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See [[Diaschismic_family#Diaschismic|Diaschismic family]].
{{Infobox Regtemp
[[Category:soft_redirect]]
| Title = Diaschismic; srutal archagall
| Subgroups = 2.3.5, 2.3.5.17
| Comma basis = [[2048/2025]] (2.3.5); <br /> [[136/135]], [[256/255]] (2.3.5.17)
| Edo join 1 = 12 | Edo join 2 = 22
| Generator = 16/15 | Generator tuning = 104.898 | Optimization method = POTE
| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
| Mapping = 2; 1 -2 1
| Pergen = (P8/2, P5)
| Color name = Saguguti
| Odd limit 1 = 5 | Mistuning 1 = 3.259 | Complexity 1 = 12
| Odd limit 2 = (2.3.5.17) 25 | Mistuning 2 = ??? | Complexity 2 = 22
}}
 
'''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]], and we also have a whole tone plus a period represent [[8/5]]. 9/8 splits in two very naturally into [[17/16]] × [[18/17]], and since we are equating half 9/8 to 16/15, it makes good sense to equate that interval to 17/16 and 18/17 as well, by tempering out [[S-expression|S16]] = [[256/255]], S17 = [[289/288]], and their product [[136/135]], leading to a 2.3.5.17 [[subgroup]] extension called '''srutal archagall'''.
 
The canonical [[extension]] to the [[7-limit]] lies where 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]], as well as the hemifamity comma, [[5120/5103]].
 
A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]), in this tuning range, 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]] and [[Diaschismic family #Septimal 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 detemperament 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 ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit prime-optimized tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~3/2 = 705.8655{{c}}
| CSEE: ~3/2 = 705.5568{{c}}
| POEE: ~3/2 = 704.9311{{c}}
|-
! Tenney
| CTE: ~3/2 = 705.1363{{c}}
| CWE: ~3/2 = 704.9585{{c}}
| POTE: ~3/2 = 704.8982{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~3/2 = 704.8398{{c}}
| CSBE: ~3/2 = 704.7309{{c}}
| POBE: ~3/2 = 704.7977{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~3/2 = 704.0394{{c}}
| CSEE: ~3/2 = 703.8161{{c}}
| POEE: ~3/2 = 703.3785{{c}}
|-
! Tenney
| CTE: ~3/2 = 704.0493{{c}}
| CWE: ~3/2 = 703.7738{{c}}
| POTE: ~3/2 = 703.6809{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~3/2 = 704.0589{{c}}
| CSBE: ~3/2 = 703.7520{{c}}
| POBE: ~3/2 = 703.7438{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 17-limit prime-optimized tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~3/2 = 704.0090{{c}}
| CSEE: ~3/2 = 703.9204{{c}}
| POEE: ~3/2 = 703.9138{{c}}
|-
! Tenney
| CTE: ~3/2 = 704.0164{{c}}
| CWE: ~3/2 = 703.8520{{c}}
| POTE: ~3/2 = 703.8121{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~3/2 = 704.0285{{c}}
| CSBE: ~3/2 = 703.7782{{c}}
| POBE: ~3/2 = 703.7642{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! 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:Starling temperaments]]
[[Category:Hemifamity temperaments]]
Retrieved from "https://en.xen.wiki/w/Diaschismic"