Diaschisma: Difference between revisions

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'''2048/2025''', the '''diaschisma''', a [[comma]] of 19.553 [[cent]]s, is the size of a [[pythagorean comma]] minus two [[schisma~~|schismas~~]], from which it derives its name. It may also be defined as the difference between four [[3/2|just perfect fifths]] plus two [[5/4|just major thirds]] and three octaves, 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, (16/15)2/(9/8), or as the difference between the 5-limit tritone [[45/32]] and its octave complement [[64/45]].

'''2048/2025''', the '''diaschisma''', a [[comma]] of 19.553 [[cent]]s, is the size of a [[pythagorean comma]] minus two [[schisma]]s, from which it derives its name. It may also be defined as the difference between four [[3/2|just perfect fifths]] plus two [[5/4|just major thirds]] and three octaves, 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, (16/15)2/(9/8), or as the difference between the 5-limit tritone [[45/32]] and its octave complement [[64/45]].

== Temperaments ==

Tempering it out leads to the [[diaschismic family]] of temperaments. See [[Diaschismic family]] for the rank-2 temperament family where it is tempered out, especially [[Srutal archagall]] which takes advantage of this comma's relation to [[256/255]] and [[289/288]] to make it as efficient and natural as possible. See [[Diaschismic rank ~~three~~ family]] for the rank-3 temperament family where it is tempered out.

Tempering it out leads to the [[diaschismic family]] of temperaments. See [[Diaschismic family]] for the rank-2 temperament family where it is tempered out, especially [[Srutal archagall]] which takes advantage of this comma's relation to [[256/255]] and [[289/288]] to make it as efficient and natural as possible. See [[Diaschismic rank-3 family]] for the rank-3 temperament family where it is tempered out.

=== Significance ===

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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.

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.

== Etymology ==

The modern sense of the term is due to {{w|Hermann von Helmholtz}} and {{w|Alexander John Ellis}} in 1875 when the English translation of ''{{w|Sensations of Tone}}'' was first published.

2048/2025 was earlier referred to as the ~~“diminished comma”~~ and ~~“comma minor”~~ by {{w|Jean-Philippe Rameau}} (1683-1764). However in modern (1875 onwards) music theory the term ~~“diaschisma”~~ is almost always used.

2048/2025 was earlier referred to as the ''diminished comma'' and ''comma minor'' by {{w|Jean-Philippe Rameau}} (1683-1764). However in modern (1875 onwards) music theory the term ''diaschisma'' is almost always used.

There have been other intervals besides 2048/2025 that were called ~~“diaschisma”~~ in the [[Ancient Greek]], Roman and [[historical temperaments|medieval]] periods, however those alternate meanings of the word fell out of use centuries ago.

There have been other intervals besides 2048/2025 that were called ''diaschisma'' in the [[Ancient Greek]], Roman and [[historical temperaments|medieval]] periods, however those alternate meanings of the word fell out of use centuries ago.

== See also ==

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 {{Wikipedia| Diaschisma }}
-'''2048/2025''', the '''diaschisma''', a [[comma]] of 19.553 [[cent]]s, is the size of a [[pythagorean comma]] minus two [[schisma|schismas]], from which it derives its name. It may also be defined as the difference between four [[3/2|just perfect fifths]] plus two [[5/4|just major thirds]] and three octaves, 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, (16/15)<sup>2</sup>/(9/8), or as the difference between the 5-limit tritone [[45/32]] and its octave complement [[64/45]].
+'''2048/2025''', the '''diaschisma''', a [[comma]] of 19.553 [[cent]]s, is the size of a [[pythagorean comma]] minus two [[schisma]]s, from which it derives its name. It may also be defined as the difference between four [[3/2|just perfect fifths]] plus two [[5/4|just major thirds]] and three octaves, 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, (16/15)<sup>2</sup>/(9/8), or as the difference between the 5-limit tritone [[45/32]] and its octave complement [[64/45]].
 == Temperaments ==
-Tempering it out leads to the [[diaschismic family]] of temperaments. See [[Diaschismic family]] for the rank-2 temperament family where it is tempered out, especially [[Srutal archagall]] which takes advantage of this comma's relation to [[256/255]] and [[289/288]] to make it as efficient and natural as possible. See [[Diaschismic rank three family]] for the rank-3 temperament family where it is tempered out.
+Tempering it out leads to the [[diaschismic family]] of temperaments. See [[Diaschismic family]] for the rank-2 temperament family where it is tempered out, especially [[Srutal archagall]] which takes advantage of this comma's relation to [[256/255]] and [[289/288]] to make it as efficient and natural as possible. See [[Diaschismic rank-3 family]] for the rank-3 temperament family where it is tempered out.
 === Significance ===
@@ Line 28: / Line 28: @@
 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.
+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.
 == Etymology ==
 The modern sense of the term is due to {{w|Hermann von Helmholtz}} and {{w|Alexander John Ellis}} in 1875 when the English translation of ''{{w|Sensations of Tone}}'' was first published.
-/2025 was earlier referred to as the “diminished comma” and “comma minor” by {{w|Jean-Philippe Rameau}} (1683-1764). However in modern (1875 onwards) music theory the term “diaschisma” is almost always used.
+/2025 was earlier referred to as the ''diminished comma'' and ''comma minor'' by {{w|Jean-Philippe Rameau}} (1683-1764). However in modern (1875 onwards) music theory the term ''diaschisma'' is almost always used.
-There have been other intervals besides 2048/2025 that were called “diaschisma” in the [[Ancient Greek]], Roman and [[historical temperaments|medieval]] periods, however those alternate meanings of the word fell out of use centuries ago.
+There have been other intervals besides 2048/2025 that were called ''diaschisma'' in the [[Ancient Greek]], Roman and [[historical temperaments|medieval]] periods, however those alternate meanings of the word fell out of use centuries ago.
 == See also ==