Sensi extensions: Difference between revisions

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VisualWikitext
-gencoms (irrelevant to tuning and trivial to derive)
Reduce the ratios
Line 296: Line 296:
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 308: Line 308:
|
|
|-
|-
| 18/13
| 13/9
| 440.846
| 440.846
|
|
Line 316: Line 316:
|
|
|-
|-
| 6/5
| 5/3
| 442.179
| 442.179
|
|
|-
|-
| 14/13
| 13/7
| 442.766
| 442.766
|
|
Line 328: Line 328:
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
| 15/8
| 443.017
| 443.017
|
|
Line 340: Line 340:
|
|
|-
|-
| 4/3
| 3/2
| 443.136
| 443.136
| 15-odd-limit minimax
| 15-odd-limit minimax
Line 348: Line 348:
|
|
|-
|-
| 12/11
| 11/6
| 443.211
| 443.211
|
|
Line 356: Line 356:
|
|
|-
|-
| 14/11
| 11/7
| 443.482
| 443.482
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
| 10/9
| 9/5
| 443.519
| 443.519
| 9- and 13-odd-limit minimax
| 9- and 13-odd-limit minimax
Line 367: Line 367:
| 443.568
| 443.568
|-
|-
| 8/7
| 7/4
| 443.756
| 443.756
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
| 16/13
| 13/8
| 444.053
| 444.053
|
|
Line 391: Line 391:
|
|
|}
|}
<nowiki/>* Besides the octave


=== Sensis ===
=== Sensis ===
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo<br>(unchanged-interval)
! Eigenmonzo<br>(unchanged-interval)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 411: Line 412:
|
|
|-
|-
| 18/13
| 13/9
| 440.846
| 440.846
|
|
Line 419: Line 420:
|
|
|-
|-
| 6/5
| 5/3
| 442.179
| 442.179
|
|
|-
|-
| 14/13
| 13/7
| 442.766
| 442.766
|
|
Line 431: Line 432:
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
| 15/8
| 443.017
| 443.017
|
|
|-
|-
| 4/3
| 3/2
| 443.136
| 443.136
|
|
|-
|-
| 10/9
| 9/5
| 443.519
| 443.519
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
| 8/7
| 7/4
| 443.756
| 443.756
| 7- and 11-odd-limit minimax
| 7- and 11-odd-limit minimax
|-
|-
| 16/13
| 13/8
| 444.053
| 444.053
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
Line 471: Line 472:
|
|
|-
|-
| 14/11
| 11/7
| 446.390
| 446.390
|
|
Line 483: Line 484:
|
|
|-
|-
| 12/11
| 11/6
| 450.212
| 450.212
|
|
Line 491: Line 492:
|
|
|}
|}
<nowiki/>* Besides the octave


=== Sensus ===
=== Sensus ===
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo<br>(unchanged-interval)
! Eigenmonzo<br>(unchanged-interval)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 507: Line 509:
|
|
|-
|-
| 18/13
| 13/9
| 440.846
| 440.846
|
|
Line 515: Line 517:
|
|
|-
|-
| 6/5
| 5/3
| 442.179
| 442.179
|
|
|-
|-
| 14/13
| 13/7
| 442.766
| 442.766
|
|
Line 527: Line 529:
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
| 15/8
| 443.017
| 443.017
|
|
|-
|-
| 4/3
| 3/2
| 443.136
| 443.136
|
|
Line 539: Line 541:
|
|
|-
|-
| 14/11
| 11/7
| 443.472
| 443.472
|
|
|-
|-
| 10/9
| 9/5
| 443.519
| 443.519
| 9-odd-limit minimax
| 9-odd-limit minimax
Line 551: Line 553:
|
|
|-
|-
| 12/11
| 11/6
| 443.723
| 443.723
|
|
|-
|-
| 8/7
| 7/4
| 443.756
| 443.756
| 7- and 11-odd-limit minimax
| 7- and 11-odd-limit minimax
Line 567: Line 569:
|
|
|-
|-
| 16/13
| 13/8
| 444.053
| 444.053
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
Line 591: Line 593:
|
|
|}
|}
<nowiki/>* Besides the octave


=== Sensa ===
=== Sensa ===
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo<br>(unchanged-interval)
! Eigenmonzo<br>(unchanged-interval)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 14/11
| 11/7
| 417.508
| 417.508
|
|
Line 619: Line 622:
|
|
|-
|-
| 18/13
| 13/9
| 440.846
| 440.846
|
|
Line 627: Line 630:
|
|
|-
|-
| 6/5
| 5/3
| 442.179
| 442.179
|
|
|-
|-
| 14/13
| 13/7
| 442.766
| 442.766
|
|
Line 639: Line 642:
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
| 15/8
| 443.017
| 443.017
|
|
|-
|-
| 4/3
| 3/2
| 443.136
| 443.136
|
|
|-
|-
| 10/9
| 9/5
| 443.519
| 443.519
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
| 8/7
| 7/4
| 443.756
| 443.756
| 7- and 11-odd-limit minimax
| 7- and 11-odd-limit minimax
|-
|-
| 16/13
| 13/8
| 444.053
| 444.053
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
Line 675: Line 678:
|
|
|-
|-
| 12/11
| 11/6
| 449.873
| 449.873
|
|
Line 691: Line 694:
|
|
|}
|}
<nowiki/>* Besides the octave


[[Category:Sensi]]
[[Category:Sensi]]

Revision as of 15:50, 2 November 2024

Sensi has multiple competing extensions to the 11-limit. The simplest 7-limit commas of sensi are starling (126/125) and sensamagic (245/243), and it can be viewed as the merge of the two corresponding rank-3 temperaments. These rank-3 temperaments are associated with distinct paths to the 11-limit. On one hand, starling strongly suggests tempering out 176/175, leading to thrush ({126/125, 176/175}), since 126/125 = (176/175)(441/440). On the other, sensamagic strongly suggests tempering out 385/384, leading to undecimal sensamagic ({245/243, 385/384}), since 245/243 = (385/384)(896/891). Taking either path for sensi leads us to one of the following entries:

  • Sensor (19 & 27) – tempering out 126/125, 245/243, and 385/384
  • Sensus (19e & 27e) – tempering out 126/125, 176/175, and 245/243

The two unite in 46et, where both 176/175 and 385/384, as well as their sum, 121/120, are tempered out. They can be extended to the 13- and 17-limit naturally by adding 91/90 and 154/153 to the comma list in this order. Then the generator represents 9/7, 13/10, and 22/17.

In addition, there are some low-complexity low-accuracy entries:

  • Sensis (19 & 27e) – tempering out 56/55, 100/99, and 245/243
  • Sensa (19e & 27) – tempering out 55/54, 77/75, and 99/98

Another possible path which relates a sense of compromise is to temper out 121/120, leading to bisensi. This has the effect of slicing the octave in two, and is supported by 38df, 46, and 54c.

Interval chain

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

# Cents* Approximate Ratios
Sensi Sensor Sensis Sensus Sensa
0 0.0 1/1
1 443.4 9/7, 13/10, 22/17 14/11, 17/13
2 886.8 5/3 17/10, 18/11, 22/13, 28/17
3 130.2 13/12, 14/13, 15/14 12/11, 17/16 11/10, 18/17
4 573.6 7/5, 18/13 11/8, 24/17 15/11, 17/12
5 1017.0 9/5 20/11 11/6, 30/17
6 260.4 7/6, 15/13 13/11, 20/17
7 703.8 3/2 26/17
8 1147.2 27/14, 35/18
9 390.6 5/4 14/11
10 834.0 13/8, 21/13 18/11, 28/17
11 77.4 21/20, 25/24 18/17 17/16
12 520.8 27/20 15/11 11/8
13 964.2 7/4 30/17
14 207.6 9/8 17/15
15 651.0 35/24 16/11 22/15
16 1094.5 15/8 32/17 17/9
17 337.9 39/32 11/9, 17/14
18 781.3 25/16 11/7
19 24.7 49/48, 65/64, 81/80
20 468.1 21/16 17/13
21 911.5 27/16 17/10, 22/13
22 154.9 35/32 12/11 11/10
23 598.3 45/32 24/17 17/12
24 1041.7 117/64 20/11 11/6
25 285.1 75/64 13/11, 20/17
26 728.5 49/32 26/17
27 1171.9 63/32
28 415.3 81/64 14/11
29 858.7 105/64 18/11, 28/17
30 102.1 135/128 18/17 17/16
31 545.5 175/128 15/11 11/8
32 988.9 225/128 30/17

* In 2.3.5.7.13.17/11 subgroup CTE tuning

Tuning spectra

Sensor

Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
9/7 435.084
15/14 439.814
13/9 440.846
15/13 441.290
5/3 442.179
13/7 442.766
5/4 442.924 5-odd-limit minimax
15/8 443.017
11/10 443.125
15/11 443.127
3/2 443.136 15-odd-limit minimax
11/9 443.193
11/6 443.211
11/8 443.245
11/7 443.482 11-odd-limit minimax
9/5 443.519 9- and 13-odd-limit minimax
13/11 443.568
7/4 443.756 7-odd-limit minimax
13/8 444.053
7/6 444.478
7/5 445.628
13/12 446.191
13/10 454.214

* Besides the octave

Sensis

Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
9/7 435.084
11/8 437.829
15/14 439.814
13/9 440.846
15/13 441.290
5/3 442.179
13/7 442.766
5/4 442.924 5-odd-limit minimax
15/8 443.017
3/2 443.136
9/5 443.519 9-odd-limit minimax
7/4 443.756 7- and 11-odd-limit minimax
13/8 444.053 13- and 15-odd-limit minimax
7/6 444.478
15/11 444.746
11/9 445.259
7/5 445.628
13/12 446.191
11/7 446.390
11/10 446.999
13/11 448.202
11/6 450.212
13/10 454.214

* Besides the octave

Sensus

Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
9/7 435.084
15/14 439.814
13/9 440.846
15/13 441.290
5/3 442.179
13/7 442.766
5/4 442.924 5-odd-limit minimax
15/8 443.017
3/2 443.136
13/11 443.371
11/7 443.472
9/5 443.519 9-odd-limit minimax
11/8 443.591
11/6 443.723
7/4 443.756 7- and 11-odd-limit minimax
11/10 443.864
11/9 443.965
13/8 444.053 13- and 15-odd-limit minimax
15/11 444.203
7/6 444.478
7/5 445.628
13/12 446.191
13/10 454.214

* Besides the octave

Sensa

Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
11/7 417.508
11/9 426.296
15/11 434.238
9/7 435.084
15/14 439.814
13/9 440.846
15/13 441.290
5/3 442.179
13/7 442.766
5/4 442.924 5-odd-limit minimax
15/8 443.017
3/2 443.136
9/5 443.519 9-odd-limit minimax
7/4 443.756 7- and 11-odd-limit minimax
13/8 444.053 13- and 15-odd-limit minimax
7/6 444.478
7/5 445.628
11/8 445.943
13/12 446.191
11/6 449.873
13/10 454.214
11/10 455.001
13/11 455.395

* Besides the octave

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