Diaschisma: Difference between revisions

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'''2048/2025''', the '''diaschisma''', an interval of 19.553 [[cent~~|cents~~]], is the difference between four perfect fifths plus two major thirds and three octaves~~. Tempering it out leads to the [[diaschismic family]] of temperaments~~. It may also be defined as the difference between a Pythagorean minor seventh ([[16/9]]) and a just augmented sixth ([[225/128]]), as the difference between two classic diatonic semitones ([[16/15]]) and the major whole tone ([[9/8]]), that is, (9/8)/(16/15)2, or as the difference between the 5-limit tritone [[45/32]] and its ~~enharmonic equivalent~~ [[64/45]].

'''2048/2025''', the '''diaschisma''', an interval of 19.553 [[cent]]s, is the difference between four [[3/2|just perfect fifths]] plus two [[5/4|just major thirds]] and three octaves. It may also be defined as the difference between a Pythagorean minor seventh ([[16/9]]) and a just augmented sixth ([[225/128]]), as the difference between two classic diatonic semitones ([[16/15]]) and the major whole tone ([[9/8]]), that is, (9/8)/(16/15)2, or as the difference between the 5-limit tritone [[45/32]] and its octave complement [[64/45]].

== Significance ==

== Temperaments ==

Tempering it out leads to the [[diaschismic family]] of temperaments.

=== Significance ===

[http://micro.soonlabel.com/petr_parizek/pp_pump_examples/pump_1.ogg parizek1] A [[comma pump]] progression that requires the diaschisma to be tempered out (i.e. equates two notes that are separated by a diaschisma).

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* the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]];

* the last three steps (cumulatively G# to D) are the same moves as the first three, moving up by the tritone 45/32 a second time.

In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma, which is (2/1) / (45/32)^2 = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that that their difference, the diaschisma, is tempered out. To carry out this tempering-out (assuming octaves are kept pure), the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1.

In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma, which is (2/1) / (45/32)2 = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that that their difference, the diaschisma, is tempered out. To carry out this tempering-out (assuming octaves are kept pure), the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1.

This also tells us that if a system tempers out the diaschisma, it has an interval that is equal to exactly half of an octave‚ namely the tempered 45/32 tritone. Thus all edos (such as [[12edo]], [[22edo]], [[34edo]] and [[46edo]]) and MOS scale structures (such as the MOS scales of [[diaschismic family|diaschismic]] and [[pajara]]) that temper out the diaschisma split the octave into two equal parts; in particular, all diaschismic edos are even-numbered edos.

== See also ==

* [[~~Comma~~]]

* [[Diaschismic family]], the rank-2 temperament family where it is tempered out

* [[Diaschismic rank three family]], the rank-3 temperament family where it is tempered out

* [[Small comma]]

[[Category:5-limit]]

[[Category:Small comma]]

[[Category:Diaschismic]]

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 }}
-'''2048/2025''', the '''diaschisma''', an interval of 19.553 [[cent|cents]], is the difference between four perfect fifths plus two major thirds and three octaves. Tempering it out leads to the [[diaschismic family]] of temperaments. It may also be defined as the difference between a Pythagorean minor seventh ([[16/9]]) and a just augmented sixth ([[225/128]]), as the difference between two classic diatonic semitones ([[16/15]]) and the major whole tone ([[9/8]]), that is, (9/8)/(16/15)<sup>2</sup>, or as the difference between the 5-limit tritone [[45/32]] and its enharmonic equivalent [[64/45]].
+'''2048/2025''', the '''diaschisma''', an interval of 19.553 [[cent]]s, is the difference between four [[3/2|just perfect fifths]] plus two [[5/4|just major thirds]] and three octaves. It may also be defined as the difference between a Pythagorean minor seventh ([[16/9]]) and a just augmented sixth ([[225/128]]), as the difference between two classic diatonic semitones ([[16/15]]) and the major whole tone ([[9/8]]), that is, (9/8)/(16/15)<sup>2</sup>, or as the difference between the 5-limit tritone [[45/32]] and its octave complement [[64/45]].
-== Significance ==
+== Temperaments ==
+Tempering it out leads to the [[diaschismic family]] of temperaments.
+=== Significance ===
 [http://micro.soonlabel.com/petr_parizek/pp_pump_examples/pump_1.ogg parizek1] A [[comma pump]] progression that requires the diaschisma to be tempered out (i.e. equates two notes that are separated by a diaschisma).
@@ Line 20: / Line 23: @@
 * the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]];
 * the last three steps (cumulatively G# to D) are the same moves as the first three, moving up by the tritone 45/32 a second time.
-In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma, which is (2/1) / (45/32)^2 = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that that their difference, the diaschisma, is tempered out. To carry out this tempering-out (assuming octaves are kept pure), the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1.
+In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma, which is (2/1) / (45/32)<sup>2</sup> = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that that their difference, the diaschisma, is tempered out. To carry out this tempering-out (assuming octaves are kept pure), the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1.
 This also tells us that if a system tempers out the diaschisma, it has an interval that is equal to exactly half of an octave‚ namely the tempered 45/32 tritone. Thus all edos (such as [[12edo]], [[22edo]], [[34edo]] and [[46edo]]) and MOS scale structures (such as the MOS scales of [[diaschismic family|diaschismic]] and [[pajara]]) that temper out the diaschisma split the octave into two equal parts; in particular, all diaschismic edos are even-numbered edos.
 == See also ==
-* [[Comma]]
+* [[Diaschismic family]], the rank-2 temperament family where it is tempered out
+* [[Diaschismic rank three family]], the rank-3 temperament family where it is tempered out
+* [[Small comma]]
 [[Category:5-limit]]
 [[Category:Small comma]]
 [[Category:Diaschismic]]