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 = 1 ⁄ 12 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	${\displaystyle 1200\cdot \log {\big (}{\tfrac {531441}{524288}}{\big )}\,{\text{Cent}}=23{,}460\,{\text{Cent}}}$
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: ${\displaystyle q}$ ${\displaystyle i}$

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

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

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

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

${\displaystyle q=2^{\frac {i}{\mathrm {Oktave} }}=2^{\frac {i}{1200\,\mathrm {Cent} }}}$

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

${\displaystyle q=2^{\frac {3\mathrm {Oktaven} }{\mathrm {Oktave} }}=2^{\frac {3600\mathrm {Cent} }{1200\,\mathrm {Cent} }}=2^{3}=8}$

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 ${\displaystyle {\mathcal {QT}}}$

${\displaystyle i=n\cdot {\text{Oktave}}+m\cdot {\text{Quinte}}+l\cdot {\text{große Terz}}}$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:

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 = 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

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

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. ${\displaystyle i<j}$ ${\displaystyle j}$ ${\displaystyle i}$

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 = 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 .

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.

Web links

• List of all Pythagorean, mean-tone and pure intervals from C-C sharp to hc

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.
7. ↑ https://eclass.uoa.gr/modules/document/index.php?course=MATH379&download=/5122bb9396op/5122bf834msr.pdf
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 = ³¹ ⁄ ₅₃  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 = ⁹ ⁄ ₂₈  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 . ${\displaystyle I}$ ${\displaystyle r\cdot Ok=\sup\{i\in I\mid {\frac {z}{n}}\leq r,n\cdot i\leq z\cdot Ok,z\in \mathbb {Z} ,n\in \mathbb {N} \}}$ ${\displaystyle {\frac {1}{2}}Ok}$