Sensi extensions: Difference between revisions

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[[Sensi]] has multiple competing [[extension]]s to the [[11-limit]]. The simplest [[7-limit]] [[comma]]s of sensi are [[126/125|starling (126/125)]] and [[245/243|sensamagic (245/243)]], and it can be viewed as the merge of the two corresponding [[rank-3 temperament]]s. 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}). Notice the factorization 126/125 = (176/175)(441/440). On the other, [[sensamagic]] strongly suggests tempering out [[385/384]], leading to undecimal sensamagic ({245/243, 385/384}). Notice the factorization 245/243 = (385/384)(896/891). Taking either path for sensi leads us to one of the following entries:  
[[Sensi]] has multiple competing [[extension]]s to the [[11-limit]]. The simplest [[7-limit]] [[comma]]s of sensi are [[126/125|starling (126/125)]] and [[245/243|sensamagic (245/243)]], and it can be viewed as the merge of the two corresponding [[rank-3 temperament]]s. 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 {{nowrap|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 {{nowrap|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, 385/384
* '''Sensor''' (19 & 27) – tempering out 126/125, 245/243, and 385/384
* '''Sensus''' (19e & 27e) – tempering out 126/125, 176/175, 245/243
* '''Sensus''' (19e & 27e) – tempering out 126/125, 176/175, and 245/243


The two unite in [[46edo|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]].
The two unite in [[46edo|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:  
In addition, there are some low-complexity low-accuracy entries:  
* '''Sensis''' (19 & 27e) – tempering out 56/55, 100/99, 245/243
* '''Sensis''' (19 & 27e) – tempering out 56/55, 100/99, and 245/243
* '''Sensa''' (19e & 27) – tempering out 55/54, 77/75, 99/98
* '''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 [[38edo|38df]], 46, and [[54edo|54c]].  
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 [[38edo|38df]], 46, and [[54edo|54c]].  
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{| class="wikitable right-1 right-2"
{| class="wikitable right-1 right-2"
|-
|-
! rowspan=2 | #
! rowspan="2" | #
! rowspan=2 | Cents*
! rowspan="2" | Cents*
! colspan=5 | Approximate Ratios
! colspan="5" | Approximate Ratios
|-
|-
! Sensi
! Sensi
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|  
|  
|}
|}
<nowiki>*</nowiki> In 2.3.5.7.13.17/11 subgroup CTE tuning
<nowiki />* In 2.3.5.7.13.17/11 subgroup CTE tuning


== Tuning spectra ==
== Tuning spectra ==
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{| 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
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{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo<br>(Unchanged-interval)
! Eigenmonzo<br />(Unchanged-interval)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo<br>(Unchanged-interval)
! Eigenmonzo<br />(Unchanged-interval)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo<br>(Unchanged-interval)
! Eigenmonzo<br />(Unchanged-interval)
! Generator (¢)
! Generator (¢)
! Comments
! Comments

Revision as of 08:59, 12 October 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

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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