Sensi extensions: Difference between revisions

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== Interval chain ==
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.
{| class="wikitable right-1 right-2"
{| class="wikitable right-1 right-2"
|-
|-
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|-
|-
| 0
| 0
| 0.000
| 0.0
| '''1/1'''
| '''1/1'''
|  
|  
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|-
|-
| 1
| 1
| 443.34
| 443.4
| 9/7, 13/10, 22/17
| 9/7, 13/10, 22/17
|  
|  
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|-
|-
| 2
| 2
| 886.69
| 886.8
| 5/3
| 5/3
|  
|  
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|-
|-
| 3
| 3
| 130.03
| 130.2
| 13/12, 14/13, 15/14
| 13/12, 14/13, 15/14
|  
|  
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|-
|-
| 4
| 4
| 573.38
| 573.6
| 7/5, 18/13
| 7/5, 18/13
|  
|  
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|-
|-
| 5
| 5
| 1016.72
| 1017.0
| 9/5
| 9/5
|  
|  
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|-
|-
| 6
| 6
| 260.07
| 260.4
| 7/6, 15/13
| 7/6, 15/13
|  
|  
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|-
|-
| 7
| 7
| 703.41
| 703.8
| '''3/2'''
| '''3/2'''
|  
|  
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|-
|-
| 8
| 8
| 1146.76
| 1147.2
|  
| 27/14, 35/18
|  
|  
|  
|  
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|-
|-
| 9
| 9
| 390.10
| 390.6
| '''5/4'''
| '''5/4'''
|  
|  
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|-
|-
| 10
| 10
| 833.45
| 834.0
| '''13/8'''
| '''13/8''', 21/13
|  
|  
| 18/11, 28/17
| 18/11, 28/17
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|-
|-
| 11
| 11
| 76.79
| 77.4
|  
| 21/20, 25/24
|  
|  
| 18/17
| 18/17
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|-
|-
| 12
| 12
| 520.14
| 520.8
|  
| 27/20
|  
|  
| 15/11
| 15/11
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|-
|-
| 13
| 13
| 963.48
| 964.2
| '''7/4'''
| '''7/4'''
|  
|  
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|-
|-
| 14
| 14
| 206.83
| 207.6
| 9/8
| '''9/8'''
|  
|  
|  
|  
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|-
|-
| 15
| 15
| 650.17
| 651.0
|  
| 35/24
| '''16/11'''
| '''16/11'''
|  
|  
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|-
|-
| 16
| 16
| 1093.51
| 1094.5
| 15/8
| '''15/8'''
| '''32/17'''
| '''32/17'''
|  
|  
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|-
|-
| 17
| 17
| 336.86
| 337.9
|  
| 39/32
|  
|  
|  
|  
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|-
|-
| 18
| 18
| 780.20
| 781.3
|  
| 25/16
|  
|  
|  
|  
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|-
|-
| 19
| 19
| 23.55
| 24.7
|  
| 49/48, 65/64, 81/80
|  
|  
|  
|  
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|-
|-
| 20
| 20
| 466.89
| 468.1
|  
| '''21/16'''
|  
|  
|  
|  
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|-
|-
| 21
| 21
| 910.24
| 911.5
|  
| 27/16
|  
|  
|  
|  
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|-
|-
| 22
| 22
| 153.58
| 154.9
|  
| 35/32
| 12/11
| 12/11
|  
|  
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|-
|-
| 23
| 23
| 596.93
| 598.3
|  
| 45/32
| 24/17
| 24/17
|  
|  
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|-
|-
| 24
| 24
| 1040.27
| 1041.7
|  
| 117/64
| 20/11
| 20/11
|  
|  
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|-
|-
| 25
| 25
| 283.62
| 285.1
|  
| 75/64
| 13/11, 20/17
| 13/11, 20/17
|  
|  
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|-
|-
| 26
| 26
| 726.96
| 728.5
|  
| 49/32
| 26/17
| 26/17
|  
|  
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|-
|-
| 27
| 27
| 1170.31
| 1171.9
|  
| 63/32
|  
|  
|  
|  
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|-
|-
| 28
| 28
| 413.65
| 415.3
|  
| 81/64
| 14/11
| 14/11
|  
|  
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|-
|-
| 29
| 29
| 857.00
| 858.7
|  
| 105/64
| 18/11, 28/17
| 18/11, 28/17
|  
|  
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|-
|-
| 30
| 30
| 100.34
| 102.1
|  
| 135/128
| 18/17
| 18/17
|  
|  
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|-
|-
| 31
| 31
| 543.68
| 545.5
|  
| 175/128
| 15/11
| 15/11
|  
|  
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|-
|-
| 32
| 32
| 987.03
| 988.9
|  
| 225/128
| 30/17
| 30/17
|  
|  
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|}
|}


: <sup>*</sup> in 2.3.5.7.13.17/11 POTE tuning
: <sup>*</sup> in 2.3.5.7.13.17/11 subgroup CTE tuning


== Tuning spectra ==
== Tuning spectra ==

Revision as of 08:20, 31 December 2023

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}). Note: 126/125 = (176/175)(441/440). On the other, sensamagic strongly suggests tempering out 385/384, leading to undecimal sensamagic ({245/243, 385/384}). Note: 245/243 = (385/384)(896/891). Taking either path for sensi leads us to the following entries:

  • Sensor (19 & 27) – tempering out 126/125, 245/243, 385/384
  • Sensus (19e & 27e) – tempering out 126/125, 176/175, 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, here are some low-complexity low-accuracy entries:

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

Another possible path which relates a sense of compromise is to temper out 121/120, leading to bisensi. This temperament 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

Gencom: [2 9/7; 91/90 126/125 169/168 385/384]

Gencom mapping: [⟨1 -1 -1 -2 9 0], ⟨0 7 9 13 -15 10]]

Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
9/7 435.084
15/14 439.814
18/13 440.846
15/13 441.290
6/5 442.179
14/13 442.766
5/4 442.924 5-odd-limit minimax
16/15 443.017
11/10 443.125
15/11 443.127
4/3 443.136 15-odd-limit minimax
11/9 443.193
12/11 443.211
11/8 443.245
14/11 443.482 11-odd-limit minimax
10/9 443.519 9- and 13-odd-limit minimax
13/11 443.568
8/7 443.756 7-odd-limit minimax
16/13 444.053
7/6 444.478
7/5 445.628
13/12 446.191
13/10 454.214

Sensis

Gencom: [2 9/7; 56/55 78/77 91/90 100/99]

Gencom mapping: [⟨1 -1 -1 -2 2 0], ⟨0 7 9 13 4 10]]

Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
9/7 435.084
11/8 437.829
15/14 439.814
18/13 440.846
15/13 441.290
6/5 442.179
14/13 442.766
5/4 442.924 5-odd-limit minimax
16/15 443.017
4/3 443.136
10/9 443.519 9-odd-limit minimax
8/7 443.756 7- and 11-odd-limit minimax
16/13 444.053 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
14/11 446.390
11/10 446.999
13/11 448.202
12/11 450.212
13/10 454.214

Sensus

Gencom: [2 9/7; 91/90 126/125 169/168 352/351]

Gencom mapping: [⟨1 -1 -1 -2 -8 0], ⟨0 7 9 13 31 10]]

Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
9/7 435.084
15/14 439.814
18/13 440.846
15/13 441.290
6/5 442.179
14/13 442.766
5/4 442.924 5-odd-limit minimax
16/15 443.017
4/3 443.136
13/11 443.371
14/11 443.472
10/9 443.519 9-odd-limit minimax
11/8 443.591
12/11 443.723
8/7 443.756 7- and 11-odd-limit minimax
11/10 443.864
11/9 443.965
16/13 444.053 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

Sensa

Gencom: [2 9/7; 55/54 66/65 77/75 143/140]

Gencom mapping: [⟨1 -1 -1 -2 -1 0], ⟨0 7 9 13 12 10]]

Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
14/11 417.508
11/9 426.296
15/11 434.238
9/7 435.084
15/14 439.814
18/13 440.846
15/13 441.290
6/5 442.179
14/13 442.766
5/4 442.924 5-odd-limit minimax
16/15 443.017
4/3 443.136
10/9 443.519 9-odd-limit minimax
8/7 443.756 7- and 11-odd-limit minimax
16/13 444.053 13- and 15-odd-limit minimax
7/6 444.478
7/5 445.628
11/8 445.943
13/12 446.191
12/11 449.873
13/10 454.214
11/10 455.001
13/11 455.395
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