Diësis ( ancient Greek δίεσις “smallest interval”, actually “passage”, to diïēmi διΐημι “through”)
- in Italy and France chromatically elevated base tones : The F sharp tone is called “fa diesis” in Italy and “fa dièse” in France ( other-language tone names ).
- a small musical interval that occurs in the two forms small and large diësis.
If three major thirds are strung together, they result in an octave in equal-tempered tuning , whereas in pure tuning they result in a slightly smaller interval. The difference to the octave is called a small diësis :
The pure major third has a frequency ratio of , so three such thirds result ; the octave with the frequency ratio is slightly larger. The two intervals therefore differ by the frequency ratio:
(Note: 1 cent = 1/1200 octave)
Occasionally the term enharmonic comma is also used for small diësis , as it makes the difference between enharmonic alternating tones in pure and mid-tone tuning , e.g. B. between G sharp and A flat .
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 :
The pure minor third has a frequency ratio of , so four such thirds result ; the octave with the frequency ratio is slightly smaller. The two intervals therefore differ by the frequency ratio:
Philolaos meant by Diesis the excess of the fourth two whole tones ( ditone ), so the later than Limma designated diatonic semitone of the Pythagoreans . Aristoxenus used the term for all intervals that are smaller than a semitone. Marchetus de Padua defines the diesis as 1/5 whole tone in his Lucidarium .
- Diesis in Willibald Gurlitt , Hans Heinrich Eggebrecht (Hrsg.): Riemann Music Lexicon (subject part) . B. Schott's Sons, Mainz 1967, p. 225 .
If one looks at the circle of fifths with the 12 fifths As – Es – B – F – C – G – D – A – E – H – F sharp – C sharp – G sharp , one sees that in the Pythagorean tuning the size of the interval As– G sharp = 12 (perfect fifths) - 7 octaves = Pythagorean comma = 23.5 cents, while in the mean-tone tuning the size of the interval G sharp-A flat = 7 octaves - 12 (mean-tone fifths) = 7 octaves - 12 (pure fifths - 1 ⁄ 4 syntonic comma) = octave −3 (pure major thirds) = minor diesis = 41 cents.
In the pure tuning, the calculation of the interval size G sharp – A flat is more complicated: With the designation apostrophe 'x and apostrophe, x of Euler's tone network the minor triad becomes the subdominant of C major C – D–, E – F – G–, A– , H-c with F-'As-c designated and the dominant of, a minor with , e - ,, G sharp, H . Thus, starting from C, with 'As = (−4 fifths + 3 octaves + syntonic comma) and “G sharp = (8 fifths - 4 octaves - 2 syntonic commas) the interval size“ G sharp –'As = (−4 fifths ) is calculated + 3 octaves + syntonic comma) - (8 fifths - 4 octaves - 2 syntonic commas) = 7 octaves - 12 fifths + 3 syntonic commas = small diesis = 41 cents.
Note: In the Pythagorean tuning, G sharp is higher than A flat , in the mean and pure tuning As higher than G sharp ( table of intervals )