Xenharmonic Wiki

Diaschisma: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>xenwolf
**Imported revision 600237838 - Original comment: added some links**
m →Temperaments: fix dead link. fix the wording in the xenpaper link
 
(45 intermediate revisions by 15 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = 2048/2025
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-11-23 07:38:07 UTC</tt>.<br>
| en = diaschisma
: The original revision id was <tt>600237838</tt>.<br>
| es =
: The revision comment was: <tt>added some links</tt><br>
| ja =
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<h4>Original Wikitext content:</h4>
{{Infobox Interval
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">2048/2025, the diaschisma, an interval of 19.553 [[cent|cents]], it is the difference between four perfect fifths plus two major thirds and three octaves, Tempering it out leads to [[Diaschismic family|Diaschismic temperament]]. It may also be defined as the difference between a just Pythagorean minor seventh and a just augmented sixth (E.A. [[16_9|16/9]] - 225/128), as the difference between two [[16_15|diatonic semitones]] and the [[9_8|major whole tone]], that is, (9/8)/(16/15)^2, or as the difference between the 5-limit tritones, [[45_32|45/32]] and its enharmonic equivalent [[64_45|64/45]].</pre></div>
| Ratio = 2048/2025
<h4>Original HTML content:</h4>
| Name = diaschisma
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;diaschisma&lt;/title&gt;&lt;/head&gt;&lt;body&gt;2048/2025, the diaschisma, an interval of 19.553 &lt;a class="wiki_link" href="/cent"&gt;cents&lt;/a&gt;, it is the difference between four perfect fifths plus two major thirds and three octaves, Tempering it out leads to &lt;a class="wiki_link" href="/Diaschismic%20family"&gt;Diaschismic temperament&lt;/a&gt;. It may also be defined as the difference between a just Pythagorean minor seventh and a just augmented sixth (E.A. &lt;a class="wiki_link" href="/16_9"&gt;16/9&lt;/a&gt; - 225/128), as the difference between two &lt;a class="wiki_link" href="/16_15"&gt;diatonic semitones&lt;/a&gt; and the &lt;a class="wiki_link" href="/9_8"&gt;major whole tone&lt;/a&gt;, that is, (9/8)/(16/15)^2, or as the difference between the 5-limit tritones, &lt;a class="wiki_link" href="/45_32"&gt;45/32&lt;/a&gt; and its enharmonic equivalent &lt;a class="wiki_link" href="/64_45"&gt;64/45&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
| 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, (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.
 
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 ==
* [[Small comma]]
 
[[Category:Diaschismic]]
[[Category:Commas named for their regular temperament properties]]
Retrieved from "https://en.xen.wiki/w/Diaschisma"