Algebra of Tonal Functions.
© 2005 Gennady Wohl.
The article is written for project COMMATOR.
The English translation of the original text has performed Mykhaylo Khramov.
Translator thanks Gennady Wohl, who kindly has agreed to explain mathematical aspects of a theme, offered to him.
Translator also thanks Vadim Bondar for the help and will be glad to any criticism which improve translation quality.
In offered notes you'll find brief description of M. Y. Khramov's ideas concerning tonal functions of higher order. 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.
1. Informal description of tonal functions.
Let's assume that in some key T means tonic, D - dominant, М - mediant, d - subdominant, m - submediant. (In conventional scale T, M, d, D, m functions correspond to I, III, IV, V, VI degrees). If frequency T conditionally equals 1, then frequencies D, d, M, m 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 T, D, d, M, m can be multiplied or divided by 2. Rejecting factor of 2 we find that T, D, d, M, m correspond to numbers 1, 3, 1/3, 5, 1/5.
Let f_{1 }be some function; the tone to which f_{1} corresponds we'll assume to be tonic; and in the new key the tone to which function f_{2} corresponds we'll assign in relation to initial tonal function f_{1}f_{2}. Obviously, f_{1}f_{2}=f_{2}f_{1}. The frequency corresponding to function f_{1}f_{2}, is obtained by multiplication of frequencies of functions f_{1} and f_{2}. For example, DD corresponds to frequency 3·3=9; MMD - frequency 5·5·3=75 and so on. Instead of DD, MMD we will write 2D, 2MD and so on.
Let's note, that Dd=Mm=T; and corresponding numerical equations are 3·(1/3)=1 and 5·(1/5)=1.
2. Formal description of tonal functions.
Let's consider free Abelian group F with two generators D and M; the unit of group we shall designate as T. Let d=D^{-1}, m=M^{-1}. The element fÎF we'll name a tonal function; f is uniquely described as D^{a}M^{b}, where a and b are integers.
In the multiplicative group Q^{+} of positive rational numbers let's look at the subgroup G generated by numbers 3 and 5. Let's define isomorphism F@G by setting it with generators: j(D)=3; j(M)=5; thereby, j(D^{a}M^{b})=3^{a}5^{b}. For each function f we'll define a set of frequencies n(f)={2^{S}j(f)n_{0}}, where S runs through integers. (Frequency n_{0} thereby corresponds to T). Besides we'll define for each function f=D^{a}M^{b} an order h(f)=|a|+|b|. There is single function of the order h=0, namely T; under h>0 there are 4h functions of the order h.
Examples:
h=1; functions D, M, D^{-1}=d, M^{-1}=m.
h=2; functions D^{2}, M^{2}, DM, D^{-2}, M^{-2}, D^{-1}M^{-1}, DM^{-1}, D^{-1}M; or using previous notation of p.1: 2D, 2M, MD, 2d, 2m, md, mD, Md.
Under h = 3, 4, 5 there are accordingly 12, 16 and 20 functions.
3. 53-tone Equal Temperament.
Let n_{0} be frequency of the tonic; let's divide an octave (i.e. the sound range between n_{0} and 2n_{0}) by 53 equal parts. Let's enumerate degrees of a given scale from 0 (corresponds to n_{0}) to 53 (corresponds to 2n_{0}). The frequency of a degree number k is equal to n_{0}·2^{k/53}.
The choice of number 53 has to do with the fact that (3/2)^{53}/2^{31} = 1,002…, i.e. 53 perfect fifths give "almost closed" scale (please note, that for 12-tone temperament (3/2)^{12}/2^{7}=1,01…). Besides, 53 degrees enable us to reproduce many other scales in use rather precisely.
We'll note that numbers 12 and 53 have a deeper mathematical meaning: they are denominators of convergents from decomposition of number log_{2}3 into continued fraction; really, log_{2}3=[1;1,1,2,2,3,1,5,…]; it is easy to see that 5-th and 7-th convergents accordingly are 19/12 and 84/53.
4. Matching Tonal Functions to 53-tone Equal Temperament Degrees. Motivations.
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 M, than its frequency 5/4=1.25 is closest to frequency of 17-th degree 2^{17./53}=1.24898…, and so it is the 17-th degree of the 53-tone scale that matches given tone.
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 2MD with frequency 5^{2}·3=75 or, within limits of one octave, 75/64 = 1.171875, it is closest to frequency of the 12-th tone of 53-degree temperament: 2^{12/53}=1.169924…; however, difficulty lies just in developing musical motivation which will allow to assign function 2MD to the given tone.
5. Matching Tonal Functions to 53-tone Equal Temperament Degrees. Formal Procedure.
Let f=D^{a}M^{b}; f¹T. Function f corresponds to frequency set n(f)={2^{S}3^{a}5^{b}}, where S runs through all integers. (we suppose that n_{0} conditionally equals 1). An inequality 1<2^{S}3^{a}5^{b}<2 (i.e. the choice of frequency within limits of an octave) uniquely defines S. If number k is a degree of 53-tone temperament corresponding to function f, it'll be defined from a condition of greatest proximity between numbers 2^{k/53} and 2^{S}3^{a}5^{b}. Since log_{2}(2^{S}3^{a}5^{b})=alog_{2}3+blog_{2}5+SÎ(0,1), then log_{2}(2^{S}3^{a}5^{b})={alog_{2}3+blog_{2}5}, where {x} designates fractional part of a number x. Thereby, k/53 should be close to {alog_{2}3+blog_{2}5} or, that same, k should be close to a number 53{alog_{2}3+blog_{2}5}. In other words, degree k of 53-tone temperament corresponding to function D^{a}M^{b} is the nearest integer to number 53{alog_{2}3+blog_{2}5}. Designating nearest integer to x numeral as [x]_{1} we shall obtain: k=[53{alog_{2}3+blog_{2}5}]_{1}
Example:
f=2MD=D^{1}M^{2}; a=1; b=2;
log_{2}3+2log_{2}5=2.2288186…;
53·{log_{2}3+2log_{2}5}=53·0.2288186=12.12739…;
k=12.
Here is a table of functions of the order h£5 and their corresponding degrees.
H | a | b | k | f | h | a | b | k | f | h | a | b | k | f |
0 | 0 | 0 | 0 | T | | 1 | 2 | 12 | 2MD | 5 | -4 | -1 | 18 | m4d |
1 | -1 | 0 | 22 | d | | 2 | -1 | 45 | m2D | | -4 | 1 | 52 | M4d |
| 0 | -1 | 36 | m | | 2 | 1 | 26 | M2D | | -3 | -2 | 32 | 2m3d |
| 0 | 1 | 17 | M | | 3 | 0 | 40 | 3D | | -3 | 2 | 47 | 2M3d |
| 1 | 0 | 31 | D | 4 | -4 | 0 | 35 | 4d | | -2 | -3 | 46 | 3m2d |
2 | -2 | 0 | 44 | 2d | | -3 | -1 | 49 | m3d | | -2 | 3 | 42 | 3M2d |
| -1 | -1 | 5 | md | | -3 | 1 | 30 | M3d | | -1 | -4 | 7 | 4md |
| -1 | 1 | 39 | Md | | -2 | -2 | 10 | 2m2d | | -1 | 4 | 37 | 4Md |
| 0 | -2 | 19 | 2m | | -2 | 2 | 25 | 2M2d | | 0 | -5 | 21 | 5m |
| 0 | 2 | 34 | 2M | | -1 | -3 | 24 | 3md | | 0 | 5 | 32 | 5M |
| 1 | -1 | 14 | mD | | -1 | 3 | 20 | 3Md | | 1 | -4 | 16 | 4mD |
| 1 | 1 | 48 | MD | | 0 | -4 | 38 | 4m | | 1 | 4 | 46 | 4MD |
| 2 | 0 | 9 | 2D | | 0 | 4 | 15 | 4M | | 2 | -3 | 11 | 3m2D |
3 | -3 | 0 | 13 | 3d | | 1 | -3 | 33 | 3mD | | 2 | 3 | 7 | 3M2D |
| -2 | -1 | 27 | m2d | | 1 | 3 | 29 | 3MD | | 3 | -2 | 6 | 2m3D |
| -2 | 1 | 8 | M2d | | 2 | -2 | 28 | 2m2D | | 3 | 2 | 21 | 2M3D |
| -1 | -2 | 41 | 2md | | 2 | 2 | 43 | 2M2D | | 4 | -1 | 1 | m4D |
| -1 | 2 | 3 | 2Md | | 3 | -1 | 23 | m3D | | 4 | 1 | 35 | M4D |
| 0 | -3 | 2 | 3m | | 3 | 1 | 4 | M3D | | 5 | 0 | 49 | 5D |
| 0 | 3 | 51 | 3M | | 4 | 0 | 18 | 4D | | | | | |
| 1 | -2 | 50 | 2mD | 5 | -5 | 0 | 4 | 5d | | | | | |
Let's note that if D^{a}M^{b} corresponds to degree k, then D^{-a}M^{-b} corresponds to degree 53-k.
Example: 2m2d corresponds to 10; 2M2D corresponds to 43. The total number of functions with h£5 is equal 61; and at least one function corresponds to each degree from 0 to 52.
6. Generalization.
Let's assume that there are n "initial" tonal functions f_{1},…, f_{n} and their corresponding frequencies are p_{1},…, p_{n}, where p_{1},…, p_{n}, - odd prime numbers (explained above corresponds to the case of n=2, f_{1}=D, f_{2}=M, p_{1}=3, p_{2}=5). Set F of tonal functions is then free Abelian group with n generators f_{1},…, f_{n}. Any element f from F is uniquely represented as f_{1}^{a1}…f_{n}^{an}. If G is a subgroup in Q^{+} generated by numbers p_{1},…, p_{n}, then exists isomorphism F@G, determined by generators of an equation j(f_{i})= p_{i}. For any f a set of frequencies {2^{S}j(f)n_{0}} is defined, where S runs through integers; n_{0 }corresponds to tonic T, which is the unit of a group. The order of function f= f_{1}^{a1}…f_{n}^{an} is determined by the equation h(f)=|a_{1}|+…+|a_{n}|. Degree k of 53-tone tempered scale corresponding to function f is calculated from formula:
k=[53{a_{1}log_{2} p_{1}+…+a_{n}log_{2} p_{n}}]_{1 }
As before, if k corresponds to function f, then 53-k corresponds to function f^{-1}. As an example we may consider case n=3; p_{1}=3; p_{2}=5; p_{3}=7. It is shown in the following table:
h | a_{1} | a_{2} | a_{3} | k | h | a_{1} | a_{2} | a_{3} | k | h | a_{1} | a_{2} | a_{3} | k |
0 | 0 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 21 | 3 | 0 | 0 | 3 | 22 |
1 | -1 | 0 | 0 | 22 | | 1 | 1 | 0 | 48 | | 0 | 1 | -2 | 37 |
| 0 | -1 | 0 | 36 | | 2 | 0 | 0 | 9 | | 0 | 1 | 2 | 50 |
| 0 | 0 | -1 | 10 | 3 | -3 | 0 | 0 | 13 | | 0 | 2 | -1 | 44 |
| 0 | 0 | 1 | 43 | | -2 | -1 | 0 | 27 | | 0 | 2 | 1 | 24 |
| 0 | 1 | 0 | 17 | | -2 | 0 | -1 | 1 | | 0 | 3 | 0 | 51 |
| 1 | 0 | 0 | 31 | | -2 | 0 | 1 | 34 | | 1 | -2 | 0 | 50 |
2 | -2 | 0 | 0 | 44 | | -2 | 1 | 0 | 8 | | 1 | -1 | -1 | 24 |
| -1 | -1 | 0 | 5 | | -1 | -2 | 0 | 41 | | 1 | -1 | 1 | 4 |
| -1 | 0 | -1 | 32 | | -1 | -1 | -1 | 15 | | 1 | 0 | -2 | 51 |
| -1 | 0 | 1 | 12 | | -1 | -1 | 1 | 48 | | 1 | 0 | 2 | 11 |
| -1 | 1 | 0 | 39 | | -1 | 1 | 0 | 42 | | 1 | 1 | -1 | 5 |
| 0 | -2 | 0 | 19 | | -1 | 0 | 2 | 2 | | 1 | 1 | 1 | 38 |
| 0 | -1 | -1 | 46 | | -1 | 1 | -1 | 49 | | 1 | 2 | 0 | 12 |
| 0 | -1 | 1 | 26 | | -1 | 1 | 1 | 29 | | 2 | -1 | 0 | 45 |
| 0 | 0 | -2 | 20 | | -1 | 2 | 0 | 3 | | 2 | 0 | -1 | 19 |
| 0 | 0 | 2 | 33 | | 0 | -3 | 0 | 2 | | 2 | 0 | 1 | 52 |
| 0 | 1 | -1 | 27 | | 0 | -2 | -1 | 29 | | 2 | 1 | 0 | 26 |
| 0 | 1 | 1 | 7 | | 0 | -2 | 1 | 9 | | 3 | 0 | 0 | 40 |
| 0 | 2 | 0 | 34 | | 0 | -1 | -2 | 3 | | | | | |
| 1 | -1 | 0 | 14 | | 0 | -1 | 2 | 16 | | | | | |
| 1 | 0 | -1 | 41 | | 0 | 0 | -3 | 31 | | | | | |
Under n=3 and h>0 there are 4h^{2}+2 functions of the order h. The total number of functions of the order £3 is 63; nevertheless no function of the order £3 corresponds to degrees 6, 18, 23, 30, 35, 47; functions of the order 4 correspond to these degrees:
h | a_{1} | a_{2} | a_{3} | k | h | a_{1} | a_{2} | a_{3} | k | h | a_{1} | a_{2} | a_{3} | k |
4 | -1 | 1 | -2 | 6 | 4 | 3 | -1 | 0 | 23 | 4 | 2 | -1 | 1 | 35 |
| -2 | 1 | -1 | 18 | | -3 | 1 | 0 | 30 | | 1 | -1 | 2 | 47 |
| 4 | 0 | 0 | 18 | | 3 | 0 | 1 | 30 | | | | | |
| -3 | 0 | -1 | 23 | | -4 | 0 | 0 | 35 | | | | | |
7. Number of Tonal Functions
Let's designate N_{n}(h) as the number of tonal functions of the order h with number of generators n (h³0, n³1). From a mathematical point of view N_{n}(h) is a number of solutions to the equation |x_{1}|+|x_{1}|+…+|x_{n}|=h in integers. Obviously, N_{n}(0)=1 and under h³1 N_{1}(h)=2 we'll have a recurrent formula:
N_{n}(h)=N_{n}(h-1)+N_{n-}_{1}(h)+N_{n-}_{1}(h-1),
which allows to calculate values N_{n}(h). When filling a table of numbers N_{n}(h) 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.
| h | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
n | | |||||||
1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 | 1 | 4 | 8 | 12 | 16 | 20 | 24 | |
3 | 1 | 6 | 18 | 38 | 66 | 102 | 146 | |
4 | 1 | 8 | 32 | 88 | 192 | 360 | 608 | |
5 | 1 | 10 | 50 | 170 | 450 | 1002 | 1970 | |
6 | 1 | 12 | 72 | 292 | 912 | 2364 | 5336 |
Under fixed n and h³1 we'll have the formulas N_{2}(h)=4h; N_{3}(h)=4h^{2}+2; (these formulas were cited in p.2 and p.6) N_{4}(h)=(8h(h^{2}+2))/3; N_{5}(h)=(4h^{2}(h^{2}+5))/3; N_{6}(h)=(4h(2h^{4}+20h^{2}+23))/15. Let's designate Ñ_{n}(h) as the number of tonal functions of the order £h with number of generators n; thereby, Ñ_{n}(h)=N_{n}(0)+N_{n}(1)+…+N_{n}(h). We have the formula: Ñ_{n}(h)=(N_{n+}_{1}(h)+N_{n}(h))/2, that allows easy calculation of values Ñ_{n}(h). For example, with the number of generators n=5 the number of tonal functions of the order £5 is equal to Ñ_{5}(5)=(N_{6}(5)+N_{5}(5))/2=(2364+1002)/2=1683.
Let's include also common formula:
_{n} | |
N_{n}(h)=S | (_{i}^{n})(_{i}^{h}_{-}^{-}_{1}^{1})2^{i} |
^{i=1} |
June, 16, 2005