This is

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

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.

Little Diësis

Formation of a small diesis

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 :

${\displaystyle {\begin{alignedat}{2}{\text{kleine Diësis}}&={\text{Oktave}}&&-{\text{3 große Terzen}}_{\text{rein}}\\&={\text{3 große Terzen}}_{\text{gleichstufig}}&&-{\text{3 große Terzen}}_{\text{rein}}\\&\approx 3\cdot {\text{400 Cent}}&&-3\cdot {\text{386,31 Cent}}\\&\approx {\text{1200 Cent}}&&-{\text{1159 Cent}}\\&\approx {\text{41 Cent}}\end{alignedat}}}$

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: ${\displaystyle {\tfrac {5}{4}}}$ ${\displaystyle \left({\tfrac {5}{4}}\right)^{3}}$ ${\displaystyle {\tfrac {2}{1}}}$

${\displaystyle {\frac {\tfrac {2}{1}}{\left({\tfrac {5}{4}}\right)^{3}}}=2\cdot {\left({\frac {4}{5}}\right)^{3}}={\frac {128}{125}}=1{,}024\,\,{\widehat {\approx }}\,\,41\,\mathrm {Cent} }$

(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 .

Great Diësis

Formation of a large 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 :

${\displaystyle {\begin{alignedat}{2}{\text{große Diësis}}&={\text{4 kleine Terzen}}_{\text{rein}}&&-{\text{Oktave}}\\&={\text{4 kleine Terzen}}_{\text{rein}}&&-{\text{4 kleine Terzen}}_{\text{gleichstufig}}\\&\approx 4\cdot {\text{ 315,64 Cent}}&&-4\cdot {\text{ 300 Cent}}\\&\approx {\text{1263 Cent}}&&-{\text{1200 Cent}}\\&\approx {\text{ 63 Cent}}\end{alignedat}}}$

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: ${\displaystyle {\tfrac {6}{5}}}$ ${\displaystyle \left({\tfrac {6}{5}}\right)^{4}}$ ${\displaystyle {\tfrac {2}{1}}}$

${\displaystyle {\frac {\left({\frac {6}{5}}\right)^{4}}{\tfrac {2}{1}}}=\left({\frac {6}{5}}\right)^{4}\cdot {\frac {1}{2}}={\frac {648}{625}}=1{,}0368\ {\widehat {\approx }}\ 63\,\mathrm {Cent} }$

history

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 .

literature

• Diesis in Willibald Gurlitt , Hans Heinrich Eggebrecht (Hrsg.): Riemann Music Lexicon (subject part) . B. Schott's Sons, Mainz 1967, p. 225 .

annotation

1. ↑ 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 )