Cotoneum: Difference between revisions

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| Title = Cotoneum
| Title = Cotoneum
| Subgroups = 2.3.5.7, 2.3.5.7.11.13, 2.3.5.7.11.13.17.19
| Subgroups = 2.3.5.7, 2.3.5.7.11.13, 2.3.5.7.11.13.17.19
| Comma basis = [[10976/10935]], [[823543/819200]] (7-limit);<br>[[364/363]], [[441/440]], [[3584/3575]], [[10976/10935]] (13-limit);<br>[[343/342]], [[364/363]], [[441/440]], [[595/594]], [[1216/1215]],<br> [[1729/1728]] (19-limit)
| Comma basis = [[10976/10935]], [[823543/819200]] (7-limit);<br>[[364/363]], [[441/440]], [[3584/3575]], <br>[[10976/10935]] (13-limit);<br>[[343/342]], [[364/363]], [[441/440]], [[595/594]], <br>[[1216/1215]], [[1729/1728]] (19-limit)
| Edo join 1 = 41 | Edo join 2 = 217
| Edo join 1 = 41 | Edo join 2 = 217
| Mapping = 1; 1 -49 -14 23 61 89 -44
| Mapping = 1; 1 -49 -14 23 61 89 -44
| Generators = 3/2
| Generators = 3/2
| Generators tuning = 702.308
| Generators tuning = 702.31
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[12L 17s]], [[12L 29s]], [[41L 12s]], [[41L 53s]]
| MOS scales = [[12L 17s]], [[12L 29s]], [[41L 12s]], [[41L 53s]]
Line 13: Line 13:
| Odd limit 2 = 21 | Mistuning 2 = 2.48 | Complexity 2 = 176
| Odd limit 2 = 21 | Mistuning 2 = 2.48 | Complexity 2 = 176
}}
}}
'''Cotoneum''' is a [[rank]]-2 [[regular temperament|temperament]] for the 7- through 19-limit. It is a member of the [[hemimage temperaments]], [[quince clan]], and [[garischismic clan]]. The generator of cotoneum is a perfect fifth sharp by about 0.4–0.5 cents, and it maps [[8/7]] to the double-augmented unison (+14 fifths), [[tempering out]] the [[garischisma]]. However, unlike in [[garibaldi]], the schisma is not tempered out, meaning 5/4 is not found at the diminished fourth. Instead, 5/4 is found at the sextuple-diminished octave (–49 fifths). It is a weak extension of the [[2.5.7 subgroup|2.5.7-subgroup]] temperament [[mercy]], with its secor-sized generator mapped to the augmented unison.
'''Cotoneum''' is a [[rank]]-2 [[regular temperament|temperament]] for the 7- through 19-limit. The generator of cotoneum is a [[3/2|perfect fifth]] sharp by about 0.3–0.4 cents, and it maps [[8/7]] to the double-augmented unison (+14 fifths), [[tempering out]] the [[garischisma]]. However, unlike in [[garibaldi]], the schisma is not tempered out, meaning 5/4 is not found as a diminished fourth. Instead, 5/4 is found as a sextuple-diminished octave (−49 fifths). It is a weak extension of the [[2.5.7 subgroup|2.5.7-subgroup]] temperament [[mercy]], with its secor-sized generator mapped to the augmented unison. It is a member of the [[hemimage temperaments]], [[quince clan]], and [[garischismic clan]].  


It can seen as a detemperament of [[41edo|41 equal temperament]], with the [[countercomp comma|41-comma]] shrunk down to about 5 cents, representing many important intervals such as the [[schisma]], [[5120/5103]], [[243/242]], [[273/272]], [[325/324]], [[352/351]], [[385/384]], [[513/512]], [[896/891]], etc.
It can seen as a detemperament of [[41edo|41 equal temperament]], with the [[countercomp comma|41-comma]] shrunk down to about 5–6 cents for a generic aberschisma, which represents the [[schisma]] and [[aberschisma]].
 
This generic aberschisma takes on more important roles from the 11-limit onwards, where it represents [[176/175]], [[243/242]], [[385/384]], [[540/539]] and [[896/891]]. In the 13-limit it represents [[352/351]], in the 17-limit [[273/272]], and in the 19-limit the undevicesimal schisma of [[513/512]].


[[217edo]] is an excellent tuning for cotoneum, with a fifth generator of 127\217, and [[mos scale]]s of 12, 17, 29, 41, 53, 94, 135, or 176 notes are available.
[[217edo]] is an excellent tuning for cotoneum, with a fifth generator of 127\217, and [[mos scale]]s of 12, 17, 29, 41, 53, 94, 135, or 176 notes are available.


The temperament was named by [[User:Xenllium|Xenllium]] in 2021. ''Cotoneum'' is Latin for "quince".  
The temperament was named by [[Xenllium]] in 2021. ''Cotoneum'' is Latin for "quince".  


For technical data, see [[Garischismic clan #Cotoneum]].
For technical data, see [[Garischismic clan #Cotoneum]].
Line 27: Line 29:


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! Fifths
! #
! Cents <br>value*
! Cents*
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
| 0.000
| 0.00
| '''1/1'''
| '''1/1'''
|-
|-
| 1
| 1
| 702.308
| 702.31
| '''3/2'''
| '''3/2'''
|-
|-
| 2
| 2
| 204.615
| 204.62
| '''9/8'''
| '''9/8'''
|-
|-
| 3
| 3
| 906.923
| 906.92
| 27/16
| 27/16
|-
|-
| 4
| 4
| 409.231
| 409.23
| 19/15
| 19/15
|-
|-
| 5
| 5
| 1111.539
| 1111.54
| 19/10
| 19/10
|-
|-
| 6
| 6
| 613.846
| 613.85
| 57/40
| 57/40
|-
|-
| 7
| 7
| 116.154
| 116.15
| 77/72
| 77/72
|-
|-
| 8
| 8
| 818.462
| 818.46
| 77/48
| 77/48
|-
|-
| 9
| 9
| 320.770
| 320.77
| 77/64
| 77/64
|-
|-
| 10
| 10
| 1023.077
| 1023.08
| 65/36
| 65/36
|-
|-
| 11
| 11
| 525.385
| 525.38
| 65/48
| 65/48
|-
|-
| 12
| 12
| 27.693
| 27.69
| 56/55, 64/63, <br>65/64, 66/65
| 56/55, 64/63, 65/64, 66/65
|-
|-
| 13
| 13
| 730.001
| 730.00
| '''32/21'''
| '''32/21'''
|-
|-
| 14
| 14
| 232.308
| 232.31
| '''8/7'''
| '''8/7'''
|-
|-
| 15
| 15
| 934.616
| 934.62
| 12/7
| 12/7
|-
|-
| 16
| 16
| 436.924
| 436.92
| 9/7
| 9/7
|-
|-
| 17
| 17
| 1139.232
| 1139.23
| 27/14
| 27/14
|-
|-
| 18
| 18
| 641.539
| 641.54
| 81/56
| 81/56
|-
|-
| 19
| 19
| 143.847
| 143.85
|  
| 88/81
|-
|-
| 20
| 20
| 846.155
| 846.15
| 44/27
| 44/27
|-
|-
| 21
| 21
| 348.463
| 348.46
| 11/9
| 11/9
|-
|-
| 22
| 22
| 1050.770
| 1050.77
| 11/6
| 11/6
|-
|-
| 23
| 23
| 553.078
| 553.08
| '''11/8'''
| '''11/8'''
|-
|-
| 24
| 24
| 55.386
| 55.38
| 33/32, 65/63
| 33/32
|-
|-
| 25
| 25
| 757.694
| 757.69
| 65/42
| 65/42
|-
|-
| 26
| 26
| 260.001
| 260.00
| 64/55, 65/56
| 64/55, 65/56
|-
|-
| 27
| 27
| 962.309
| 962.31
| 68/39, 96/55
| 68/39, 96/55
|-
|-
| 28
| 28
| 464.617
| 464.62
| 17/13
| 17/13
|-
|-
| 29
| 29
| 1166.925
| 1166.92
| 51/26, 96/49,<br>108/55, 112/57
| 51/26, 96/49, 108/55, 112/57
|-
|-
| 30
| 30
| 669.232
| 669.23
| 28/19
| 28/19
|-
|-
| 31
| 31
| 171.540
| 171.54
| 21/19
| 21/19
|-
|-
| 32
| 32
| 873.848
| 873.85
| 63/38
| 63/38
|-
|-
| 33
| 33
| 376.156
| 376.15
| 56/45
| 56/45
|-
|-
| 34
| 34
| 1078.463
| 1078.46
| 28/15
| 28/15
|-
|-
| 35
| 35
| 580.771
| 580.77
| 7/5
| 7/5
|-
|-
| 36
| 36
| 83.079
| 83.08
| 21/20, 22/21
| 21/20, 22/21
|-
|-
| 37
| 37
| 785.387
| 785.38
| 11/7
| 11/7
|-
|-
| 38
| 38
| 287.694
| 287.69
| 13/11
| 13/11
|-
|-
| 39
| 39
| 990.002
| 990.00
| 39/22
| 39/22
|-
|-
| 40
| 40
| 492.310
| 492.31
|  
| 117/88
|-
|-
| 41
| 41
| 1194.618
| 1194.62
| 351/176, 891/448
| 351/176, 484/243, 539/270
|}
|}
<nowiki>*</nowiki> in 19-limit POTE tuning
<nowiki/>* In 19-limit CWE tuning, octave reduced


== Tuning spectrum ==
== Notation ==
Gencom: [2 4/3; 343/342 364/363 441/440 595/594 1216/1215 1729/1728]
Cotoneum can be notated just like [[cassaschismic]], with accidentals for the generic comma and the generic aberschisma. As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step. The only difference is that the aberschisma step which is independent in cassaschismic is equated with the 41-comma here. In other words, we have C–^↑↑E ~ C–↓↓E, implying ~11/9 (double-comma-up minor third) + an aberschisma-up = ~27/22 (double-comma-down major third).


Gencom mapping: [{{val| 1 2 -18 -3 13 29 41 -14 }}, {{val| 0 -1 49 14 -23 -61 -89 44 }}]
== Tunings ==
=== Norm-based 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 = 702.3149{{C}}
| CWE: ~3/2 = 702.3164{{C}}
| POTE: ~3/2 = 702.3170{{C}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.3063{{C}}
| CWE: ~3/2 = 702.3061{{C}}
| POTE: ~3/2 = 702.3060{{C}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 19-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.3069{{C}}
| CWE: ~3/2 = 702.3077{{C}}
| POTE: ~3/2 = 702.3077{{C}}
|}


{| class="wikitable center-1 center-2"
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-Interval)]]
! Edo generator
! Generator<br>(¢)
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator (¢)
! Comments
! Comments
|-
|-
| 4/3
| '''[[53edo|31\53]]'''
|
| '''701.8868'''
| '''Lower bound of 9-odd-limit [[diamond monotone]]'''<br>53cffgggh val
|-
|
| [[4/3]]
| 701.9550
| 701.9550
|  
|  
|-
|-
| 9/7
| '''[[94edo|55\94]]'''
|
| '''702.1277'''
| '''Lower bound of 11-odd-limit diamond monotone'''<br>94cfggh val
|-
|
| [[9/7]]
| 702.1928
| 702.1928
|  
|  
|-
|-
| 7/6
|  
| [[7/6]]
| 702.2086
| 702.2086
|  
|  
|-
|-
| 8/7
| '''[[135edo|79\135]]'''
|
| '''702.2222'''
| '''Lower bound of 13- and 15-odd-limit diamnod monotone''' <br>135cfgh val
|-
|
| [[8/7]]
| 702.2267
| 702.2267
|  
|  
|-
|-
| 14/11
|  
| [[14/11]]
| 702.2295
| 702.2295
|  
|  
|-
|-
| 11/8
|  
| [[11/8]]
| 702.2312
| 702.2312
|  
|  
|-
|-
| 22/21
|  
| [[22/21]]
| 702.2371
| 702.2371
|  
|  
|-
|-
| 20/19
|  
| [[20/19]]
| 702.2399
| 702.2399
|  
|  
|-
|-
| 12/11
|  
| [[12/11]]
| 702.2438
| 702.2438
|  
|  
|-
|-
| 21/16
|  
| [[21/16]]
| 702.2476
| 702.2476
|  
|  
|-
|-
| 11/9
|  
| [[11/9]]
| 702.2575
| 702.2575
|  
|  
|-
|-
| 14/13
| '''[[176edo|103\176]]'''
|
| '''702.2727'''
| '''Lower bound of 17- through 21-odd-limit diamond monotone'''
|-
|
| [[14/13]]
| 702.2894
| 702.2894
|  
|  
|-
|-
| 11/10
|  
| [[11/10]]
| 702.2917
| 702.2917
| 11 and 13-odd-limit minimax
| 11- and 13-odd-limit minimax
|-
|-
| 17/14
|  
| [[17/14]]
| 702.2925
| 702.2925
|  
|  
|-
|-
| 26/21
|  
| [[26/21]]
| 702.2939
| 702.2939
|  
|  
|-
|-
| 22/19
|  
| [[22/19]]
| 702.2956
| 702.2956
|  
|  
|-
|-
| 21/17
|  
| [[21/17]]
| 702.2958
| 702.2958
|  
|  
|-
|-
| 15/11
|  
| [[15/11]]
| 702.2965
| 702.2965
| 15, 17, 19, and 21-odd-limit minimax  
| 15- through 21-odd-limit minimax
|-
|-
| 17/13
|  
| [[17/13]]
| 702.3010
| 702.3010
|  
|  
|-
|-
| 17/16
|  
| [[17/16]]
| 702.3029
| 702.3029
|  
|  
|-
|-
| 16/13
|  
| [[16/13]]
| 702.3037
| 702.3037
|  
|  
|-
|-
| 10/9
| [[217edo|127\217]]
|
| 702.3041
|
|-
|
| [[10/9]]
| 702.3058
| 702.3058
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
| 24/17
|  
| [[24/17]]
| 702.3068
| 702.3068
|  
|  
|-
|-
| 20/17
|  
| [[20/17]]
| 702.3090
| 702.3090
|  
|  
|-
|-
| 13/12
|  
| [[13/12]]
| 702.3095
| 702.3095
|  
|  
|-
|-
| 18/17
|  
| [[18/17]]
| 702.3109
| 702.3109
|  
|  
|-
|-
| 13/10
|  
| [[13/10]]
| 702.3110
| 702.3110
|  
|  
|-
|-
| 19/15
|  
| [[19/15]]
| 702.3111
| 702.3111
|  
|  
|-
|-
| 17/15
|  
| [[17/15]]
| 702.3116
| 702.3116
|  
|  
|-
|-
| 19/17
|  
| [[19/17]]
| 702.3116
| 702.3116
|  
|  
|-
|-
| 6/5
|  
| [[6/5]]
| 702.3128
| 702.3128
| 5 and 7-odd-limit minimax
| 5- and 7-odd-limit minimax
|-
|-
| 19/18
|  
| [[19/18]]
| 702.3130
| 702.3130
|  
|  
|-
|-
| 15/13
|  
| [[15/13]]
| 702.3143
| 702.3143
|  
|  
|-
|-
| 26/19
|  
| [[26/19]]
| 702.3144
| 702.3144
|  
|  
|-
|-
| 18/13
|  
| [[18/13]]
| 702.3156
| 702.3156
|  
|  
|-
|-
| 5/4
|  
| [[5/4]]
| 702.3201
| 702.3201
|  
|  
|-
|-
| 24/19
|  
| [[24/19]]
| 702.3209
| 702.3209
|  
|  
|-
|-
| 16/15
| [[258edo|151\258]]
|
| 702.3256
|
|-
|
| [[16/15]]
| 702.3277
| 702.3277
|  
|  
|-
|-
| 22/17
|  
| [[22/17]]
| 702.3278
| 702.3278
|  
|  
|-
|-
| 19/16
|  
| [[19/16]]
| 702.3292
| 702.3292
|  
|  
|-
|-
| 21/20
|  
| [[21/20]]
| 702.3463
| 702.3463
|  
|  
|-
|-
| 13/11
|  
| [[13/11]]
| 702.3476
| 702.3476
|  
|  
|-
|-
| 7/5
|  
| [[7/5]]
| 702.3575
| 702.3575
|  
|  
|-
|-
| 21/19
|  
| [[21/19]]
| 702.3635
| 702.3635
|  
|  
|-
|-
| 15/14
|  
| [[15/14]]
| 702.3693
| 702.3693
|  
|  
|-
|-
| 19/14
|  
| [[19/14]]
| 702.3771
| 702.3771
|  
|  
|-
| '''[[41edo|24\41]]'''
|
| '''702.4390'''
| '''Upper bound of 11- through 21-odd-limit diamond monotone'''
|}
|}


Latest revision as of 08:51, 22 May 2026

Cotoneum
Subgroups 2.3.5.7, 2.3.5.7.11.13, 2.3.5.7.11.13.17.19
Comma basis 10976/10935, 823543/819200 (7-limit);
364/363, 441/440, 3584/3575,
10976/10935 (13-limit);
343/342, 364/363, 441/440, 595/594,
1216/1215, 1729/1728 (19-limit)
Reduced mapping ⟨1; 1 -49 -14 23 61 89 -44]
ET join 41 & 217
Generators (CWE) ~3/2 = 702.31 ¢
MOS scales 12L 17s, 12L 29s, 41L 12s, 41L 53s
Ploidacot monocot
Pergen (P8, P5)
Minimax error 15-odd-limit: 2.48 ¢;
21-odd-limit: 2.48 ¢
Target scale size 15-odd-limit: 135 notes;
21-odd-limit: 176 notes

Cotoneum is a rank-2 temperament for the 7- through 19-limit. The generator of cotoneum is a perfect fifth sharp by about 0.3–0.4 cents, and it maps 8/7 to the double-augmented unison (+14 fifths), tempering out the garischisma. However, unlike in garibaldi, the schisma is not tempered out, meaning 5/4 is not found as a diminished fourth. Instead, 5/4 is found as a sextuple-diminished octave (−49 fifths). It is a weak extension of the 2.5.7-subgroup temperament mercy, with its secor-sized generator mapped to the augmented unison. It is a member of the hemimage temperaments, quince clan, and garischismic clan.

It can seen as a detemperament of 41 equal temperament, with the 41-comma shrunk down to about 5–6 cents for a generic aberschisma, which represents the schisma and aberschisma.

This generic aberschisma takes on more important roles from the 11-limit onwards, where it represents 176/175, 243/242, 385/384, 540/539 and 896/891. In the 13-limit it represents 352/351, in the 17-limit 273/272, and in the 19-limit the undevicesimal schisma of 513/512.

217edo is an excellent tuning for cotoneum, with a fifth generator of 127\217, and mos scales of 12, 17, 29, 41, 53, 94, 135, or 176 notes are available.

The temperament was named by Xenllium in 2021. Cotoneum is Latin for "quince".

For technical data, see Garischismic clan #Cotoneum.

Interval chain

Odd harmonics and subharmonics 1–21 are in bold.

# Cents* Approximate ratios
0 0.00 1/1
1 702.31 3/2
2 204.62 9/8
3 906.92 27/16
4 409.23 19/15
5 1111.54 19/10
6 613.85 57/40
7 116.15 77/72
8 818.46 77/48
9 320.77 77/64
10 1023.08 65/36
11 525.38 65/48
12 27.69 56/55, 64/63, 65/64, 66/65
13 730.00 32/21
14 232.31 8/7
15 934.62 12/7
16 436.92 9/7
17 1139.23 27/14
18 641.54 81/56
19 143.85 88/81
20 846.15 44/27
21 348.46 11/9
22 1050.77 11/6
23 553.08 11/8
24 55.38 33/32
25 757.69 65/42
26 260.00 64/55, 65/56
27 962.31 68/39, 96/55
28 464.62 17/13
29 1166.92 51/26, 96/49, 108/55, 112/57
30 669.23 28/19
31 171.54 21/19
32 873.85 63/38
33 376.15 56/45
34 1078.46 28/15
35 580.77 7/5
36 83.08 21/20, 22/21
37 785.38 11/7
38 287.69 13/11
39 990.00 39/22
40 492.31 117/88
41 1194.62 351/176, 484/243, 539/270

* In 19-limit CWE tuning, octave reduced

Notation

Cotoneum can be notated just like cassaschismic, with accidentals for the generic comma and the generic aberschisma. As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step. The only difference is that the aberschisma step which is independent in cassaschismic is equated with the 41-comma here. In other words, we have C–^↑↑E ~ C–↓↓E, implying ~11/9 (double-comma-up minor third) + an aberschisma-up = ~27/22 (double-comma-down major third).

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 702.3149 ¢ CWE: ~3/2 = 702.3164 ¢ POTE: ~3/2 = 702.3170 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 702.3063 ¢ CWE: ~3/2 = 702.3061 ¢ POTE: ~3/2 = 702.3060 ¢
19-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 702.3069 ¢ CWE: ~3/2 = 702.3077 ¢ POTE: ~3/2 = 702.3077 ¢

Tuning spectrum

Edo generator Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
31\53 701.8868 Lower bound of 9-odd-limit diamond monotone
53cffgggh val
4/3 701.9550
55\94 702.1277 Lower bound of 11-odd-limit diamond monotone
94cfggh val
9/7 702.1928
7/6 702.2086
79\135 702.2222 Lower bound of 13- and 15-odd-limit diamnod monotone
135cfgh val
8/7 702.2267
14/11 702.2295
11/8 702.2312
22/21 702.2371
20/19 702.2399
12/11 702.2438
21/16 702.2476
11/9 702.2575
103\176 702.2727 Lower bound of 17- through 21-odd-limit diamond monotone
14/13 702.2894
11/10 702.2917 11- and 13-odd-limit minimax
17/14 702.2925
26/21 702.2939
22/19 702.2956
21/17 702.2958
15/11 702.2965 15- through 21-odd-limit minimax
17/13 702.3010
17/16 702.3029
16/13 702.3037
127\217 702.3041
10/9 702.3058 9-odd-limit minimax
24/17 702.3068
20/17 702.3090
13/12 702.3095
18/17 702.3109
13/10 702.3110
19/15 702.3111
17/15 702.3116
19/17 702.3116
6/5 702.3128 5- and 7-odd-limit minimax
19/18 702.3130
15/13 702.3143
26/19 702.3144
18/13 702.3156
5/4 702.3201
24/19 702.3209
151\258 702.3256
16/15 702.3277
22/17 702.3278
19/16 702.3292
21/20 702.3463
13/11 702.3476
7/5 702.3575
21/19 702.3635
15/14 702.3693
19/14 702.3771
24\41 702.4390 Upper bound of 11- through 21-odd-limit diamond monotone

Scales

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