Sensi extensions: Difference between revisions

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[[Sensi]] temperament has various extensions to the 11-limit. These are all members of the [[sensipent family]], but in addition they are linear temperaments:  
[[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}). 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
* '''Sensor''' (19 & 27) – tempering out 126/125, 245/243, 385/384
* Sensis (19 & 27e) - tempering out 56/55, 100/99, 245/243
* '''Sensus''' (19e & 27e) – tempering out 126/125, 176/175, 245/243
* Sensus (19e & 27e) - tempering out 126/125, 176/175, 245/243
* Sensa (19e & 27) - tempering out 55/54, 77/75, 99/98


These can be extended 13 and 17 prime limits 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, 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 [[38edo|38df]], 46, and [[54edo|54c]].  


== Interval chain ==
== Interval chain ==
{| class="wikitable right-1 right-2"
{| class="wikitable right-1 right-2"
|-
|-
! rowspan=2 | Generators
! rowspan=2 | #
! rowspan=2 | Cents<sup>*</sup>
! rowspan=2 | Cents<sup>*</sup>
! colspan=4 | Approximate ratios
! colspan=4 | Approximate ratios
Line 244: Line 248:
Gencom: [2 9/7; 91/90 126/125 169/168 385/384]
Gencom: [2 9/7; 91/90 126/125 169/168 385/384]


Gencom mapping: [{{val| 1 -1 -1 -2 9 0 }}, {{val| 0 7 9 13 -15 10 }}]
Gencom mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}


{| class="wikitable center-all"
{| class="wikitable center-1 center-2"
|-
|-
! [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval]])
! supermajor<br>third (¢)
! Generator (¢)
! comments
! Comments
|-
|-
| 9/7
| 9/7
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Gencom: [2 9/7; 56/55 78/77 91/90 100/99]
Gencom: [2 9/7; 56/55 78/77 91/90 100/99]


Gencom mapping: [{{val| 1 -1 -1 -2 2 0 }}, {{val| 0 7 9 13 4 10 }}]
Gencom mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}


{| class="wikitable center-all"
{| class="wikitable center-1 center-2"
|-
|-
! eigenmonzo<br>(unchanged-interval)
! Eigenmonzo<br>(Unchanged-interval)
! supermajor<br>third (¢)
! Generator (¢)
! comments
! Comments
|-
|-
| 9/7
| 9/7
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Gencom: [2 9/7; 91/90 126/125 169/168 352/351]
Gencom: [2 9/7; 91/90 126/125 169/168 352/351]


Gencom mapping: [{{val| 1 -1 -1 -2 -8 0 }}, {{val| 0 7 9 13 31 10 }}]
Gencom mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}


{| class="wikitable center-all"
{| class="wikitable center-1 center-2"
|-
|-
! eigenmonzo<br>(unchanged-interval)
! Eigenmonzo<br>(Unchanged-interval)
! supermajor<br>third (¢)
! Generator (¢)
! comments
! Comments
|-
|-
| 9/7
| 9/7
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Gencom: [2 9/7; 55/54 66/65 77/75 143/140]
Gencom: [2 9/7; 55/54 66/65 77/75 143/140]


Gencom mapping: [{{val| 1 -1 -1 -2 -1 0 }}, {{val| 0 7 9 13 12 10 }}]
Gencom mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10 }}


{| class="wikitable center-all"
{| class="wikitable center-1 center-2"
|-
|-
! eigenmonzo<br>(unchanged-interval)
! Eigenmonzo<br>(Unchanged-interval)
! supermajor<br>third (¢)
! Generator (¢)
! comments
! Comments
|-
|-
| 14/11
| 14/11
Line 656: Line 660:
|}
|}


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

Revision as of 07:52, 6 October 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

# Cents* Approximate ratios
Sensor Sensis Sensus Sensa
0 0.000 1/1
1 443.34 9/7, 13/10, 22/17 9/7, 13/10,
14/11, 17/13,
22/17
2 886.69 5/3 5/3, 17/10,
18/11, 22/13,
28/17
3 130.03 13/12, 14/13,
15/14 		12/11, 13/12,
14/13, 15/14,
17/16 		13/12, 14/13,
15/14 		11/10, 13/12,
14/13, 15/14,
18/17
4 573.38 7/5, 18/13 7/5, 11/8,
18/13, 24/17 		7/5, 18/13 7/5, 15/11,
17/12, 18/13
5 1016.72 9/5 9/5, 20/11 9/5 9/5, 11/6,
30/17
6 260.07 7/6, 15/13 7/6, 13/11,
15/13, 20/17 		7/6, 15/13
7 703.41 3/2 3/2, 26/17 3/2
8 1146.76
9 390.10 5/4 5/4, 14/11 5/4
10 833.45 13/8 13/8, 18/11,
28/17 		13/8
11 76.79 18/17 17/16
12 520.14 15/11 11/8
13 963.48 7/4 7/4, 30/17 7/4
14 206.83 9/8 9/8, 17/15 9/8
15 650.17 16/11 22/15
16 1093.51 15/8, 32/17 15/8 15/8, 17/9 15/8
17 336.86 11/9, 17/14
18 780.20 11/7
19 23.55
20 466.89 17/13
21 910.24 17/10, 22/13
22 153.58 12/11 11/10
23 596.93 24/17 17/12
24 1040.27 20/11 11/6
25 283.62 13/11, 20/17
26 726.96 26/17
27 1170.31
28 413.65 14/11
29 857.00 18/11, 28/17
30 100.34 18/17 17/16
31 543.68 15/11 11/8
32 987.03 30/17
* in 2.3.5.7.13.17/11 POTE 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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