# 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 ). ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle f_ {1}}$${\ displaystyle f_ {2}}$ ${\ displaystyle xy}$${\ displaystyle q = f_ {2}: f_ {1}}$

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 ) . ${\ displaystyle x}$${\ displaystyle f_ {1}}$${\ displaystyle i}$${\ displaystyle q}$${\ displaystyle y}$${\ displaystyle f_ {2} = f_ {1} \ cdot q}$${\ displaystyle i = xy}$

Example: If a 'has the frequency , then the tone c' ', which sounds a minor third with the frequency ratio higher, has the frequency .${\ displaystyle f_ {1} = 440 \, \ mathrm {Hz}}$${\ displaystyle q = 6: 5}$${\ displaystyle \ textstyle f_ {2} = 440 \, \ mathrm {Hz} \ cdot {\ frac {6} {5}} = 528 \, \ mathrm {Hz}}$

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 ce-gis-his ), 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 .${\ displaystyle \ textstyle {\ frac {6} {5}} \ cdot {\ frac {5} {4}} = {\ frac {3} {2}}}$
The frequency ratio of the Pythagorean comma is calculated to and that of the small Diësis to .${\ displaystyle \ textstyle {\ big (} {\ frac {3} {2}} {\ big)} ^ {12}: 2 ^ {7} = 531441: 524288}$${\ displaystyle \ textstyle 2: {\ big (} {\ frac {5} {4}} {\ big)} ^ {3} = 128: 125}$

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.

example
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 sub-unit 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.

example
interval Frequency ratio size
1 octave 2 1200 cents
2 octaves 4th 2400 cents
3 octaves 8th 3600 cents
...
${\ displaystyle k}$ Octaves ${\ displaystyle 2 ^ {k}}$ ${\ displaystyle 1200 \ cdot k \, {\ text {Cent}}}$
${\ displaystyle \ log _ {2} (q)}$ Octaves ${\ displaystyle q}$  ${\ displaystyle 1200 \ cdot \ log _ {2} (q) \ {\ text {Cent}}}$
equal semitone = 112 octave ${\ displaystyle {\ sqrt [{12}] {2}}}$ 100 cents
minor third 6: 5 ${\ displaystyle 1200 \ cdot \ log _ {2} {\ big (} {\ tfrac {6} {5}} {\ big)} \, {\ text {Cent}} = 315 {,} 641 \, { \ text {Cent}}}$
major third 5: 4 ${\ displaystyle 1200 \ cdot \ log _ {2} {\ big (} {\ tfrac {5} {4}} {\ big)} \, {\ text {Cent}} = 386 {,} 314 \, { \ text {Cent}}}$
Fifth 3: 2 ${\ displaystyle 1200 \ cdot \ log _ {2} {\ big (} {\ tfrac {3} {2}} {\ big)} \, {\ text {Cent}} = 701 {,} 955 \, { \ text {Cent}}}$
Pythagorean comma 531441: 524288 ${\ displaystyle 1200 \ cdot \ log {\ big (} {\ tfrac {531441} {524288}} {\ big)} \, {\ text {cents}} = 23 {,} 460 \, {\ text {cents }}}$
little Diësis 128: 125 ${\ displaystyle 1200 \ cdot \ log _ {2} {\ big (} {\ tfrac {128} {125}} {\ big)} \, {\ text {Cent}} = 41 {,} 059 \, { \ text {Cent}}}$
${\ displaystyle \ log _ {2} x = {\ frac {\ log x} {\ log 2}}}$( = Logarithm to base 10, = logarithm to base 2).${\ displaystyle \ log}$${\ displaystyle \ log _ {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: ${\ displaystyle q}$${\ displaystyle i}$

${\ displaystyle i = \ log _ {2} (q) {\ text {octave}} = 1200 \ cdot \ log _ {2} (q) {\ text {Cent}}}$

Example: The fifth has the frequency ratio of . Then their size is calculated ${\ displaystyle {\ tfrac {3} {2}}}$

${\ displaystyle i = \ log _ {2} ({\ tfrac {3} {2}}) {\ text {octave}} = 1200 \ cdot \ log _ {2} ({\ tfrac {3} {2} }) {\ text {Cent}} = 702 {\ text {Cent}}}$

On the other hand, if the interval is, the frequency ratio is calculated as: ${\ displaystyle i}$${\ displaystyle q}$

${\ displaystyle q = 2 ^ {\ frac {i} {\ mathrm {octave}}} = 2 ^ {\ frac {i} {1200 \, \ mathrm {cent}}}}$

Example 1: The interval of the size has the frequency ratio of: ${\ displaystyle 3 \ mathrm {octaves} = 3600 \ mathrm {cent}}$

${\ displaystyle q = 2 ^ {\ frac {3 \ mathrm {octaves}} {\ mathrm {octave}}} = 2 ^ {\ frac {3600 \ mathrm {cents}} {1200 \, \ mathrm {cents}} } = 2 ^ {3} = 8}$

Example 2: The fifth is 702 cents, exactly . The frequency ratio is then calculated as: ${\ displaystyle i = 1200 \ cdot \ log _ {2} ({\ tfrac {3} {2}}) {\ text {Cent}}}$${\ displaystyle q}$

${\ displaystyle q = 2 ^ {\ frac {i} {1200 \, \ mathrm {Cent}}} = 2 ^ {\ frac {1200 \ cdot \ log _ {2} ({\ tfrac {3} {2} }) {\ text {Cent}}} {1200 \, \ mathrm {Cent}}} = {\ tfrac {3} {2}}}$

## 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 ),${\ displaystyle 2 {\ mathrel {\ hat {=}}} 1200 \, {\ text {Cent}}}$
• H = semitone (frequency ratio ),${\ displaystyle {\ sqrt [{12}] {2}} {\ mathrel {\ hat {=}}} 100 \, {\ text {Cent}}}$
• Q = fifth (frequency ratio ),${\ displaystyle {\ tfrac {3} {2}} {\ mathrel {\ hat {\ approx}}} 702 \, {\ text {Cent}}}$
• Q m = ¼ decimal point mean fifth (frequency ratio ),${\ displaystyle {\ sqrt [{4}] {5}} {\ mathrel {\ hat {\ approx}}} 697 \, {\ text {Cent}}}$
• T = third octave (frequency ratio ).${\ displaystyle {\ tfrac {5} {4}} {\ mathrel {\ hat {\ approx}}} 386 \, {\ text {Cent}}}$
Name of the interval space Interval space
The system of fifths
interval space in Pythagorean tuning
${\ displaystyle {\ mathcal {Q}} = \ {x \ cdot \ mathrm {Ok} + y \ cdot \ mathrm {Q} \ mid x, y \ in \ mathbb {Z} \}}$
The ¼ decimal point mean-tone system of fifths
Interval space of the mean-tone tuning
${\ displaystyle {\ mathcal {Q}} _ {m} = \ {x \ cdot \ mathrm {Ok} + y \ cdot \ mathrm {Q_ {m}} \ mid x, y \ in \ mathbb {Z} \ }}$
The fifth-third system
interval space of pure tuning
${\ displaystyle {\ mathcal {QT}} = \ {x \ cdot \ mathrm {Ok} + y \ cdot \ mathrm {Q} + z \ cdot \ mathrm {T} \ mid x, y, z \ in \ mathbb {Z} \}}$
The twelve-level interval space = interval space of equal tuning (error on pure tuning up to 23 cents) ${\ displaystyle {\ mathcal {H}} = \ {x \ cdot \ mathrm {H} \ mid x \ in \ mathbb {Z} \}}$
The 53-step interval space (error on pure tuning up to 2 cents) ${\ displaystyle \ textstyle {\ mathcal {I}} _ {53} = \ {{\ frac {n} {53}} \ cdot \ mathrm {Ok} \ mid n \ in \ mathbb {Z} \}}$
The all-encompassing interval space
(all intervals can be divided as required.)
${\ displaystyle {\ mathcal {A}} = \ {r \ cdot \ mathrm {Ok} \ mid r \ in \ mathbb {R} \}}$

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: ${\ displaystyle {\ mathcal {Q}}}$

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 ) Quart-Ditonos = 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

## Mid-tone mood

The basis of the ¼-point mean-tone tuning is the ¼-point-mean-tone 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 fifth-third system , which consists of the intervals of the form ${\ displaystyle {\ mathcal {QT}}}$

${\ displaystyle i = n \ cdot {\ text {octave}} + m \ cdot {\ text {fifth}} + l \ cdot {\ text {major third}}}$with the frequency relationships.${\ displaystyle q = 2 ^ {n} \ cdot ({\ frac {3} {2}}) ^ {m} \ cdot ({\ frac {5} {4}}) ^ {l}}$

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 fifth-third 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 fifth-third system, the numerator and denominator of these fractions are only products of 2, 3 and 5. ${\ displaystyle {\ frac {n + 1} {n}}}$

It is important in this context: Intervals whose frequency ratio is super-particulate cannot be divided (in particular not halved).

In order to find out from a frequency ratio of the fifth-third system of which basic intervals the interval is composed, one must calculate the triple logarithm.

Example:

the equation

${\ displaystyle {\ frac {81} {80}} = 2 ^ {x} \ cdot \ left ({\ frac {3} {2}} \ right) ^ {y} \ cdot \ left ({\ frac { 5} {4}} \ right) ^ {z}, \ quad x, y, z \ in \ mathbb {Z}}$

has the unique solution, called "triple logarithm": and . ${\ displaystyle x = -2, \ y = 4}$${\ displaystyle z = -1}$

The relationship (see syntonic comma ) therefore applies to the interval with the frequency ratio 81:80 . ${\ displaystyle i}$${\ displaystyle i = -2 \ mathrm {Ok} +4 \ mathrm {Q} - \ mathrm {T}}$

## 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, E-flat 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, D-flat 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:

volume Cent value Occurrence Adjacent tones that differ only by the schism (2 cents) are marked with *. 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 = 300-3p b = 1000-2p f = 500-pc = 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 = 100-5p
200 d = 200 + 2p eses = 200-10p
300 dis = 300 + 9p es = 300-3p
400 e = 400 + 4p fes = 400-8p
500 f = 500-p ice = 500 + 11p
600 f sharp = 600 + 6p tot = 600-6p
700 g = 700 + p asas = 700-11p
800 g sharp = 800 + 8p as = 800-4p
900 a = 900 + 3p heses = 900-9p
1000 ais = 1000 + 10p b = 1000-2p
1100 h = 1100 + 5p ces = 1100-7p
1200 c = 1200 deses = 1200-12p

With p = 1 / 12 Pythagorean comma ≈ 2.0 cents and K = syntonic point ≈ 21.5 cents is calculated as:

• ,, cis = (100 + 7p-2K) Cent = 71 cents (= interval c ,, cis = interval from c to ,, cis)
• 'as = 800-4p + K = 814 cents (= interval from c' as )
• Interval ,, cis' as = (700-11p + 3K) cents = 743 cents.
Frequency ratio 2 (700-11p + 3K) / 1200 = 192 / 125

## Equal mood

The basis of equal tuning is the 12-step interval space with the following intervals: ${\ displaystyle {\ mathcal {H}}}$

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 53-step interval space . The octave is divided into 53 equal parts. ${\ displaystyle \ textstyle {\ mathcal {I}} _ {53} = \ {{\ frac {n} {53}} \ cdot \ mathrm {Ok} \ mid n \ in \ mathbb {Z} \}}$

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):

C-Dur:     c(0)    d(9)    ,e(17)    f(22)    g(31)   ,a(39)   ,h(48)     c(53)
G-Dur:     c(0)    d(9)    ,e(17) ,fis(26)    g(31)    a(40)   ,h(48)     c(53)
D-Dur:  ,cis(4)    d(9)     e(18) ,fis(26)    g(31)    a(40)   ,h(48)  ,cis(57)
A-Dur:  ,cis(4)    d(9)     e(18) ,fis(26) ,gis(35)    a(40)    h(49)  ,cis(57)
E-Dur:  ,cis(4) ,dis(13)    e(18)  fis(27) ,gis(35)    a(40)    h(49)  ,cis(57)
H-Dur:   cis(5) ,dis(13)    e(18)  fis(27) ,gis(35) ,ais(44)    h(49)   cis(58)
Fis-Dur: cis(5) ,dis(13) ,eis(22)  fis(27)  gis(36) ,ais(44)    h(49) cisis(61)
Cis-Dur: cis(5)  dis(14) ,eis(22)  fis(27)  gis(36) ,ais(44) ,his(53)   cis(58)

C-Dur:     c(0)     d(9)   ,e(17)    f(22)    g(31)   ,a(39)   ,h(48)     c(53)
F-Dur:     c(0)    ,d(8)   ,e(17)    f(22)    g(31)   ,a(39)    b(44)     c(53)
B-Dur:     c(0)    ,d(8)    es(13)   f(22)   ,g(30)   ,a(39)    b(44)     c(53)
Es-dur:   ,c(52)   ,d(8)    es(13)   f(22)   ,g(30)   as(35)    b(44)    ,c(52)
As-dur:   ,c(52)    des(4)  es(13)  ,f(21)   ,g(30)   as(35)    b(44)    ,c(52)
Des-dur:  ,c(52)    des(4)  es(13)  ,f(21)  ges(26)   as(35)   ,b(43)    ,c(52)
Ges-dur: ces(48)    des(4) ,es(12)  ,f(21)  ges(26)   as(35)   ,b(43)   ces(48)
Ces-dur: 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 53-level 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 mix-up

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 53-point 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

The 12-step keyboard
Frequency ratio Interval size in cents Interval designation
1 0 Prim
${\ displaystyle {\ sqrt [{12}] {2 ^ {1}}}}$ 100 equal semitone
${\ displaystyle {\ sqrt [{12}] {2 ^ {2}}}}$ 200 equal whole tone
${\ displaystyle {\ sqrt [{12}] {2 ^ {3}}}}$ 300 equal minor third
${\ displaystyle {\ sqrt [{12}] {2 ^ {4}}}}$ 400 equal major third
${\ displaystyle {\ sqrt [{12}] {2 ^ {5}}}}$ 500 equal fourths
${\ displaystyle {\ sqrt [{12}] {2 ^ {6}}}}$ 600 equal tritone
${\ displaystyle {\ sqrt [{12}] {2 ^ {7}}}}$ 700 equal fifth
${\ displaystyle {\ sqrt [{12}] {2 ^ {8}}}}$ 800 equal minor sixth
${\ displaystyle {\ sqrt [{12}] {2 ^ {9}}}}$ 900 equal major sixth
${\ displaystyle {\ sqrt [{12}] {2 ^ {10}}}}$ 1000 equal minor seventh
${\ displaystyle {\ sqrt [{12}] {2 ^ {11}}}}$ 1100 equal major seventh
2 1200 octave

### Pythagorean tuning intervals

Around 1270 there were instruments with 12-step keyboards. On this one had to decide how the black keys were tuned. Either as D flat or C sharp, D flat or E flat and so on. s. w.

The following table gives an overview of the intervals that can occur in Pythagorean tuning. Each of the intervals was calculated: C -Cis, C-Des *, CD, C-Dis *, C-Es, CE, ..., Cis -Dis *, Cis-Es, Cis-E, Cis-F, Cis-Fis ,…, Des * -Es, Des * -E,…, D -Dis *, D-Es, 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 * -Es-BFCGDAEH-F sharp-C sharp-G sharp-D sharp * -A sharp * are pure (frequency ratio 3: 2).

Note: The tones Gb *, Db *, A-flat *, D-sharp * and A sharp * are not available on a 12-point 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
C-Cis 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
Eb-F 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
Eb-G sharp ice 177147/131072 521.505 11Q - 6Ok Pythagorean excessive third
EB Ges 1024/729 588.270 −6Q + 4Ok Pythagorean diminished fifth
C-F 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
Eb-Ais * 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
Es-cis 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

Mid-tone keyboard

The following table gives an overview of the intervals that can occur in the mid-tone 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

${\ displaystyle w = {\ frac {3} {2}} \ cdot {\ sqrt [{4}] {\ frac {80} {81}}} = {\ sqrt [{4}] {5}}. }$

Note: The tones (Ges *), (Des *), (As *), (Dis *) and (Ais *) are not available on a 12-point 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 mix-ups.

The frequency ratio in the third column is often algebraic-irrational. Here means

${\ displaystyle w = {\ sqrt [{4}] {5}}, \ w ^ {2} = {\ sqrt [{4}] {5 ^ {2}}} {\ text {and}} w ^ {3} = {\ sqrt [{4}] {5 ^ {3}}}.}$

Each interval can be clearly represented as the sum of the two basic intervals of the mean-tone fifth system.

• Ok = octave
• Q m = mid-tone 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 mid-tone semitone
(E) - (F) (Of) (8:25) w 3 117.108 −5Q m + 3Ok = −T - Q m + Ok diatonic mid-tone semitone
(Des *) - (Dis *) (Cisis) (125: 256) w 2 152.098 14Q m - 8Ok = 4T - 2Q m medium-tone double excessive prim
(CD) (D) (1: 2) w 2 193.157 2Q m - Ok mid-tone whole tone
(Cis) - (Es) (Eses) (64: 125) w 2 234,216 −10Q m + 6Ok = −3T + 2Q m mid-tone diminished third
(Es) - (F sharp) (Dis) (25:32) w 269.206 9Q m - 5Ok = 2T + Q m - Ok mean-tone excessive second
(D) - (F) (It) (4: 5) w 310.265 −3Q m + 2Ok = −T + Q m medium-tone minor third
(Ges *) - (Ais *) (Disis) 625: 512 345.255 16Q m - 9Ok = 4T - Ok mid-tone double excessive second
(Dis *) - (Ges *) (Feses) (512: 625) w 351,324 −15Q m + 9Ok = −4T + Q m + Ok mid-tone 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 medium-tone excessive third
(C) - (F) (F) (2: 5) w 3 503,422 −Q m + Ok medium-tone fourth
(Cis) - (Gb *) (Geses) (256: 625) w 3 544,480 −13Q m + 8Ok = −3T - Q m + 2Ok mid-tone doubly diminished fifth
(F) - (H) (F sharp) (5: 8) w 2 579,471 6Q m - 3Ok = T + 2Q m - Ok medium-tone excessive fourth, medium-tone tritone
(Cis) - (G) (Ges) (16:25) w 2 620.529 −6Q m + 4Ok = −2T + 2Q m medium-tone diminished fifth
(Des *) - (G sharp) (Fisis) (125: 128) w 655.520 13Q m - 7Ok = 3T + Q m - Ok mid-tone 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 mid-tone 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 mid-tone doubly excessive fifth
(Ais *) - (ges *) (Broom) 1024: 625 854.745 −16Q m + 10Ok = 4T + 2Ok mid-tone double diminished seventh
(C) - (A) (A) (1: 2) w 3 889.735 3Q m - Ok = T - Q m + Ok medium-tone major sixth
(Cis) - (B) (Bes) (64: 125) w 3 930.794 −9Q m + 6Ok = −2T - Q m + 2Ok mid-tone diminished seventh
(Es) - (cis) (Ais) (25:32) w 2 965.784 10Q m - 5Ok = 2T + 2Q m - Ok medium-tone excessive sixth
(D) - (c) (B) (4: 5) w 2 1006,843 −2Q m + 2Ok medium-tone minor seventh
(G sharp) - (total *) (ceses) (512: 625) w 2 1047.902 −14Q m + 9Ok = −4T + 2Q m + Ok mid-tone 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 mid-tone 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 /, Cis-F /, 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-'Es-G / F-, Ac / F-'As-c / G-, HD and G-'Bd / supplemented by more Intermediate tones with the diatonic semitone steps (frequency ratio 16/15) C-'Des /, C sharp-D / ,, D-sharp, E / F-'Ges /, F sharp-G / ,, G sharp, A and ,, A sharp, H.

Each interval can be clearly represented as the sum of the three basic intervals of the fifth-third 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)
, E-total '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
D-Total '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 sharp-c 'ces 256: 135 1107.821 −T - 3Q + 3Ok (smaller) probably octave
,, Dis-d '' 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:

C-Cis-Des * -D-D-flat * -Es-E ... Pythagorean scale supplemented by semitones, based on perfect fifths.

(C) - (Cis) - (Des *) - (D) - (Dis *) - (Es) - (E) - (F) -… ¼-point mean-tone scale supplemented by semitones, based on mean-tone fifths ( 696.6 cents).

C-, Cis-'Des-D - ,, 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 = mid-tone fifth (frequency ratio )${\ displaystyle w = {\ sqrt [{4}] {5}} {\ mathrel {\ hat {\ approx}}} 696 {,} 6 \, {\ text {Cent}}}$
• 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 mid-tone 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
C-Cis 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 mid-tone 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 medium-tone 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 mid-tone whole tone
200 (2:12) Ok large equal second
CD D. 9: 8 203.910 2Q - Ok large whole tone = Pythagorean second
, E-total '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 mid-tone 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 mean-tone 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 medium-tone minor third
C-'Es 'It 6: 5 315.641 −T + Q minor third
Eb-F 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 mid-tone double excessive second. (Disis) = ,,,, Disis.
(Dis *) - (Ges *) (Feses) (512: 625) w 351,324 −15Q m + 9Ok mid-tone 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
D-Total '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 medium-tone 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 medium-tone 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)
Eb-G 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 mid-tone 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 medium-tone excessive fourth, medium-tone 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
C-F 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 medium-tone 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 medium-tone 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
Eb-Ais * Fisis 1594323: 1048576 725,415 13Q - 7Ok Pythagorean double augmented fourth
(G sharp) - (es) (Asas) (128: 125) w 737.637 −11Q m + 7Ok mid-tone 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 medium-tone doubly excessive fifth
(Ais *) - (ges *) (Broom) 1024: 625 854.745 −16Q m + 10Ok = −4T + 2Ok mid-tone 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 medium-tone 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 mid-tone 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 medium-tone 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 medium-tone minor seventh
C-'B 'B 9: 5 1017,596 −T + 2Q minor seventh
Es-cis 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 mid-tone 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 sharp-c '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 mid-tone diminished octave
,, Dis-d '' 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 auditory-psychological 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 . ${\ displaystyle (x, y)}$${\ displaystyle i = {\ overrightarrow {xy}}}$${\ displaystyle x}$${\ displaystyle y}$

Conversely, if the fundamental tone and the interval are known, the final tone is uniquely determined. ${\ displaystyle x}$${\ displaystyle i}$${\ displaystyle i = {\ overrightarrow {xy}}}$${\ displaystyle y}$

The successive execution of intervals defines an addition: is and , then is . ${\ displaystyle i = {\ overrightarrow {xy}}}$${\ displaystyle j = {\ overrightarrow {yz}}}$${\ displaystyle i + j = {\ overrightarrow {xz}}}$

Intervals can be compared: We write when the final note of is higher than the final note of with the same root note. ${\ displaystyle i ${\ displaystyle j}$${\ displaystyle i}$

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 = 712  octave = 700 cents . Corresponding:
• If you go up three major thirds, you get (roughly) an octave. So major third = 13  octave = 400 cents . You can now continue to calculate :
• Minor third = fifth - major third = 14  octave = 300 cents and
• Semitone = major third - minor third = 112  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 .

diminished third B-G sharp = 10 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.

## Remarks

1. 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 historical-systematic 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 3-8204-9492-8
2. The information relates to the pure tuning , with which intervals can be assigned whole-number relationships.
3. Euclid calculated with proportions, namely with string ratios that correspond to the reciprocal of the frequency ratios.
4. Except for powers of two (integer multiples of the octave) logarithms of two of pure ( rational ) frequency ratios are irrational and even transcendent .
5. 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”.
6. 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.
8. Note: 700-11p has the frequency ratio: (2/3) 11 • 2 7 (11 fifths octaved down, see asas) ⇒ 2 (700-11p + 3K) / 1200 = (2/3) 11 • 2 7 • (81/80) 3 = 192 / 125
9. 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.
10. 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.
11. Hermann von Helmholtz : The theory of tone sensations as a physiological basis for the theory of music . Vieweg, Braunschweig 1863, p. 531 (reprint: Minerva-Verlag, Frankfurt am Main 1981), ISBN 3-8102-0715-2 , ( 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] ”.
12. In contrast to the pure or medium-tone 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 Cis-Deses * or C-Deses is notated negatively. The interval Cis-des * or C-deses, 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.
13. 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 = 3153  octave = 702 cents . So you can determine very precise values ​​for interval sizes without acoustics.
14. ↑ 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 historical-systematic 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 3-8204-9492-8
15. The next better approximation would be: 28 major thirds = 9 octaves (hardly comprehensible to the ear), so major third = 928  octaves = 386 cents .
16. 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
17. 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
18. 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 .${\ displaystyle I}$${\ displaystyle r \ cdot Ok = \ sup \ {i \ in I \ mid {\ frac {z} {n}} \ leq r, n \ cdot i \ leq z \ cdot Ok, z \ in \ mathbb {Z }, n \ in \ mathbb {N} \}}$${\ displaystyle {\ frac {1} {2}} Ok}$