Lehmerismic temperaments: Difference between revisions

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* ''[[Spectacle]]'' (+225/224 or 243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]

* ''[[Portent]]'' (+385/384 or 441/440) → [[Gamelismic family #Portent|Gamelismic family]]

* ''[[Indra]]'' (+540/539) → [[~~Mirkwai family~~ #Indra|~~Mirkwai family~~]]

* ''[[Indra]]'' (+540/539) → [[Swetismic temperaments #Indra|Swetismic temperaments]]

* ''[[Tolerant]]'' (+896/891) → [[Pentacircle clan #Tolerant|Pentacircle clan]]

* ''[[Freya]]'' (+2401/2400) → [[Breed family #Freya|Breed family]]

Line 13:

* ''[[Thor]]'' (+4375/4374) → [[Ragismic family #Thor|Ragismic family]]

* ''[[Kapo]]'' (+5120/5103) → [[Hemifamity family #Kapo|Hemifamity family]]

* ''[[Tyr]]'' (+102487/102400) → [[Landscape family #Tyr|Landscape family]]

Considered below are ~~tyr,~~ skadi, ganesha, hanuman, lux, and galaxy. For the rank-4 temperament, see [[Rank-4 temperament #Lehmerismic (3025/3024)]].

Considered below are skadi, ganesha, hanuman, lux, and galaxy, in the order of increasing [[badness]]. For the rank-4 temperament, see [[Rank-4 temperament #Lehmerismic (3025/3024)]].

~~== Tyr ==~~

~~{{See also| Landscape family }}~~

~~[[Subgroup]]: 2.3.5.7.11~~

~~[[Comma list]]: 3025/3024, 102487/102400~~

~~{{Mapping|legend=1| 3 0 0 4 8 | 0 2 0 -4 1 | 0 0 1 2 0 }}~~

~~: mapping generators: ~63/50, ~400/231, ~5~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~63/50 = 400.0162{{c}}, ~400/231 = 950.9828{{c}}, ~5/4 = 386.2400{{c}}

~~: [[error map]]: {{val| +0.049 +0.011 +0.024 -0.017 -0.205 }}~~

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~400/231 = 950.9591{{c}}, ~5/4 = 386.2764{{c}}

~~: error map: {{val| 0.000 -0.037 -0.037 -0.109 -0.359 }}~~

~~{{Optimal ET sequence|legend=1| 15, 24d, 33cd, 39d, 54c, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee }}~~

~~[[Badness]] (Sintel): 0.311~~

~~=== Borneo ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 676/675, 1001/1000, 3025/3024~~

~~Mapping: {{mapping| 3 0 0 4 8 -3 | 0 2 0 -4 1 3 | 0 0 1 2 0 1 }}~~

~~Optimal tunings:~~

* WE: ~63/50 = 400.0002{{c}}, ~26/15 = 951.0850{{c}}, ~5/4 = 386.5817{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.0846{{c}}, ~5/4 = 386.5821{{c}}

~~{{Optimal ET sequence|legend=0| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }}~~

~~Badness (Sintel): 0.514~~

~~Complexity spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7~~

== Skadi ==

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Line 88:

== Lux ==

The last generator of lux, represented by 55/48, exceeds 8/7 by 385/384, which is equated with a number of important superparticular ~~ratios~~ in the 13-limit: 325/324, 352/351, 364/363, and 441/440.

The last generator of lux, represented by [[55/48]], exceeds [[8/7]] by [[385/384]], which is equated with a number of important [[superparticular ratio]]s in the [[13-limit]]: [[325/324]], [[352/351]], [[364/363]], and [[441/440]].

This ultra-efficient full 13-limit temperament was first considered by [[Flora Canou]] in 2021. After a few attempts to come up with a memorable name, ''lux'', suggested by [[Godtone]], who associated certain intervals of 13 with light, was adopted.

[[Subgroup]]: 2.3.5.7.11

@@ Line 7: / Line 7: @@
 * ''[[Spectacle]]'' (+225/224 or 243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
 * ''[[Portent]]'' (+385/384 or 441/440) → [[Gamelismic family #Portent|Gamelismic family]]
-* ''[[Indra]]'' (+540/539) → [[Mirkwai family #Indra|Mirkwai family]]
+* ''[[Indra]]'' (+540/539) → [[Swetismic temperaments #Indra|Swetismic temperaments]]
 * ''[[Tolerant]]'' (+896/891) → [[Pentacircle clan #Tolerant|Pentacircle clan]]
 * ''[[Freya]]'' (+2401/2400) → [[Breed family #Freya|Breed family]]
@@ Line 13: / Line 13: @@
 * ''[[Thor]]'' (+4375/4374) → [[Ragismic family #Thor|Ragismic family]]
 * ''[[Kapo]]'' (+5120/5103) → [[Hemifamity family #Kapo|Hemifamity family]]
+* ''[[Tyr]]'' (+102487/102400) → [[Landscape family #Tyr|Landscape family]]
-Considered below are tyr, skadi, ganesha, hanuman, lux, and galaxy. For the rank-4 temperament, see [[Rank-4 temperament #Lehmerismic (3025/3024)]].
+Considered below are skadi, ganesha, hanuman, lux, and galaxy, in the order of increasing [[badness]]. For the rank-4 temperament, see [[Rank-4 temperament #Lehmerismic (3025/3024)]].
-== Tyr ==
-{{See also| Landscape family }}
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 3025/3024, 102487/102400
-{{Mapping|legend=1| 3 0 0 4 8 | 0 2 0 -4 1 | 0 0 1 2 0 }}
-: mapping generators: ~63/50, ~400/231, ~5
-[[Optimal tuning]]s:
-* [[WE]]: ~63/50 = 400.0162{{c}}, ~400/231 = 950.9828{{c}}, ~5/4 = 386.2400{{c}}
-: [[error map]]: {{val| +0.049 +0.011 +0.024 -0.017 -0.205 }}
-* [[CWE]]: ~63/50 = 400.0000{{c}}, ~400/231 = 950.9591{{c}}, ~5/4 = 386.2764{{c}}
-: error map: {{val| 0.000 -0.037 -0.037 -0.109 -0.359 }}
-{{Optimal ET sequence|legend=1| 15, 24d, 33cd, 39d, 54c, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee }}
-[[Badness]] (Sintel): 0.311
-=== Borneo ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1001/1000, 3025/3024
-Mapping: {{mapping| 3 0 0 4 8 -3 | 0 2 0 -4 1 3 | 0 0 1 2 0 1 }}
-Optimal tunings:
-* WE: ~63/50 = 400.0002{{c}}, ~26/15 = 951.0850{{c}}, ~5/4 = 386.5817{{c}}
-* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.0846{{c}}, ~5/4 = 386.5821{{c}}
-{{Optimal ET sequence|legend=0| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }}
-Badness (Sintel): 0.514
-Complexity spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7
 == Skadi ==
@@ Line 124: / Line 88: @@
 == Lux ==
-The last generator of lux, represented by 55/48, exceeds 8/7 by 385/384, which is equated with a number of important superparticular ratios in the 13-limit: 325/324, 352/351, 364/363, and 441/440.
+The last generator of lux, represented by [[55/48]], exceeds [[8/7]] by [[385/384]], which is equated with a number of important [[superparticular ratio]]s in the [[13-limit]]: [[325/324]], [[352/351]], [[364/363]], and [[441/440]].
+This ultra-efficient full 13-limit temperament was first considered by [[Flora Canou]] in 2021. After a few attempts to come up with a memorable name, ''lux'', suggested by [[Godtone]], who associated certain intervals of 13 with light, was adopted.
 [[Subgroup]]: 2.3.5.7.11