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 ==

{| class="wikitable right-1 right-2"

|-

! rowspan=2 | ~~Generators~~

! rowspan=2 | #

! rowspan=2 | Cents*

! colspan=4 | Approximate ratios

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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~~ (~~unchanged~~-interval]])

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval]])

! ~~supermajor third~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

| 9/7

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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~~ (~~unchanged~~-interval)

! Eigenmonzo (Unchanged-interval)

! ~~supermajor third~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

| 9/7

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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~~ (~~unchanged~~-interval)

! Eigenmonzo (Unchanged-interval)

! ~~supermajor third~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

| 9/7

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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~~ (~~unchanged~~-interval)

! Eigenmonzo (Unchanged-interval)

! ~~supermajor third~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

| 14/11

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|}

~~[[Category:Sensipent]]~~

[[Category:Sensi]]

@@ Line 1: / Line 1: @@
-[[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 &amp; 27) - tempering out 126/125, 245/243, 385/384
+* '''Sensor''' (19 &amp; 27) – tempering out 126/125, 245/243, 385/384
-* Sensis (19 &amp; 27e) - tempering out 56/55, 100/99, 245/243
+* '''Sensus''' (19e &amp; 27e) – tempering out 126/125, 176/175, 245/243
-* Sensus (19e &amp; 27e) - tempering out 126/125, 176/175, 245/243
-* Sensa (19e &amp; 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 &amp; 27e) – tempering out 56/55, 100/99, 245/243
+* '''Sensa''' (19e &amp; 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 ==
 {| class="wikitable right-1 right-2"
 |-
-! rowspan=2 | Generators
+! rowspan=2 | #
 ! rowspan=2 | Cents<sup>*</sup>
 ! colspan=4 | Approximate ratios
@@ Line 244: / Line 248: @@
 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
@@ Line 347: / Line 351: @@
 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
@@ Line 451: / Line 455: @@
 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
@@ Line 555: / Line 559: @@
 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
@@ Line 656: / Line 660: @@
 |}
-[[Category:Sensipent]]
 [[Category:Sensi]]