Tone structure (mathematical description)
A tone structure describes a tone system with the help of tones and intervals . Since ancient times, the sound supply of a musical culture has been reproduced on the one hand by specifying pitches and on the other hand by the term interval.
Nowadays, heights and intervals are described using frequencies and frequency relationships. The music theory of Pythagoras is known with the help of proportions (= string ratios on the monochord = reciprocal of the frequency ratios).
The mathematical teaching of tones and intervals is, however, also possible without these physical terms (see description of hearing psychology ). The first known hearing psychological descriptions of a sound system come from Aristoxenos .
The ordered pitch space
A frequency can be assigned to each tone .
 Example: c ' (the dashed c ) has the frequency 264 Hz , e' the frequency 330 Hz, g ' the frequency 396 Hz and c' ' the frequency 528 Hz.
Tones can be distinguished in height. The following applies: The higher a tone sounds, the greater its frequency. From a mathematical point of view, it is a (transitive and trichotomic) strict total order .
 Transitive means: From a higher than b and b higher than c follows a higher than c .
 Trichotomic means: For tones a and b the following applies: Either a = b or a higher than b or b higher than a .
The ordered additive interval space
Every two tones and (with the frequencies and ) is clearly assigned an interval (with the frequency ratio ).
 For example, the octave c'c '' has the frequency ratio 528: 264 = 2, the pure fifth C'G 'the frequency ratio 396: 264 = 3: 2, the major third c'e' the frequency ratio 330: 264 = 5: 4 and the minor third e'g 'the frequency ratio 396: 330 = 6: 5.
An end tone (with the frequency ) of the interval is clearly assigned to each initial tone (with the frequency ) and to each interval (with the frequency ratio ) .
 Example: If a 'has the frequency , then the tone c' ', which sounds a minor third with the frequency ratio higher, has the frequency .
In the language of musicians, intervals are added when performing one after the other. In this sense, the interval space has an additive structure.
 Example: major third + minor third = fifth .
 12 fifths are roughly equal to 7 octaves . The difference is called the Pythagorean comma . One writes: Pythagorean comma = 12 fifths  7 octaves . If you perform three pure major thirds one after the other (for example cegishis ), you get an interval (from c to his ) that is slightly smaller than the octave . The difference is called little Diësis . Thus: minor Diësis = octave  3 major thirds .
The addition of intervals corresponds to the multiplication of the frequency ratios and the subtraction of intervals corresponds to the division of the frequency ratios.
 Example: The addition of the minor third + major third = fifth corresponds to multiplication .
 The frequency ratio of the Pythagorean comma is calculated to and that of the small Diësis to .
Intervals can be compared in terms of size. The following applies: the larger the interval, the greater its frequency ratio .
The frequency ratio increases exponentially.
interval  Frequency ratio 

1 octave  2 
2 octaves  4th 
3 octaves  8th 
•••  
1 fifth  ^{3} / _{2} 
2 fifths  ^{9} / _{4} 
3 fifths  ^{27} / _{8} 
••• 
From a mathematical point of view, an interval space is an Archimedean ordered commutative group .
Measure the size of intervals
Intervals can be given as multiples of an octave. Mostly, however, the subunit cents is used.
It is a logarithmic measure of the frequency relationships. The subunit cents with the definition 1200 cents = 1 octave (or 1 semitone with equal steps = 100 cents ) enables a very precise comparison of the interval size in whole numbers. This octave or cent measure is then proportional to the size of the interval.
interval  Frequency ratio  size 

1 octave  2  1200 cents 
2 octaves  4th  2400 cents 
3 octaves  8th  3600 cents 
...  
Octaves  
Octaves  
equal semitone = ^{1} ⁄ _{12} octave  100 cents  
minor third  6: 5  
major third  5: 4  
Fifth  3: 2  
Pythagorean comma  531441: 524288  
little Diësis  128: 125 
 ( = Logarithm to base 10, = logarithm to base 2).
By using the logarithm when calculating cents, the multiplicative structure of the frequency ratios becomes the additive structure of the intervals again.
 Example:
 Quinte = minor third + major third = 315.641 + 386.314 cents cents = 701.955 cents.
 Pythagorean comma = 12 fifths  7 octaves = 12 701.955 cents  7 1200 cents = 23.460 cents.
 Small Diësis = octave  3 major thirds = 1200 cents  3 386.3137 cents = 41.059 cents (rounded).
Calculation of the interval size and the frequency ratio
If the frequency ratio of the interval, the size of the interval is calculated as:
Example: The fifth has the frequency ratio of . Then their size is calculated
On the other hand, if the interval is, the frequency ratio is calculated as:
Example 1: The interval of the size has the frequency ratio of:
Example 2: The fifth is 702 cents, exactly . The frequency ratio is then calculated as:
Examples of interval spaces
An interval space consists of the set of all intervals of the tone structure to be considered combined with the combination of the addition of the associated intervals.
In the following tables:
 Ok = octave (frequency ratio ),
 H = semitone (frequency ratio ),
 Q = fifth (frequency ratio ),
 Q _{m} = ¼ decimal point mean fifth (frequency ratio ),
 T = third octave (frequency ratio ).
Name of the interval space  Interval space 

The system of fifths interval space in Pythagorean tuning 

The ¼ decimal point meantone system of fifths Interval space of the meantone tuning 

The fifththird system interval space of pure tuning 

The twelvelevel interval space = interval space of equal tuning (error on pure tuning up to 23 cents)  
The 53step interval space (error on pure tuning up to 2 cents)  
The allencompassing interval space (all intervals can be divided as required.) 
Divisibility of intervals
In general, you cannot “divide” intervals by hearing. The "half fifth" ^{1} / _{2} Q would be somewhere between small and large third and is in the tuning system of the Pythagorean neither nor on the mean, pure or equal to stage a mood occurring interval. Also, the half octave ^{1} / _{2} Ok (600 cents) does not exist in the tuning system of the Pythagorean, on the mean or just intonation.
Pythagorean mood
The basis of the Pythagorean tuning is the fifth system with the following intervals:
interval  presentation  Frequency ratio  Size in cents 

octave  Ok (basic interval)  2: 1  1200 
Fifth  Q (basic interval)  3: 2  702 
Whole tone  2Q  Ok  9: 8  204 
Pythagorean major third ( Ditonos )  2 whole tones = 4Q  2Ok  81:64  408 
Fourth  Ok  Q  4: 3  498 
Pythagorean semitone ( Limma )  QuartDitonos = 3Ok  5Q  256: 243  90 
pythagorean chromatic semitone ( apotome )  Whole tone Limma = 7Q  4Ok  2187: 2048  114 
Pythagorean comma  12Q  7Ok  531441: 524288  23 
detailed table 
Midtone mood
The basis of the ¼point meantone tuning is the ¼pointmeantone fifth system with the following intervals:
interval  presentation  Frequency ratio  Indicated in cents 

Octave ok  Ok (basic interval)  2: 1  1200 
Fifth Q _{m}  Q _{m} (basic interval)  697  
Major third  4Q _{m}  2OK = T  5: 4  386 
Fourth  Ok  Q _{m}  503  
Little sext  3Ok  4Q _{m} = Ok  T  8: 5  814 
Minor third  2ok  3Q _{m}  310  
Great sext  3Q _{m}  Ok  890  
Whole tone  2Q _{m}  Ok  193  
Minor seventh  2ok  2Q _{m}  1007  
halftone  3OK  5Q _{m}  117  
Major seventh  5Q _{m}  2Ok  1083  
detailed table 
Names of the Euler's Tonnetz
In the pure tuning, it is not enough just to specify the tone name after the note image. A designation has to be added that shows whether the occurring fifths and thirds are pure. The names of Euler's clay net are helpful for this:
Pure fifths in the circle of fifths: ... es bfcgdae ...
A syntonic comma lower ..., es, b, c, g, d, a, e ... (deep comma before the tone name)
One syntonic comma higher ... 'es' b' c 'g' d 'a' e ... (apostrophe in front of the tone name)
Example: pure major third: c, e and perfect fifth c g.
Example: pure C major scale: cd, efg, a, h c.
Example: pure, A minor scale:, a, hc, d, efg, a.
Each major key is of the form: 1 2, 3 4 5, 6, 7 8 or '1' 2 3 '4' 5 6 7 '8 etc.
Each minor key is of the form: 1 2 '3 4 5' 6 '7 8 or, 1, 2 3, 4, 5 6 7, 8 etc., with "1" for the first tone and "2" for the second tone etc. the scale is up.
Pure mood
The basis of the pure tuning is the fifththird system , which consists of the intervals of the form
 with the frequency relationships.
The main intervals are:
Interval (example)  presentation  Frequency ratio  Indicated in cents 

Octave c c '  Ok (basic interval)  2: 1  1200 
Fifth cg  Q (basic interval)  3: 2  702 
Major third c, e  T (basic interval)  5: 4  386 
Fourth cf.  Ok  Q  4: 3  498 
Small sext c'as  Ok  T  8: 5  814 
Minor third c 'es  Q  T  6: 5  316 
Major sixth c, a  Ok + T  Q  5: 3  884 
Big whole tone cd  2Q  Ok  9: 8  204 
Small whole tone d, e  T  (large whole tone) = Ok + T  2Q  10: 9  182 
Minor seventh gf (1st possibility)  Ok  (large whole tone) = 2Ok  2Q  16: 9  996 
Minor seventh, ag (2nd possibility)  Ok  (small whole tone) = 2Q  T  9: 5  1018 
diatonic semitone, ef  Fourth  T = Ok  Q  T  16:15  112 
chromatic semitone c, cis or d ,, dis 
large whole tone  diatonic semitone = T + 3Q  2Ok small whole tone  diatonic semitone = 2T  Q 
135: 128 25:24 
92 71 
Major seventh ch  Ok  semitone = Q + T  15: 8  1088 
Syntonic comma , ee  2 (large whole tones)  T = 4Q  2Ok  T  81:80  22nd 
detailed table 
Super particulate ratios (engl. Super Particular ratios ) are of the form (n = 1,2,3, ...). The single intervals with such frequency receives nits are in the fifththird system: octave ( ^{2} / _{1} ), fifth ( ^{3} / _{2} ), fourth ( ^{4} / _{3} ), major third ( ^{5} / _{4} ), minor third ( ^{6} / _{5} ) , large whole tone ( ^{9} / _{8} ), small whole tone ( ^{10} / _{9} ), diatonic halftone ( ^{16} / _{15} ), chromatic halftone ( ^{25} / _{24} ) and syntonic comma ( ^{81} / _{80} ). In the fifththird system, the numerator and denominator of these fractions are only products of 2, 3 and 5. ^{}_{}^{}_{}^{}_{}^{}_{}^{}_{}^{}_{}^{}_{}^{}_{}^{}_{}^{}_{}
It is important in this context: Intervals whose frequency ratio is superparticulate cannot be divided (in particular not halved).
In order to find out from a frequency ratio of the fifththird system of which basic intervals the interval is composed, one must calculate the triple logarithm.
Example:
the equation
has the unique solution, called "triple logarithm": and .
The relationship (see syntonic comma ) therefore applies to the interval with the frequency ratio 81:80 .
The scales of pure tuning in the circle of fifths
When modulating into a neighboring key, two tones change, one of which is recognizable with a change in sign, the other slightly by a syntonic comma . This can best be represented with the names of Euler's Tone Network : For the tone that sounds a syntonic comma lower than x, the term, x (deep point x) is used. Correspondingly, 'x (apostrophe x) denotes the tone that is one syntonic comma higher than x. The fifths in the circle of fifths… as es bfcgda… are all pure (frequency ratio 3: 2).
The pure scales in the circle of fifths always have the same appearance:
scale  Scale tones listed in a table  
C flat major  ces  of  ,it  fes  total  , as  , b  ces  , A flat minor  , as  , b  ces  ,of  ,it  fes  total  , as 
G flat major  total  as  , b  ces  of  ,it  , f  total  , E flat minor  ,it  , f  total  , as  , b  ces  of  ,it 
D flat major  of  it  , f  total  as  , b  , c  of  , B flat minor  , b  , c  of  ,it  , f  total  as  , b 
A flat major  as  b  , c  of  it  , f  ,G  as  , F minor  , f  ,G  as  , b  , c  of  it  , f 
E flat major  it  f  ,G  as  b  , c  , d  it  , C minor  , c  , d  it  , f  ,G  as  b  , c 
B flat major  b  c  , d  it  f  ,G  , a  b  ,G minor  ,G  , a  b  , c  , d  it  f  ,G 
F major  f  G  , a  b  c  , d  , e  f  , D minor  , d  , e  f  ,G  , a  b  c  , d 
C major  c  d  , e  f  G  , a  ,H  c  ,A minor  , a  ,H  c  , d  , e  f  G  , a 
G major  G  a  ,H  c  d  , e  , fis  G  , E minor  , e  , fis  G  , a  ,H  c  d  , e 
D major  d  e  , fis  G  a  ,H  , cis  d  , B minor  ,H  , cis  d  , e  , fis  G  a  ,H 
A major  a  H  , cis  d  e  , fis  , g sharp  a  , F sharp minor  , fis  , g sharp  a  ,H  , cis  d  e  , fis 
E major  e  f sharp  , g sharp  a  H  , cis  , dis  e  , c sharp minor  , cis  , dis  e  , fis  , g sharp  a  H  , cis 
B major  H  cis  , dis  e  f sharp  , g sharp  , ais  H  , G sharp minor  , g sharp  , ais  H  , cis  , dis  e  f sharp  , g sharp 
F sharp major  f sharp  g sharp  , ais  H  cis  , dis  ,ice  f sharp  , D flat minor  , dis  ,ice  f sharp  , g sharp  , ais  H  cis  , dis 
C sharp major  cis  dis  ,ice  f sharp  g sharp  , ais  , his  cis  , A sharp minor  , ais  , his  cis  , dis  ,ice  f sharp  g sharp  , ais 
The specified minor keys are natural minor. In the harmonic minor, the 6th and 7th degrees are still to be considered:
 For, Eflat minor ,, c and ,, d
 at, b minor ,, g and ,, a
 at, F minor ,, d and ,, e
 in, C minor ,, a and ,, h
 at, G minor ,, e and ,, f sharp
 with, D minor ,, B and ,, C sharp
 in, A minor ,, f sharp and ,, g sharp
 at, E minor ,, c sharp and ,, d flat
 with, B minor ,, G sharp and ,, a sharp
 with, F sharp minor ,, d flat and ,, ice
 in, c sharp minor ,, a sharp and ,, his
 in, G sharp minor ,, eis and ,, fisis
 at, Dflat minor ,, his and ,, cisis
 at, a sharp minor,, f sharp and, g sharp
When transitioning to the minor key, the following tones should be added
 in ces minor the notes' eses / 'asas /' heses
 in G minor the notes' heses / 'eses /' fes
 in D flat minor the notes' fes / 'heses /' ces
 in A flat minor the notes' ces / 'fes /' ges
 in E flat minor the tones' gt / 'ces /' des
 in B flat minor the notes' des / 'gt /' a flat
 in F minor the notes' a flat / 'des /' es
 in c minor the tones' es / 'a flat /' b
 in G minor the notes' b / 'es /' f
 in D minor the notes' f / 'b /' c
 in A minor the notes' c / 'f /' g
 in E minor the notes' g / 'c /' d
 in B minor the notes' d / 'g /' a
 in F sharp minor the notes' a / 'd /' e
 in c sharp minor the notes' e / 'a /' h
The cent values of the tones are calculated as follows:
Adjacent tones that differ only by the schism (2 cents) are marked with *.  
volume  Cent value  Occurrence  

c  0 *  in C major  
, his  2 *  from C sharp major  
'c  22nd  from D minor  
,, cis  71  ab, E minor  
of  90 *  from A flat major  
, cis  92 *  from D major  
'of  112 *  from F minor  
cis  114 *  from B major  
,, d  161  from, F minor  
, d  182  from F major  
'eses  202 *  in G minor  
d  204 *  in C major  
'd  225  from E minor  
,it  273 *  from G flat major  
,, dis  275 *  from, F sharp minor  
it  294 *  from B flat major  
, dis  296 *  from E major  
'it  316 *  in c minor  
dis  318 *  from C sharp major  
,, e  365  from, G minor  
fes  384 *  from C flat major  
, e  386 *  in C major  
'fes  406 *  from a flat minor  
e  408 *  from D major  
'e  429  from F sharp minor  
, f  477 *  from A flat major  
,,ice  478 *  off, G sharp minor  
f  498 *  in C major  
,ice  500 *  from F sharp major  
'f  520  from g minor  
,, f sharp  569  from, A minor  
total  588 *  from D flat major  
, fis  590 *  from G major  
'total  610 *  from B flat minor  
f sharp  612 *  from E major  
,,G  659  ab, B flat minor  
,G  680 *  from B flat major  
,, fisis  682 *  ab, a sharp minor  
G  702  in C major  
'G  723  from A minor  
, as  771 *  from C flat major  
,, g sharp  773 *  from, B minor  
as  792 *  from E flat major  
, g sharp  794 *  from A major  
'as  814 *  in c minor  
g sharp  816 *  from F sharp major  
,, a  863  ab, C minor  
, a  884 *  in C major  
,, gisis  886 *  ab, a sharp minor  
'heses  904 *  from d flat minor  
a  906 *  from G major  
'a  927  from B minor  
, b  975 *  from D flat major  
,, ais  977 *  ab, C sharp minor  
b  996 *  from F major  
, ais  998 *  from B major  
'b  1018  in c minor  
,,H  1067  ab, D minor  
ces  1086 *  from G flat major  
,H  1088 *  in C major  
'ces  1108 *  from G flat major  
H  1110 *  from A major  
'H  1131  from c sharp minor  
,, c  1157  from, E flat minor  
, c  1178 *  from E flat major  
,, his  1180 *  ab, D flat minor  
c '  1200  in C major 
The calculation of the cent values here can be carried out according to the following scheme. With p = 1/12 Pythagorean comma ≈ 2.0 cents, the Pythagorean circle of fifths results in ... es = 3003p b = 10002p f = 500pc = 0 g = 700 + pd = 200 + 2p a = 900 + 3p ... arranged according to semitones:
Equal  Pythagorean  enharmonic 

0  c = 0  his = 12p 
100  cis = 100 + 7p  des = 1005p 
200  d = 200 + 2p  eses = 20010p 
300  dis = 300 + 9p  es = 3003p 
400  e = 400 + 4p  fes = 4008p 
500  f = 500p  ice = 500 + 11p 
600  f sharp = 600 + 6p  tot = 6006p 
700  g = 700 + p  asas = 70011p 
800  g sharp = 800 + 8p  as = 8004p 
900  a = 900 + 3p  heses = 9009p 
1000  ais = 1000 + 10p  b = 10002p 
1100  h = 1100 + 5p  ces = 11007p 
1200  c = 1200  deses = 120012p 
With p = ^{1} / _{12} Pythagorean comma ≈ 2.0 cents and K = syntonic point ≈ 21.5 cents is calculated as:
 ,, cis = (100 + 7p2K) Cent = 71 cents (= interval c ,, cis = interval from c to ,, cis)
 'as = 8004p + K = 814 cents (= interval from c' as )
 Interval ,, cis' as = (70011p + 3K) cents = 743 cents.
Frequency ratio 2 ^{(70011p + 3K) / 1200} = ^{192} / _{125}
Equal mood
The basis of equal tuning is the 12step interval space with the following intervals:
interval  presentation  Size in cents 

halftone  H  100 cents 
Whole tone  2H  200 cents 
minor third  3H  300 cents 
major third  4H  400 cents 
...  
detailed table 
The division of the octave into 53 pitches
The basis of this mood is the 53step interval space . The octave is divided into 53 equal parts.
At the time of Zarlinus (1517–1590) it was taught in music schools that the major third can be intone only and that there are deviations from the Pythagorean tuning. It has been taught that the scale should be intoned in such a way that parts can be assigned to the following intervals.
 cd = fg = 9 parts (large whole tone)
 de = ga = 8 parts (small whole tone)
 ef = hc = 5 parts (diatonic semitone)
If you note the distance of the scale from C in brackets and the distance between the notes written lower down, the C major scale reads:
c(0) _{9} d(9) _{8} ,e(17) _{5} f(22) _{9} g(31) _{8} ,a(39) _{9} ,h(48) _{5} c(53)
, e ("low point e") means here in a modification of Euler's notation : ", e sounds 1/53 octave lower than e" etc.
The scale is divided into 53 parts here, with
große Terz c,e = 17 Teile Quinte = cg = 31 Teile
The scales in the circle of fifths are then written as follows (for the sake of clarity, all notated from c):
CDur: c(0) d(9) ,e(17) f(22) g(31) ,a(39) ,h(48) c(53) GDur: c(0) d(9) ,e(17) ,fis(26) g(31) a(40) ,h(48) c(53) DDur: ,cis(4) d(9) e(18) ,fis(26) g(31) a(40) ,h(48) ,cis(57) ADur: ,cis(4) d(9) e(18) ,fis(26) ,gis(35) a(40) h(49) ,cis(57) EDur: ,cis(4) ,dis(13) e(18) fis(27) ,gis(35) a(40) h(49) ,cis(57) HDur: cis(5) ,dis(13) e(18) fis(27) ,gis(35) ,ais(44) h(49) cis(58) FisDur: cis(5) ,dis(13) ,eis(22) fis(27) gis(36) ,ais(44) h(49) cisis(61) CisDur: cis(5) dis(14) ,eis(22) fis(27) gis(36) ,ais(44) ,his(53) cis(58)
CDur: c(0) d(9) ,e(17) f(22) g(31) ,a(39) ,h(48) c(53) FDur: c(0) ,d(8) ,e(17) f(22) g(31) ,a(39) b(44) c(53) BDur: c(0) ,d(8) es(13) f(22) ,g(30) ,a(39) b(44) c(53) Esdur: ,c(52) ,d(8) es(13) f(22) ,g(30) as(35) b(44) ,c(52) Asdur: ,c(52) des(4) es(13) ,f(21) ,g(30) as(35) b(44) ,c(52) Desdur: ,c(52) des(4) es(13) ,f(21) ges(26) as(35) ,b(43) ,c(52) Gesdur: ces(48) des(4) ,es(12) ,f(21) ges(26) as(35) ,b(43) ces(48) Cesdur: ces(48) des(4) ,es(12) fes(17) ges(26) ,as(34) ,b(43) ces(48)
Hermann von Helmholtz writes the following in his theory of tone sensations: “If you want to produce a scale in an almost exact natural tuning, which allows unlimited modulation, ... this can be achieved by dividing the octave into 53 equal intervals, as suggested by Mercator . "
The 53level uniform tuning
step  Distance from c in cents  Tones of the circle of fifths in cents 

00  0  c = 0, his = 2 
01  23  'c = 22 his = 23 
02  45  
03  68  ,, cis = 71 
04  91  des = 90, cis = 92 
05  113  'des = 112 cis = 114 
06  136  '' des = 133 
07  158  ,, d = 161 
08  181  , d = 182 ,, cisis = 184 
09  204  'eses = 202 d = 204, cisis = 206 
10  226  'd = 225 cisis = 227 
11  249  
12  272  , es = 273 ,, dis275 
13  294  es = 294, dis = 296 
14th  317  'es = 316 dis = 318 
15th  340  '' es = 337 
16  362  ,, e = 365 
17th  385  fes = 384, e = 386 
18th  408  'fes = 406 e = 408 
19th  430  'e = 429 
20th  453  
21st  475  , f = 477 ,, ice = 478 
22nd  498  f = 498, ice = 500 
23  521  'f = 520 ice = 522 
24  543  
25th  566  ,, f sharp = 569 
26th  589  ges = 588, f sharp = 590 
27  611  'ges = 610 f sharp = 612 
28  634  
29  657  ,, g = 659 
30th  679  , g = 680 ,, fisis = 682 
31  702  g = 702, fisis = 704 
32  725  'g = 723 fisis = 725 
33  747  
34  770  , as = 771 ,, g sharp = 772 
35  792  as = 792, g sharp = 794 
36  815  'as = 814 g sharp = 816 
37  838  
38  860  ,, a = 863 
39  883  , a = 884 ,, g sharp = 886 
40  906  'heses = 904 a = 906 
41  928  'a = 927 g sharp = 929 
42  951  
43  974  , b = 975 ,, ais = 977 
44  996  b = 996, ais = 998 
45  1019  'b = 1018 ais = 1020 
46  1042  
47  1064  ,, h = 1067 
48  1087  ces = 1086, h = 1088 
49  1109  'ces = 1108 h = 1110 
50  1132  'h = 1131 
51  1155  ,, c = 1157 
52  1177  , c = 1178 ,, his1180 
53  1200  c = 1200 
Interval table with comparison with the pure mood
interval  Size in cents  Level in the 53 system  Size in cents  Difference exactly 

diet. halftone  112  05  113  −1.48 
small whole tone  182  08  181  +1.29 
large whole tone  204  09  204  +0.13 
minor third  316  14th  317  −1.34 
major third  386  17th  385  +1.40 
Fourth  498  22nd  498  −0.07 
Tritone  590  26th  589  +0.07 
Fifth  702  31  702  −1.41 
little sext  814  36  815  −1.01 
great sext  884  39  883  +1.34 
Minor Seventh I.  996  44  996  −0.14 
Minor seventh II  1018  45  1019  −1.27 
major seventh  1088  48  1087  +1.47 
octave  1200  53  1200  0.00 
Schismatic mixup
You can see here: All notes of the circle of fifths are reached with a tolerance of a schism . The tones c and, his, des and, cis etc. differ by the schism of 1.95 cents (see third column in the first table with two tones each).
If you look at all the tones that you get when you look at the circle of fifths to infinity, then these tones are recorded on the 53point scale with a maximum deviation from a Kleisma. According to Tanaka , a Kleisma (Greek: κλεῖσμα "closure") is the distance from '' 'fes to ,,, eis or' '' ges to ,,, fisis or '' 'ces to ,,, his etc. With' ' 'fes = 448.89 cents and ,,, eis = 456.99 cents results in: Kleisma = 8.11 cents .
Examples in detail
Intervals of equal tuning
Frequency ratio  Interval size in cents  Interval designation 

1  0  Prim 
100  equal semitone  
200  equal whole tone  
300  equal minor third  
400  equal major third  
500  equal fourths  
600  equal tritone  
700  equal fifth  
800  equal minor sixth  
900  equal major sixth  
1000  equal minor seventh  
1100  equal major seventh  
2  1200  octave 
Pythagorean tuning intervals
The following table gives an overview of the intervals that can occur in Pythagorean tuning. Each of the intervals was calculated: C Cis, CDes *, CD, CDis *, CEs, CE, ..., Cis Dis *, CisEs, CisE, CisF, CisFis ,…, Des * Es, Des * E,…, D Dis *, DEs, DE,… The intervals were then sorted according to size in cents. At the same intervals, only one representative was selected.
In the Pythagorean tuning, the fifths of the sequence Gb * Des * As * EsBFCGDAEHF sharpC sharpG sharpD sharp * A sharp * are pure (frequency ratio 3: 2).
Note: The tones Gb *, Db *, Aflat *, Dsharp * and A sharp * are not available on a 12point scale. They differ from their enharmonically confused ones by the Pythagorean comma.
Each interval can be clearly represented as the sum of the two basic intervals octave and fifth.
 Ok = octave (frequency ratio 2: 1)
 Q = fifth (frequency ratio 3: 2).
interval  from C to  Frequency ratio  in cents  calculation  Interval designation 

C sharp *  Deses  524288/531441  −23.460  −12Q + 7Ok  Pythagorean diminished second =  Pythagorean comma 
EF  Of  256/243  90.225  −5Q + 3Ok  Pythagorean Limma = Pythagorean small second 
CCis  Cis  2187/2048  113.685  7Q  4Ok  Pythagorean apotome = Pythagorean excessive prim 
C # Eb  Eses  65536/59049  180.450  −10Q + 6Ok  Pythagorean minor third 
CD  D.  9/8  203.910  2Q  Ok  large whole tone = Pythagorean second 
Des * Dis *  Cisis  4782969/4194304  227,370  14Q  8Ok  Pythagorean double excessive prim 
Dis * Ges *  Feses  16777216/14348907  270.675  −15Q + 9Ok  Pythagorean double diminished fourth 
DF  It  32/27  294.135  −3Q + 2Ok  Pythagorean minor third 
EbF sharp  Dis  19683/16384  317,595  9Q  5Ok  Pythagorean excessive second 
C sharp F  Fes  8192/6561  384,360  −8Q + 5Ok  Pythagorean diminished fourth 
CE  E.  81/64  407.820  4Q  2Ok  Pythagorean major third = Ditonos 
Ges * Ais *  Disis  43046721/33554432  431.280  16Q  9Ok  Pythagorean double excessive second 
C sharp Ges *  Geses  2097152/1594323  474,585  −13Q + 8Ok  Pythagorean double diminished fifth 
CF  F.  4/3  498.045  −Q + Ok  Fourth 
EbG sharp  ice  177147/131072  521.505  11Q  6Ok  Pythagorean excessive third 
EB  Ges  1024/729  588.270  −6Q + 4Ok  Pythagorean diminished fifth 
CF sharp  F sharp  729/512  611.730  6Q  3Ok  Pythagorean excessive fourth = Pythagorean tritone 
G sharp es  Asas  262144/177147  678.495  −11Q + 7Ok  Pythagorean diminished sixth 
CG  G  3/2  701.955  Q  Fifth 
EbAis *  Fisis  1594323/1048576  725,415  13Q  7Ok  Pythagorean double augmented fourth 
Ais * ges *  Heseses  67108864/43046721  768.720  −16Q + 10Ok  Pythagorean double diminished seventh 
Ec  As  128/81  792.180  −4Q + 3Ok  Pythagorean minor sext 
C G sharp  G sharp  6561/4096  815.640  8Q  4Ok  Pythagorean excessive fifth 
C sharp B  Heses  32768/19683  882,405  −9Q + 6Ok  Pythagorean diminished seventh 
CA  A.  27/16  905.865  3Q  Ok  Pythagorean major sixth 
Des * Ais *  Gisis  14348907/8388608  929,325  15Q  8Ok  Pythagorean double superficial fifth 
Dis * des *  ceses  8388608/4782969  972.630  −14Q + 9Ok  Pythagorean double diminished octave 
CB  B.  16/9  996.090  −2Q + 2Ok  Pythagorean minor seventh 
Escis  Ais  59049/32768  1019,550  10Q  5Ok  Pythagorean excessive sixth 
C sharp c  ces  4096/2187  1086.315  −7Q + 5Ok  Pythagorean diminished octave 
CH  H  243/128  1109.775  5Q  2Ok  Pythagorean major seventh 
C sharp *  deses  1048576/531441  1176.540  −12Q + 8Ok  Pythagorean diminished ninth (= Ok  Pythagorean diminished second) 
Cc  c  2/1  1200  OK  octave 
Intervals of the ¼ decimal point mean pitch
The following table gives an overview of the intervals that can occur in the midtone tuning. Each of the intervals was calculated: (C)  (Cis), (C)  (Des *), (C)  (D), (C)  (Dis *), (C)  (Es), (C )  (E), ..., (Cis)  (Dis *), (Cis)  (Eb), (Cis)  (E), (Cis)  (F), (Cis)  (Fis), ... , (Des *)  (Es), (Des *)  (E),…, (D)  (Dis *), (D)  (Es), (D)  (E), ... The intervals became then sorted by size (in cents). At the same intervals, only one representative was selected.
The fifths of the sequence (Gb *)  (Db *)  (As *)  (Es)  (B)  (F)  (C)  (G)  (D )  (A)  (E)  (H)  (F sharp)  (C sharp)  (G sharp)  (D sharp *)  (A sharp *) by a quarter of the syntonic comma (frequency ratio 81:80) smaller (or closer) than the perfect fifth. So these fifths have the frequency ratio
Note: The tones (Ges *), (Des *), (As *), (Dis *) and (Ais *) are not available on a 12point scale. They differ from their enharmonically confused ones by the small Diësis (41 cents). Intervals of the form, for example (Cis)  (Des *), however, give an impression of the impurities that occur in enharmonic mixups.
The frequency ratio in the third column is often algebraicirrational. Here means
Each interval can be clearly represented as the sum of the two basic intervals of the meantone fifth system.
 Ok = octave
 Q _{m} = midtone fifth.
The major third T = (C)  (E) can be represented here as T = 4Q _{m}  2Ok. The respective calculation appears in the 4th column.
interval  from C to  Frequency ratio  in cents  calculation  Interval designation 

(Cis)  (Des *)  (Deses)  128: 125  41.059  −12Q _{m} + 7Ok = −3T + Ok  (larger) diminished second = small Diësis 
(C)  (Cis)  (Cis)  (5:16) w ^{3}  76.049  7Q _{m}  4Ok = 2T  Q _{m}  chromatic midtone semitone 
(E)  (F)  (Of)  (8:25) w ^{3}  117.108  −5Q _{m} + 3Ok = −T  Q _{m} + Ok  diatonic midtone semitone 
(Des *)  (Dis *)  (Cisis)  (125: 256) w ^{2}  152.098  14Q _{m}  8Ok = 4T  2Q _{m}  mediumtone double excessive prim 
(CD)  (D)  (1: 2) w ^{2}  193.157  2Q _{m}  Ok  midtone whole tone 
(Cis)  (Es)  (Eses)  (64: 125) w ^{2}  234,216  −10Q _{m} + 6Ok = −3T + 2Q _{m}  midtone diminished third 
(Es)  (F sharp)  (Dis)  (25:32) w  269.206  9Q _{m}  5Ok = 2T + Q _{m}  Ok  meantone excessive second 
(D)  (F)  (It)  (4: 5) w  310.265  −3Q _{m} + 2Ok = −T + Q _{m}  mediumtone minor third 
(Ges *)  (Ais *)  (Disis)  625: 512  345.255  16Q _{m}  9Ok = 4T  Ok  midtone double excessive second 
(Dis *)  (Ges *)  (Feses)  (512: 625) w  351,324  −15Q _{m} + 9Ok = −4T + Q _{m} + Ok  midtone double diminished fourth 
(C)  (E)  (E)  5: 4  386,314  4Q _{m}  2Ok = T  major third 
(Cis)  (F)  (Fes)  32:25  427,373  −8Q _{m} + 5Ok = −2T + Ok  diminished fourth 
(Es)  (G sharp)  (Ice)  (25:64) w ^{3}  462,363  11Q _{m}  6Ok = 3T  Q _{m}  mediumtone excessive third 
(C)  (F)  (F)  (2: 5) w ^{3}  503,422  −Q _{m} + Ok  mediumtone fourth 
(Cis)  (Gb *)  (Geses)  (256: 625) w ^{3}  544,480  −13Q _{m} + 8Ok = −3T  Q _{m} + 2Ok  midtone doubly diminished fifth 
(F)  (H)  (F sharp)  (5: 8) w ^{2}  579,471  6Q _{m}  3Ok = T + 2Q _{m}  Ok  mediumtone excessive fourth, mediumtone tritone 
(Cis)  (G)  (Ges)  (16:25) w ^{2}  620.529  −6Q _{m} + 4Ok = −2T + 2Q _{m}  mediumtone diminished fifth 
(Des *)  (G sharp)  (Fisis)  (125: 128) w  655.520  13Q _{m}  7Ok = 3T + Q _{m}  Ok  midtone double excessive fourth 
(C)  (G)  (G)  w  696,578  Q _{m}  mean fifth 
(G sharp)  (es)  (Asas)  (128: 125) w  737.637  −11Q _{m} + 7Ok = −3T + Q _{m} + Ok  midtone diminished sixth 
(C)  (G sharp)  (G sharp)  25:16  772.627  8Q _{m}  4Ok = 2T  small excessive fifth, double third 
(E)  (c)  (As)  8: 5  813.686  −4Q _{m} + 3Ok = −T + Ok  small sixth 
(Des *)  (Ais *)  (Gisis)  (125: 256) w ^{3}  848.676  15Q _{m}  8Ok = 4T  Q _{m}  midtone doubly excessive fifth 
(Ais *)  (ges *)  (Broom)  1024: 625  854.745  −16Q _{m} + 10Ok = 4T + 2Ok  midtone double diminished seventh 
(C)  (A)  (A)  (1: 2) w ^{3}  889.735  3Q _{m}  Ok = T  Q _{m} + Ok  mediumtone major sixth 
(Cis)  (B)  (Bes)  (64: 125) w ^{3}  930.794  −9Q _{m} + 6Ok = −2T  Q _{m} + 2Ok  midtone diminished seventh 
(Es)  (cis)  (Ais)  (25:32) w ^{2}  965.784  10Q _{m}  5Ok = 2T + 2Q _{m}  Ok  mediumtone excessive sixth 
(D)  (c)  (B)  (4: 5) w ^{2}  1006,843  −2Q _{m} + 2Ok  mediumtone minor seventh 
(G sharp)  (total *)  (ceses)  (512: 625) w ^{2}  1047.902  −14Q _{m} + 9Ok = −4T + 2Q _{m} + Ok  midtone double diminished octave 
(C)  (H)  (H)  (5: 4) w  1082.892  5Q _{m}  2Ok = T + Q _{m}  mean major seventh 
(Cis)  (c)  (ces)  (32:25) w  1123,951  −7Q _{m} + 5Ok = −2T + Q _{m} + Ok  midtone diminished octave 
(Es)  (dis *)  (his)  125: 64  1158.941  12Q _{m}  6Ok = 3T  excessive seventh 
(C)  (c)  (c)  2: 1  1200  OK  octave 
Intervals of pure mood
The following table gives an overview of the intervals that can occur with pure tuning. Starting from the chromatic scale C 'D' D 'E flat , EF, F sharp G' A flat , A 'B, HC, each of the intervals is calculated: C , C sharp / C' Des / CD / C  ,, Dis / C 'Es / C, E /… / , Cis  ,, Dis /, Cis'Es /, Cis, E /, CisF /, Cis, Fis /… / D  ,, Dis / D 'Es / D, E / ... (for the designations see Euler's Tonnetz : "Low point x" with the designation ", x" means ", x" is a syntonic comma lower than "x". "Quotation x" with the designation "' x "is a syntonic comma higher than" x ". The pure C major scale is written as" CD, EFG, A, B c ". The pure C minor scale is written as" CD 'Es FG' A 'B c'). The intervals were then sorted by size (in cents). At the same intervals, only one representative was selected.
The interval reference is C major and C minor with the pure chords C, EG / C'EsG / F, Ac / F'Asc / G, HD and G'Bd / supplemented by more Intermediate tones with the diatonic semitone steps (frequency ratio 16/15) C'Des /, C sharpD / ,, Dsharp, E / F'Ges /, F sharpG / ,, G sharp, A and ,, A sharp, H.
Each interval can be clearly represented as the sum of the three basic intervals of the fifththird system.
 Ok = octave
 Q = fifth and
 T = major third.
The respective calculation appears in the 5th column.
interval  from C to  Frequency ratio  in cents  calculation  Interval designation 

Des, Cis  , His  32805: 32768  1.954  T + 8Q  5Ok  minor augmented seventh  octave, schism 
, Cis'Des  'Deses  2048: 2025  19,553  −2T  4Q + 3Ok  (smaller) diminished second, diaschism 
,, Dis'Es  '' 'Deses  128: 125  41.059  −3T + Ok  (larger) diminished second, little Diësis 
D  ,, Dis  ,, Cis  25:24  70.672  2T  Q  (minor) excessive prim, minor chromatic halftone , minor chroma 
C, Cis  , Cis  135: 128  92.179  T + 3Q  2Ok  (larger) excessive prim, large chromatic halftone , large chroma 
, EF  'Of  16:15  111.731  −T  Q + Ok  small seconds , diatonic semitone 
,FROM  ''Of  27:25  133.238  −2T + 3Q  Ok  (bigger) small seconds, big Limma, 
'Des  ,, Dis  ,,, Cisis  1125: 1024  162.851  3T + 2Q  2Ok  double excessive prim 
D, E  , D  10: 9  182,404  T  2Q + Ok  small whole tone (smaller major second) 
CD  D.  9: 8  203.910  2Q  Ok  large whole tone = Pythagorean whole tone (larger major second) 
, Etotal  'Eses  256: 225  223,463  −2T  2Q + 2Ok  (minor) diminished third 
,, G sharp'B  '' 'Eses  144: 125  244,969  −3T + 2Q  (major) diminished third 
C  ,, Dis  ,, Dis  75:64  274,582  2T + Q  Ok  excessive second 
DF  It  32:27  294.135  −3Q + 2Ok  Pythagorean minor third (impure minor third of the 2nd degree) 
C'Es  'It  6: 5  315.641  −T + Q  minor third 
,, Dis'Ges  '' 'Feses  4096: 3375  335.194  −3T  3Q + 3Ok  double diminished fourth 
'Ges  ,, Ais  ,,, Disis  10125: 8192  366.761  3T + 4Q  3Ok  double excessive second 
C, E  , E  5: 4  386,314  T  major third 
DTotal  'Fes  512: 405  405.866  −T  4Q + 3Ok  (smaller) diminished fourth 
, A, cis  E.  81:64  407.820  4Q  2Ok  Pythagorean major third = Ditonos 
, E'As  '' Fes  32:25  427,373  −2T + Ok  diminished fourth 
'Es  ,, G sharp  ,,,Ice  125: 96  456.986  3T  Q  (minor) excessive third 
F  ,, Ais  ,,Ice  675: 512  478,492  2T + 3Q  2Ok  (major) excessive third 
CF  F.  4: 3  498.045  −Q + Ok  Fourth 
, C sharp 'total  'Geses  8192: 6075  517,598  −2T  5Q + 4Ok  doubly diminished fifth 
, Ad  'F  27:20  519,551  −T + 3Q  Ok  impure fourths (in C major, 2nd degree ad) 
,, Dis'As  '' 'Geses  512: 375  539.104  −3T  Q + 2Ok  doubly diminished fifth 
D  ,, G sharp  ,, Fis  25:18  568.717  2T  2Q + Ok  (minor) excessive fourths 
'Ges, cis  ,, Fisis  6075: 4096  682,402  2T + 5Q  3Ok  double diminished fourth 
C, Fis  , Fis  45:32  590.224  T + 2Q  Ok  Tritone , excessive fourth 
, F sharp c  'Ges  64:45  609.776  −T  2Q + 2Ok  (minor) diminished fifth 
, A'es  '' Ges  36:25  631.283  −2T + 2Q  (major) diminished fifth 
'It  ,, Ais  ,,, Fisis  375: 256  660.896  3T + Q  Ok  double excessive fourth 
THERE  ,G  40:27  680,449  T  3Q + 2Ok  impure fifth (in C major because of the second degree chord) 
CG  G  3: 2  701.955  Q  Fifth 
, H'ges  'Asas  1024: 675  721.508  −2T  3Q + 3Ok  (smaller) diminished sixth 
,, Dis'B  '' 'Asas  192: 125  743.014  −3T + Q + Ok  (larger) diminished sixth 
C  ,, G sharp  ,, G sharp  25:16  772.627  2T  small excessive fifth, double third 
, C sharp, A  As  128: 81  792.180  −4Q + 3Ok  Pythagorean minor sixth 
F, cis  , G sharp  405: 256  794.134  T + 4Q  2Ok  (larger) excessive fifth 
, Ec  'As  8: 5  813.686  −T + Ok  small sixth 
,, Ais'ges  '' 'Broom  16384: 10125  833.239  −3T  4Q + 4Ok  double diminished seventh 
'Des  ,, Ais  ,,, Gisis  3375: 2048  864,806  3T + 3Q  2Ok  double excessive fifth 
C, A  , A  5: 3  884.359  T  Q + Ok  major sixth 
Fd  A.  27:16  905.865  3Q  Ok  pyth. major sixth (in 2nd chord) 
, E'des  '' Bes  128: 75  925,418  −2T  Q + 2Ok  (major) diminished seventh 
'B  ,, g sharp  ,,, Ais  125: 72  955.031  3T  2Q + Ok  (minor) excessive sixth 
C  ,, Ais  ,, Ais  225: 128  976,537  2T + 2Q  Ok  (larger) excessive sixth 
Dc  B.  16: 9  996.090  −2Q + 2Ok  minor minor seventh (= octave  major whole tone) 
C'B  'B  9: 5  1017,596  −T + 2Q  major minor seventh (= octave  small whole tone) 
,, Dis'des  '' 'ceses  2048: 1125  1037.149  −3T  2Q + 3Ok  double diminished octave 
'B  ,, ais  ,,, his  125: 64  1158.941  3T  excessive seventh 
'B, a  ,,H  50:27  1066,762  2T  3Q + 2Ok  (minor) major seventh 
C, H  ,H  15: 8  1088.269  T + Q  major seventh 
, C sharpc  'ces  256: 135  1107.821  −T  3Q + 3Ok  (smaller) probably octave 
,, Disd  '' ces  48:25  1129,328  −2T + Q + Ok  (larger) diminished octave 
'Des, cis  ,, his  2025: 1024  1180,447  2T + 4Q  2Ok  (larger) overm. Seventh 
Cc  c  2: 1  1200  OK  octave 
Intervals sorted by size
Designations:
CCisDes * DDflat * EsE ... Pythagorean scale supplemented by semitones, based on perfect fifths.
(C)  (Cis)  (Des *)  (D)  (Dis *)  (Es)  (E)  (F) … ¼point meantone scale supplemented by semitones, based on meantone fifths ( 696.6 cents).
C, Cis'DesD  ,, D'Es, E ... Pure scale supplemented by semitones (for names see Euler's Tonnetz : "Low point x" with the name ", x" means ", x" is a syntonic one Comma lower than "x". "Quotation x" with the designation "'x" is a syntonic comma higher than "x").
 Ok = octave (frequency ratio 2)
 Q = fifth (frequency ratio 3: 2)
 Q _{m} = midtone fifth (frequency ratio )
 T = major third (frequency ratio 5: 4).
Intervals  from C to 
Frequency ratio  in cents  calculation  Interval designation 

CC  C.  1: 1  0  Prim  
, His  32805: 32768  1.954  8Q + T  5Ok  Schism = difference between Pythagorean and syntonic commas  
, Cis'Des  'Deses  2048: 2025  19,553  −2T  4Q + 3Ok  (smaller) diminished second, diaschism 
'C  81:80  21.506  4Q  T  2Ok  syntonic comma : difference d (C major) and, d (F major)  
Des * Cis  His  531441: 524288  23,460  12Q  7Ok  Pythagorean comma 
(Dis)  (Es) = ,, Dis'Es 
(Deses) = '' 'Deses 
128: 125  41.059  −12Q _{m} + 7Ok = −3T + Ok  (in the pure tuning: larger) diminished second = minor diësis (difference from octave to 3 major thirds). 
'' '' Deses  648: 625  62.565  4Q  4T  Ok  major Diësis = difference of four minor thirds to the octave  
D  ,, Dis  ,, Cis  25:24  70.672  2T  Q  (minor) excessive prim, minor chromatic halftone , minor chroma 
(C)  (Cis)  (Cis)  (5:16) w ^{3}  76.049  7Q _{m}  4Ok  chromatic midtone semitone 
EF  Of  256: 243  90.225  −5Q + 3Ok  Pythagorean Limma = Pythagorean small second 
C, Cis  , Cis  135: 128  92.179  T + 3Q  2Ok  (larger) excessive prim, large chromatic halftone , large chroma 
100  (1:12) Ok  small equal second  
, EF  'Of  16:15  111.731  −T  Q + Ok  small seconds, diatonic semitone 
CCis  Cis  2187: 2048  113.685  7Q  4Ok  Pythagorean apotome = Pythagorean excessive prim 
(E)  (F)  (Of)  (8:25) w ^{3}  117.108  −5Q _{m} + 3Ok  diatonic midtone semitone 
,FROM  ''Of  27:25  133.238  −2T + 3Q  Ok  (bigger) small seconds, big Limma, 
(Des *)  (Dis *)  (Cisis)  (125: 256) w ^{2}  152.098  14Q _{m}  8Ok  mediumtone double excessive prim 
'Des  ,, Dis  ,,, Cisis  1125: 1024  162.851  3T + 2Q  2Ok  double excessive prim 
C # Eb  Eses  65536: 59049  180.450  −10Q + 6Ok  Pythagorean minor third 
D, E  , D  10: 9  182,404  T  2Q + Ok  small whole tone 
(CD)  (D)  (1: 2) w ^{2}  193.157  2Q _{m}  Ok  midtone whole tone 
200  (2:12) Ok  large equal second  
CD  D.  9: 8  203.910  2Q  Ok  large whole tone = Pythagorean second 
, Etotal  'Eses  256: 225  223,463  −2T  2Q + 2Ok  (minor) diminished third 
Des * Dis *  Cisis  4782969: 4194304  227,370  14Q  8Ok  Pythagorean double excessive prim 
(Cis)  (Es)  (Eses)  (64: 125) w ^{2}  234,216  −10Q _{m} + 6Ok  midtone diminished third 
,, G sharp'B  '' 'Eses  144: 125  244,969  −3T + 2Q  (major) diminished third 
(Es)  (F sharp)  (Dis)  (25:32) w  269.206  9Q _{m}  5Ok  meantone excessive second 
Dis * Ges *  Feses  16777216: 14348907  270.675  −15Q + 9Ok  Pythagorean double diminished fourth 
C  ,, Dis  ,, Dis  75:64  274,582  2T + Q  Ok  excessive second 
DF  It  32:27  294.135  −3Q + 2Ok  Pythagorean minor third (impure minor third of the 2nd degree) 
300  (3:12) Ok  minor equal third  
(D)  (F)  (It)  (4: 5) w  310.265  −3Q _{m} + 2Ok  mediumtone minor third 
C'Es  'It  6: 5  315.641  −T + Q  minor third 
EbF sharp  Dis  19683: 16384  317,595  9Q  5Ok  Pythagorean excessive second 
,, Dis'Ges  '' 'Feses  4096: 3375  335.194  −3T  3Q + 3Ok  double diminished fourth 
(Ges *)  (Ais *)  (Disis)  625: 512  345.255  16Q _{m}  9Ok = 4T  Ok  midtone double excessive second. (Disis) = ,,,, Disis. 
(Dis *)  (Ges *)  (Feses)  (512: 625) w  351,324  −15Q _{m} + 9Ok  midtone double diminished fourth 
'Ges  ,, Ais  ,,, Disis  10125: 8192  366.761  3T + 4Q  3Ok  double excessive second 
C sharp F  Fes  8192: 6561  384,360  −8Q + 5Ok  Pythagorean diminished fourth 
(C)  (E) = C, E 
(E) =, E. 
5: 4  386,314  4Q _{m}  2Ok = T  major third 
400  (4:12) Ok  major equal third  
DTotal  'Fes  512: 405  405.866  −T  4Q + 3Ok  (smaller) diminished fourth 
, A, cis  E.  81:64  407.820  4Q  2Ok  Pythagorean major third = Ditonos 
(Cis)  (F) =, E'As 
(Fes) = '' Fes 
32:25  427,373  −8Q _{m} + 5Ok = Ok  2T  diminished fourth 
Ges * Ais *  Disis  602409: 469571  431.280  16Q  9Ok  Pythagorean double excessive second 
'Es  ,, G sharp  ,,,Ice  125: 96  456.986  3T  Q  (minor) excessive third 
(Es)  (G sharp)  (Ice)  (25:64) w ^{3}  462,363  11Q _{m}  6Ok  mediumtone excessive third 
C sharp Ges *  Geses  2097152: 1594323  474,585  −13Q + 8Ok  Pythagorean double diminished fifth 
F  ,, Ais  ,,Ice  675: 512  478,492  2T + 3Q  2Ok  (major) excessive third 
CF  F.  4: 3  498.045  −Q + Ok  Fourth 
500  (5:12) Ok  equal fourths  
(C)  (F)  (F)  (2: 5) w ^{3}  503,422  −Q _{m} + Ok  mediumtone fourth 
, C sharp 'total  'Geses  8192: 6075  517,598  −2T  5Q + 4Ok  doubly diminished fifth 
, Ad  'F  27:20  519,551  −T + 3Q  Ok  impure fourths (in C major, 2nd degree ad) 
EbG sharp  ice  177147: 131072  521.505  11Q  6Ok  Pythagorean excessive third 
,, Dis'As  '' 'Geses  512: 375  539.104  −3T  Q + 2Ok  doubly diminished fifth 
(Cis)  (Gb *)  (Geses)  (256: 625) w ^{3}  544,480  −13Q _{m} + 8Ok  midtone doubly diminished fifth 
11: 8  551,318  Just as a supplement: The Alphorn Fa (the 11th natural tone)  
D  ,, G sharp  ,, Fis  25:18  568.717  2T  2Q + Ok  (minor) excessive fourths 
(F)  (H)  (F sharp)  (5: 8) w ^{2}  579,471  6Q _{m}  3Ok  mediumtone excessive fourth, mediumtone tritone 
EB  Ges  1024: 729  588.270  −6Q + 4Ok  Pythagorean diminished fifth 
C, Fis  , Fis  45:32  590.224  T + 2Q  Ok  Tritone, excessive fourth 
600  (6:12) Ok  equal tritone, excessive equal fourth, diminished equal fifth  
, F sharp c  'Ges  64:45  609.776  −T  2Q + 2Ok  (minor) diminished fifth 
CF sharp  F sharp  729: 512  611.730  6Q  3Ok  Pythagorean excessive fourth = Pythagorean tritone 
(Cis)  (G)  (Ges)  (16:25) w ^{2}  620.529  −6Q _{m} + 4Ok  mediumtone diminished fifth 
, A'es  '' Ges  36:25  631.283  −2T + 2Q  (major) diminished fifth 
(Des *)  (G sharp)  (Fisis)  (125: 128) w  655.520  13Q _{m}  7Ok  mediumtone double excessive fourth 
'It  ,, Ais  ,,, Fisis  375: 256  660.896  3T + Q  Ok  double excessive fourth 
G sharp es  Asas  262144: 177147  678.495  −11Q + 7Ok  Pythagorean diminished sixth 
THERE  ,G  40:27  680,449  T  3Q + 2Ok  impure fifth (in C major because of the second degree chord) 
'Ges, cis  ,, Fisis  6075: 4096  682,402  2T + 5Q  3Ok  double diminished fourth 
(C)  (G)  (G)  w  696,578  Q _{m}  mean fifth 
700  (7:12) Ok  equal fifth  
CG  G  3: 2  701.955  Q  Fifth 
, H'ges  'Asas  1024: 675  721.508  −2T  3Q + 3Ok  (smaller) diminished sixth 
EbAis *  Fisis  1594323: 1048576  725,415  13Q  7Ok  Pythagorean double augmented fourth 
(G sharp)  (es)  (Asas)  (128: 125) w  737.637  −11Q _{m} + 7Ok  midtone diminished sixth 
,, Dis'B  '' 'Asas  192: 125  743.014  −3T + Q + Ok  (larger) diminished sixth 
Ais * ges *  Broom  67108864: 43046721  768.720  −16Q + 10Ok  Pythagorean double diminished seventh 
(C)  (G sharp) = C  ,, G sharp 
(G sharp) = ,, G sharp 
25:16  772.627  8Q _{m}  4Ok = 2T  (In the pure tuning smaller) excessive fifth, double third 
Ec  As  128: 81  792.180  −4Q + 3Ok  Pythagorean minor sixth 
F, cis  , G sharp  405: 256  794.134  T + 4Q  2Ok  (larger) excessive fifth 
800  (8:12) Ok  small equal sixth  
, Ec  'As  8: 5  813.686  −T + Ok  small sixth 
C G sharp  G sharp  6561: 4096  815.640  8Q  4Ok  Pythagorean excessive fifth 
,, Ais'ges  '' 'Broom  16384: 10125  833.239  −3T  4Q + 4Ok  double diminished seventh 
(Des *)  (Ais *)  (Gisis)  (125: 256) w ^{3}  848.676  15Q _{m}  8Ok  mediumtone doubly excessive fifth 
(Ais *)  (ges *)  (Broom)  1024: 625  854.745  −16Q _{m} + 10Ok = −4T + 2Ok  midtone double diminished seventh. (Broom) = '' '' broom. 
'Des  ,, Ais  ,,, Gisis  3375: 2048  864,806  3T + 3Q  2Ok  double excessive fifth 
C sharp B  Bes  32768: 19683  882,405  −9Q + 6Ok  Pythagorean diminished seventh 
C, A  , A  5: 3  884.359  T  Q + Ok  major sixth 
(C)  (A)  (A)  (1: 2) w ^{3}  889.735  3Q _{m}  Ok  mediumtone major sixth 
900  (9:12) Ok  large equal sixth  
CA  A.  27:16  905.865  3Q  Ok  Pythagorean major sixth 
, E'des  '' Bes  128: 75  925,418  −2T  Q + 2Ok  (major) diminished seventh 
Des * Ais *  Gisis  14348907: 8388608  929,325  15Q  8Ok  Pythagorean double superficial fifth 
(Cis)  (B)  (Bes)  (64: 125) w ^{3}  930.794  −9Q _{m} + 6Ok  midtone diminished seventh 
'B  ,, g sharp  ,,, Ais  125: 72  955.031  3T  2Q + Ok  (minor) excessive sixth 
(Es)  (cis)  (Ais)  (25:32) w ^{2}  965.784  10Q _{m}  5Ok  mediumtone excessive sixth 
7: 4  968.826  i  Just to complement: The natural septime , the 7th natural tone, sometimes referred to as i.  
Dis * des *  Ceses  8388608: 4782969  972.630  −14Q + 9Ok  Pythagorean double diminished octave 
C  ,, Ais  ,, Ais  225: 128  976,537  2T + 2Q  Ok  (larger) excessive sixth 
Dc  B.  16: 9  996.090  −2Q + 2Ok  Pythagorean minor seventh 
1000  (10:12) Ok  small equal seventh  
(D)  (c)  (B)  (4: 5) w ^{2}  1006,843  −2Q _{m} + 2Ok  mediumtone minor seventh 
C'B  'B  9: 5  1017,596  −T + 2Q  minor seventh 
Escis  Ais  59049: 32768  1019,550  10Q  5Ok  Pythagorean excessive sixth 
,, Dis'des  '' 'ceses  2048: 1125  1037.149  −3T  2Q + 3Ok  double diminished octave 
(G sharp)  (total *)  (ceses)  (512: 625) w ^{2}  1047.902  −14Q _{m} + 9Ok  midtone double diminished octave 
'B, a  ,,H  50:27  1066,762  2T  3Q + 2Ok  (minor) major seventh 
(C)  (H)  (H)  (5: 4) w  1082.892  5Q _{m}  2Ok  mean major seventh 
C sharp c  Ces  4096: 2187  1086.315  −7Q + 5Ok  Pythagorean diminished octave 
C, H  ,H  15: 8  1088.269  T + Q  major seventh 
1100  (11:12) Ok  major seventh of the same order  
, C sharpc  'ces  256: 135  1107.821  −T  3Q + 3Ok  (smaller) diminished octave 
CH  H  243: 128  1109.775  5Q  2Ok  Pythagorean major seventh 
(Cis)  (c)  (ces)  (32:25) w  1123,951  −7Q _{m} + 5Ok  midtone diminished octave 
,, Disd  '' ces  48:25  1129,328  −2T + Q + Ok  (larger) diminished octave 
(Es)  (dis *) = 'B  ,, ais 
(his) = ,,, his 
125: 64  1158.941  12Q _{m}  6Ok = 3T  excessive seventh 
C sharp *  deses  1048576: 531441  1176.540  −12Q + 8Ok  Pythagorean diminished ninth (= Ok  Pythagorean diminished sec.) 
'Des, cis  ,, his  2025: 1024  1180,447  2T + 4Q  2Ok  (major) excessive seventh 
Cc  2: 1  1200  OK  octave 
Description of the tone structure in terms of hearing psychology without acoustics
The understanding of tones and intervals can be conveyed without physical terms. The first known auditorypsychological mathematical descriptions of a sound system come from Aristoxenus . The pitch of a certain tone can be determined and passed on by a "original" tuning fork without specifying its frequency (similar to how the unit meter can be determined by the original meter ). A teacher can “show” his student what an octave, a fifth, a major third, etc. is without going into the frequency relationship of the vibrations. The underlying theory is explained below.
Description of the tone structure as an algebraic structure
With a tone structure you have a set of tones on the one hand and a set of intervals on the other , for which the following rules apply:
A unique interval of is assigned to each tone pair .
Conversely, if the fundamental tone and the interval are known, the final tone is uniquely determined.
The successive execution of intervals defines an addition: is and , then is .
Intervals can be compared: We write when the final note of is higher than the final note of with the same root note.
Everyday calculating with quantities applies to intervals on the additive musical level . From a mathematical point of view, the interval space is an Archimedean ordered commutative group . From a purely auditory psychological perspective, this results from the experience of musical practice.
To measure the interval size suitable as a unit , the octave with the subunit cents 1200 cents = 1 octave.
Example 1 (octave = 12 semitones)
 If you go up 12 fifths, you get the starting tone again with an octave (approximately): 12 fifths = 7 octaves. The result is a fifth = ^{7} ⁄ _{12} octave = 700 cents . Corresponding:
 If you go up three major thirds, you get (roughly) an octave. So major third = ^{1} ⁄ _{3} octave = 400 cents . You can now continue to calculate :
 Minor third = fifth  major third = ^{1} ⁄ _{4} octave = 300 cents and
 Semitone = major third  minor third = ^{1} ⁄ _{12} octave = 100 cents .
 From a purely psychological point of view, you can divide the octave (approximately) into 12 semitones and represent each interval as a multiple of semitones.
Example 2 (octave = 53 commas)
At the time of Zarlino (16th century) one taught in music schools: The large whole tone has a size of 9 parts , the small whole tone of 8 parts and the diatonic semitone of 5 parts .
It follows from this:
 Octave = 1200 cents = 3 large whole tones + 2 small whole tones + 2 diatonic semitones = 53 parts
 Major third = major whole tone + small whole tone = 17 parts = 385 cents
 minor third = major whole tone + diatonic semitone = 14 parts = 317 cents
 Fifth = major third + minor third = 31 parts = 702 cents
With this classification, the proportions for the pure intonation of tone steps can be easily described.
 diatonic semitone = 5 parts
 small whole tone = 8 parts
 Large whole tone = 9 parts
 diminished third (see example B  G sharp = BA (5 parts) + A G sharp (5 parts) = 10 parts
This division of the octave into 53 parts can be derived purely mathematically from two integer relationships for the three intervals Ok = octave, Q = fifth and gT = major third without reference to the frequency relationships. (Confirmed on the spinet by Neumaier)
 53 Q = 31 Ok (no difference between initial tone and octaves audible after 53 fifths)
 12 Q  7Ok = 4Q  2Ok gT (no difference between syntonic comma and Pythagorean comma audible)
Dissolved This system of equations with k = ^{1} / _{53} Ok:
 Ok = 53k
 Q = 31k
 gT = 17k
Now you can define further intervals and represent them as multiples of k: For example:
 Fourth = Ok  Q = 22k
 minor third = Q  gT = 14k
 large whole tone = 2Q  Ok = 9k
 small whole tone = gT  large whole tone = 8k
 diatonic semitone = gT  minor third = 5k
Example 3 (the fifth third system)
Axiom : There is a homomorphism f from the additive group of the interval space with the intervals Ok = octave, Q = fifth and gT = major third into the multiplicative group of real numbers, for which applies:
 f (Ok) = 2
 f (Q) = ^{3} / _{2} and
 f (GT) = ^{5} / _{4}
Homomorphism says: f (i _{1} + i _{2} ) = f (i _{1} ) • f (i _{2} ) and f (r • i) = f (i) ^{r} for intervals i _{1} , i _{2} and i as well as for a real number r.
For the calculation of r and s for Q = r • Ok and gT = s • Ok it follows with the subunit Ok = 1200 cents:
 f (r • Ok) = 2 ^{r} = ^{3} / _{2} ie Q = log _{2} ( ^{3} / _{2} ) Ok = 701.955 cents
 f (s • Ok) = 2 ^{s} = ^{5} / _{4} so gT = log _{2} ( ^{5} / _{4} ) Ok = 386.314 cents.
Web links
Remarks
 ↑ Sources: Rudolf Wille : "Mathematics and Music Theory", in Music and Numbers, Bonn  Bad Godesberg 1976, pp. 233–264 and "Mathematical Language in Music Theory", in the Yearbook Overviews Mathematics 1980, pp. 167–184. Wilfried Neumaier: "What is a tone system? A historicalsystematic theory of the western tone systems, based on the ancient theorists Aristoxenus, Eucleides and Ptolemaios, represented by means of modern algebra." Verlag Peter Lang, Frankfurt / Main ISBN 3820494928
 ↑ The information relates to the pure tuning , with which intervals can be assigned wholenumber relationships.
 ↑ Euclid calculated with proportions, namely with string ratios that correspond to the reciprocal of the frequency ratios.
 ↑ Except for powers of two (integer multiples of the octave) logarithms of two of pure ( rational ) frequency ratios are irrational and even transcendent .
 ↑ The cent unit is so small that the difference from, for example, a schism (2 cents) is to be located at the “limit of perceptible tone differences”.
 ↑ The Pythagorean Archytas of Taranto (approx. 400 BC) proved that the octave, the fifth and fourth, etc., cannot be halved if commensurable quantities are used as a basis.
 ↑ https://eclass.uoa.gr/modules/document/index.php?course=MATH379&download=/5122bb9396op/5122bf834msr.pdf
 ↑ Note: 70011p has the frequency ratio: (2/3) ^{11} • 2 ^{7} (11 fifths ^{octaved} down, see asas) ⇒ 2 ^{(70011p + 3K) / 1200} = (2/3) ^{11} • 2 ^{7} • (81/80) ^{3} = ^{192} / _{125}
 ↑ In Euler's notation  a notation for the pure mood , the deep decimal means a decrease by the syntonic comma = 21.5 cents. Here the low point means a decrease of 1200/53 cents = 22.6 cents. A deviation of 1 cent cannot be distinguished by hearing.
 ↑ The approximation of the octave by fifths (12 fifths corresponds to approximately 7 octaves) led to equal tempering by dividing the octave into 12 equal intervals. It has the disadvantage of very rough major thirds. The closest approximation (41 fifths corresponds to approximately 24 octaves) is better for an equal division of the octave into 41 parts, but not satisfactory with regard to the major third and the displacements around a syntonic comma. The following approximation of the octave (53 fifths corresponds to almost exactly 31 octaves) has a convincing advantage: If you divide the octave into 53 equal intervals, then the 31st degree (701.887 cents) corresponds very precisely to the perfect fifth (701.955 cents) and  that is particularly important and not to be expected  the 17th step (384.906 cents) of the major third (386.314 cents) and the shift by a syntonic comma (21.506 cents) by almost exactly one step (22.642 cents) of this tempering.
 ↑ Hermann von Helmholtz : The theory of tone sensations as a physiological basis for the theory of music . Vieweg, Braunschweig 1863, p. 531 (reprint: MinervaVerlag, Frankfurt am Main 1981), ISBN 3810207152 , ( excerpt ). Helmholtz continues: “Mr. Bosanquet has recently used such a tuning for a harmonium with a symmetrium arranged keyboard. [An elementary Treatease on Musical Intervals and Temperament by. RHM Bosanquet, London. Macmillan 1875] ”.
 ↑ In contrast to the pure or mediumtone tuning, in the Pythagorean tuning the tone C sharp is higher than Des or  better known  His is higher than c. Therefore the note Deses is lower than C and the interval CisDeses * or CDeses is notated negatively. The interval Cisdes * or Cdeses, increased by one octave, is notated here as a Pythagorean reduced ninth. To get from C sharp to D flat or from His to C you have to go twelve fifths down and seven octaves up. The Pythagorean comma is known as an interval = twelve fifths up and seven octaves down.

↑ ^{a } ^{b } ^{c} Winfried Neumaier p. 64ff shows: Already Aristoxenus calculated in the 3rd century BC as described in this section. He calculated with octave, fifth, fourth = octave  fifth, whole tone = fifth  fourth and with the help of the axiom that the whole tone can still be divided, with semitones and even with quarter tones (but not with pure major thirds). As an empirical value he “heard”: fourth = 2½ whole tones and based on this a coherent theory. (Euclid recognized: 2½ whole tones are slightly smaller than the fourth.)
According to Neumaier, for example, on the spinet you can still verify: 53 fifths = 31 octaves (no more hearing difference) and this then results in: fifth = ^{31} ⁄ _{53} octave = 702 cents . So you can determine very precise values for interval sizes without acoustics.  ↑ In addition to clarity, this is important for the interpretation of historical descriptions of the pitch system. According to Wilfried Neumaier What is a sound system. A historicalsystematic theory of the occidental sound systems, based on the ancient theorists Aristoxenus, Eucleides and Ptolemaios, presented with the means of modern algebra (= sources and studies on the history of music from antiquity to the present. Vol. 9). Peter Lang, Frankfurt am Main a. a. 1986, ISBN 3820494928
 ↑ The next better approximation would be: 28 major thirds = 9 octaves (hardly comprehensible to the ear), so major third = ^{9} ⁄ _{28} octaves = 386 cents .

↑ The exact values of the intervals in the pure tuning , which are calculated with the help of the frequency ratios, differ only slightly from the values determined here:
 major third (neat) = 1200 • log _{2} ( ^{5} / _{4} ) = 386 cents
 minor third (neat) = 1200 • log _{2} ( ^{6} / _{5} ) = 316 cents
 Fifth (neat) = 1200 • log _{2} ( ^{3} / _{2} ) = 702 cents

↑ The deviation from the pure mood is smaller than a schism (2 cents ).
 Ok = 1200 cents (So k = ^{1,200} / _{53} cents = 22.642 cents)
 Q = 1200 * log _{2} ( ^{3} / _{2} ) = 701.955 cent cents. 31k = 701.887 cents
 gT = 1200 * log _{2} ( ^{5} / _{4} )) = 386.3137 cent cents. 17k = 384.906 cents
 ↑ If no scalar multiplication is assumed in the interval space, the definition applies . This smallest upper bound does not always have to exist. For example, the interval space of all multiples of Ok, Q and gT does not contain .