Trisected: Difference between revisions

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


{{Clear}}<!-- remove when no longer needed -->
== Intervals ==
== Intervals ==
In the following table, odd harmonics 1–21 are in '''bold'''.
In the following table, odd harmonics 1–21 are in '''bold'''.
Line 165: Line 164:


=== Target tunings ===
=== Target tunings ===
{{Todo|inline=1|complete section}}
{| 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 ===
=== Tuning spectrum ===
{{Todo|inline=1|complete section}}
{| 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


[[Category:Trisected| ]] <!-- main article -->
[[Category:Trisected| ]] <!-- main article -->

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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