Pele: Difference between revisions

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| Comma basis = [[441/440]], [[896/891]]; <br>[[196/195]], [[352/351]], [[364/363]]

| Edo join 1 = 41 | Edo join 2 = 46 | Edo join 3 = 58

| Mapping = 1; 1 0 -6 -10; 0 1 1 1

| Mapping = 1; 1 0 -6 -10 -13; 0 1 1 1 1

| Generators = 3/2; 5/4 | Generators tuning = 703.4; 387.8

| Optimization method = CWE

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| Odd limit 2 = 13-limit 21 | Mistuning 2 = 4.43 | Complexity 2 = ?

}}

'''~~Pele~~''' ~~is a~~ [[rank-3 temperament]] ~~generated by a perfect fifth~~ of ~~[[~]][[3/2]] and a step for~~ the [[~~81/80|syntonic~~]]~[[~~64/63|septimal comma~~]] ~~to reach the interval classes~~ of [[~~5/1|5~~]], [[~~7/1~~|7]], and ~~higher [[prime harmonic|primes]]. Using an arrow to represent this~~ comma step~~, we have [[5/4]] at~~ the ~~down major third (C–vE),~~ [[7/4]] ~~at the down minor seventh (C–vBb),~~ and [[11/8]] ~~at the down diminished fifth (C–vGb),~~ [[tempering out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]].

The '''pele''' [[rank-3 temperament]] is one of the [[11-limit]] [[extension]]s of [[aberschismic]], inheriting the [[chain of fifths|chain-of-fifths-based]] structure and the generic comma step that results from equating the [[81/80|syntonic]] and [[64/63|septimal commas]]. It [[tempering out|tempers out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]], and as a consequence, [[11/8]] is mapped to the comma-down diminished fifth (C–vG♭),

The canonical [[extension]] to the [[13-limit]] ~~finds~~ [[13/8]] at the down diminished seventh (~~C–vBbb~~)~~, tempering out~~ [[196/195]], [[352/351]] and [[847/845]], ~~and~~ a [[17-limit]] extension is available by recognizing [[17/16]] at the up minor second (C–^Db), tempering out [[256/255]].

The canonical [[extension]] to the [[13-limit]] maps [[13/11]] to the diatonic minor third, which is the exact mean of [[6/5]] and [[7/6]], and implies [[13/8]] is the comma-down diminished seventh (C–vB𝄫) and that [[196/195]], [[352/351]] and [[847/845]] are tempered out. Additionally, a [[17-limit]] extension is available by recognizing [[17/16]] at the comma-up minor second (C–^D♭), the same interval as [[16/15]], and thus tempering out [[256/255]].

Another way to view this temperament is to look at it relative to [[parapyth]], for which it is an extension that addresses the missing prime 5. If we use an arrow to represent the quartertone spacer of parapyth, we have 5/4 at the up augmented second (C–^D#).

Another way to view this temperament is to look at it relative to [[parapyth]], for which it is an extension that addresses the missing prime 5. If we use an arrow to represent the quartertone spacer of parapyth, we have 5/4 at the quartertone-up augmented second (C–^D♯).

See [[~~Hemifamity~~ family #Pele]] for technical ~~details~~.

See [[Aberschismic family #Pele]] for technical data.

== Interval lattice ==

@@ Line 4: / Line 4: @@
 | Comma basis = [[441/440]], [[896/891]]; <br>[[196/195]], [[352/351]], [[364/363]]
 | Edo join 1 = 41 | Edo join 2 = 46 | Edo join 3 = 58
-| Mapping = 1; 1 0 -6 -10; 0 1 1 1
+| Mapping = 1; 1 0 -6 -10 -13; 0 1 1 1 1
 | Generators = 3/2; 5/4 | Generators tuning = 703.4; 387.8
 | Optimization method = CWE
@@ Line 10: / Line 10: @@
 | Odd limit 2 = 13-limit 21 | Mistuning 2 = 4.43 | Complexity 2 = ?
 }}
-'''Pele''' is a [[rank-3 temperament]] generated by a perfect fifth of [[~]][[3/2]] and a step for the [[81/80|syntonic]]~[[64/63|septimal comma]] to reach the interval classes of [[5/1|5]], [[7/1|7]], and higher [[prime harmonic|primes]]. Using an arrow to represent this comma step, we have [[5/4]] at the down major third (C–vE), [[7/4]] at the down minor seventh (C–vBb), and [[11/8]] at the down diminished fifth (C–vGb), [[tempering out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]].
+The '''pele''' [[rank-3 temperament]] is one of the [[11-limit]] [[extension]]s of [[aberschismic]], inheriting the [[chain of fifths|chain-of-fifths-based]] structure and the generic comma step that results from equating the [[81/80|syntonic]] and [[64/63|septimal commas]]. It [[tempering out|tempers out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]], and as a consequence, [[11/8]] is mapped to the comma-down diminished fifth (C–vG♭),
-The canonical [[extension]] to the [[13-limit]] finds [[13/8]] at the down diminished seventh (C–vBbb), tempering out [[196/195]], [[352/351]] and [[847/845]], and a [[17-limit]] extension is available by recognizing [[17/16]] at the up minor second (C–^Db), tempering out [[256/255]].
+The canonical [[extension]] to the [[13-limit]] maps [[13/11]] to the diatonic minor third, which is the exact mean of [[6/5]] and [[7/6]], and implies [[13/8]] is the comma-down diminished seventh (C–vB𝄫) and that [[196/195]], [[352/351]] and [[847/845]] are tempered out. Additionally, a [[17-limit]] extension is available by recognizing [[17/16]] at the comma-up minor second (C–^D♭), the same interval as [[16/15]], and thus tempering out [[256/255]].
-Another way to view this temperament is to look at it relative to [[parapyth]], for which it is an extension that addresses the missing prime 5. If we use an arrow to represent the quartertone spacer of parapyth, we have 5/4 at the up augmented second (C–^D#).
+Another way to view this temperament is to look at it relative to [[parapyth]], for which it is an extension that addresses the missing prime 5. If we use an arrow to represent the quartertone spacer of parapyth, we have 5/4 at the quartertone-up augmented second (C–^D♯).
-See [[Hemifamity family #Pele]] for technical details.
+See [[Aberschismic family #Pele]] for technical data.
 == Interval lattice ==