Trisected: Difference between revisions

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| Title = Trisected
| Title = Trisected
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[128/125]], [[1029/1024]] (7-limit);<br>[[56/55]], [[128/125]], [[1029/1000]] (11-limit);<br>[[56/55]], [[91/90]], [[128/125]], [[1029/1000]] (13-limit)
| Comma basis = [[128/125]], [[1029/1000]] (7-limit);<br>[[56/55]], [[128/125]], [[1029/1000]] (11-limit);<br>[[56/55]], [[91/90]], [[128/125]], [[1029/1000]] (13-limit)
| Edo join 1 = 15 | Edo join 2 = 36
| Edo join 1 = 15 | Edo join 2 = 36
| Mapping = 3; 3 0 -1 -1 7
| Mapping = 3; 3 0 -1 -1 7
| Generators = 8/7
| Generators = 10/7
| Generators tuning = 235.0
| Generators tuning = 635.0
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[6L 9s]], [[15L 6s]], [[15L 21s]]
| MOS scales = [[6L 9s]], [[15L 6s]], [[15L 21s]]
| Pergen = (P8/3, P5/3)
| Pergen = (P8/3, P5/3)
| Odd limit 1 = 9 | Mistuning 1 = 16.1 | Complexity 1 =  
| Odd limit 1 = 9 | Mistuning 1 = 16.1 | Complexity 1 = 36
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 17.5 | Complexity 2 =  
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 17.5 | Complexity 2 = 36
}}
}}
'''Trisected''' is the [[rank-2 temperament]] tempering out [[128/125]], [[1029/1000]], and [[1029/1024]] in the [[7-limit]], making it a member of the [[augmented family]], [[keegic temperaments]], and [[gamelismic clan]].
'''Trisected''' is the [[rank-2 temperament]] tempering out [[128/125]], [[1029/1000]], and [[1029/1024]] in the [[7-limit]], making it a member of the [[augmented family]], [[keegic temperaments]], and [[gamelismic clan]]. Since it tempers out 128/125, the [[2/1|octave]] is split into 3 ~[[5/4]]'s, each tuned to 400{{C}} if the octave is pure. Since it tempers out 1029/1024, the [[3/2|perfect fifth]] is split into three intervals of ~[[8/7]].Since it tempers out [[1029/1000]], the [[3/1|tritave]] is split into three intervals of [[10/7]]. This means that every [[Pythagorean tuning|Pythagorean]] interval is split into three equal parts.
 
In the [[11-limit]], the [[4/3|perfect fourth]] is split into three ~[[11/10]]'s, thus tempering out [[4000/3993]]. Additionally, the 1/3-octave period represents [[14/11]], tempering out [[56/55]] and [[176/175]]. The [[13-limit]] extension equates the ~10/7 with [[13/9]], tempering out [[91/90]] and [[2197/2187]].
 
The 2.3.7.11/5 subgroup [[restriction]], known as [[trisect]], removes the individual mappings for 5 and 11 while still tempering out 1029/1024 and 4000/3993, and is much more accurate.


For technical data, see [[Augmented family #Trisected]].
For technical data, see [[Augmented family #Trisected]].


{{Clear}}<!-- remove when no longer needed -->
== Intervals ==
== Intervals ==
{{Todo|complete section|inline=1}}
In the following table, odd harmonics 1–21 are in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4 right-6"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 2
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| 0
| 0.0
| '''1/1'''
| 400.0
| '''5/4''', 14/11
| 800.0
| '''8/5''', 11/7
|-
| 1
| 235.0
| '''8/7'''
| 635.0
| 10/7, '''16/11''', 13/9
| 1035.0
| 20/11
|-
| 2
| 470.0
| '''21/16'''
| 870.0
| 33/20
| 70.0
| 21/20, 33/32
|-
| 3
| 705.0
| '''3/2'''
| 1105.0
| '''15/8''', 21/11
| 305.0
| 6/5
|-
| 4
| 940.0
| 12/7, 26/15
| 140.0
| 12/11, 13/12, 15/14
| 540.0
| 15/11
|-
| 5
| 1175.0
| 63/32
| 375.0
| 26/21
| 775.0
| 52/33
|-
| 6
| 209.9
| 9/8
| 609.9
| 45/32, 63/44
| 1009.9
| 9/5
|-
| 7
| 444.9
| 9/7, 13/10
| 844.9
| 18/11, '''13/8'''
| 44.9
| 36/35
|-
| 8
| 679.9
| 52/35, 72/49
| 1079.9
| 13/7
| 279.9
| 13/11
|}
<nowiki/>* In 13-limit CWE tuning, octave reduced


== Tunings ==
== Tunings ==
{{Todo|complete section|categorize|inline=1}}
=== Norm-based tunings ===
{{Todo|review}}
{| 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: ~10/7 = 633.889{{C}}
| CWE: ~10/7 = 634.339{{C}}
| POTE: ~10/7 = 634.476{{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: ~10/7 = 634.215{{C}}
| CWE: ~10/7 = 634.769{{C}}
| POTE: ~10/7 = 634.893{{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: ~10/7 = 634.286{{C}}
| CWE: ~10/7 = 634.991{{C}}
| POTE: ~10/7 = 635.144{{C}}
|}
 
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~10/7 = 633.282{{C}}
| 7/6
|-
| 9-odd-limit
| ~10/7 = 633.583{{C}}
| 9/7
|-
| 11-odd-limit
| ~10/7 = 633.760{{C}}
| 77/45
|-
| 13-odd-limit
| ~10/7 = 633.962{{C}}
| 13/7
|-
| 15-odd-limit
| ~10/7 = 633.962{{C}}
| 13/7
|-
| 13-limit 21-odd-limit
| ~10/7 = 634.129{{C}}
| 45/44
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| [[7/5]]
| 617.488
|
|-
| [[21edo|11\21]]
|
| 628.571
| Lower bound of 7-odd-limit diamond monotone<br>21f val
|-
|
| [[15/8]]
| 629.423
|
|-
|
| [[15/14]]
| 629.861
|
|-
|
| [[7/4]]
| 631.174
|
|-
|
| [[7/6]]
| 633.282
| 7-odd-limit minimax
|-
| [[36edo|19\36]]
|
|
| Lower bound of 9- through 13-odd-limit diamond monotone<br>15-odd-limit diamond monotone (singleton)
|-
|
| [[9/7]]
| 633.583
| 9-odd-limit minimax
|-
|
| [[21/13]]
| 633.949
|
|-
|
| [[13/7]]
| 633.962
| 13- and 15-odd-limit minimax
|-
|
| [[3/2]]
| 633.985
|
|-
|
| [[15/11]]
| 634.238
|
|-
| [[87edo|46\87]]
|
| 634.483
| 87cee val
|-
|
| [[13/8]]
| 634.361
|
|-
|
| [[13/12]]
| 634.643
|
|-
|
| [[11/10]]
| 634.996
|
|-
| [[51edo|27\51]]
|
| 635.294
| 51ce val
|-
|
| [[21/16]]
| 635.390
|
|-
|
| [[11/9]]
| 636.085
|
|-
|
| [[13/11]]
| 636.151
|
|-
|
| [[9/5]]
| 636.266
|
|-
|
| [[13/10]]
| 636.316
|
|-
| [[66edo|35\66]]
|
| 636.364
| 66cef val
|-
|
| [[13/9]]
| 636.618
|
|-
|
| [[11/6]]
| 637.659
|
|-
|
| [[15/13]]
| 638.065
|
|-
|
| [[5/3]]
| 638.547
|
|-
|
| [[21/11]]
| 639.821
|
|-
| [[15edo|8\15]]
|
| 640.000
| Upper bound of 7- through 13-odd-limit diamond monotone
|-
|
| [[21/20]]
| 642.234
|
|}
<nowiki/>* Besides the octave


{{Stub}}
[[Category:Trisected| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Augmented family]]
[[Category:Gamelismic clan]]
[[Category:Keegic temperaments]]

Latest revision as of 18:31, 22 March 2026

Trisected
Subgroups 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis 128/125, 1029/1000 (7-limit);
56/55, 128/125, 1029/1000 (11-limit);
56/55, 91/90, 128/125, 1029/1000 (13-limit)
Reduced mapping ⟨3; 3 0 -1 -1 7]
ET join 15 & 36
Generators (CWE) ~10/7 = 635.0 ¢
MOS scales 6L 9s, 15L 6s, 15L 21s
Ploidacot triploid tricot
Pergen (P8/3, P5/3)
Minimax error 9-odd-limit: 16.1 ¢;
13-limit 21-odd-limit: 17.5 ¢
Target scale size 9-odd-limit: 36 notes;
13-limit 21-odd-limit: 36 notes

Trisected is the rank-2 temperament tempering out 128/125, 1029/1000, and 1029/1024 in the 7-limit, making it a member of the augmented family, keegic temperaments, and gamelismic clan. Since it tempers out 128/125, the octave is split into 3 ~5/4's, each tuned to 400 ¢ if the octave is pure. Since it tempers out 1029/1024, the perfect fifth is split into three intervals of ~8/7.Since it tempers out 1029/1000, the tritave is split into three intervals of 10/7. This means that every Pythagorean interval is split into three equal parts.

In the 11-limit, the perfect fourth is split into three ~11/10's, thus tempering out 4000/3993. Additionally, the 1/3-octave period represents 14/11, tempering out 56/55 and 176/175. The 13-limit extension equates the ~10/7 with 13/9, tempering out 91/90 and 2197/2187.

The 2.3.7.11/5 subgroup restriction, known as trisect, removes the individual mappings for 5 and 11 while still tempering out 1029/1024 and 4000/3993, and is much more accurate.

For technical data, see Augmented family #Trisected.

Intervals

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

# Period 0 Period 1 Period 2
Cents* Approx. ratios Cents* Approx. ratios Cents* Approx. ratios
0 0.0 1/1 400.0 5/4, 14/11 800.0 8/5, 11/7
1 235.0 8/7 635.0 10/7, 16/11, 13/9 1035.0 20/11
2 470.0 21/16 870.0 33/20 70.0 21/20, 33/32
3 705.0 3/2 1105.0 15/8, 21/11 305.0 6/5
4 940.0 12/7, 26/15 140.0 12/11, 13/12, 15/14 540.0 15/11
5 1175.0 63/32 375.0 26/21 775.0 52/33
6 209.9 9/8 609.9 45/32, 63/44 1009.9 9/5
7 444.9 9/7, 13/10 844.9 18/11, 13/8 44.9 36/35
8 679.9 52/35, 72/49 1079.9 13/7 279.9 13/11

* In 13-limit CWE tuning, octave reduced

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/7 = 633.889 ¢ CWE: ~10/7 = 634.339 ¢ POTE: ~10/7 = 634.476 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/7 = 634.215 ¢ CWE: ~10/7 = 634.769 ¢ POTE: ~10/7 = 634.893 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/7 = 634.286 ¢ CWE: ~10/7 = 634.991 ¢ POTE: ~10/7 = 635.144 ¢

Target tunings

Odd-limit-based target tunings
Target Minimax
Generator Eigenmonzo*
7-odd-limit ~10/7 = 633.282 ¢ 7/6
9-odd-limit ~10/7 = 633.583 ¢ 9/7
11-odd-limit ~10/7 = 633.760 ¢ 77/45
13-odd-limit ~10/7 = 633.962 ¢ 13/7
15-odd-limit ~10/7 = 633.962 ¢ 13/7
13-limit 21-odd-limit ~10/7 = 634.129 ¢ 45/44

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
7/5 617.488
11\21 628.571 Lower bound of 7-odd-limit diamond monotone
21f val
15/8 629.423
15/14 629.861
7/4 631.174
7/6 633.282 7-odd-limit minimax
19\36 Lower bound of 9- through 13-odd-limit diamond monotone
15-odd-limit diamond monotone (singleton)
9/7 633.583 9-odd-limit minimax
21/13 633.949
13/7 633.962 13- and 15-odd-limit minimax
3/2 633.985
15/11 634.238
46\87 634.483 87cee val
13/8 634.361
13/12 634.643
11/10 634.996
27\51 635.294 51ce val
21/16 635.390
11/9 636.085
13/11 636.151
9/5 636.266
13/10 636.316
35\66 636.364 66cef val
13/9 636.618
11/6 637.659
15/13 638.065
5/3 638.547
21/11 639.821
8\15 640.000 Upper bound of 7- through 13-odd-limit diamond monotone
21/20 642.234

* Besides the octave

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