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{{interwiki
| de = 2048/2025
| en = diaschisma
| es =
| ja =
}}
{{Infobox Interval
{{Infobox Interval
| JI glyph =
| Ratio = 2048/2025
| Ratio = 2048/2025
| Monzo = 11 -4 -2
| Cents = 19.55257
| Name = diaschisma
| Name = diaschisma
| Color name =  
| Color name = sgg2, sagugu 2nd, <br>Sagugu comma
| FJS name = d2<sub>25</sub>
| Comma = yes
| Sound =  
}}
}}
{{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]].
== 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.
=== Significance ===
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 (*).
If we ignore octaves,
* 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)<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.


'''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]].
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.


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


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-(up 4/3)-G-(up 3/2)-D (*). If we ignore octaves, the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]], and the last three steps (cumulatively G# to D) are the same moves as the first three, thus it moves us 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. The fact that the D we come back to is exactly the same as the first D, indicates that 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 temperament contexts, we see this as equivalent to saying that their difference, which is (2/1) / (45/32)^2 = 2048/2025 is tempered out.
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.


This also implies that there is 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 temperaments (such as [[diaschismic family|diaschismic]] and [[pajara]]) splits the octave into two equal parts; in particular, all diaschismic edos are even-numbered edos.
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 ==
== See also ==
* [[Comma]]
* [[Small comma]]


[[Category:5-limit]]
[[Category:Small comma]]
[[Category:Diaschismic]]
[[Category:Diaschismic]]
[[Category:Commas named for their regular temperament properties]]
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