Sensi extensions: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Analyse the problem in the intro. Cleanup
→Interval chain: was a readability hell
Line 16: Line 16:
! rowspan=2 | #
! rowspan=2 | #
! rowspan=2 | Cents<sup>*</sup>
! rowspan=2 | Cents<sup>*</sup>
! colspan=4 | Approximate ratios
! colspan=5 | Approximate Ratios
|-
|-
! Sensi
! Sensor
! Sensor
! Sensis
! Sensis
Line 25: Line 26:
| 0
| 0
| 0.000
| 0.000
| colspan=4 | 1/1
| '''1/1'''
|
|
|
|
|-
|-
| 1
| 1
| 443.34
| 443.34
| colspan=3 | 9/7, 13/10, 22/17
| 9/7, 13/10, 22/17
| 9/7, 13/10, <br>14/11, 17/13, <br>22/17
|  
|
|
| 14/11, 17/13
|-
|-
| 2
| 2
| 886.69
| 886.69
| colspan=3 | 5/3
| 5/3
| 5/3, 17/10, <br>18/11, 22/13, <br>28/17
|  
|
|
| 17/10, 18/11, 22/13, 28/17
|-
|-
| 3
| 3
| 130.03
| 130.03
| 13/12, 14/13, <br>15/14
| 13/12, 14/13, 15/14
| 12/11, 13/12, <br>14/13, 15/14, <br>17/16
|
| 13/12, 14/13, <br>15/14
| 12/11, '''17/16'''
| 11/10, 13/12, <br>14/13, 15/14, <br>18/17
|  
| 11/10, 18/17
|-
|-
| 4
| 4
| 573.38
| 573.38
| 7/5, 18/13
| 7/5, 18/13
| 7/5, 11/8, <br>18/13, 24/17
|  
| 7/5, 18/13
| '''11/8''', 24/17
| 7/5, 15/11, <br>17/12, 18/13
|  
| 15/11, 17/12
|-
|-
| 5
| 5
| 1016.72
| 1016.72
| 9/5
| 9/5
| 9/5, 20/11
|  
| 9/5
| 20/11
| 9/5, 11/6, <br>30/17
|  
| 11/6, 30/17
|-
|-
| 6
| 6
| 260.07
| 260.07
| 7/6, 15/13
| 7/6, 15/13
| 7/6, 13/11, <br>15/13, 20/17
|  
| colspan=2 | 7/6, 15/13
| 13/11, 20/17
|  
|  
|-
|-
| 7
| 7
| 703.41
| 703.41
| 3/2
| '''3/2'''
| 3/2, 26/17
|
| colspan=2 | 3/2
| 26/17
|  
|  
|-
|-
| 8
| 8
| 1146.76
| 1146.76
|
|  
|  
|  
|  
Line 79: Line 98:
| 9
| 9
| 390.10
| 390.10
| 5/4
| '''5/4'''
| 5/4, 14/11
|
| colspan=2 | 5/4
| 14/11
|  
|  
|-
|-
| 10
| 10
| 833.45
| 833.45
| 13/8
| '''13/8'''
| 13/8, 18/11, <br>28/17
|
| colspan=2 | 13/8
| 18/11, 28/17
|  
|  
|-
|-
| 11
| 11
| 76.79
| 76.79
|
|  
|  
| 18/17
| 18/17
|  
|  
| 17/16
| '''17/16'''
|-
|-
| 12
| 12
| 520.14
| 520.14
|
|  
|  
| 15/11
| 15/11
|  
|  
| 11/8
| '''11/8'''
|-
|-
| 13
| 13
| 963.48
| 963.48
| 7/4
| '''7/4'''
| 7/4, 30/17
|
| colspan=2 | 7/4
| 30/17
|  
|  
|-
|-
| 14
| 14
| 206.83
| 206.83
| colspan=2 | 9/8
| 9/8, 17/15
| 9/8
| 9/8
|
|
| 17/15
|
|-
|-
| 15
| 15
| 650.17
| 650.17
| 16/11
|  
| '''16/11'''
|  
|  
| 22/15
| 22/15
Line 124: Line 154:
| 16
| 16
| 1093.51
| 1093.51
| 15/8, 32/17
| 15/8
| 15/8, 17/9
| 15/8
| 15/8
| '''32/17'''
|
| 17/9
|
|-
|-
| 17
| 17
| 336.86
| 336.86
|
|  
|  
|  
|  
Line 138: Line 170:
| 18
| 18
| 780.20
| 780.20
|
|  
|  
|  
|  
Line 145: Line 178:
| 19
| 19
| 23.55
| 23.55
|
|  
|  
|  
|  
Line 152: Line 186:
| 20
| 20
| 466.89
| 466.89
|
|  
|  
|  
|  
Line 159: Line 194:
| 21
| 21
| 910.24
| 910.24
|
|  
|  
|  
|  
Line 166: Line 202:
| 22
| 22
| 153.58
| 153.58
|
| 12/11
| 12/11
|  
|  
Line 173: Line 210:
| 23
| 23
| 596.93
| 596.93
|
| 24/17
| 24/17
|  
|  
Line 180: Line 218:
| 24
| 24
| 1040.27
| 1040.27
|
| 20/11
| 20/11
|  
|  
Line 187: Line 226:
| 25
| 25
| 283.62
| 283.62
|
| 13/11, 20/17
| 13/11, 20/17
|  
|  
Line 194: Line 234:
| 26
| 26
| 726.96
| 726.96
|
| 26/17
| 26/17
|  
|  
Line 201: Line 242:
| 27
| 27
| 1170.31
| 1170.31
|
|  
|  
|  
|  
Line 208: Line 250:
| 28
| 28
| 413.65
| 413.65
|
| 14/11
| 14/11
|  
|  
Line 215: Line 258:
| 29
| 29
| 857.00
| 857.00
|
| 18/11, 28/17
| 18/11, 28/17
|  
|  
Line 222: Line 266:
| 30
| 30
| 100.34
| 100.34
|
| 18/17
| 18/17
|  
|  
| 17/16
| '''17/16'''
|  
|  
|-
|-
| 31
| 31
| 543.68
| 543.68
|
| 15/11
| 15/11
|  
|  
| 11/8
| '''11/8'''
|  
|  
|-
|-
| 32
| 32
| 987.03
| 987.03
|
| 30/17
| 30/17
|  
|  

Revision as of 03:35, 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

# Cents* Approximate Ratios
Sensi Sensor Sensis Sensus Sensa
0 0.000 1/1
1 443.34 9/7, 13/10, 22/17 14/11, 17/13
2 886.69 5/3 17/10, 18/11, 22/13, 28/17
3 130.03 13/12, 14/13, 15/14 12/11, 17/16 11/10, 18/17
4 573.38 7/5, 18/13 11/8, 24/17 15/11, 17/12
5 1016.72 9/5 20/11 11/6, 30/17
6 260.07 7/6, 15/13 13/11, 20/17
7 703.41 3/2 26/17
8 1146.76
9 390.10 5/4 14/11
10 833.45 13/8 18/11, 28/17
11 76.79 18/17 17/16
12 520.14 15/11 11/8
13 963.48 7/4 30/17
14 206.83 9/8 17/15
15 650.17 16/11 22/15
16 1093.51 15/8 32/17 17/9
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
Retrieved from "https://en.xen.wiki/index.php?title=Sensi_extensions&oldid=130125"