Archytas clan: Difference between revisions

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These all use the same generators as archy.

[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a ~~semioctave~~ period. [[392/375]] gives progress~~, splitting~~ the twelfth in three. [[250/243]] gives porcupine, splitting the fourth in three. [[126/125]] gives augene with a 1/3-octave period. [[4375/4374]] gives modus, splitting the fifth in four. [[3125/3024]] gives brightstone. [[9604/9375]] gives fervor. [[3125/2916]] gives sixix. [[3125/3087]] gives passion. Those split the generator in five in various ways. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.

[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semi-octave period. [[392/375]] gives progress. [[6125/5832]] gives kangaroo. Those split the twelfth in three. [[250/243]] gives porcupine, splitting the fourth in three. [[126/125]] gives augene with a 1/3-octave period. [[4375/4374]] gives modus, splitting the fifth in four. [[3125/3024]] gives brightstone. [[9604/9375]] gives fervor. [[3125/2916]] gives sixix. [[3125/3087]] gives passion. Those split the generator in five in various ways. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.

Temperaments discussed elsewhere are:

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: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Ultrapyth (5-limit)]].''

Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[~~the Biosphere~~ #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).

Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[#Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).

[[Subgroup]]: 2.3.5.7

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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''

Progress tempers out 392/375 and may be described as {{nowrap| 15 & 17c }}. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 32c-edo gives an obvious tuning.

Progress tempers out [[392/375]] and may be described as {{nowrap| 15 & 17c }}. It splits the [[3/1|perfect twelfth]] into three generators of [[~]][[10/7]]; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]], and the 13-limit extension further equates it to [[13/9]]. [[32edo|32c-edo]] gives an obvious tuning.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~10/7

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Badness (Sintel): 1.35

== Kangaroo ==

: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''

Kangaroo splits the [[3/1|perfect twelfth]] into three [[35/24]][[~]][[40/27]] generators; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]]. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat.

Originally the canonical extension of 5-limit kangaroo was [[trienstonic clan #Parakangaroo|parakangaroo]], but it was decanonicalized in 2024 in favor of this extension because parakangaroo is an [[exotemperament]] that only supports the [[patent val]] of [[15edo]], while this extension is a bit more practical since it is at least low-medium accuracy and supports the [[patent val]]s of 15edo, [[17edo]] and [[32edo]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 64/63, 6125/5832

: mapping generators: ~2, ~35/24

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1198.1767{{c}}, ~35/24 = 637.2914{{c}}

: [[error map]]: {{val| -1.823 +9.919 -7.930 -3.514 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 638.1977{{c}}

: error map: {{val| 0.000 +12.638 -4.336 +1.988 }}

[[Badness]] (Sintel): 2.91

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 245/242

Optimal tunings:

* WE: ~2 = 1197.8893{{c}}, ~16/11 = 637.1922{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 638.2605{{c}}

Badness (Sintel): 1.65

== Fervor ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''

Fervor tempers out ~~9704~~/9375 and may be described as {{nowrap| 25 & 27 }}. It splits the 6th harmonic into five generators of ~10/7; its ploidacot is beta-pentacot. 27edo is about as accurate as it can be tuned.

Fervor tempers out [[9604/9375]] and may be described as {{nowrap| 25 & 27 }}. It splits the [[6/1|6th harmonic]] into five generators of [[~]][[10/7]]; its ploidacot is beta-pentacot. The 11-limit extension simply equates the generator to [[16/11]]. [[27edo]] is about as accurate as it can be tuned.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~10/7

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Scales: [[supra7]], [[supra12]]

=== Oceanfront (2.3.7.13/5) ===

This is the [[restriction]] of [[#Ultrapyth|ultrapyth]] to the 2.3.7.13/5 subgroup, mapping [[13/10]] to the major third.

[[Subgroup]]: 2.3.7.13/5

[[Comma list]]: 64/63 ({{monzo| 6 -2 -1 0 }}), 91/90 ({{monzo| -1 -2 1 1 }})

[[Optimal tuning]]s:

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1197.2656{{c}}, ~3/2 = 710.1993{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.7413{{c}}

[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents

Scales: [[Oceanfront scales]]

[[Category:Archytas clan| ]]

@@ Line 41: / Line 41: @@
 These all use the same generators as archy.
-[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semioctave period. [[392/375]] gives progress, splitting the twelfth in three. [[250/243]] gives porcupine, splitting the fourth in three. [[126/125]] gives augene with a 1/3-octave period. [[4375/4374]] gives modus, splitting the fifth in four. [[3125/3024]] gives brightstone. [[9604/9375]] gives fervor. [[3125/2916]] gives sixix. [[3125/3087]] gives passion. Those split the generator in five in various ways. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.
+[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semi-octave period. [[392/375]] gives progress. [[6125/5832]] gives kangaroo. Those split the twelfth in three. [[250/243]] gives porcupine, splitting the fourth in three. [[126/125]] gives augene with a 1/3-octave period. [[4375/4374]] gives modus, splitting the fifth in four. [[3125/3024]] gives brightstone. [[9604/9375]] gives fervor. [[3125/2916]] gives sixix. [[3125/3087]] gives passion. Those split the generator in five in various ways. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.
 Temperaments discussed elsewhere are:
@@ Line 250: / Line 250: @@
 : ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Ultrapyth (5-limit)]].''
-Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[the Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).
+Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[#Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).
 [[Subgroup]]: 2.3.5.7
@@ Line 503: / Line 503: @@
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
-Progress tempers out 392/375 and may be described as {{nowrap| 15 & 17c }}. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 32c-edo gives an obvious tuning.
+Progress tempers out [[392/375]] and may be described as {{nowrap| 15 & 17c }}. It splits the [[3/1|perfect twelfth]] into three generators of [[~]][[10/7]]; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]], and the 13-limit extension further equates it to [[13/9]]. [[32edo|32c-edo]] gives an obvious tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 510: / Line 510: @@
 {{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
 : mapping generators: ~2, ~10/7
@@ Line 567: / Line 566: @@
 Badness (Sintel): 1.35
+== Kangaroo ==
+: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
+Kangaroo splits the [[3/1|perfect twelfth]] into three [[35/24]][[~]][[40/27]] generators; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]]. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat.
+Originally the canonical extension of 5-limit kangaroo was [[trienstonic clan #Parakangaroo|parakangaroo]], but it was decanonicalized in 2024 in favor of this extension because parakangaroo is an [[exotemperament]] that only supports the [[patent val]] of [[15edo]], while this extension is a bit more practical since it is at least low-medium accuracy and supports the [[patent val]]s of 15edo, [[17edo]] and [[32edo]].
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 64/63, 6125/5832
+{{Mapping|legend=1| 1 0 -3 6 | 0 3 10 -6 }}
+: mapping generators: ~2, ~35/24
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1198.1767{{c}}, ~35/24 = 637.2914{{c}}
+: [[error map]]: {{val| -1.823 +9.919 -7.930 -3.514 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 638.1977{{c}}
+: error map: {{val| 0.000 +12.638 -4.336 +1.988 }}
+{{Optimal ET sequence|legend=1| 15, 32, 47b }}
+[[Badness]] (Sintel): 2.91
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 55/54, 64/63, 245/242
+{{Mapping|legend=0| 1 0 -3 6 4 | 0 3 10 -6 -1 }}
+Optimal tunings:
+* WE: ~2 = 1197.8893{{c}}, ~16/11 = 637.1922{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 638.2605{{c}}
+{{Optimal ET sequence|legend=0| 15, 32, 47b }}
+Badness (Sintel): 1.65
 == Fervor ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
-Fervor tempers out 9704/9375 and may be described as {{nowrap| 25 & 27 }}. It splits the 6th harmonic into five generators of ~10/7; its ploidacot is beta-pentacot. 27edo is about as accurate as it can be tuned.
+Fervor tempers out [[9604/9375]] and may be described as {{nowrap| 25 & 27 }}. It splits the [[6/1|6th harmonic]] into five generators of [[~]][[10/7]]; its ploidacot is beta-pentacot. The 11-limit extension simply equates the generator to [[16/11]]. [[27edo]] is about as accurate as it can be tuned.
 [[Subgroup]]: 2.3.5.7
@@ Line 578: / Line 616: @@
 {{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
 : mapping generators: ~2, ~10/7
@@ Line 728: / Line 765: @@
 Scales: [[supra7]], [[supra12]]
+=== Oceanfront (2.3.7.13/5) ===
+This is the [[restriction]] of [[#Ultrapyth|ultrapyth]] to the 2.3.7.13/5 subgroup, mapping [[13/10]] to the major third.
+[[Subgroup]]: 2.3.7.13/5
+[[Comma list]]: 64/63 ({{monzo| 6 -2 -1 0 }}), 91/90 ({{monzo| -1 -2 1 1 }})
+{{Mapping|legend=3| 1 0 6 -5 | 0 1 -2 4 }}
+[[Optimal tuning]]s:
+* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1197.2656{{c}}, ~3/2 = 710.1993{{c}}
+* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.7413{{c}}
+{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 32, 59 }}
+[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
+Scales: [[Oceanfront scales]]
 [[Category:Archytas clan| ]] <!-- main article -->