# 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”)

## 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 {small diësis}} & = {\ text {octave}} && - {\ text {3 major thirds}} _ {\ text {pure}} \\ & = {\ text {3 major thirds}} _ {\ text {equal}} && - {\ text {3 major thirds}} _ {\ text {pure}} \\ & \ approx 3 \ cdot {\ text { 400 cents}} && - 3 \ cdot {\ text {386.31 cents}} \\ & \ approx {\ text {1200 cents}} && - {\ text {1159 cents}} \\ & \ approx {\ text {41 cents}} \ 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 {major diësis}} & = {\ text {4 minor thirds}} _ {\ text {pure}} && - {\ text {octave}} \\ & = {\ text {4 minor thirds}} _ {\ text {pure}} && - {\ text {4 minor thirds}} _ {\ text {equal steps}} \\ & \ approx 4 \ cdot {\ text { 315.64 cents}} && - 4 \ cdot {\ text {300 cents}} \\ & \ approx {\ text {1263 cents}} && - {\ text {1200 cents}} \\ & \ approx {\ text {63 cents}} \ 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 .

## 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 - 14  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 )