Sonantometry.
© 2007 Mykhaylo Khramov.
The article is writing during the project COMMATOR.
The last modifications are introduced on Jule, 25, 2007.
The English translation of the original text has performed Mykhaylo Khramov.
In offered notes you'll find developing of ideas concerning tonal functions of higher order. Is talking on practice of application these tonal functions as sonants to the analysis and synthesis of music.
Sonantometry, or the algebra of tonal functions, 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: R. Wagner's Tristan opening. 7limit 53EDO approximation. Also the MIDI  model of this fragment sounding is given.
The following table can explain the essence of sonantometry:
5limit Just Intonation Diatonic Scale  Tonal Function  Sonant  
Grade  Frequency Ratio of Grade and 1st Grade  Name  Symbol  Formulas by Order of Affinity to Tonant  Name  
No.  Symbol  Fraction  Factorization  0  1  2  
8  VIII ~ I  2/1 ~ 1  2^{1 }~ 1  Tonic  T  :T 
 :Dd  Tonant and Dedant 
7  VII<  15/8~15  5^{1}×3^{1}×2^{3 }~ 5^{1}×3^{1}  Leading 


 :MD or :DM  MeDent or DeMent 
6  VI<  5/3  5^{1}×3^{1}×2^{0 }~ 5^{1}×3^{1}  Submediant 


 :Md or :dM  Medant or daMent 
5  V  3/2~3  3^{1}×2^{1 }~ 3^{1}  Dominant  D 
 :D 
 Dominant or Dent 
4  IV  4/3~1/3  3^{1}×2^{2 }~ 3^{1}  Subdominant  S 
 :d 
 Subdominant or dant 
3  III<  5/4~5  5^{1}×2^{2 }~ 5^{1}  Mediant  M 
 :M 
 Mediant or Ment 
2  II  9/8~9  3^{2}×2^{3 }~ 3^{2}  Double Dominant  DD or 2D 

 :2D or :DD  TwoDent or DeDent 
1  I ~ VIII  1/1=1 ~ 2/1  2^{0}=1 ~ 2^{1}  Tonic  T  :T 
 :dD  Tonant and daDent 
This table shows relations among wellknown 5limit just intonation major diatonic scale, also wellknown tonal functions and those ones that are named here sonants and underlies discipline named here sonantometry.
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.
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^{1}, 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.
In 5limit just intonation major diatonic scale there is one more sonant of the order 1. It corresponds to number 5^{1} 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 5limit 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 5limit just intonation major diatonic scale substitutes the grade VI< by the grade VIb> (8/5 ~ 51) and the correspondence of submediant to mant is restored.
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.
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 5limit and order of 2 just intonation. The full set of these sonants is reflected in a table below.
5limit and Order of 2 Just Intonation Scale  Sonant  
Grade  Frequency Ratio of Grade and 1st Grade  Formulas by Order of Affinity to Tonant  Name  
No.  Symbol  Fraction  Factorization  0  1  2  
14  VIII ~ I  2/1 ~ 1  2^{1}~1  :T 
 :Dd and :Mm  Tonant and Dedant and Medant 
13  VII<  15/8 ~ 15  5^{1}×3^{1}×2^{3 }~ 5^{1}×3^{1} 

 :MD or :DM  MeDent or DeMent 
12  VIIb>  16/9 ~ 1/9  3^{2}×2^{4 }~ 3^{2} 

 :2d or :dd  Twodant or dadant 
11  VI<  5/3  5^{1}×3^{1}×2^{0 }~ 5^{1}×3^{1} 

 :Md or :dM  Medant or daMent 
10  VIb>  8/5 ~ 1/5  5^{1}×2^{3 }~ 5^{1} 
 :m 
 Submediant or mant 
9  V#(  25/16 ~ 25  5^{2}×2^{4 }~ 5^{2} 

 :2M or :MM  TwoMent or MeMent 
8  V  3/2 ~ 3  3^{1}×2^{1 }~ 3^{1} 
 :D 
 Dominant or Dent 
7  IV  4/3 ~ 1/3  3^{1}×2^{2 }~ 3^{1} 
 :d 
 Subdominant or dant 
6  IVb)  32/25 ~ 1/25  5^{2}×2^{5 }~ 5^{2} 

 :2m or :mm  Twomant or mamant 
5  III<  5/4 ~ 5  5^{1}×2^{2 }~ 5^{1} 
 :M 
 Mediant or Ment 
4  IIIb>  6/5  5^{1}×3^{1}×2^{1 }~ 5^{1}×3^{1} 

 :mD or :Dm  maDent or Demant 
3  II  9/8 ~ 9  3^{2}×2^{3 }~ 3^{2} 

 :2D or :DD  TwoDent or DeDent 
2  IIb<  16/15 ~ 1/15  5^{1}×3^{1}×2^{4 }~ 5^{1}×3^{1} 

 :md or :dm  madant or damant 
1  I ~ VIII  1/1=1 ~ 2/1  2^{0}=1 ~ 2^{1}  :T 
 :dD and :mM  Tonant and daDent and maMent 
At the beginning of clearing up the essence of signs ">", "<", ")", "(", let’s recollect, that the scale of 5limit just intonation can be expressed through a 3limit 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 3limit 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.
In the following table the scale of 5limit just intonation (sonants :T, :D, :d, :M, :m and their combination) is expressed through interval sonants (intersonants) to Pythagorean grades:
Pythagorean 3limit and Order of 8 Just Intonation Scale  5limit and Order of 2 Just Intonation Scale  
Grade  Sonants by Order of Affinity to Tonant  Grade  Sonants by Order of Affinity to Tonant  Pitch Bend in Intersonants by Order of Affinity to Pythagorean Scale Grades.  
№  Symbol  0  1  2  3  4  5  …  8  №  Symbol  0  1  2  5  10 
14  VIII ~ I  :T 






 14  VIII ~ I  :T 




13  VII 




 :5D 






 :δ[5D = :M4d[5D 










 13  VII< 

 :MD 
 
12  VIIb 

 :2d 




 12  VIIb 

 :2d 


11  VI 


 :3D 








 :δ[3D = :M4d[3D 










 11  VI< 

 :Md 
 
10  V# 






 :8D  :2δ[8D = :2M8d[8D  









 10  VIb> 
 :m 
 :Δ[4d = :m4D[4d  
9  VIb 



 :4d 







 









 9  V#( 

 :2M  
8  V 
 :D 





 8  V 
 :D 

 
7  IV 
 :d 





 7  IV 
 :d 












 6  IVb) 

 :2m 
 :2Δ[8d = :2m8D[8d 
6  III 



 :4D 







 :δ[4D = :M4d[4D  









 5  III< 
 :M 
 
5  IVb 






 :8d  









 4  IIIb> 

 :mD  :Δ[3d = :m4D[3d 

4  IIIb 


 :3d 









 
3  II 

 :2D 




 3  II 

 :2D 











 2  IIb< 

 :md  :Δ[5d = :m4D[5d 

2  IIb 




 :5d 







 
1  I ~ VIII  :T 






 1  I ~ VIII  :T 



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.
:md = :md 5d 5D = :md5D[5d = :m4D[5d;
:mD = :mD 3d 3D = :mD3D[3d = :m4D[3d;
:M = :M 4D 4d = :M4d[4D;
:2m = :2m 8d 8D = 2m8D[8d;
:2M = :2M 8D 8d = 2M8d[8D;
:m = :m 4d 4D = :m4D[4d;
:Md = :Md 3D 3d = :Md3d[3D = :M4d[3D;
:MD = :MD 5D 5d = :MD5d[5D = :M4d[5D.
Restoring numerical values of the intersonant :m4D by the formula 5^{1}×3^{4}×2^{Z}, Z ={n,…, 0,… n}, n Î N, N = {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.
The table shows, that ">" sharply bends a grade pitch on Didiment, and "<" flatly. Sign ")" order TwoDidiment sharp, and "(" means Twodidimant flat.
Let's remark, if the note sonant include:
M, then arises "<" and the grade is flattened by the intersonant :δ;
2M, arises "(" and the grade is flattened by the intersonant :2δ;
m, arises ">" and the grade is sharpened by the intersonant :Δ;
2m, arises ")" and the grade is sharpened by the intersonant :2Δ.
In other words, in 5limit 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 :δ or :Δ, how many M, or m are in the note makeup.
This observation gives the following, consisting of two phases, rule of placing of additional accidentals for a writing notes of the 5limit just intonation in scores of Pithagorean scale:
 to appropriate to each note head proper to it sonant depending on a score context;
 if in sonant are present M, 2M, 3M, to bend a pitch of the note accordingly by :δ,:2δ,:3δ, and to designate these flattening accordingly by additional accidentals "<", "(", "{". The presence m, 2m, 3m requires bends accordingly by :Δ,:2Δ,:3Δ, and designating of a sharping accordingly by additional accidentals ">", ")" and "}".