Comma (music)

Diatonic intervals
Prime second third fourth fifth sixth seventh octave none decime undezime duodecime tredezime semitone / whole tone
Special intervals
Microinterval Comma Diësis Limma Apotome Ditone Tritone Wolf fifth Natural septime
units
Cent Millioctave Octave Savart

In music theory, a comma is understood to be a small interval (much smaller than a semitone ) that is the difference between different combinations of pure intervals . The term is closely related to the mood systems. When trying to obtain the largest possible number of musically usable tones and intervals, one or more commas are always balanced.

The Pythagorean and syntonic commas are particularly important . The Pythagorean comma becomes clear in the circle of fifths: The stringing together of 12 perfect fifths leads to an octave tone that is slightly higher than the starting tone. The syntonic comma is the difference between the pure and Pythagorean thirds : The stringing together of 4 pure fifths leads to an octave tone that is slightly higher than the tone at a distance of a pure third.

Overview

The most popular commas

Surname	Emergence	Interval in Euler notation	size
Pythagorean comma	12 fifths - 7 octaves	Ges-F sharp	23.46 cents
Syntonic comma	4 fifths - major third - 2 Oct.	, F sharp-F sharp	21.51 cents
schism	8 fifths + major third - 5 Oct = pyth. K. - snyt. K.	Ges-, Fis	1.954 cents
Diaschism	3 octaves - 4 fifths - 2 major thirds = 2 * synth. K. - pyth. K.	, F sharp 'total	19.55 cents
Little diesis	Octave - 3 major thirds = 3 * synth. K. - pyth. K.	,, F sharp 'total	41.06 cents
Big diesis	4 small Thirds - octave = 4 * synth. K. - pyth. K.	,, F sharp - '' Ges	62.57 cents

Pythagorean comma

→ Main article : Pythagorean comma

Twelve perfect fifths placed one on top of the other achieve a tone that is about a quarter of a semitone apart from the seventh octave of the fundamental , the Pythagorean comma:

${\displaystyle {\frac {\left({\frac {3}{2}}\right)^{12}}{2^{7}}}={\frac {3^{12}}{2^{19}}}={\frac {531441}{524288}}\,\,{\widehat {\approx }}\,{\text{ 23,46 Cent}}}$

Syntonic comma

→ Main article : Syntonic comma

Four pure fifths ( ³ / ₂ ) superimposed reach a tone that a spacing of a (large) of the second octave of the fundamental Pythagorean third has. This third is about a fifth of a semitone , the syntonic comma or didymic comma , larger than the pure third.

The Pythagorean third compared to the pure third:

${\displaystyle {\text{pyth. Terz: }}{\frac {\left({\frac {3}{2}}\right)^{4}}{2^{2}}}={\frac {3^{4}}{2^{6}}}={\frac {81}{64}}\,\,{\widehat {\approx }}\,{\text{ 407,82 Cent;}}\quad {\text{ reine Terz: }}{\frac {5}{4}}\,\,{\widehat {\approx }}\,{\text{ 386,31 Cent.}}}$

The syntonic comma:

${\displaystyle {\frac {81}{64}}\cdot {\frac {4}{5}}={\frac {81}{80}}\,\,{\widehat {\approx }}\,\,(407{,}82-386{,}31){\text{ Cent}}={\text{21,51 Cent.}}}$

The large whole tone ( ⁹ / ₈ ) differs from the small whole tone ( ¹⁰ / ₉ ) to the syntonic comma:

${\displaystyle {\frac {9}{8}}\cdot {\frac {9}{10}}={\frac {81}{80}}\,\,{\widehat {\approx }}{\text{ 21,51 Cent.}}}$

schism

The schism (Greek: σχίσμα "separation") is the difference between the Pythagorean comma and the syntonic comma :

${\displaystyle 23{,}46\;\mathrm {Cent} -21{,}51\;\mathrm {Cent} =1{,}95\;\mathrm {Cent.} }$

The exact frequency ratio is

${\displaystyle {\frac {\left({\frac {3}{2}}\right)^{12}}{2^{7}}}:{\frac {81}{80}}={\frac {32\,805}{32\,768}}\,\,{\widehat {\approx }}\,\,1{,}9537\;\mathrm {Cent} }$.

Andreas Werckmeister ( Musicalische Temperatur, Quedlinburg 1691) considers the schism in the construction of his well-tempered moods: If you go down a series of perfect fifths from b to ces, the last note - octaves - is a Pythagorean comma lower than b. On the other hand, if you go down a syntonic comma from h, you get a tone, h (low point h) , which occurs in the pure major chord g-, hd and which differs from Ces only by the schism. This difference is at the "limit of the perceptible tone differences" (see Das Reinharmonium ). So, h can be identified with ces:, h = ces, also des =, cis; es =, dis; ges =, f sharp; as =, g sharp; b =, ais etc.

The schism should not be confused with the twelfth part of the Pythagorean comma (which is relevant for mood systems ), even if the numerical values ​​in cents are similar:

${\displaystyle {\sqrt[{12}]{\frac {531441}{524288}}}\,\,{\widehat {\approx }}\,\,1{,}9550\;\mathrm {Cent} }$.

Small Diesis (enharmonic comma)

In pure tuning, for example, D flat has a lower pitch than E flat.

• Frequency ratio D – E = (small whole tone) ${\displaystyle {\frac {10}{9}}}$
• Frequency ratio D – Es = ( diatonic semitone ) ${\displaystyle {\frac {16}{15}}}$
• Frequency ratio Dis – E = (diatonic semitone) ${\displaystyle {\frac {16}{15}}}$
• Frequency ratio Dis – Es = ${\displaystyle {\frac {16}{15}}\cdot {\frac {9}{10}}\cdot {\frac {16}{15}}={\frac {128}{125}}\,\,{\widehat {\approx }}\,\,41{,}06\;\mathrm {Cent} \approx {\frac {1}{5}}\;\mathrm {Ganzton} }$

This interval is called a small diësis (less often an enharmonic comma) = 1 octave - 3 major thirds = 7 octaves - 12 mid-tone fifths.

Around the small Diësis there is a difference between D-sharp and E-flat, G sharp and A flat, and in mid-tone tuning, C sharp and D flat, D flat and E flat, F sharp and G flat, G sharp and A flat and A sharp and B flat you then decide on one assignment.)

Big diesis

If four minor thirds are strung together, they result in an octave in equal-tempered tuning, but a slightly larger interval in pure tuning. The difference to the octave is called a large diësis :

In C major: C-Es-Ges-Heses-deses:

${\displaystyle {\begin{alignedat}{2}{\text{große Diësis}}&={\text{4 kleine Terzen}}_{\text{rein}}&&-{\text{Oktave}}\\&\approx {\text{ 63 Cent}}\end{alignedat}}}$

Diaschism

In pure tuning, for example, C sharp has a lower pitch than D flat.

• Frequency ratio C – D = (large whole tone) ${\displaystyle {\frac {9}{8}}}$
• Frequency ratio C – Des = ( diatonic semitone ) ${\displaystyle {\frac {16}{15}}}$
• Frequency ratio C sharp – D = (diatonic semitone) ${\displaystyle {\frac {16}{15}}}$
• Frequency ratio Cis – Des = ${\displaystyle {\frac {16}{15}}\cdot {\frac {8}{9}}\cdot {\frac {16}{15}}={\frac {2048}{2025}}\,\,{\widehat {\approx }}\,\,19{,}55\;\mathrm {Cent} \approx {\frac {1}{5}}\;\mathrm {Halbton} }$

This interval is called diaschism = 3 octaves - 4 fifths - 2 major thirds.

Around the diaschism there are also F sharp and G flat and A sharp and B.

7th comma

The approx. 27.26 cent large interval with the oscillation ratio 64:63 is called the seventh or Leipzig comma, that between

• the natural septime (7: 4 approx. 968.82 cents) and
• the minor seventh (16: 9 approx. 996.08 cents)

the pure mood lies.

Historical classification

In Euclid's division of the canon , in which the theoretical knowledge about music of the time (approx. 3rd century BC) is summarized, one can read as sentence 14: "The octave is smaller than 6 whole tones." Octave the interval with the proportion (today's interpretation: frequency ratio) 2: 1 and the whole tone the interval with the proportion 9: 8. The difference (six whole tones - octave) is called the Pythagorean comma. In Euclid, its proportion is given as 531441: 524288 (however, the term κόμμα does not appear in Euclid).

It was only with the advent of polyphonic music in the Renaissance and Baroque periods that commas played a decisive role, especially for tuning keyboard instruments, where there were only 12 pitches in the octave. A variety of tuning systems were developed in which the commas were distributed differently to the pitches.

See also

• cent
• Pure mood
• Mid-tone mood
• Pythagorean mood
• Equal mood

Remarks

1. ↑ Exactly: equal whole tone = 200 cents. 1/5 semitone = 40 cents.