Diaschisma: Difference between revisions

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{{interwiki

| de = 2048/2025

| en = diaschisma

| es =

| ja =

}}

{{Infobox Interval

| Ratio = 2048/2025

| Name = diaschisma

| Color name = sgg2, sagugu 2nd, Sagugu comma

| Color name = sgg2, sagugu 2nd, Sagugu comma

| Comma = yes

}}

'''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, ~~(9/8)/~~(16/15)2, 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|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]].

== 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. 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 three family]] for the rank-3 temperament family where it is tempered out.

=== 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).

Pařízek's diaschisma pump [https://web.archive.org/web/20201127014513/http://micro.soonlabel.com/petr_parizek/pp_pump_examples/pump_1.ogg play] [https://luphoria.com/xenpaper/#(osc%3Asawtooth8)(bpm%3A90)_%7Br220hz%7D%0A%23_This_is_a_transcription_of_Pařízek's_diaschisma_comma_pump.png%0A%23_Just_intonation%2C_with_pitch_drifting.%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D_%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D_%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D_%0A2%3A3%3A5--%0A%23_We_are_now_4_diaschismas_from_where_we_started.%0A...%0A2%3A3%3A5-_%7Br220hz%7D_2%3A3%3A5-%0A..%0A%23_12edo%2C_the_comma_is_tempered_out_therefore_there_is_no_pitch_drifting.%0A%0A%7B12edo%7D%0A%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%0A%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%0A%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%0A%5B0%2C7%2C16%5D--_%0A%23The_root_hasn't_changed_from_where_we_started. xenpaper] – a [[comma pump]] progression that requires the diaschisma to be tempered out (i.e. equates two notes that are separated by a diaschisma).

[[File:Parizek diaschisma comma pump.png|thumb|Pařízek's diaschisma comma pump example in JI (notated with HEJI) and 12edo.]]

In the progression, the bassline moves as follows:

D (up 5/4) F# (down 4/3) C# (down 4/3) G# (up 5/4) C (down 4/3) G (up 3/2) D (*).

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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.

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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[[Category:Diaschismic]]

[[Category:Commas named for their regular temperament properties]]

@@ Line 1: / Line 1: @@
+{{interwiki
+| de = 2048/2025
+| en = diaschisma
+| es =
+| ja =
+}}
 {{Infobox Interval
 | Ratio = 2048/2025
 | Name = diaschisma
-| Color name = sgg2, sagugu 2nd,<br>Sagugu comma
+| Color name = sgg2, sagugu 2nd, <br>Sagugu comma
 | Comma = yes
 }}
 {{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, (9/8)/(16/15)<sup>2</sup>, 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|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]].
 == 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. 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 three family]] for the rank-3 temperament family where it is tempered out.
 === 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).
+Pařízek's diaschisma pump [https://web.archive.org/web/20201127014513/http://micro.soonlabel.com/petr_parizek/pp_pump_examples/pump_1.ogg play] [https://luphoria.com/xenpaper/#(osc%3Asawtooth8)(bpm%3A90)_%7Br220hz%7D%0A%23_This_is_a_transcription_of_Pařízek's_diaschisma_comma_pump.png%0A%23_Just_intonation%2C_with_pitch_drifting.%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D_%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D_%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D_%0A2%3A3%3A5--%0A%23_We_are_now_4_diaschismas_from_where_we_started.%0A...%0A2%3A3%3A5-_%7Br220hz%7D_2%3A3%3A5-%0A..%0A%23_12edo%2C_the_comma_is_tempered_out_therefore_there_is_no_pitch_drifting.%0A%0A%7B12edo%7D%0A%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%0A%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%0A%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%0A%5B0%2C7%2C16%5D--_%0A%23The_root_hasn't_changed_from_where_we_started. xenpaper] – a [[comma pump]] progression that requires the diaschisma to be tempered out (i.e. equates two notes that are separated by a diaschisma).
+[[File:Parizek diaschisma comma pump.png|thumb|Pařízek's diaschisma comma pump example in JI (notated with HEJI) and 12edo.]]
 In the progression, the bassline moves as follows:
   D (up 5/4) F# (down 4/3) C# (down 4/3) G# (up 5/4) C (down 4/3) G (up 3/2) D (*).
@@ Line 23: / Line 29: @@
 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.
+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 ==
@@ Line 28: / Line 41: @@
 [[Category:Diaschismic]]
+[[Category:Commas named for their regular temperament properties]]