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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}). 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:  
{{Breadcrumb|Sensi}}
* '''Sensor''' (19 & 27) – tempering out 126/125, 245/243, 385/384
 
* '''Sensus''' (19e & 27e) – tempering out 126/125, 176/175, 245/243
[[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 ({{nowrap|{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 ({{nowrap|{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''' ({{nowrap| 19 & 27 }}) – tempering out 126/125, 245/243, and 385/384
* '''Sensus''' ({{nowrap| 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, here 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''' ({{nowrap| 19 & 27e }}) – tempering out 56/55, 100/99, and 245/243
* '''Sensa''' (19e & 27) – tempering out 55/54, 77/75, 99/98
* '''Sensa''' ({{nowrap| 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 temperament 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]].
 
See [[Sensipent family #Sensor]], [[Sensipent family #Sensus|#Sensus]], [[Sensipent family #Sensis|#Sensis]], and [[Sensipent family #Sensa|#Sensa]] for technical data.  


== 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"
|-
|-
! rowspan=2 | #
! rowspan="2" | #
! rowspan=2 | Cents<sup>*</sup>
! rowspan="2" | Cents*
! colspan=5 | Approximate Ratios
! colspan="5" | Approximate ratios
|-
|-
! Sensi
! Sensi
Line 25: Line 31:
|-
|-
| 0
| 0
| 0.000
| 0.0
| '''1/1'''
| '''1/1'''
|  
|  
Line 33: Line 39:
|-
|-
| 1
| 1
| 443.34
| 443.4
| 9/7, 13/10, 22/17
| 9/7, 13/10, 22/17
|  
|  
Line 41: Line 47:
|-
|-
| 2
| 2
| 886.69
| 886.8
| 5/3
| 5/3
|  
|  
Line 49: Line 55:
|-
|-
| 3
| 3
| 130.03
| 130.2
| 13/12, 14/13, 15/14
| 13/12, 14/13, 15/14
|  
|  
Line 57: Line 63:
|-
|-
| 4
| 4
| 573.38
| 573.6
| 7/5, 18/13
| 7/5, 18/13
|  
|  
Line 65: Line 71:
|-
|-
| 5
| 5
| 1016.72
| 1017.0
| 9/5
| 9/5
|  
|  
Line 73: Line 79:
|-
|-
| 6
| 6
| 260.07
| 260.4
| 7/6, 15/13
| 7/6, 15/13
|  
|  
Line 81: Line 87:
|-
|-
| 7
| 7
| 703.41
| 703.8
| '''3/2'''
| '''3/2'''
|  
|  
Line 89: Line 95:
|-
|-
| 8
| 8
| 1146.76
| 1147.2
|  
| 27/14, 35/18
|  
|  
|  
|  
Line 97: Line 103:
|-
|-
| 9
| 9
| 390.10
| 390.6
| '''5/4'''
| '''5/4'''
|  
|  
Line 105: Line 111:
|-
|-
| 10
| 10
| 833.45
| 834.0
| '''13/8'''
| '''13/8''', 21/13
|  
|  
| 18/11, 28/17
| 18/11, 28/17
Line 113: Line 119:
|-
|-
| 11
| 11
| 76.79
| 77.4
|  
| 21/20, 25/24
|  
|  
| 18/17
| 18/17
Line 121: Line 127:
|-
|-
| 12
| 12
| 520.14
| 520.8
|  
| 27/20
|  
|  
| 15/11
| 15/11
Line 129: Line 135:
|-
|-
| 13
| 13
| 963.48
| 964.2
| '''7/4'''
| '''7/4'''
|  
|  
Line 137: Line 143:
|-
|-
| 14
| 14
| 206.83
| 207.6
| 9/8
| '''9/8'''
|  
|  
|  
|  
Line 145: Line 151:
|-
|-
| 15
| 15
| 650.17
| 651.0
|  
| 35/24
| '''16/11'''
| '''16/11'''
|  
|  
Line 153: Line 159:
|-
|-
| 16
| 16
| 1093.51
| 1094.5
| 15/8
| '''15/8'''
| '''32/17'''
| '''32/17'''
|  
|  
Line 161: Line 167:
|-
|-
| 17
| 17
| 336.86
| 337.9
|  
| 39/32
|  
|  
|  
|  
Line 169: Line 175:
|-
|-
| 18
| 18
| 780.20
| 781.3
|  
| 25/16
|  
|  
|  
|  
Line 177: Line 183:
|-
|-
| 19
| 19
| 23.55
| 24.7
|  
| 49/48, 65/64, 81/80
|  
|  
|  
|  
Line 185: Line 191:
|-
|-
| 20
| 20
| 466.89
| 468.1
|  
| '''21/16'''
|  
|  
|  
|  
Line 193: Line 199:
|-
|-
| 21
| 21
| 910.24
| 911.5
|  
| 27/16
|  
|  
|  
|  
Line 201: Line 207:
|-
|-
| 22
| 22
| 153.58
| 154.9
|  
| 35/32
| 12/11
| 12/11
|  
|  
Line 209: Line 215:
|-
|-
| 23
| 23
| 596.93
| 598.3
|  
| 45/32
| 24/17
| 24/17
|  
|  
Line 217: Line 223:
|-
|-
| 24
| 24
| 1040.27
| 1041.7
|  
| 117/64
| 20/11
| 20/11
|  
|  
Line 225: Line 231:
|-
|-
| 25
| 25
| 283.62
| 285.1
|  
| 75/64
| 13/11, 20/17
| 13/11, 20/17
|  
|  
Line 233: Line 239:
|-
|-
| 26
| 26
| 726.96
| 728.5
|  
| 49/32
| 26/17
| 26/17
|  
|  
Line 241: Line 247:
|-
|-
| 27
| 27
| 1170.31
| 1171.9
|  
| 63/32
|  
|  
|  
|  
Line 249: Line 255:
|-
|-
| 28
| 28
| 413.65
| 415.3
|  
| 81/64
| 14/11
| 14/11
|  
|  
Line 257: Line 263:
|-
|-
| 29
| 29
| 857.00
| 858.7
|  
| 105/64
| 18/11, 28/17
| 18/11, 28/17
|  
|  
Line 265: Line 271:
|-
|-
| 30
| 30
| 100.34
| 102.1
|  
| 135/128
| 18/17
| 18/17
|  
|  
Line 273: Line 279:
|-
|-
| 31
| 31
| 543.68
| 545.5
|  
| 175/128
| 15/11
| 15/11
|  
|  
Line 281: Line 287:
|-
|-
| 32
| 32
| 987.03
| 988.9
|  
| 225/128
| 30/17
| 30/17
|  
|  
Line 288: Line 294:
|  
|  
|}
|}
 
<nowiki/>* In 2.3.5.7.13.17/11 subgroup CTE tuning
: <sup>*</sup> in 2.3.5.7.13.17/11 POTE tuning


== Tuning spectra ==
== Tuning spectra ==
=== Sensor ===
=== Sensor ===
Gencom: [2 9/7; 91/90 126/125 169/168 385/384]
{| class="wikitable center-all left-4"
 
Gencom mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
 
{| class="wikitable center-1 center-2"
|-
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval]])
! Edo<br>generators
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|
| 9/7
| 9/7
| 435.084
| 435.084
|
|  
|-
|-
| 4\11
|
| 436.364
| 11cdefgg val
|-
|
| 15/14
| 15/14
| 439.814
| 439.814
|
|  
|-
|-
| 18/13
|  
| 13/9
| 440.846
| 440.846
|
|  
|-
|-
|
| 15/13
| 15/13
| 441.290
| 441.290
|
|  
|-
|-
| 6/5
| 7\19
|
| 442.105
|
|-
|
| 5/3
| 442.179
| 442.179
|
|  
|-
|-
| 14/13
|  
| 13/7
| 442.766
| 442.766
|
|  
|-
|-
|
| 5/4
| 5/4
| 442.924
| 442.924
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
|  
| 15/8
| 443.017
| 443.017
|
|  
|-
|-
|
| 11/10
| 11/10
| 443.125
| 443.125
|
|  
|-
|-
|
| 15/11
| 15/11
| 443.127
| 443.127
|
|  
|-
|-
| 4/3
|  
| 3/2
| 443.136
| 443.136
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|-
|
| 11/9
| 11/9
| 443.193
| 443.193
|
|  
|-
|-
| 12/11
|  
| 11/6
| 443.211
| 443.211
|
|  
|-
|-
|
| 11/8
| 11/8
| 443.245
| 443.245
|
|
|-
| 17\46
|
| 443.478
|  
|-
|-
| 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
|-
|-
|
| 13/11
| 13/11
| 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
|
|  
|-
|-
| 10\27
|
| 444.444
|
|-
|
| 7/6
| 7/6
| 444.478
| 444.478
|
|
|-
|
| 117/70
| 444.649
|  
|-
|-
|
| 7/5
| 7/5
| 445.628
| 445.628
|
|  
|-
|-
|
| 13/12
| 13/12
| 446.191
| 446.191
|
|  
|-
| 3\8
|
| 450.000
| 8deg val
|-
|-
|
| 13/10
| 13/10
| 454.214
| 454.214
|
|  
|}
|}
<nowiki/>* Besides the octave


=== Sensis ===
=== Sensis ===
Gencom: [2 9/7; 56/55 78/77 91/90 100/99]
{| class="wikitable center-all left-4"
 
Gencom mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
 
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo<br>(Unchanged-interval)
! Edo<br>generators
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|
| 9/7
| 9/7
| 435.084
| 435.084
|
|  
|-
|-
| 4\11
|
| 436.364
| 11cdf val
|-
|
| 11/8
| 11/8
| 437.829
| 437.829
|
|  
|-
|-
|
| 15/14
| 15/14
| 439.814
| 439.814
|
|  
|-
|-
| 18/13
|  
| 13/9
| 440.846
| 440.846
|
|  
|-
|-
|
| 15/13
| 15/13
| 441.290
| 441.290
|
|
|-
| 7\19
|
| 442.105
|  
|-
|-
| 6/5
|  
| 5/3
| 442.179
| 442.179
|
|  
|-
|-
| 14/13
|  
| 13/7
| 442.766
| 442.766
|
|  
|-
|-
|
| 5/4
| 5/4
| 442.924
| 442.924
| 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
| 17\46
|
| 443.478
| 46e val
|-
|
| 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
|-
|-
| 10\27
|
| 444.444
| 27e val
|-
|
| 7/6
| 7/6
| 444.478
| 444.478
|
|  
|-
|-
|
| 117/70
| 444.649
|
|-
|
| 15/11
| 15/11
| 444.746
| 444.746
|
|  
|-
|-
|
| 11/9
| 11/9
| 445.259
| 445.259
|
|  
|-
|-
|
| 7/5
| 7/5
| 445.628
| 445.628
|
|  
|-
|-
|
| 13/12
| 13/12
| 446.191
| 446.191
|
|  
|-
|-
| 14/11
|  
| 11/7
| 446.390
| 446.390
|
|  
|-
|-
|
| 11/10
| 11/10
| 446.999
| 446.999
|
|  
|-
|-
|
| 13/11
| 13/11
| 448.202
| 448.202
|
|  
|-
|-
| 12/11
| 3\8
|
| 450.000
| 8d val
|-
|
| 11/6
| 450.212
| 450.212
|
|  
|-
|-
|
| 13/10
| 13/10
| 454.214
| 454.214
|
|  
|}
|}
<nowiki/>* Besides the octave


=== Sensus ===
=== Sensus ===
Gencom: [2 9/7; 91/90 126/125 169/168 352/351]
{| class="wikitable center-all left-4"
 
Gencom mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
 
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo<br>(Unchanged-interval)
! Edo<br>generators
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|
| 9/7
| 9/7
| 435.084
| 435.084
|
|  
|-
| 4\11
|
| 436.364
| 11cdeeeefggg val
|-
|-
|
| 15/14
| 15/14
| 439.814
| 439.814
|
|  
|-
|-
| 18/13
|  
| 13/9
| 440.846
| 440.846
|
|  
|-
|-
|
| 15/13
| 15/13
| 441.290
| 441.290
|
|  
|-
| 7\19
|
| 442.105
| 19eg val
|-
|-
| 6/5
|  
| 5/3
| 442.179
| 442.179
|
|  
|-
|-
| 14/13
|  
| 13/7
| 442.766
| 442.766
|
|  
|-
|-
|
| 5/4
| 5/4
| 442.924
| 442.924
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
|  
| 15/8
| 443.017
| 443.017
|
|  
|-
|-
| 4/3
|  
| 3/2
| 443.136
| 443.136
|
|  
|-
|-
|
| 13/11
| 13/11
| 443.371
| 443.371
|
|  
|-
|-
| 14/11
|  
| 11/7
| 443.472
| 443.472
|
|
|-
| 17\46
|
| 443.478
|  
|-
|-
| 10/9
|  
| 9/5
| 443.519
| 443.519
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
|
| 11/8
| 11/8
| 443.591
| 443.591
|
|  
|-
|-
| 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
|-
|-
|
| 11/10
| 11/10
| 443.864
| 443.864
|
|  
|-
|-
|
| 11/9
| 11/9
| 443.965
| 443.965
|
|  
|-
|-
| 16/13
|  
| 13/8
| 444.053
| 444.053
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|
| 15/11
| 15/11
| 444.203
| 444.203
|
|  
|-
| 10\27
|
| 444.444
| 27eg val
|-
|-
|
| 7/6
| 7/6
| 444.478
| 444.478
|
|  
|-
|-
|
| 117/70
| 444.649
|
|-
|
| 7/5
| 7/5
| 445.628
| 445.628
|
|  
|-
|-
|
| 13/12
| 13/12
| 446.191
| 446.191
|
|  
|-
|-
| 3\8
|
| 450.000
| 8deegg val
|-
|
| 13/10
| 13/10
| 454.214
| 454.214
|
|  
|}
|}
<nowiki/>* Besides the octave


=== Sensa ===
=== Sensa ===
Gencom: [2 9/7; 55/54 66/65 77/75 143/140]
{| class="wikitable center-all left-4"
 
Gencom mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10 }}
 
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo<br>(Unchanged-interval)
! Edo<br>generators
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 14/11
|  
| 11/7
| 417.508
| 417.508
|
|  
|-
|-
|
| 11/9
| 11/9
| 426.296
| 426.296
|
|  
|-
|-
|
| 15/11
| 15/11
| 434.238
| 434.238
|
|  
|-
|-
|
| 9/7
| 9/7
| 435.084
| 435.084
|
|  
|-
| 4\11
|
| 436.364
| 11cdeef val
|-
|-
|
| 15/14
| 15/14
| 439.814
| 439.814
|
|  
|-
|-
| 18/13
|  
| 13/9
| 440.846
| 440.846
|
|  
|-
|-
|
| 15/13
| 15/13
| 441.290
| 441.290
|
|  
|-
| 7\19
|
| 442.105
| 19e val
|-
|-
| 6/5
|  
| 5/3
| 442.179
| 442.179
|
|  
|-
|-
| 14/13
|  
| 13/7
| 442.766
| 442.766
|
|  
|-
|-
|
| 5/4
| 5/4
| 442.924
| 442.924
| 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
| 17\46
|
| 443.478
| 46ee val
|-
|
| 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
|-
|-
| 10\27
|
| 444.444
|
|-
|
| 7/6
| 7/6
| 444.478
| 444.478
|
|  
|-
|-
|
| 117/70
| 444.649
|
|-
|
| 7/5
| 7/5
| 445.628
| 445.628
|
|  
|-
|-
|
| 11/8
| 11/8
| 445.943
| 445.943
|
|  
|-
|-
|
| 13/12
| 13/12
| 446.191
| 446.191
|
|  
|-
|-
| 12/11
|  
| 11/6
| 449.873
| 449.873
|
|  
|-
|-
| 3\8
|
| 450.000
| 8d val
|-
|
| 13/10
| 13/10
| 454.214
| 454.214
|
|  
|-
|-
|
| 11/10
| 11/10
| 455.001
| 455.001
|
|  
|-
|-
|
| 13/11
| 13/11
| 455.395
| 455.395
|
|  
|}
|}
<nowiki/>* Besides the octave


[[Category:Sensi]]
[[Category:Sensi]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Sensi_extensions"