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See [[Archytas_clan#Quasisuper|Archytas clan]].
{{Infobox regtemp
| Title = Quasisuper; quasisupra
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[64/63]], [[2430/2401]] (7-limit);<br>[[64/63]], [[99/98]], [[121/120]] (11-limit)
| Edo join 1 = 17c | Edo join 2 = 22
| Mapping = 1; 1 -13 -2 -6
| Generators = 3/2
| Generators tuning = 708.3
| Optimization method = CWE
| MOS scales = [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[5L&nbsp;12s]], [[17L&nbsp;5s]]
| Pergen = (P8, P5)
| Color name = Sasaguti
| Odd limit 1 = 9 | Mistuning 1 = 13.7 | Complexity 1 = 17
| Odd limit 2 = 11-limit 15 | Mistuning 2 = 14.9 | Complexity 2 = 17
}}
 
'''Quasisuper''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], [[tempering out]] [[64/63]]. The difference is that this extension maps [[prime interval|prime]] [[5/1|5]] to −13 [[generator]]s, as a double-diminished fifth (C–G𝄫). This extension works in the range [[17edo|17c-edo]] to [[22edo|22-edo]]. In contrast, full 7-limit [[superpyth]] does not work in this range, as tunings with a flatter fifth than 22edo swap the sizes of [[7/5]] and [[10/7]]. This extension may be preferred over superpyth due to having a softer [[5L 2s|diatonic]] scale, with a small step of around 60 [[cent]]s compared to about 50 cents in regular 7-limit superpyth.
 
The best extension to the [[11-limit]], '''quasisupra''', maps prime [[11/1|11]] to −6 generators as a diminished fifth (C–G♭), tempering out [[99/98]] as well as [[121/120]] and [[540/539]]. Removing prime 5 from quasisupra results in a 2.3.7.11-subgroup restriction, called '''supra''', which is notable for its simplicity. Finally, taking every other step of supra gives a 2.9.7.11-subgroup restriction, called [[machine]].
 
For technical data see [[Archytas clan #Quasisuper]] and [[Archytas clan #Supra|#Supra]].
 
== Interval chain ==
In the following tables, odd harmonics and subharmonics 1–11 are in '''bold'''.
 
<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2 right-4"
|+ style="font-size: 105%;" | Supra (2.3.7.11)
|-
! # !! Cents* !! Approximate ratios
|-
| 0 || 0.0 || '''1/1'''
|-
| 1 || 707.5 || '''3/2'''
|-
| 2 || 215.0 || '''8/7''', '''9/8'''
|-
| 3 || 922.5 || 12/7
|-
| 4 || 430.0 || 9/7, 14/11
|-
| 5 || 1137.5 || 21/11, 27/14, 64/33
|-
| 6 || 645.0 || '''16/11'''
|-
| 7 || 152.5 || 12/11
|}
<nowiki/>* In 2.3.7.11-subgroup [[CWE]] tuning, <br>octave reduced
</div></div>
<div><div style="display: inline-grid;">
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Quasisuper/quasisupra
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 708.3
| '''3/2'''
|-
| 2
| 216.6
| '''8/7''', '''9/8'''
|-
| 3
| 925.0
| 12/7
|-
| 4
| 433.3
| 9/7, 14/11
|-
| 5
| 1141.6
| 21/11, 27/14
|-
| 6
| 649.9
| '''16/11''', 22/15
|-
| 7
| 158.2
| 11/10, 12/11
|-
| 8
| 866.6
| 18/11
|-
| 9
| 374.9
| 27/22, 56/45
|-
| 10
| 1083.2
| 28/15
|-
| 11
| 591.5
| 7/5
|-
| 12
| 99.8
| 16/15
|-
| 13
| 808.2
| '''8/5'''
|-
| 14
| 316.5
| 6/5
|-
| 15
| 1024.8
| 9/5
|-
| 16
| 533.1
| 27/20
|-
| 17
| 41.4
| 81/80, 56/55
|}
<nowiki/>* in 11-limit CWE tuning, octave reduced
</div></div>
 
== Scales ==
=== Scala files ===
* [[Supra7]] – in 56edo tuning
* [[Supra12]] – in 56edo tuning
* [[12-22a]] – in 22edo tuning
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 708.7690{{c}}
| CWE: ~3/2 = 708.3716{{c}}
| POTE: ~3/2 = 708.2385{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 708.7131{{c}}
| CWE: ~3/2 = 708.3200{{c}}
| POTE: ~3/2 = 708.2046{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| [[12edo|7\12]]
|
| 700.000
| 12cc val
|-
|
| 3/2
| 701.955
| Pythagorean tuning
|-
|
| 21/11
| 703.893
|
|-
|
| 11/7
| 704.377
|
|-
| '''[[17edo|10\17]]'''
|
| '''705.882'''
| 17c val, '''lower bound of 7-, 9-, and 11-odd-limit diamond monotone'''
|-
|
| 11/9
| 706.574
|
|-
|
| 21/20
| 707.039
|
|-
| [[56edo|33\56]]
|
| 707.143
| 56cd val
|-
|
| 11/6
| 707.234
|
|-
|
| 7/5
| 707.501
|
|-
| [[39edo|23\39]]
|
| 707.692
| 39d val
|-
|
| 9/5
| 707.840
|
|-
|
| 15/14
| 708.056
|
|-
|
| 11/8
| 708.114
|
|-
| [[61edo|36\61]]
|
| 708.197
| 61d val
|-
|
| 5/3
| 708.260
|
|-
|
| 5/4
| 708.745
|
|-
|
| 9/7
| 708.771
|
|-
| '''[[22edo|13\22]]'''
|
| '''709.091'''
| '''Upper bound of 7-, 9-, and 11-odd-limit diamond monotone'''
|-
|
| 11/10
| 709.286
|
|-
|
| 15/8
| 709.311
|
|-
|
| 15/11
| 710.508
|
|-
|
| 7/6
| 711.043
|
|-
| [[27edo|16\27]]
|
| 711.111
| 27c val
|-
|
| 7/4
| 715.587
|
|-
| [[5edo|3\5]]
|
| 720.000
| 5c val
|-
|
| 21/16
| 729.219
|
|}
<nowiki/>* Besides the octave
 
[[Category:Quasisuper| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Archytas clan]]
[[Category:Nuwell temperaments]]
[[Category:Hemimage temperaments]]

Latest revision as of 13:07, 19 February 2026

Quasisuper; quasisupra
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 64/63, 2430/2401 (7-limit);
64/63, 99/98, 121/120 (11-limit)
Reduced mapping ⟨1; 1 -13 -2 -6]
ET join 17c & 22
Generators (CWE) ~3/2 = 708.3 ¢
MOS scales 5L 2s, 5L 7s, 5L 12s, 17L 5s
Ploidacot monocot
Pergen (P8, P5)
Color name Sasaguti
Minimax error 9-odd-limit: 13.7 ¢;
11-limit 15-odd-limit: 14.9 ¢
Target scale size 9-odd-limit: 17 notes;
11-limit 15-odd-limit: 17 notes

Quasisuper is an alternative extension of the archy chain of fifths to superpyth. Like superpyth, it is a temperament generated by a perfect fifth, where stacking two of them reaches the interval of 8/7~9/8, tempering out 64/63. The difference is that this extension maps prime 5 to −13 generators, as a double-diminished fifth (C–G𝄫). This extension works in the range 17c-edo to 22-edo. In contrast, full 7-limit superpyth does not work in this range, as tunings with a flatter fifth than 22edo swap the sizes of 7/5 and 10/7. This extension may be preferred over superpyth due to having a softer diatonic scale, with a small step of around 60 cents compared to about 50 cents in regular 7-limit superpyth.

The best extension to the 11-limit, quasisupra, maps prime 11 to −6 generators as a diminished fifth (C–G♭), tempering out 99/98 as well as 121/120 and 540/539. Removing prime 5 from quasisupra results in a 2.3.7.11-subgroup restriction, called supra, which is notable for its simplicity. Finally, taking every other step of supra gives a 2.9.7.11-subgroup restriction, called machine.

For technical data see Archytas clan #Quasisuper and #Supra.

Interval chain

In the following tables, odd harmonics and subharmonics 1–11 are in bold.

Supra (2.3.7.11)
# Cents* Approximate ratios
0 0.0 1/1
1 707.5 3/2
2 215.0 8/7, 9/8
3 922.5 12/7
4 430.0 9/7, 14/11
5 1137.5 21/11, 27/14, 64/33
6 645.0 16/11
7 152.5 12/11

* In 2.3.7.11-subgroup CWE tuning,
octave reduced

Quasisuper/quasisupra
# Cents* Approximate ratios
0 0.0 1/1
1 708.3 3/2
2 216.6 8/7, 9/8
3 925.0 12/7
4 433.3 9/7, 14/11
5 1141.6 21/11, 27/14
6 649.9 16/11, 22/15
7 158.2 11/10, 12/11
8 866.6 18/11
9 374.9 27/22, 56/45
10 1083.2 28/15
11 591.5 7/5
12 99.8 16/15
13 808.2 8/5
14 316.5 6/5
15 1024.8 9/5
16 533.1 27/20
17 41.4 81/80, 56/55

* in 11-limit CWE tuning, octave reduced

Scales

Scala files

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 708.7690 ¢ CWE: ~3/2 = 708.3716 ¢ POTE: ~3/2 = 708.2385 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 708.7131 ¢ CWE: ~3/2 = 708.3200 ¢ POTE: ~3/2 = 708.2046 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
7\12 700.000 12cc val
3/2 701.955 Pythagorean tuning
21/11 703.893
11/7 704.377
10\17 705.882 17c val, lower bound of 7-, 9-, and 11-odd-limit diamond monotone
11/9 706.574
21/20 707.039
33\56 707.143 56cd val
11/6 707.234
7/5 707.501
23\39 707.692 39d val
9/5 707.840
15/14 708.056
11/8 708.114
36\61 708.197 61d val
5/3 708.260
5/4 708.745
9/7 708.771
13\22 709.091 Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
11/10 709.286
15/8 709.311
15/11 710.508
7/6 711.043
16\27 711.111 27c val
7/4 715.587
3\5 720.000 5c val
21/16 729.219

* Besides the octave

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