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Sensi extensions: Difference between revisions

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


== Interval chain ==
== Interval chain ==
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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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== Tuning spectra ==
== Tuning spectra ==
=== Sensor ===
=== Sensor ===
{| class="wikitable center-1 center-2"
{| class="wikitable center-all left-4"
|-
|-
! [[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
|
|
|-
| 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
|
|  
|-
|-
|
| 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 ===
{| class="wikitable center-1 center-2"
{| class="wikitable center-all left-4"
|-
|-
! 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
|
|  
|-
| 17\46
|
| 443.478
| 46e val
|-
|-
| 10/9
|  
| 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 ===
{| class="wikitable center-1 center-2"
{| class="wikitable center-all left-4"
|-
|-
! 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 ===
{| class="wikitable center-1 center-2"
{| class="wikitable center-all left-4"
|-
|-
! 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
|
|  
|-
| 17\46
|
| 443.478
| 46ee val
|-
|-
| 10/9
|  
| 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"