Ragismic microtemperaments: Difference between revisions

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~~The ragisma~~ is [[~~4375/4374~~]] ~~with a~~ [[~~monzo~~]] of {{monzo| -1 -7 4 1 }}, the smallest 7-limit [[superparticular]] ratio~~. Since (10/9)4 = 4375/4374 × 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity~~]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 × (27/25)2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the ragisma, [[4375/4374]] = {{monzo| -1 -7 4 1 }}. The ragisma is the smallest [[7-limit]] [[superparticular ratio]].

~~Temperaments discussed~~ elsewhere ~~include~~:

Since (10/9)4 = 4375/4374 × 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 × (27/25)2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:

* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]

* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]

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* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]

* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]]

Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium.

~~Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci.~~

== Ennealimmal ==

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[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}

: mapping generators: ~83349/81920, ~3

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Comma list: 3025/3024, 4375/4374, 134775333/134217728

Mapping: {{mapping| 46 ~~73 107 129 159~~ | 0 -1 -~~2 1~~ 1 }}

Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}

Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951

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Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364

Mapping: {{mapping| 46 ~~73 107 129 159 170~~ | 0 -1 -~~2 1~~ 1 2 }}

Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}

Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419

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Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224

Mapping: {{mapping| 46 ~~73 107 129 159 170~~ 188 | 0 -1 -~~2 1~~ 1 2 0 }}

Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}

Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425

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Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

Mapping: {{mapping| 16 ~~26 38 46 56 59~~ | 0 -3 -4 -5 -3 1 }}

Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}

: mapping generators: ~448/429, ~7/5

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Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: {{mapping| 16 ~~26 38 46 56 59 65~~ | 0 -3 -4 -5 -3 1 2 }}

Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}

Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932

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Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: {{mapping| 16 ~~26 38 46 56 59 65~~ 68 | 0 -3 -4 -5 -3 1 2 0 }}

Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}

Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896

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=== 7-limit ===

Subgroup: 2.3.5.7

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 4375/4374

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[[Comma list]]: 4375/4374, 83349/81920

[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410

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Comma list: 245/242, 385/384, 4375/4374

Mapping: {{mapping| 13 ~~21 31 36~~ 45 | 0 -1 -2 1 0 }}

Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}

Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179

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Comma list: 169/168, 245/242, 325/324, 385/384

Mapping: {{mapping| 13 ~~21 31 36~~ 45 48 | 0 -1 -2 1 0 0 }}

Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}

Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969

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-The ragisma is [[4375/4374]] with a [[monzo]] of {{monzo| -1 -7 4 1 }}, the smallest 7-limit [[superparticular]] ratio. Since (10/9)<sup>4</sup> = 4375/4374 × 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 × (27/25)<sup>2</sup>, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
+This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the ragisma, [[4375/4374]] = {{monzo| -1 -7 4 1 }}. The ragisma is the smallest [[7-limit]] [[superparticular ratio]].
-Temperaments discussed elsewhere include:
+Since (10/9)<sup>4</sup> = 4375/4374 × 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 × (27/25)<sup>2</sup>, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
+Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
 * ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
 * ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
@@ Line 24: / Line 26: @@
 * ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
 * ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]]
-Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium.
-Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci.
 == Ennealimmal ==
@@ Line 1,506: / Line 1,505: @@
 [[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
-{{Mapping|legend=1| 46 73 107 129 | 0 -1 -2 1 }}
+{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
 : mapping generators: ~83349/81920, ~3
@@ Line 1,523: / Line 1,522: @@
 Comma list: 3025/3024, 4375/4374, 134775333/134217728
-Mapping: {{mapping|  46 73 107 129 159 | 0 -1 -2 1 1  }}
+Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
 Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951
@@ Line 1,536: / Line 1,535: @@
 Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
-Mapping: {{mapping| 46 73 107 129 159 170 | 0 -1 -2 1 1 2 }}
+Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
 Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419
@@ Line 1,549: / Line 1,548: @@
 Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
-Mapping: {{mapping| 46 73 107 129 159 170 188 | 0 -1 -2 1 1 2 0 }}
+Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
 Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425
@@ Line 1,725: / Line 1,724: @@
 Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
-Mapping: {{mapping| 16 26 38 46 56 59 | 0 -3 -4 -5 -3 1 }}
+Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
 : mapping generators: ~448/429, ~7/5
@@ Line 1,740: / Line 1,739: @@
 Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
-Mapping: {{mapping| 16 26 38 46 56 59 65 | 0 -3 -4 -5 -3 1 2 }}
+Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
 Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932
@@ Line 1,753: / Line 1,752: @@
 Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
-Mapping: {{mapping| 16 26 38 46 56 59 65 68 | 0 -3 -4 -5 -3 1 2 0 }}
+Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}
 Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896
@@ Line 1,781: / Line 1,780: @@
 === 7-limit ===
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 3136/3125, 4375/4374
@@ Line 2,164: / Line 2,163: @@
 [[Comma list]]: 4375/4374, 83349/81920
-{{Mapping|legend=1| 13 21 31 36 | 0 -1 -2 1 }}
+{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
 [[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410
@@ Line 2,177: / Line 2,176: @@
 Comma list: 245/242, 385/384, 4375/4374
-Mapping: {{mapping| 13 21 31 36 45 | 0 -1 -2 1 0 }}
+Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
 Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179
@@ Line 2,190: / Line 2,189: @@
 Comma list: 169/168, 245/242, 325/324, 385/384
-Mapping: {{mapping| 13 21 31 36 45 48 | 0 -1 -2 1 0 0 }}
+Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
 Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969