2> </FONT></P></FONT> <P> </P></FONT><A name=0.1_table02></A> <CENTER> <TABLE cellSpacing=0 width=0 border=2> <TBODY> <TR vAlign=top> <TD width="47%" colSpan=4 height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>5-limit and Order of 2 Just Intonation Scale </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="52%" colSpan=4> <P align=center><FONT face="Times New Roman"><FONT size=2>Sonant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="14%" colSpan=2 height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>Grade </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="33%" colSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>Frequency Ratio of Grade and 1st Grade </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="20%" colSpan=3> <P align=center><FONT face="Times New Roman"><FONT size=2>Formulas by Order of Affinity to Tonant </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%" rowSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>Name </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>No. </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Symbol </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Fraction </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Factorization </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>0 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>2 </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>14 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VIII ~ I </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>2/1 ~ 1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>2<SUP>1</SUP>~1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:Dd and :Mm </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Tonant and Dedant and Medant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>13 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VII< </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>15/8 ~ 15 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>1</SUP>×3<SUP>1</SUP>×2<SUP>-3 </SUP>~ 5<SUP>1</SUP>×3<SUP>1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:MD or :DM </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>MeDent or DeMent </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>12 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VIIb> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>16/9 ~ 1/9 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3<SUP>-2</SUP>×2<SUP>4 </SUP>~ 3<SUP>-2</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:2d or :dd </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Twodant or dadant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>11 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VI<</FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5/3 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>1</SUP>×3<SUP>-1</SUP>×2<SUP>0 </SUP>~ 5<SUP>1</SUP>×3<SUP>-1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:Md or :dM </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Medant or daMent </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>10 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VIb></FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>8/5 ~ 1/5 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>-1</SUP>×2<SUP>3 </SUP>~ 5<SUP>-1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:m </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Submediant or mant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>9 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>V#( </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>25/16 ~ 25 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>2</SUP>×2<SUP>-4 </SUP>~ 5<SUP>2</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:2M or :MM </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>TwoMent or MeMent </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>8 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>V </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3/2 ~ 3 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3<SUP>1</SUP>×2<SUP>-1 </SUP>~ 3<SUP>1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:D </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Dominant or Dent </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>7 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>IV </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>4/3 ~ 1/3 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3<SUP>-1</SUP>×2<SUP>2 </SUP>~ 3<SUP>-1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:d </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Subdominant or dant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>6 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>IVb) </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>32/25 ~ 1/25 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>-2</SUP>×2<SUP>5 </SUP>~ 5<SUP>-2</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:2m or :mm </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Twomant or mamant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>5 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>III< </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5/4 ~ 5 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>1</SUP>×2<SUP>-2 </SUP>~ 5<SUP>1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:M </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Mediant or Ment </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>4 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>IIIb></FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>6/5 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>-1</SUP>×3<SUP>1</SUP>×2<SUP>1 </SUP>~ 5<SUP>-1</SUP>×3<SUP>1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:mD or :Dm </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>maDent or Demant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>3 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>II </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>9/8 ~ 9 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3<SUP>2</SUP>×2<SUP>-3 </SUP>~ 3<SUP>2</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:2D or :DD </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>TwoDent or DeDent </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>2 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>IIb< </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>16/15 ~ 1/15 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5<SUP>-1</SUP>×3<SUP>-1</SUP>×2<SUP>4 </SUP>~ 5<SUP>-1</SUP>×3<SUP>-1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:md or :dm </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>madant or damant </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="4%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman"><FONT size=2>I ~ VIII </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="13%"> <P align=center><FONT face="Times New Roman"><FONT size=2>1/1=1 ~ 2/1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="19%"> <P align=center><FONT face="Times New Roman"><FONT size=2>2<SUP>0</SUP>=1 ~ 2<SUP>1</SUP> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="14%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:dD and :mM </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="31%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Tonant and daDent and maMent </FONT><FONT size=2> </FONT></FONT></P></TD></TR></TBODY></TABLE></CENTER> <P><FONT face="Times New Roman"> <P><FONT size=3> <FONT face="Times New Roman" size=3>At the beginning of clearing up the essence of signs ">", "<", ")", "(", let’s recollect, that the scale of 5-limit just intonation can be expressed through a 3-limit just intonation scale. The last one has a name of Pythagor, and to this day underlies the musical notation and concept 'fifth spiral', that through tuning of the 12 equal divisions of the octave (12 EDO) is degenerated in 'fifth circle'. In the Pythagorean scale, as 3-limit just intonation scale, we shall find universal sonant :T, sonants :D and :d, which are prime to it, and also all complex ones which it possible to combine from :T, :D and :d. Other sonants in this scale are impossible. </FONT><FONT face=Verdana size=3> </FONT></P> <P><FONT face="Times New Roman" size=3>In the following table the scale of 5-limit just intonation (sonants :T, :D, :d, :M, :m and their combination) is expressed through interval sonants (intersonants) to Pythagorean grades:</FONT><FONT face=Verdana size=2> </FONT></P></FONT> <P> </P></FONT><A name=0.1_table03></A> <DIV align=center> <CENTER> <TABLE cellSpacing=0 width=0 border=2> <TBODY> <TR vAlign=top> <TD width="39%" colSpan=10 height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>Pythagorean 3-limit and Order of 8 Just Intonation Scale </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="60%" colSpan=7> <P align=center><FONT face="Times New Roman"><FONT size=2>5-limit and Order of 2 Just Intonation Scale </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="10%" colSpan=2 height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>Grade </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="28%" colSpan=8> <P align=center><FONT face="Times New Roman"><FONT size=2>Sonants by Order of Affinity to Tonant </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="12%" colSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>Grade </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="14%" colSpan=3> <P align=center><FONT face="Times New Roman"><FONT size=2>Sonants by Order of Affinity to Tonant </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="33%" colSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>Pitch Bend in Intersonants by Order of Affinity to Pythagorean Scale Grades. </FONT><FONT size=2> </FONT></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>№ </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman"><FONT size=2>Symbol </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>0 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2>2 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2>3 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2>4 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2>5 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>… </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>8 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD 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</FONT></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VIII ~ I </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>14 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman"><FONT size=2>VIII ~ I </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Times 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size=2>III< </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:M </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>5 </FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman"><FONT size=2>IVb </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:8d </FONT><FONT size=2> 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size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>4 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman"><FONT size=2>IIIb> </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:mD </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%" rowSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>:Δ[3d = :m4D[3d 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align=center><FONT face="Times New Roman"><FONT size=2>2 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman"><FONT size=2>IIb<</FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:md </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="16%" rowSpan=2> <P align=center><FONT face="Times New Roman"><FONT size=2>:Δ[5d = :m4D[5d </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="17%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD></TR> <TR 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</FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="17%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman"><FONT size=2>1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman"><FONT size=2>I ~ VIII </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman"><FONT size=2>1 </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman"><FONT size=2>I ~ VIII </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2>:T </FONT><FONT size=2> </FONT></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </P></TD> <TD width="17%"><FONT face="Times New Roman"><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=2> </FONT><FONT size=3> </FONT></FONT> </TD></TR></TBODY></TABLE></CENTER></DIV></FONT></DIV> <P><FONT face="Times New Roman" size=3>How it is possible to notice, the scales exactly coincide in grades I, II, IV, V, VIIb. In other grades we find a fifth order divergence with maFourDent or its inverse MeFourdant. The grades IVb and V# with TwomaEightDent and its inverse TwoMeEightdant, both tenth order. Here is given a simple rule to determine intersonant between any pair of sonants. It is enough to write first sonant, to add to it all elements of the second, all elements of the inverse of the second, and to factor out the elements of the second, as origin. The required intersonant remains before a bracket.</FONT><FONT face=Verdana size=2> </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:md = :md 5d 5D = :md5D[5d = :m4D[5d; </FONT><FONT face=Verdana size=2> </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:mD = :mD 3d 3D = :mD3D[3d = :m4D[3d; </FONT><FONT face=Verdana size=2> </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:M = :M 4D 4d = :M4d[4D; </FONT><FONT face=Verdana size=2> </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:2m = :2m 8d 8D = 2m8D[8d; </FONT><FONT face=Verdana size=2> </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:2M = :2M 8D 8d = 2M8d[8D; </FONT><FONT face=Verdana size=2> </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:m = :m 4d 4D = :m4D[4d; </FONT><FONT face=Verdana size=2> </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:Md = :Md 3D 3d = :Md3d[3D = :M4d[3D; </FONT><FONT face=Verdana size=2> </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:MD = :MD 5D 5d = :MD5d[5D = :M4d[5D. </FONT><FONT face=Verdana size=2> </FONT></P> <P><FONT face="Times New Roman" size=3>Restoring numerical values of the intersonant :m4D by the formula 5<SUP>-1</SUP>×3<SUP>4</SUP>×2<SUP><B>Z</B></SUP>, <B>Z</B> ={-n,…, 0,… n}, n </FONT><FONT face=Symbol size=3><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol">Î </SPAN></FONT><FONT face="Times New Roman" size=3><B>N</B>,<B> N </B>= {1, 2, 3,…, n} gives a set of the octave equivalents of 81/80 fraction and sharply bends a grade pitch on a comma of Didymus. Inverse sonant :M4d (the set of the octave equivalents of 80/81 fraction) flatly bends a grade pitch on a comma of Didymus. Therefore it is possible to name :m4D as Didiment, with designation :Δ. Thus an inversion we shall name didimant, and designate :δ.</FONT><FONT face=Verdana size=2> </FONT></P> <P><FONT face="Times New Roman" size=3>The table shows, that ">" sharply bends a grade pitch on Didiment, and "<" flatly. Sign ")" order TwoDidiment sharp, and "(" means Twodidimant flat.</FONT><FONT face=Verdana size=2> </FONT></P> <P><FONT face="Times New Roman"><FONT size=3>Let's remark, if the note sonant include:</FONT> </FONT></P> <P align=center><FONT face="Times New Roman"><FONT size=3>M, then arises "<" and the grade is flattened by the intersonant </FONT><FONT size=3>:δ</FONT><FONT size=3>;</FONT> </FONT></P> <P align=center><FONT face="Times New Roman"><FONT size=3>2M, arises "(" and the grade is flattened by the intersonant :2</FONT><FONT size=3>δ</FONT><FONT size=3>;</FONT> </FONT></P> <P align=center><FONT face="Times New Roman"><FONT size=3>m, arises ">" and the grade is sharpened by the intersonant :</FONT><FONT size=3>Δ</FONT><FONT size=3>;</FONT> </FONT></P> <P align=center><FONT face="Times New Roman"><FONT size=3>2m, arises ")" and the grade is sharpened by the intersonant :2</FONT><FONT size=3>Δ</FONT><FONT size=3>.</FONT> </FONT></P> <P><FONT face="Times New Roman"><FONT size=3>In other words, in 5-limit just intonation, the pitch bend of a Pithagorean grade, arises only at the notes containing M, or m. The size of bend, accordingly is exactly so many intersonants :</FONT><FONT size=3>δ</FONT><FONT size=3> or</FONT><FONT size=3> </FONT><FONT size=3>:</FONT><FONT size=3>Δ</FONT><FONT size=3>, how many M, or m are in the note make-up.</FONT> </FONT></P> <P><FONT face="Times New Roman"><FONT size=3>This observation gives the following, consisting of two phases, rule of placing of additional accidentals for a writing notes of the 5-limit just intonation in scores of Pithagorean scale:</FONT> </FONT></P> <OL type=1> <LI><FONT face="Times New Roman"><FONT size=3>to appropriate to each note head proper to it sonant depending on a score context;</FONT> </FONT> <LI><FONT face="Times New Roman"><FONT size=3>if in sonant are present M, 2M, 3M, to bend a pitch of the note accordingly by :</FONT><FONT size=3>δ</FONT><FONT size=3>,:2</FONT><FONT size=3>δ</FONT><FONT size=3>,:3</FONT><FONT size=3>δ</FONT><FONT size=3>, and to designate these flattening accordingly by additional accidentals "<", "(", "{". The presence m, 2m, 3m requires bends accordingly by :</FONT><FONT size=3>Δ</FONT><FONT size=3>,:2</FONT><FONT size=3>Δ</FONT><FONT size=3>,:3</FONT><FONT size=3>Δ</FONT><FONT size=3>, and designating of a sharping accordingly by additional accidentals ">", ")" and "}".</FONT> </FONT></LI></OL>Mykhaylo Khramovhttp://www.blogger.com/profile/00483331671036933638noreply@blogger.com5tag:blogger.com,1999:blog-3624839272690505593.post-34541393361714417922007-05-30T06:52:00.000-07:002007-06-05T05:02:27.578-07:00<P align=center> <FONT size=5><B>Algebra of Tonal Functions.</B></FONT> </P> <P class=western align=center><BR></P> <P class=western align=center><B><FONT color=#000000>© 2005 Gennady Wohl.</FONT></B> </P> <P class=western align=center><BR></P> <P class=western align=center><EM><B><SPAN style="FONT-STYLE: normal">The article is written for </SPAN></B></EM><A href="http://geocities.com/commator/"><B><FONT color=#999999>project COMMATOR.</FONT></B></A> </P> <P class=western align=center><BR></P> <P class=western align=center><B><FONT color=#000000>The English translation of </FONT></B><A title="the original text" href="http://sonantometria.blogspot.com/2007/05/blog-post_30.html"><B><FONT color=#999999>the original text</FONT></B></A><B><FONT color=#000000> has performed </FONT></B><A href="http://geocities.com/kmy180753/"><B><FONT color=#999999>Mykhaylo Khramov.</FONT></B></A> </P> <P class=western align=center> </P> <P class=western align=center><B>Translator thanks Gennady Wohl, who kindly has agreed to explain mathematical aspects of a theme, offered to him.</B> </P> <P class=western align=center><B>Translator also thanks Vadim Bondar for the help and will be glad to any criticism which improve translation quality.</B> </P> <P class=western align=center><BR></P> <P class=western align=center><FONT color=#000000>In offered notes you'll find brief description of </FONT><A title="M. Y. Khramov's ideas concerning tonal functions of higher order" href="http://sonantometry.blogspot.com/2007/06/sonantometry.html" target=_blank><FONT color=#999999>M. Y. Khramov's ideas concerning tonal functions of higher order</FONT></A><FONT color=#000000>. Also included elementary mathematical formulas concerning formal part of the problem of finding functions corresponding to different tones as well as formulas for counting number of tonal functions.</FONT> </P> <P class=western align=center><BR></P> <P class=western align=center><BR></P> <P class=western><B>1. Informal description of tonal functions.</B> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>Let's assume that in some key <I>T</I> means tonic, <I>D</I> - dominant, <I>М</I> - mediant, <I>d</I> - subdominant, <I>m</I> - submediant. (In conventional scale <I>T, M, d, D, m </I>functions correspond to I, III, IV, V, VI degrees). If frequency <I>T</I> conditionally equals 1, then frequencies <I>D, d, M, m </I>are equal accordingly to 3/2, 4/3, 5/4, 8/5 (within one octave). When transposing a tone one octave up or down its function does not change; therefore, the frequencies corresponding to <I>T, D, d, M, m</I> can be multiplied or divided by 2. Rejecting factor of 2 we find that <I>T, D, d, M, m</I> correspond to numbers 1, 3, 1/3, 5, 1/5.</FONT> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>Let </FONT><I>f</I><SUB>1<I> </I></SUB><FONT color=#000000>be some function; the tone to which </FONT><I>f</I><SUB>1</SUB><FONT color=#000000> corresponds we'll assume to be tonic; and in the new key the tone to which function </FONT><I>f</I><SUB>2</SUB><FONT color=#000000> corresponds we'll assign in relation to initial tonal function </FONT><I>f</I><SUB>1</SUB><I>f</I><SUB>2</SUB><FONT color=#000000>. Obviously, </FONT><I>f</I><SUB>1</SUB><I>f</I><SUB>2</SUB><FONT color=#000000>=</FONT><I>f</I><SUB>2</SUB><I>f</I><SUB>1</SUB><FONT color=#000000>. The frequency corresponding to function </FONT><I>f</I><SUB>1</SUB><I>f</I><SUB>2</SUB><FONT color=#000000>, is obtained by multiplication of frequencies of functions </FONT><I>f</I><SUB>1</SUB><FONT color=#000000> and </FONT><I>f</I><SUB>2</SUB><FONT color=#000000>. For example, <I>DD</I> corresponds to frequency 3·3=9; <I>MMD </I>- frequency 5·5·3=75 and so on. Instead of <I>DD, </I></FONT><I>MMD</I><FONT color=#000000> we will write 2<I>D</I>, 2<I>MD</I> and so on.</FONT> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>Let's note, that <I>Dd</I>=<I>Mm</I>=<I>T</I>; and corresponding numerical equations are 3·(1/3)=1 and 5·(1/5)=1.</FONT> </P> <P class=western><BR></P> <P class=western><B>2. Formal description of tonal functions.</B> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>Let's consider free Abelian group </FONT><FONT face="Symbol, serif">F</FONT><FONT color=#000000> with two generators <I>D</I> and <I>M</I>; the unit of group we shall designate as <I>T</I>. Let </FONT><I>d=D</I><SUP>-1</SUP>, <I>m=M</I><SUP>-1</SUP><FONT color=#000000>. The element </FONT><I>f</I><FONT face="Symbol, serif">ÎF</FONT> <FONT color=#000000>we'll name a tonal function; <I>f </I>is uniquely described as </FONT><I>D</I><FONT face="Symbol, serif"><SUP>a</SUP></FONT><I>M</I><FONT face="Symbol, serif"><SUP>b</SUP></FONT>, <FONT color=#000000>where </FONT><FONT face="Symbol, serif">a</FONT> <FONT color=#000000>and </FONT><FONT face="Symbol, serif">b</FONT><FONT color=#000000> are integers.</FONT> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>In the multiplicative group </FONT><I>Q<SUP>+</SUP></I> of <FONT color=#000000>positive rational numbers let's look at the subgroup <I>G</I> generated by numbers 3 and 5. Let's define isomorphism </FONT><FONT face="Symbol, serif">F@</FONT><I>G</I><FONT color=#000000> by setting it with generators: </FONT><FONT face="Symbol, serif">j</FONT>(<I>D</I>)=3; <FONT face="Symbol, serif">j</FONT>(<I>M</I>)=5;<FONT color=#000000> thereby, </FONT><FONT face="Symbol, serif">j</FONT><I>(D</I><FONT face="Symbol, serif"><SUP>a</SUP></FONT><I>M</I><FONT face="Symbol, serif"><SUP>b</SUP></FONT><I>)</I>=3<FONT face="Symbol, serif"><SUP>a</SUP></FONT>5<FONT face="Symbol, serif"><SUP>b</SUP></FONT>. <FONT color=#000000>For each function <I>f</I> we'll define a set of frequencies </FONT><FONT face="Symbol, serif">n</FONT>(<I>f</I>)={2<I><SUP>S</SUP></I><FONT face="Symbol, serif">j</FONT>(<I>f</I>)<FONT face="Symbol, serif">n</FONT><SUB>0</SUB>},<FONT color=#000000> where <I>S</I> runs through integers. (Frequency </FONT><FONT face="Symbol, serif">n</FONT><SUB>0</SUB><FONT color=#000000> thereby corresponds to <I>T</I>). Besides we'll define for each function </FONT><I>f</I>=<I>D</I><FONT face="Symbol, serif"><SUP>a</SUP></FONT><I>M</I><FONT face="Symbol, serif"><SUP>b</SUP></FONT> <FONT color=#000000>an order </FONT><I>h(f)</I>=|<FONT face="Symbol, serif">a</FONT>|+|<FONT face="Symbol, serif">b</FONT>|. <FONT color=#000000>There is single function of the order <I>h</I>=0, namely <I>T</I>; under <I>h</I>>0 there are 4<I>h </I>functions of the order <I>h</I>.</FONT> </P> <P class=western><BR></P> <P class=western>Examples: </P> <P class=western><I>h</I>=1<FONT color=#000000>; functions </FONT><I>D</I>, <I>M</I>, <I>D</I><SUP>-1</SUP>=<I>d</I>, <I>M</I><SUP>-1</SUP>=<I>m</I><FONT color=#000000>.</FONT> </P> <P class=western><I>h</I>=2<FONT color=#000000>; functions </FONT><I>D</I><SUP>2</SUP>,<I> M</I><SUP>2</SUP>,<I> DM</I>,<I> D</I><SUP>-2</SUP>,<I> M</I><SUP>-2</SUP>,<I> D</I><SUP>-1</SUP><I>M</I><SUP>-1</SUP>,<I> DM</I><SUP>-1</SUP>,<I> D<SUP>-1</SUP>M</I><FONT color=#000000>; or using previous notation of p.1: </FONT>2<I>D</I>, 2<I>M</I>, <I>MD</I>, 2<I>d</I>, 2<I>m</I>, <I>md</I>, <I>mD</I>, <I>Md</I><FONT color=#000000>.</FONT> </P> <P class=western><FONT color=#000000>Under <I>h</I> = 3, 4, 5 there are accordingly 12, 16 and 20 functions.</FONT> </P> <P class=western><BR></P> <P class=western><B>3. 53-tone Equal Temperament.</B> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>Let </FONT><FONT face="Symbol, serif">n</FONT><SUB>0</SUB> <FONT color=#000000>be frequency of the tonic; let's divide an octave (i.e. the sound range between </FONT><FONT face="Symbol, serif">n</FONT><SUB>0</SUB> <FONT color=#000000>and </FONT>2<FONT face="Symbol, serif">n</FONT><SUB>0</SUB><FONT color=#000000>) by 53 equal parts. Let's enumerate degrees of a given scale from 0 (corresponds to </FONT><FONT face="Symbol, serif">n</FONT><SUB>0</SUB><FONT color=#000000>) to 53 (corresponds to </FONT>2<FONT face="Symbol, serif">n</FONT><SUB>0</SUB><FONT color=#000000>). The frequency of a degree number </FONT><I>k</I><FONT color=#000000> is equal to </FONT><FONT face="Symbol, serif">n</FONT><SUB>0</SUB>·2<SUP><I>k</I>/53</SUP><FONT color=#000000>.</FONT> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>The choice of number 53 has to do with the fact that </FONT>(3/2)<SUP>53</SUP>/2<SUP>31</SUP><FONT color=#000000> = 1,002…, i.e. 53 perfect fifths give "almost closed" scale (please note, that for 12-tone temperament </FONT>(3/2)<SUP>12</SUP>/2<SUP>7</SUP>=1,01…<FONT color=#000000>). Besides, 53 degrees enable us to reproduce many other scales in use rather precisely.</FONT> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>We'll note that numbers 12 and 53 have a deeper mathematical meaning: they are denominators of convergents from decomposition of number </FONT>log<SUB>2</SUB>3 <FONT color=#000000>into continued fraction; really, l</FONT>og<SUB>2</SUB>3=[1;1,1,2,2,3,1,5,…]; <FONT color=#000000>it is easy to see that 5-th and 7-th convergents accordingly are 19/12 and 84/53. </FONT></P> <P class=western><BR></P> <P class=western><B>4. Matching Tonal Functions to 53-tone Equal Temperament Degrees. Motivations.</B> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>When interpreting musical pieces from conventional notation to 53-tone tempered scale we are confronted with ambiguity of choosing one degree among 53 that corresponds to the given note. However this ambiguity disappears if we assign certain function to the tone designated by note. For example, if tone corresponds to mediant <I>M</I>, than its frequency 5/4=1.25 is closest to frequency of 17-th degree </FONT>2<SUP>17./53</SUP>=1.24898…, <FONT color=#000000>and so it is the 17-th degree of the 53-tone scale that matches given tone.</FONT> </P> <P class=western><BR></P> <P class=western><FONT color=#000000>The informal part of the problem (and probably a very complicated one) consists of the development of rules based on which from a specific musical context we may assign this or that function to a tone. Thus if tone corresponds to function </FONT>2<I>MD</I><FONT color=#000000> with frequency </FONT>5<SUP>2</SUP>·3=75 <FONT color=#000000>or, within limits of one octave, 75/64 = 1.171875, it is closest to frequency of the 12-th tone of 53-degree temperament: </FONT>2<SUP>12/53</SUP>=1.169924…; <FONT color=#000000>however, difficulty lies just in developing <U>musical</U> motivation which will allow to assign function </FONT>2<I>MD</I><FONT color=#000000> to the given tone. </FONT></P> <P class=western><BR></P> <P class=western><B>5. Matching Tonal Functions to 53-tone Equal Temperament Degrees. Formal Procedure.</B> </P> <P class=western><BR></P> <P class=western><FONT color=#000000></FONT>Let <I>f</I>=<I>D</I><FONT face="Symbol, serif"><SUP>a</SUP></FONT><I>M</I><FONT face="Symbol, serif"><SUP>b</SUP></FONT>; <I>f</I><FONT face="Symbol, serif">¹</FONT><I>T</I>. Function <I>f</I> corresponds to frequency set <I><FONT face="Symbol, serif">n</FONT>(f)</I>={2<I><SUP>S</SUP></I>3<FONT face="Symbol, serif"><SUP>a</SUP></FONT>5<FONT face="Symbol, serif"><SUP>b</SUP></FONT>}, where <I>S</I> runs through all integers. (we suppose that <FONT face="Symbol, serif">n</FONT><SUB>0</SUB> conditionally equals 1). An inequality 1<2<I><SUP>S</SUP></I>3<FONT face="Symbol, serif"><SUP>a</SUP></FONT>5<FONT face="Symbol, serif"><SUP>b</SUP></FONT><2 (i.e. the choice of frequency within limits of an octave) uniquely defines <I>S</I>. If number <I>k</I> is a degree of 53-tone temperament corresponding to function <I>f</I>, it'll be defined from a condition of greatest proximity between numbers 2<SUP><I>k</I>/53</SUP> and 2<I><SUP>S</SUP></I>3<FONT face="Symbol, serif"><SUP>a</SUP></FONT>5<FONT face="Symbol, serif"><SUP>b</SUP></FONT>. Since log<SUB>2</SUB>(2<I><SUP>S</SUP></I>3<FONT face="Symbol, serif"><SUP>a</SUP></FONT>5<FONT face="Symbol, serif"><SUP>b</SUP></FONT>)=<FONT face="Symbol, serif">a</FONT>log<SUB>2</SUB>3+<FONT face="Symbol, serif">b</FONT>log<SUB>2</SUB>5+<I>S</I><FONT face="Symbol, serif">Î</FONT>(0,1), then log<SUB>2</SUB>(2<I><SUP>S</SUP></I>3<FONT face="Symbol, serif"><SUP>a</SUP></FONT>5<FONT face="Symbol, serif"><SUP>b</SUP></FONT>)={<FONT face="Symbol, serif">a</FONT>log<SUB>2</SUB>3+<FONT face="Symbol, serif">b</FONT>log<SUB>2</SUB>5}, where {<I>x</I>} designates fractional part of a number <I>x</I>. Thereby, <I>k</I>/53 should be close to {<FONT face="Symbol, serif">a</FONT>log<SUB>2</SUB>3+<FONT face="Symbol, serif">b</FONT>log<SUB>2</SUB>5} or, that same, <I>k</I> should be close to a number 53{<FONT face="Symbol, serif">a</FONT>log<SUB>2</SUB>3+<FONT face="Symbol, serif">b</FONT>log<SUB>2</SUB>5}. In other words, degree <I>k </I>of 53-tone temperament corresponding to function <I>D</I><FONT face="Symbol, serif"><SUP>a</SUP></FONT><I>M</I><FONT face="Symbol, serif"><SUP>b</SUP></FONT> is the nearest integer to number 53{<FONT face="Symbol, serif">a</FONT>log<SUB>2</SUB>3+<FONT face="Symbol, serif">b</FONT>log<SUB>2</SUB>5}. Designating nearest integer to <I>x</I> numeral as [<I>x</I>]<SUB>1</SUB> we shall obtain: <I>k</I>=[53{<FONT face="Symbol, serif">a</FONT>log<SUB>2</SUB>3+<FONT face="Symbol, serif">b</FONT>log<SUB>2</SUB>5}]<SUB>1</SUB> </P> <P><BR></P> <P><FONT face="Arial, sans-serif"><FONT size=3>Example: </FONT></FONT></P> <P><BR></P> <P class=western><I>f</I>=2<I>MD</I>=<I>D</I><SUP>1</SUP><I>M</I><SUP>2</SUP>; <FONT face="Symbol, serif">a</FONT>=1; <FONT face="Symbol, serif">b</FONT>=2; </P> <P class=western>log<SUB>2</SUB>3+2log<SUB>2</SUB>5=2.2288186…; </P> <P class=western>53·{log<SUB>2</SUB>3+2log<SUB>2</SUB>5}=53·0.2288186=12.12739…; </P> <P class=western><I>k</I>=12. </P> <P class=western><BR></P> <P class=western>Here is a table of functions of the order <I>h</I><FONT face="Symbol, serif">£</FONT>5 and their corresponding degrees. </P> <P class=western><BR></P> <CENTER> <TABLE borderColor=#000000 cellSpacing=0 cellPadding=1 rules=groups width=541 border=1> <COLGROUP> <COL width=29> <COL width=30> <COL width=30> <COL width=30> <COL width=52></COLGROUP> <COLGROUP> <COL width=30> <COL width=30> <COL width=30> <COL width=30> <COL width=52></COLGROUP> <COLGROUP> <COL width=29> <COL width=29> <COL width=29> <COL width=29> <COL width=52></COLGROUP> <TBODY> <TR> <TD width=29 height=15> <P class=western align=center><I>H</I> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a</FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">b</FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD> <TD width=52> <P class=western align=center><I>f</I> </P></TD> <TD width=30> <P class=western align=center><I>h</I> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a</FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">b</FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD> <TD width=52> <P class=western align=center><I>f</I> </P></TD> <TD width=29> <P class=western align=center><I>h</I> </P></TD> <TD width=29> <P class=western align=center><FONT face="Symbol, serif">a</FONT> </P></TD> <TD width=29> <P class=western align=center><FONT face="Symbol, serif">b</FONT> </P></TD> <TD width=29> <P class=western align=center><I>k</I> </P></TD> <TD width=52> <P class=western align=center><I>f</I> </P></TD></TR></TBODY> <TBODY> <TR> <TD width=29 height=16> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=52> <P class=western align=center><I>T</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>12 </P></TD> <TD width=52> <P class=western align=center>2<I>MD</I> </P></TD> <TD width=29> <P class=western align=center>5 </P></TD> <TD width=29> <P class=western align=center>-4 </P></TD> <TD width=29> <P class=western align=center>-1 </P></TD> <TD width=29> <P class=western align=center>18 </P></TD> <TD width=52> <P class=western align=center><I>m</I>4<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>22 </P></TD> <TD width=52> <P class=western align=center><I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>45 </P></TD> <TD width=52> <P class=western align=center><I>m</I>2<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-4 </P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=29> <P class=western align=center>52 </P></TD> <TD width=52> <P class=western align=center><I>M</I>4<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>36 </P></TD> <TD width=52> <P class=western align=center><I>m</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>26 </P></TD> <TD width=52> <P class=western align=center><I>M</I>2<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-3 </P></TD> <TD width=29> <P class=western align=center>-2 </P></TD> <TD width=29> <P class=western align=center>32 </P></TD> <TD width=52> <P class=western align=center><I>2m</I>3<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>17 </P></TD> <TD width=52> <P class=western align=center><I>M</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>40 </P></TD> <TD width=52> <P class=western align=center>3<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-3 </P></TD> <TD width=29> <P class=western align=center>2 </P></TD> <TD width=29> <P class=western align=center>47 </P></TD> <TD width=52> <P class=western align=center><I>2M</I>3<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>31 </P></TD> <TD width=52> <P class=western align=center><I>D</I> </P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=30> <P class=western align=center>-4 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>35 </P></TD> <TD width=52> <P class=western align=center>4<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-2 </P></TD> <TD width=29> <P class=western align=center>-3 </P></TD> <TD width=29> <P class=western align=center>46 </P></TD> <TD width=52> <P class=western align=center>3<I>m</I>2<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>44 </P></TD> <TD width=52> <P class=western align=center>2<I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>49 </P></TD> <TD width=52> <P class=western align=center><I>m</I>3<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-2 </P></TD> <TD width=29> <P class=western align=center>3 </P></TD> <TD width=29> <P class=western align=center>42 </P></TD> <TD width=52> <P class=western align=center>3<I>M</I>2<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>5 </P></TD> <TD width=52> <P class=western align=center><I>md</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>30 </P></TD> <TD width=52> <P class=western align=center><I>M</I>3<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-1 </P></TD> <TD width=29> <P class=western align=center>-4 </P></TD> <TD width=29> <P class=western align=center>7 </P></TD> <TD width=52> <P class=western align=center>4<I>md</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>39 </P></TD> <TD width=52> <P class=western align=center><I>Md</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>10 </P></TD> <TD width=52> <P class=western align=center>2<I>m</I>2<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-1 </P></TD> <TD width=29> <P class=western align=center>4 </P></TD> <TD width=29> <P class=western align=center>37 </P></TD> <TD width=52> <P class=western align=center>4<I>Md</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>19 </P></TD> <TD width=52> <P class=western align=center>2<I>m</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>25 </P></TD> <TD width=52> <P class=western align=center>2<I>M</I>2<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>0 </P></TD> <TD width=29> <P class=western align=center>-5 </P></TD> <TD width=29> <P class=western align=center>21 </P></TD> <TD width=52> <P class=western align=center>5<I>m</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>34 </P></TD> <TD width=52> <P class=western align=center>2<I>M</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>24 </P></TD> <TD width=52> <P class=western align=center>3<I>md</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>0 </P></TD> <TD width=29> <P class=western align=center>5 </P></TD> <TD width=29> <P class=western align=center>32 </P></TD> <TD width=52> <P class=western align=center>5<I>M</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>14 </P></TD> <TD width=52> <P class=western align=center><I>mD</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>20 </P></TD> <TD width=52> <P class=western align=center>3<I>Md</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=29> <P class=western align=center>-4 </P></TD> <TD width=29> <P class=western align=center>16 </P></TD> <TD width=52> <P class=western align=center>4<I>mD</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>48 </P></TD> <TD width=52> <P class=western align=center><I>MD</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-4 </P></TD> <TD width=30> <P class=western align=center>38 </P></TD> <TD width=52> <P class=western align=center>4<I>m</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=29> <P class=western align=center>4 </P></TD> <TD width=29> <P class=western align=center>46 </P></TD> <TD width=52> <P class=western align=center>4<I>MD</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>9 </P></TD> <TD width=52> <P class=western align=center>2<I>D</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=30> <P class=western align=center>15 </P></TD> <TD width=52> <P class=western align=center>4<I>M</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>2 </P></TD> <TD width=29> <P class=western align=center>-3 </P></TD> <TD width=29> <P class=western align=center>11 </P></TD> <TD width=52> <P class=western align=center>3<I>m</I>2<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>13 </P></TD> <TD width=52> <P class=western align=center>3<I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>33 </P></TD> <TD width=52> <P class=western align=center>3<I>mD</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>2 </P></TD> <TD width=29> <P class=western align=center>3 </P></TD> <TD width=29> <P class=western align=center>7 </P></TD> <TD width=52> <P class=western align=center>3<I>M</I>2<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>27 </P></TD> <TD width=52> <P class=western align=center><I>m</I>2<I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>29 </P></TD> <TD width=52> <P class=western align=center>3<I>MD</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>3 </P></TD> <TD width=29> <P class=western align=center>-2 </P></TD> <TD width=29> <P class=western align=center>6 </P></TD> <TD width=52> <P class=western align=center>2<I>m</I>3<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>8 </P></TD> <TD width=52> <P class=western align=center><I>M</I>2<I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>28 </P></TD> <TD width=52> <P class=western align=center>2<I>m</I>2<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>3 </P></TD> <TD width=29> <P class=western align=center>2 </P></TD> <TD width=29> <P class=western align=center>21 </P></TD> <TD width=52> <P class=western align=center>2<I>M3D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>41 </P></TD> <TD width=52> <P class=western align=center>2<I>md</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>43 </P></TD> <TD width=52> <P class=western align=center>2<I>M</I>2<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>4 </P></TD> <TD width=29> <P class=western align=center>-1 </P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=52> <P class=western align=center><I>m</I>4<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=52> <P class=western align=center>2<I>Md</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>23 </P></TD> <TD width=52> <P class=western align=center><I>m</I>3<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>4 </P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=29> <P class=western align=center>35 </P></TD> <TD width=52> <P class=western align=center><I>M</I>4<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=52> <P class=western align=center><I>3m</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=52> <P class=western align=center><I>M</I>3<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>5 </P></TD> <TD width=29> <P class=western align=center>0 </P></TD> <TD width=29> <P class=western align=center>49 </P></TD> <TD width=52> <P class=western align=center>5<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>51 </P></TD> <TD width=52> <P class=western align=center>3<I>M</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>18 </P></TD> <TD width=52> <P class=western align=center>4<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=52> <P class=western align=center><BR></P></TD></TR> <TR> <TD width=29 height=15> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>50 </P></TD> <TD width=52> <P class=western align=center>2<I>mD</I> </P></TD> <TD width=30> <P class=western align=center>5 </P></TD> <TD width=30> <P class=western align=center>-5 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=52> <P class=western align=center>5<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=52> <P class=western align=center><BR></P></TD></TR></TBODY></TABLE></CENTER> <P class=western><BR></P> <P class=western>Let's note that if <I>D</I><FONT face="Symbol, serif"><SUP>a</SUP></FONT><I>M</I><FONT face="Symbol, serif"><SUP>b</SUP></FONT> corresponds to degree <I>k</I>, then <I>D</I><SUP>-<FONT face="Symbol, serif">a</FONT></SUP><I>M</I><SUP>-<FONT face="Symbol, serif">b</FONT></SUP> corresponds to degree 53-k. </P> <P class=western>Example: 2<I>m</I>2<I>d</I> corresponds to 10; 2<I>M</I>2<I>D</I> corresponds to 43. The total number of functions with <I>h</I><FONT face="Symbol, serif">£</FONT>5 is equal 61; and at least one function corresponds to each degree from 0 to 52. </P> <P class=western><BR></P> <P class=western><B>6.<FONT size=1><FONT face="Times New Roman, serif"> </FONT></FONT>Generalization</B>. </P> <P class=western><BR></P> <P class=western>Let's assume that there are <I>n</I> "initial" tonal functions <I>f</I><SUB>1</SUB><I>,…, f<SUB>n</SUB></I> and their corresponding frequencies are <I>p</I><SUB>1</SUB><I>,…, p<SUB>n</SUB></I>, where <I>p</I><SUB>1</SUB><I>,…, p<SUB>n</SUB></I>, - odd prime numbers (explained above corresponds to the case of <I>n</I>=2,<I> f</I><SUB>1</SUB>=<I>D, f</I><SUB>2</SUB>=<I>M, p</I><SUB>1</SUB>=3<I>, p</I><SUB>2</SUB>=5). Set <FONT face="Symbol, serif">F</FONT> of tonal functions is then free Abelian group with <I>n</I> generators <I>f</I><SUB>1</SUB><I>,…, f<SUB>n</SUB></I>. Any element <I>f</I> from <FONT face="Symbol, serif">F</FONT><I> </I>is uniquely represented as <I>f</I><SUB>1</SUB><SUP><FONT face="Symbol, serif">a</FONT>1</SUP><I>…f<SUB>n</SUB></I><SUP><FONT face="Symbol, serif">a</FONT><I>n</I></SUP>.<I> </I>If <I>G</I> is a subgroup in <I>Q<SUP>+</SUP></I> generated by numbers <I>p</I><SUB>1</SUB><I>,…, p<SUB>n</SUB></I>, then exists isomorphism <FONT face="Symbol, serif">F@</FONT><I>G</I>, determined by generators of an equation <FONT face="Symbol, serif">j</FONT>(<I>f<SUB>i</SUB></I>)=<I> p<SUB>i</SUB></I>. For any <I>f</I> a set of frequencies {2<I><SUP>S</SUP></I><FONT face="Symbol, serif">j</FONT>(<I>f</I>)<FONT face="Symbol, serif">n</FONT><SUB>0</SUB>} is defined, where <I>S</I> runs through integers; <FONT face="Symbol, serif">n</FONT><SUB>0<I> </I></SUB>corresponds to tonic <I>T</I>, which is the unit of a group. The order of function <I>f=</I> <I>f</I><SUB>1</SUB><SUP><FONT face="Symbol, serif">a</FONT>1</SUP><I>…f<SUB>n</SUB></I><SUP><FONT face="Symbol, serif">a</FONT><I>n</I></SUP> is determined by the equation <I>h</I>(<I>f</I>)=|<FONT face="Symbol, serif">a</FONT><SUB>1</SUB>|+<I>…</I>+|<FONT face="Symbol, serif">a</FONT><I><SUB>n</SUB></I>|. Degree <I>k</I> of 53-tone tempered scale corresponding to function <I>f</I> is calculated from formula: </P> <P class=western><BR></P> <P class=western align=center><I>k</I>=[53{<FONT face="Symbol, serif">a</FONT><SUB>1</SUB>log<SUB>2</SUB><I> p</I><SUB>1</SUB>+…+<FONT face="Symbol, serif">a</FONT><I><SUB>n</SUB></I>log<SUB>2</SUB><I> p<SUB>n</SUB></I>}]<SUB>1 </SUB></P> <P class=western><BR></P> <P class=western>As before, if <I>k</I> corresponds to function <I>f</I>, then 53-<I>k</I> corresponds to function <I>f</I><SUP>-1</SUP>. As an example we may consider case <I>n</I>=3; <I>p</I><SUB>1</SUB>=3; <I>p</I><SUB>2</SUB>=5; <I>p</I><SUB>3</SUB>=7. It is shown in the following table: </P> <P class=western><BR></P> <CENTER> <TABLE borderColor=#000000 cellSpacing=0 cellPadding=1 rules=groups width=480 border=1> <COLGROUP> <COL width=29> <COL width=30> <COL width=30> <COL width=30> <COL width=30></COLGROUP> <COLGROUP> <COL width=30> <COL width=30> <COL width=30> <COL width=30> <COL width=30></COLGROUP> <COLGROUP> <COL width=30> <COL width=30> <COL width=30> <COL width=30> <COL width=30></COLGROUP> <TBODY> <TR> <TD width=29 height=17> <P class=western align=center><I>h</I> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD> <TD width=30> <P class=western align=center><I>h</I> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD> <TD width=30> <P class=western align=center><I>h</I> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD></TR></TBODY> <TBODY> <TR> <TD width=29 height=16> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>21 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>22 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>22 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>48 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>37 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>36 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>9 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>50 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>10 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>13 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>44 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>43 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>27 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>24 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>17 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>51 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>31 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>34 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>50 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>44 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>8 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>24 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>5 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>41 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>4 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>32 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>15 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>51 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>12 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>48 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>11 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>39 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>42 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>5 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>19 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>38 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>46 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>49 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>12 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>26 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>29 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>45 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>20 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>19 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>33 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>52 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>27 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>29 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>26 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>7 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>9 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>40 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>34 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>14 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>16 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD></TR> <TR> <TD width=29 height=15> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>41 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>31 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD></TR></TBODY></TABLE></CENTER> <P class=western><BR></P> <P class=western><FONT color=#000000>Under</FONT><I> n</I>=3 and<FONT color=#000000> <I>h</I>>0 there are </FONT>4<I>h</I><SUP>2</SUP>+2 <FONT color=#000000>functions of the order <I>h</I>. </FONT>The total number of functions of the order <FONT face="Symbol, serif">£</FONT>3 is 63; nevertheless no function of the order <FONT face="Symbol, serif">£</FONT>3 corresponds to degrees 6, 18, 23, 30, 35, 47; functions of the order 4 correspond to these degrees: </P> <P class=western><BR></P> <CENTER> <TABLE borderColor=#000000 cellSpacing=0 cellPadding=1 rules=groups width=542 border=1> <COLGROUP> <COL width=33> <COL width=34> <COL width=34> <COL width=34> <COL width=34></COLGROUP> <COLGROUP> <COL width=34> <COL width=34> <COL width=34> <COL width=34> <COL width=34></COLGROUP> <COLGROUP> <COL width=34> <COL width=34> <COL width=34> <COL width=34> <COL width=34></COLGROUP> <TBODY> <TR> <TD width=33 height=17> <P class=western align=center><I>h</I> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><I>k</I> </P></TD> <TD width=34> <P class=western align=center><I>h</I> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><I>k</I> </P></TD> <TD width=34> <P class=western align=center><I>h</I> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><I>k</I> </P></TD></TR></TBODY> <TBODY> <TR> <TD width=33 height=16> <P class=western align=center>4 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>-2 </P></TD> <TD width=34> <P class=western align=center>6 </P></TD> <TD width=34> <P class=western align=center>4 </P></TD> <TD width=34> <P class=western align=center>3 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>23 </P></TD> <TD width=34> <P class=western align=center>4 </P></TD> <TD width=34> <P class=western align=center>2 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>35 </P></TD></TR> <TR> <TD width=33 height=16> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>-2 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>18 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>-3 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>30 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>2 </P></TD> <TD width=34> <P class=western align=center>47 </P></TD></TR> <TR> <TD width=33 height=16> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>4 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>18 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>3 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>30 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD></TR> <TR> <TD width=33 height=15> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>-3 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>23 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>-4 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>35 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD></TR></TBODY></TABLE></CENTER> <P class=western><BR></P> <P class=western><B>7.<FONT size=1><FONT face="Times New Roman, serif"> </FONT></FONT>Number of Tonal Functions</B> </P> <P class=western><BR></P> <P class=western>Let's designate <I>N<SUB>n</SUB></I>(<I>h</I>) as the number of tonal functions of the order <I>h</I> with number of generators <I>n </I>(<I>h</I><FONT face="Symbol, serif">³</FONT>0, <I>n</I><FONT face="Symbol, serif">³</FONT>1). From a mathematical point of view <I>N<SUB>n</SUB></I>(<I>h</I>) is a number of solutions to the equation |<I>x</I><SUB>1</SUB>|+|<I>x</I><SUB>1</SUB>|+<I>…</I>+|<I>x<SUB>n</SUB></I>|=<I>h</I> in integers. Obviously, <I>N<SUB>n</SUB></I>(0)=1 and under <I>h</I><FONT face="Symbol, serif">³</FONT>1 <I>N</I><SUB>1</SUB>(<I>h</I>)=2 we'll have a recurrent formula: </P> <P class=western><BR></P> <P class=western align=center><I>N<SUB>n</SUB></I>(<I>h</I>)=<I>N<SUB>n</SUB></I>(<I>h-</I>1)+<I>N<SUB>n-</SUB></I><SUB>1</SUB>(<I>h</I>)+<I>N<SUB>n-</SUB></I><SUB>1</SUB>(<I>h-</I>1), </P> <P class=western align=center><BR></P> <P class=western>which allows to calculate values <I>N<SUB>n</SUB></I>(<I>h</I>). When filling a table of numbers <I>N<SUB>n</SUB></I>(<I>h</I>) shown below it is convenient to use the recurrent formula as the following rule: each "interior" number is equal to the sum of numbers located to the left, upper left and above from it. </P> <P class=western><BR></P> <CENTER> <TABLE borderColor=#000000 cellSpacing=0 cellPadding=1 rules=groups width=434 border=1> <COLGROUP> <COL width=19> <COL width=20></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=53></COLGROUP> <TBODY> <TR> <TD width=19 height=15> <P class=western align=center><BR></P></TD> <TD width=20> <P class=western align=center>h </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>0 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>1 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>2 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>3 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>4 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>5 </P></TD> <TD width=53 rowSpan=2> <P class=western align=center>6 </P></TD></TR> <TR> <TD width=19 height=16> <P class=western align=center>n </P></TD> <TD width=20> <P class=western align=center><BR></P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=53> <P class=western align=center>2 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>4 </P></TD> <TD width=54> <P class=western align=center>8 </P></TD> <TD width=54> <P class=western align=center>12 </P></TD> <TD width=54> <P class=western align=center>16 </P></TD> <TD width=54> <P class=western align=center>20 </P></TD> <TD width=53> <P class=western align=center>24 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>3 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>6 </P></TD> <TD width=54> <P class=western align=center>18 </P></TD> <TD width=54> <P class=western align=center>38 </P></TD> <TD width=54> <P class=western align=center>66 </P></TD> <TD width=54> <P class=western align=center>102 </P></TD> <TD width=53> <P class=western align=center>146 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>4 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>8 </P></TD> <TD width=54> <P class=western align=center>32 </P></TD> <TD width=54> <P class=western align=center>88 </P></TD> <TD width=54> <P class=western align=center>192 </P></TD> <TD width=54> <P class=western align=center>360 </P></TD> <TD width=53> <P class=western align=center>608 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>5 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>10 </P></TD> <TD width=54> <P class=western align=center>50 </P></TD> <TD width=54> <P class=western align=center>170 </P></TD> <TD width=54> <P class=western align=center>450 </P></TD> <TD width=54> <P class=western align=center>1002 </P></TD> <TD width=53> <P class=western align=center>1970 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=15> <P class=western align=center>6 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>12 </P></TD> <TD width=54> <P class=western align=center>72 </P></TD> <TD width=54> <P class=western align=center>292 </P></TD> <TD width=54> <P class=western align=center>912 </P></TD> <TD width=54> <P class=western align=center>2364 </P></TD> <TD width=53> <P class=western align=center>5336 </P></TD></TR></TBODY></TABLE></CENTER> <P class=western><BR></P> <P class=western>Under fixed <I>n</I> and <I>h</I><FONT face="Symbol, serif">³</FONT>1 we'll have the formulas <I>N</I><SUB>2</SUB>(<I>h</I>)=4<I>h</I>; <I>N</I><SUB>3</SUB>(<I>h</I>)=4<I>h</I><SUP>2</SUP>+2; (these formulas were cited in p.2 and p.6) <I>N</I><SUB>4</SUB>(<I>h</I>)=(8<I>h</I>(<I>h</I><SUP>2</SUP>+2))/3; <I>N</I><SUB>5</SUB>(<I>h</I>)=(4<I>h</I><SUP>2</SUP>(<I>h</I><SUP>2</SUP>+5))/3; <I>N</I><SUB>6</SUB>(<I>h</I>)=(4<I>h</I>(2<I>h</I><SUP>4</SUP>+20<I>h</I><SUP>2</SUP>+23))/15. Let's designate <I>Ñ<SUB>n</SUB></I>(<I>h</I>) as the number of tonal functions of the order <FONT face="Symbol, serif">£</FONT><I>h </I>with number of generators <I>n</I>; <FONT color=#000000>thereby, </FONT><I>Ñ<SUB>n</SUB></I>(<I>h</I>)=<I>N<SUB>n</SUB></I>(0)+<I>N<SUB>n</SUB></I>(1)+<I>…</I>+<I>N<SUB>n</SUB></I>(<I>h</I>). We have the formula: <I>Ñ<SUB>n</SUB></I>(<I>h</I>)=(<I>N<SUB>n+</SUB></I><SUB>1</SUB>(<I>h</I>)+<I>N<SUB>n</SUB></I>(<I>h</I>))/2, that allows easy calculation of values <I>Ñ<SUB>n</SUB></I>(<I>h</I>). For example, with the number of generators <I>n</I>=5 the number of tonal functions of the order <FONT face="Symbol, serif">£</FONT>5 is equal to <I>Ñ</I><SUB>5</SUB>(5)=(<I>N</I><SUB>6</SUB>(5)+<I>N</I><SUB>5</SUB>(5))/2=(2364+1002)/2=1683. </P> <P class=western><BR></P> <P class=western>Let's include also common formula: </P> <P class=western> </P> <P class=western> </P> <DIV align=left> <TABLE cellSpacing=0 cellPadding=3 width=300 border=0> <TBODY> <TR> <TD width="50%"> <P align=right><EM><SUB><FONT face="Courier New">n</FONT></SUB></EM></P></TD> <TD width="50%"></TD></TR> <TR> <TD width="50%"> <P align=right><FONT face="Courier New"><EM>N</EM><SUB>n</SUB><EM>(h)<FONT face="Courier New">=</FONT></EM><FONT face="Symbol, serif">S</FONT></P></FONT></TD> <TD width="50%"> <P align=left><FONT face="Courier New, monospace"><EM>(<SUB>i</SUB><SUP>n</SUP>)(<SUB>i</SUB><SUP>h</SUP><SUB>-</SUB><SUP>-</SUP><SUB>1</SUB><SUP>1</SUP>)</EM>2<SUP><EM>i</EM></SUP><FONT face="Times New Roman"> </FONT></FONT></P></TD></TR> <TR> <TD width="50%"> <P align=right><EM><SUP><FONT face="Courier New">i=1</FONT></SUP></EM> </P></TD> <TD width="50%"></TD></TR></TBODY></TABLE></DIV> <DIV align=left><FONT face="Courier New, monospace"><I><SUB></SUB></I></FONT><BR></DIV> <P class=western>June, 16, 2005 </P>Mykhaylo Khramovhttp://www.blogger.com/profile/00483331671036933638noreply@blogger.com2