Metric microtemperaments: Difference between revisions

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These are rank-2 temperaments that temper out the [[meter comma]], 703125/702464 = {{monzo| -11 2 7 -3 }}.

Temperaments ~~not dicussed here~~ include:

Temperaments discussed elsewhere include:

* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]

* ''[[Triwell]]'', {1029/1024, 235298/234375} → [[Semicomma family #Triwell|Semicomma family]]

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POTE generator: ~80/77 = 65.5881

~~Vals~~: {{val list| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}

Optimal GPV sequence: {{val list| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}

Badness: 0.032816

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POTE generator: ~27/26 = 65.5883

~~Vals~~: {{val list| 128, 183, 311, 494, 1299, 1793 }}

Optimal GPV sequence: {{val list| 128, 183, 311, 494, 1299, 1793 }}

Badness: 0.016849

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POTE generator: ~27/26 = 65.5878

~~Vals~~: {{val list| 128, 183, 311, 494 }}

Optimal GPV sequence: {{val list| 128, 183, 311, 494 }}

Badness: 0.014674

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POTE generator: ~5/4 = 385.396

~~Vals~~: {{Val list| 50, 90, 140, 190, 330e, 520de }}

Optimal GPV sequence: {{Val list| 50, 90, 140, 190, 330e, 520de }}

Badness: 0.056915

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POTE generator: ~5/4 = 385.419

~~Vals~~: {{Val list| 50, 90, 140, 190, 330e }}

Optimal GPV sequence: {{Val list| 50, 90, 140, 190, 330e }}

Badness: 0.029297

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 385/384, 561/560, 595/594, 625/624, 847/845

Mapping: [{{val| 10 3 20 12 41 37 28 }}, {{val| 0 4 1 5 -2 0 4 }}]

POTE generator: ~5/4 = 385.435

Optimal GPV sequence: {{Val list| 50, 90, 140, 190g, 330eg }}

Badness: 0.021949

== Stockhausenic ==

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POTE generator: ~16/15 = 111.460

~~Vals~~: {{Val list| 43, 97e, 140, 183, 323, 506, 829de, 1335ddee }}

Optimal GPV sequence: {{Val list| 43, 97e, 140, 183, 323, 506, 829de, 1335ddee }}

Badness: 0.056043

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POTE generator: ~16/15 = 111.458

~~Vals~~: {{Val list| 43, 97e, 140, 183, 323, 829def, 1152deef }}

Optimal GPV sequence: {{Val list| 43, 97e, 140, 183, 323, 829def, 1152deef }}

Badness: 0.029947

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POTE generator: ~16/15 = 111.459

~~Vals~~: {{Val list| 43, 97e, 140, 183, 323, 506, 829defg }}

Optimal GPV sequence: {{Val list| 43, 97e, 140, 183, 323, 506, 829defg }}

Badness: 0.019205

== Geb ==

The ''geb'' ~~temperaent~~ (171&506, named after the earth god of ancient Egyptian religion) tempers out {{monzo|-31 43 -16}} (quadla-quadquadgu) in the 5-limit; the meter and the euzenius comma, 78125000/78121827 as well as skeetsma (14348907/14336000, lalarutrigu) in the 7-limit. [[Breedsmic temperaments #Osiris|Osiris temperament]] (171&670, named after the eldest son of Geb) tempers out the same 5-limit comma as the geb, but with the [[2401/2400|breedsma]] rather than the meter tempered out.

The ''geb'' temperament (171&506, named after the earth god of ancient Egyptian religion) tempers out {{monzo|-31 43 -16}} (quadla-quadquadgu) in the 5-limit; the meter and the euzenius comma, 78125000/78121827 as well as skeetsma (14348907/14336000, lalarutrigu) in the 7-limit. [[Breedsmic temperaments #Osiris|Osiris temperament]] (171&670, named after the eldest son of Geb) tempers out the same 5-limit comma as the geb, but with the [[2401/2400|breedsma]] rather than the meter tempered out.

Subgroup: 2.3.5

@@ Line 1: / Line 1: @@
 These are rank-2 temperaments that temper out the [[meter comma]], 703125/702464 = {{monzo| -11 2 7 -3 }}.
-Temperaments not dicussed here include:
+Temperaments discussed elsewhere include:
 * [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]
 * ''[[Triwell]]'', {1029/1024, 235298/234375} → [[Semicomma family #Triwell|Semicomma family]]
@@ Line 40: / Line 40: @@
 POTE generator: ~80/77 = 65.5881
-Vals: {{val list| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}
+Optimal GPV sequence: {{val list| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}
 Badness: 0.032816
@@ Line 53: / Line 53: @@
 POTE generator: ~27/26 = 65.5883
-Vals: {{val list| 128, 183, 311, 494, 1299, 1793 }}
+Optimal GPV sequence: {{val list| 128, 183, 311, 494, 1299, 1793 }}
 Badness: 0.016849
@@ Line 66: / Line 66: @@
 POTE generator: ~27/26 = 65.5878
-Vals: {{val list| 128, 183, 311, 494 }}
+Optimal GPV sequence: {{val list| 128, 183, 311, 494 }}
 Badness: 0.014674
@@ Line 96: / Line 96: @@
 POTE generator: ~5/4 = 385.396
-Vals: {{Val list| 50, 90, 140, 190, 330e, 520de }}
+Optimal GPV sequence: {{Val list| 50, 90, 140, 190, 330e, 520de }}
 Badness: 0.056915
@@ Line 109: / Line 109: @@
 POTE generator: ~5/4 = 385.419
-Vals: {{Val list| 50, 90, 140, 190, 330e }}
+Optimal GPV sequence: {{Val list| 50, 90, 140, 190, 330e }}
 Badness: 0.029297
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 385/384, 561/560, 595/594, 625/624, 847/845
+Mapping: [{{val| 10 3 20 12 41 37 28 }}, {{val| 0 4 1 5 -2 0 4 }}]
+POTE generator: ~5/4 = 385.435
+Optimal GPV sequence: {{Val list| 50, 90, 140, 190g, 330eg }}
+Badness: 0.021949
 == Stockhausenic ==
@@ Line 139: / Line 152: @@
 POTE generator: ~16/15 = 111.460
-Vals: {{Val list| 43, 97e, 140, 183, 323, 506, 829de, 1335ddee }}
+Optimal GPV sequence: {{Val list| 43, 97e, 140, 183, 323, 506, 829de, 1335ddee }}
 Badness: 0.056043
@@ Line 152: / Line 165: @@
 POTE generator: ~16/15 = 111.458
-Vals: {{Val list| 43, 97e, 140, 183, 323, 829def, 1152deef }}
+Optimal GPV sequence: {{Val list| 43, 97e, 140, 183, 323, 829def, 1152deef }}
 Badness: 0.029947
@@ Line 165: / Line 178: @@
 POTE generator: ~16/15 = 111.459
-Vals: {{Val list| 43, 97e, 140, 183, 323, 506, 829defg }}
+Optimal GPV sequence: {{Val list| 43, 97e, 140, 183, 323, 506, 829defg }}
 Badness: 0.019205
 == Geb ==
-The ''geb'' temperaent (171&amp;506, named after the earth god of ancient Egyptian religion) tempers out {{monzo|-31 43 -16}} (quadla-quadquadgu) in the 5-limit; the meter and the euzenius comma, 78125000/78121827 as well as skeetsma (14348907/14336000, lalarutrigu) in the 7-limit. [[Breedsmic temperaments #Osiris|Osiris temperament]] (171&amp;670, named after the eldest son of Geb) tempers out the same 5-limit comma as the geb, but with the [[2401/2400|breedsma]] rather than the meter tempered out.
+The ''geb'' temperament (171&amp;506, named after the earth god of ancient Egyptian religion) tempers out {{monzo|-31 43 -16}} (quadla-quadquadgu) in the 5-limit; the meter and the euzenius comma, 78125000/78121827 as well as skeetsma (14348907/14336000, lalarutrigu) in the 7-limit. [[Breedsmic temperaments #Osiris|Osiris temperament]] (171&amp;670, named after the eldest son of Geb) tempers out the same 5-limit comma as the geb, but with the [[2401/2400|breedsma]] rather than the meter tempered out.
 Subgroup: 2.3.5