Archytas clan: Difference between revisions

Revision as of 11:44, 26 September 2021

The archytas clan (or archy family) tempers out the Archytas comma, 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to 81/80 in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of 22EDO.

Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.

Discussed under their respective 5-limit families are:

Father → Father family #Mother
Dominant → Meantone family #Dominant
Augene → Augmented family #Augene
Porcupine → Porcupine family #Septimal porcupine
Pajara → Diaschismic family #Pajara
Blacksmith → Limma family #Blacksmith
Catalan → Kleismic family #Catalan
Modus → Tetracot family #Modus
Passion → Passion family #Septimal passion
Immunized → Immunity family #Immunized

The rest are considered below.

Archy

Subgroup: 2.3.7

Comma list: 64/63

Sval mapping: [⟨1 0 6], ⟨0 1 -2]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4], ⟨0 1 0 -2]]

Gencom: [2 3/2; 64/63]

POTE generator: ~3/2 = 709.321

Template:Val list

Scales: archy5, archy7, archy12

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4 7], ⟨0 1 0 -2 -6]]

Gencom: [2 3/2; 64/63 99/98]

POTE generator: ~3/2 = 707.192

@@ Line 3: / Line 3: @@
 Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.
-Discussed under their respective 5-limit families are [[Father family #Mother|mother]], [[Meantone family #Dominant|dominant]], [[Augmented family #Augene|augene]], [[Porcupine family #Porcupine|porcupine]], [[Diaschismic family #Pajara|pajara]], [[Kleismic family #Catalan|catalan]], [[Tetracot family #Modus|modus]], [[Passion family #Septimal passion|passion]], and [[Immunity family #Immunized|immunized]]. The rest are considered below.
+Discussed under their respective 5-limit families are:
+* ''[[Father]]'' → [[Father family #Mother]]
+* ''[[Dominant]]'' → [[Meantone family #Dominant]]
+* ''[[Augene]]'' → [[Augmented family #Augene]]
+* [[Porcupine]] → [[Porcupine family #Septimal porcupine]]
+* [[Pajara]] → [[Diaschismic family #Pajara]]
+* ''[[Blacksmith]]'' → [[Limma family #Blacksmith]]
+* ''[[Catalan]]'' → [[Kleismic family #Catalan]]
+* ''[[Modus]]'' → [[Tetracot family #Modus]]
+* ''[[Passion]]'' → [[Passion family #Septimal passion]]
+* ''[[Immunized]]'' → [[Immunity family #Immunized]]
+The rest are considered below.
 == Archy ==
@@ Line 323: / Line 335: @@
 Badness: 0.037482
-== Blacksmith ==
-[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]
-=== 5-limit (blackwood) ===
-Subgroup: 2.3.5
-[[Comma]]: 256/243
-[[Mapping]]: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
-Mapping generators: ~9/8, ~5
-[[POTE generator]]: ~5/4 = 399.594
-{{Val list|legend=1| 5, 10, 15 }}
-[[Badness]]: 0.063760
-=== 7-limit ===
-Subgroup: 2.3.5.7
-[[Comma list]]: 28/27, 49/48
-[[Mapping]]: [{{val| 5 8 0 14 }}, {{val| 0 0 1 0 }}]
-Mapping generators: ~7/6, ~5
-{{Multival|legend=1| 0 5 0 8 0 -14 }}
-[[POTE generator]]: ~5/4 = 392.767
-{{Val list|legend=1| 5, 10, 15, 40b, 55b }}
-[[Badness]]: 0.025640
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 28/27, 49/48, 55/54
-Mapping: [{{val| 5 8 0 14 29 }}, {{val| 0 0 1 0 -1 }}]
-POTE generator: ~5/4 = 394.948
-Vals: {{Val list| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}
-Badness: 0.024641
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 28/27, 40/39, 49/48, 55/54
-Mapping: [{{val| 5 8 0 14 29 7 }}, {{val| 0 0 1 0 -1 1 }}]
-POTE generator: ~5/4 = 391.037
-Vals: {{Val list| 5, 10, 15, 25e, 40bef}}
-Badness: 0.020498
-=== Farrier ===
-Subgroup: 2.3.5.7.11
-Comma list: 28/27, 49/48, 77/75
-Mapping: [{{val| 5 8 0 14 -6 }}, {{val| 0 0 1 0 2 }}]
-POTE generator: ~5/4 = 398.070
-Vals: {{Val list| 5e, 10e, 15 }}
-Badness: 0.029200
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 28/27, 40/39, 49/48, 66/65
-Mapping: [{{val| 5 8 0 14 -6 7 }}, {{val| 0 0 1 0 2 1 }}]
-POTE generator: ~5/4 = 396.812
-Vals: {{Val list| 5e, 10e, 15 }}
-Badness: 0.022325
-=== Ferrum ===
-Subgroup: 2.3.5.7.11
-Comma list: 28/27, 35/33, 49/48
-Mapping: [{{val| 5 8 0 14 6 }}, {{val| 0 0 1 0 1 }}]
-POTE generator: ~5/4 = 374.763
-Vals: {{Val list| 5e, 10 }}
-Badness: 0.030883
 == Beatles ==

Archytas clan: Difference between revisions

Revision as of 11:44, 26 September 2021

Archy

Supra

Supraphon

Suhajira

2.3.7.11.13

Superpyth

11-limit

13-limit

Suprapyth

13-limit

Quasisuper

Quasisupra

13-limit

Quasisoup

Ultrapyth

11-limit

13-limit

Counterultrapyth

13-limit

Schism

11-limit

Beatles

7-limit

11-limit

13-limit

Ringo

13-limit

Beetle

13-limit

Fervor

7-limit

11-limit

13-limit

Progress

7-limit

11-limit

13-limit

Progressive

Sixix

7-limit

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