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{{Infobox Regtemp
{{Infobox regtemp
| Title = Diaschismic; Srutal archagall
| Title = Diaschismic
| Subgroups = 2.3.5, 2.3.5.17
| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17
| Comma basis = [[2048/2025]] (2.3.5); <br /> [[136/135]], [[256/255]] (2.3.5.17)
| Comma basis = [[2048/2025]] (5-limit); <br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)
| Edo join 1 = 12 | Edo join 2 = 22
| Edo join 1 = 46 | Edo join 2 = 58
| Generator = 16/15 | Generator tuning = 104.898 | Optimization method = POTE
| Mapping = 2; 1 -2 -8 -12 -15 1
| Generators = 3/2 | Generators tuning = 703.9 | Optimization method = CWE
| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
| Mapping = 2; 1 -2 1
| Pergen = (P8/2, P5)
| Pergen = (P8/2, P5)
| Color name = Saguguti
| Color name = Saguguti
| Odd limit 1 = 5 | Mistuning 1 = 3.259 | Complexity 1 = 12
| Odd limit 1 = 5 | Mistuning 1 = 3.259 | Complexity 1 = 12
| Odd limit 2 = (2.3.5.17) 25 | Mistuning 2 = ??? | Complexity 2 = 22
| Odd limit 2 = 17-limit 21 | Mistuning 2 = ??? | Complexity 2 = 46
}}
}}
'''Diaschismic''', sometimes known as ''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]].


'''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 argent comma, [[5120/5103]]. This equates the [[64/63|septimal comma]] with the [[81/80|syntonic comma]] and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals.  


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 argent 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]]. The mappings of primes [[11/1|11]] and [[13/1|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp.  


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]]. The mappings of primes [[11/1|11]] and [[13/1|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp. Finally, since the whole tone has been split in two, each can be used to represent [[17/16]]~[[18/17]], leading to the same mapping of prime 17 as in srutal archagall. 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. This extension works best between [[46edo]] and [[58edo]].
Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S|16}}), [[289/288]] ({{S|17}}), and their product [[136/135]]. 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. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup|2.3.5.17-]][[subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.  


An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is mapped as in srutal archagall. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]].
An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is mapped as in srutal archagall. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]]. Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler representation of the 7-limit.  


Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler representation of the 7-limit.
See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic|#Septimal diaschismic]] for technical data.  
 
See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic]] for technical data.  


== Interval chain ==
== Interval chain ==
Line 152: Line 151:
[[File: Diaschismic 12et Detempering.png|thumb|Diaschismic as a 58-tone 12et detempering]]
[[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.  
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 generic comma step. 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.  
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.
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 comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise.
{{Clear}}


== Chords ==
== Chords and harmony ==
{{Main| Chords of diaschismic }}
{{See also| Chords of diaschismic }}


== Tunings ==
== Tunings ==
Line 319: Line 319:
|-
|-
|  
|  
| 12/7
| 7/6
| 703.681
| 703.681
|  
|  
|-
| [[162edo|95\162]]
|
| 703.704
| 162cef val
|-
|-
|  
|  
Line 372: Line 377:
| 703.965
| 703.965
|  
|  
|-
| [[150edo|88\150]]
|
| 704.000
| 150c val
|-
|-
|  
|  

Latest revision as of 10:18, 9 February 2026

Diaschismic
Subgroups 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17
Comma basis 2048/2025 (5-limit);
126/125, 2048/2025 (7-limit);
126/125, 136/135, 176/175,
196/195, 256/255 (17-limit)
Reduced mapping ⟨2; 1 -2 -8 -12 -15 1]
ET join 46 & 58
Generators (CWE) ~3/2 = 703.9 ¢
MOS scales 2L 8s, 10L 2s, 12L 10s
Ploidacot diploid monocot
Pergen (P8/2, P5)
Color name Saguguti
Minimax error 5-odd-limit: 3.259 ¢;
17-limit 21-odd-limit: ??? ¢
Target scale size 5-odd-limit: 12 notes;
17-limit 21-odd-limit: 46 notes

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, and we also have a whole tone plus a period represent 8/5.

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 argent comma, 5120/5103. This equates the septimal comma with the syntonic comma and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals.

A stack of twelve perfect fifths octave reduced (a 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. The mappings of primes 11 and 13 can also be characterized by parapyth, where the major third at +4 fifths represents 14/11, and the minor third at -3 fifths represents 13/11, which makes sense as the fifth is tuned slightly sharp.

Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to 17/16 and 18/17 as well by tempering out 256/255 (S16), 289/288 (S17), and their product 136/135. 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. This extension works best between 46edo and 58edo. The restriction to the 2.3.5.17-subgroup, called srutal archagall, is simpler and more flexible with its tunings (any tuning between 10edo and 12edo works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.

An alternative extension to the full 17-limit is srutal, which has a more complex mapping of prime 7 at +15 fifths, tempering out 4375/4374. The mappings for primes 11 and 13 follow through parapyth, and 17 is mapped as in srutal archagall. This works best for a sharper tuning, between 34edo (with the 34d val) and 46edo. Another option for extension is pajara, which equates the semioctave to 7/5 and 10/7. This inherits the inaccuracy of archy, while providing a much simpler representation of the 7-limit.

See Diaschismic family #Diaschismic and #Septimal 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 detemperament 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 generic comma step. 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 comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise.

Chords and harmony

See also: Chords of diaschismic

Tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~3/2 = 705.8655 ¢ CSEE: ~3/2 = 705.5568 ¢ POEE: ~3/2 = 704.9311 ¢
Tenney CTE: ~3/2 = 705.1363 ¢ CWE: ~3/2 = 704.9585 ¢ POTE: ~3/2 = 704.8982 ¢
Benedetti,
Wilson 		CBE: ~3/2 = 704.8398 ¢ CSBE: ~3/2 = 704.7309 ¢ POBE: ~3/2 = 704.7977 ¢
7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~3/2 = 704.0394 ¢ CSEE: ~3/2 = 703.8161 ¢ POEE: ~3/2 = 703.3785 ¢
Tenney CTE: ~3/2 = 704.0493 ¢ CWE: ~3/2 = 703.7738 ¢ POTE: ~3/2 = 703.6809 ¢
Benedetti,
Wilson 		CBE: ~3/2 = 704.0589 ¢ CSBE: ~3/2 = 703.7520 ¢ POBE: ~3/2 = 703.7438 ¢
17-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~3/2 = 704.0090 ¢ CSEE: ~3/2 = 703.9204 ¢ POEE: ~3/2 = 703.9138 ¢
Tenney CTE: ~3/2 = 704.0164 ¢ CWE: ~3/2 = 703.8520 ¢ POTE: ~3/2 = 703.8121 ¢
Benedetti,
Wilson 		CBE: ~3/2 = 704.0285 ¢ CSBE: ~3/2 = 703.7782 ¢ POBE: ~3/2 = 703.7642 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		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
7/6 703.681
95\162 703.704 162cef val
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
88\150 704.000 150c val
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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