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#redirect [[Diaschismic family #Diaschismic]]
'''Diaschismic''', sometimes known as [[srutal vs diaschismic|srutal]] in the [[5-limit]], is a half-octave [[regular temperament|temperament]] [[generator|generated]] by a [[3/2|perfect fifth]] or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]], is [[tempering out|tempered out]]. We also have a whole tone plus a period represent [[8/5]], and in the canonical [[extension]] to the [[7-limit]], the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate [[8/7]], tempering out the starling comma, [[126/125]].


[[Category:Temperaments]]
A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]) is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. Finally, since the whole tone has been split in two, each can be used to represent [[17/16]]~[[18/17]]. Therefore, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125, [[136/135]], [[176/175]], [[196/195]], and [[256/255]].
 
See [[Diaschismic family #Diaschismic]] for technical data.
 
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
! Cents*
! Approximate ratios
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
| 600.0
| 17/12, 24/17
|-
| 1
| 703.9
| '''3/2'''
| 103.9
| '''16/15''', '''17/16''', 18/17
|-
| 2
| 207.7
| '''9/8'''
| 807.7
| '''8/5'''
|-
| 3
| 911.6
| 17/10, 22/13
| 311.6
| 6/5
|-
| 4
| 415.4
| 14/11
| 1015.4
| 9/5
|-
| 5
| 1119.5
| 21/11, 40/21, 48/25
| 519.5
| 27/20
|-
| 6
| 623.1
| 10/7
| 23.1
| 56/55, 64/63, 81/80
|-
| 7
| 127.0
| 14/13, 15/14
| 727.0
| '''32/21'''
|-
| 8
| 830.8
| 21/13, 34/21
| 230.8
| '''8/7'''
|-
| 9
| 334.7
| 17/14, 40/33
| 934.7
| 12/7
|-
| 10
| 1038.5
| 20/11
| 438.5
| 9/7
|-
| 11
| 542.4
| 15/11
| 1142.4
| 27/14, 64/33
|-
| 12
| 46.2
| 36/35, 40/39, 45/44, 50/49
| 646.2
| '''16/11'''
|-
| 13
| 750.1
| 17/11, 20/13
| 150.1
| 12/11
|-
| 14
| 253.9
| 15/13
| 853.9
| 18/11
|-
| 15
| 957.8
| 45/26, 68/39
| 357.8
| '''16/13'''
|-
| 16
| 461.6
| 17/13
| 1061.6
| 24/13
|-
| 17
| 1165.5
| 51/26, 96/49, 108/55
| 565.5
| 18/13
|}
<nowiki/>* In 17-limit CWE tuning, octave-reduced
 
=== As a detempereament of 12et ===
[[File: Diaschismic 12et Detempering.png|thumb|Diaschismic as a 58-tone 12et detempering]]
 
Diaschismic is naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by 6 generator steps, which represent the syntonic comma. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.
 
The 13th harmonic is just beyond the specified generator range, so the diagram does not show it.
 
Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the syntonic comma in 58edo. 104edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.
 
== Chords ==
{{Main| Chords of diaschismic }}
 
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator (¢)
! Comments
|-
|
| 17/9
| 698.955
|
|-
| [[12edo|7\12]]
|
| 700.000
| 12f val, lower bound of 7-, 9- and 11-odd-limit diamond monotone
|-
|
| 3/2
| 701.955
|
|-
|
| 15/14
| 702.778
|
|-
| [[70edo|41\70]]
|
| 702.857
| 70ef val
|-
|
| 7/5
| 702.915
|
|-
|
| 21/20
| 703.107
|
|-
|
| 15/11
| 703.359
|
|-
|
| 15/13
| 703.410
|
|-
| [[58edo|34\58]]
|
| 703.448
| Lower bound of 13-, 15-, 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 11/10
| 703.500
|
|-
|
| 9/7
| 703.508
|
|-
|
| 13/10
| 703.522
|
|-
|
| 13/11
| 703.597
|
|-
|
| 12/7
| 703.681
|
|-
|
| 13/9
| 703.728
|
|-
|
| 11/9
| 703.757
|
|-
|
| 21/13
| 703.782
|
|-
|
| 49/48
| 703.783
|
|-
|
| 13/12
| 703.839
|
|-
| [[104edo|61\104]]
|
| 703.846
| 104c val
|-
|
| 21/11
| 703.893
|
|-
|
| 11/6
| 703.895
|
|-
|
| 7/4
| 703.897
|
|-
|
| 13/8
| 703.965
|
|-
|
| 17/14
| 704.014
|
|-
|
| 17/13
| 704.027
|
|-
|
| 13/7
| 704.043
|
|-
|
| 11/8
| 704.057
|
|-
|
| 36/35
| 704.064
|
|-
|
| 17/11
| 704.126
|
|-
|
| 21/16
| 704.174
|
|-
|
| 21/17
| 704.272
|
|-
| [[46edo|27\46]]
|
| 704.348
| Upper bound of 11-, 13-, 15-, 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 11/7
| 704.377
|
|-
|
| 9/5
| 704.399
|
|-
|
| 17/16
| 704.955
|
|-
|
| 5/3
| 705.214
|
|-
|
| 25/24
| 705.866
|
|-
| [[34edo|20\34]]
|
| 705.882
| 34ef val, upper bound of 7- and 9-odd-limit diamond monotone
|-
|
| 17/10
| 706.214
|
|-
|
| 5/4
| 706.843
|
|-
|
| 17/15
| 708.343
|
|-
|
| 15/8
| 711.731
|
|}
<nowiki/>* Besides the octave
 
[[Category:Diaschismic| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Diaschismic family]]
[[Category:Diaschismic family]]
[[Category:Starling temperaments]]
[[Category:Hemifamity temperaments]]

Revision as of 15:16, 27 April 2025

Diaschismic, sometimes known as srutal in the 5-limit, is a half-octave temperament generated by a perfect fifth or that minus a half-octave period, which is a semitone representing 16/15. Two of these semitones give a whole tone of 9/8, so the diaschisma, 2048/2025, is tempered out. We also have a whole tone plus a period represent 8/5, and in the canonical extension to the 7-limit, the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate 8/7, tempering out the starling comma, 126/125.

A stack of twelve perfect fifths octave reduced (a diesis) is close in size to a quartertone, and that plus a period can be used to represent 16/11. Three more fifths on top of 16/11 give 16/13. Finally, since the whole tone has been split in two, each can be used to represent 17/16~18/17. Therefore, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125, 136/135, 176/175, 196/195, and 256/255.

See Diaschismic family #Diaschismic for technical data.

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are in bold.

# Period 0 Period 1
Cents* Approximate ratios Cents* Approximate ratios
0 0.0 1/1 600.0 17/12, 24/17
1 703.9 3/2 103.9 16/15, 17/16, 18/17
2 207.7 9/8 807.7 8/5
3 911.6 17/10, 22/13 311.6 6/5
4 415.4 14/11 1015.4 9/5
5 1119.5 21/11, 40/21, 48/25 519.5 27/20
6 623.1 10/7 23.1 56/55, 64/63, 81/80
7 127.0 14/13, 15/14 727.0 32/21
8 830.8 21/13, 34/21 230.8 8/7
9 334.7 17/14, 40/33 934.7 12/7
10 1038.5 20/11 438.5 9/7
11 542.4 15/11 1142.4 27/14, 64/33
12 46.2 36/35, 40/39, 45/44, 50/49 646.2 16/11
13 750.1 17/11, 20/13 150.1 12/11
14 253.9 15/13 853.9 18/11
15 957.8 45/26, 68/39 357.8 16/13
16 461.6 17/13 1061.6 24/13
17 1165.5 51/26, 96/49, 108/55 565.5 18/13

* In 17-limit CWE tuning, octave-reduced

As a detempereament of 12et

Diaschismic as a 58-tone 12et detempering

Diaschismic is naturally considered as a detemperament of the 12 equal temperament. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by 6 generator steps, which represent the syntonic comma. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.

The 13th harmonic is just beyond the specified generator range, so the diagram does not show it.

Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the syntonic comma in 58edo. 104edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.

Chords

Main article: Chords of diaschismic

Tunings

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
17/9 698.955
7\12 700.000 12f val, lower bound of 7-, 9- and 11-odd-limit diamond monotone
3/2 701.955
15/14 702.778
41\70 702.857 70ef val
7/5 702.915
21/20 703.107
15/11 703.359
15/13 703.410
34\58 703.448 Lower bound of 13-, 15-, 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
11/10 703.500
9/7 703.508
13/10 703.522
13/11 703.597
12/7 703.681
13/9 703.728
11/9 703.757
21/13 703.782
49/48 703.783
13/12 703.839
61\104 703.846 104c val
21/11 703.893
11/6 703.895
7/4 703.897
13/8 703.965
17/14 704.014
17/13 704.027
13/7 704.043
11/8 704.057
36/35 704.064
17/11 704.126
21/16 704.174
21/17 704.272
27\46 704.348 Upper bound of 11-, 13-, 15-, 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
11/7 704.377
9/5 704.399
17/16 704.955
5/3 705.214
25/24 705.866
20\34 705.882 34ef val, upper bound of 7- and 9-odd-limit diamond monotone
17/10 706.214
5/4 706.843
17/15 708.343
15/8 711.731

* Besides the octave

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