Pele: Difference between revisions

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

Latest revision as of 06:59, 26 July 2026

Pele
Subgroups 2.3.5.7.11; 2.3.5.7.11.13
Comma basis 441/440, 896/891;
196/195, 352/351, 364/363
Reduced mapping ⟨1; 1 0 -6 -10 -13; 0 1 1 1 1]
ET join 41 & 46 & 58
Generators (CWE) ~3/2 = 703.4 ¢, ~5/4 = 387.8 ¢
MOS scales n/a
Ploidacot n/a
Minimax error 11-odd-limit: 3.51 ¢;
13-limit 21-odd-limit: 4.43 ¢
Target scale size 11-odd-limit: ? notes;
13-limit 21-odd-limit: ? notes

The pele rank-3 temperament is one of the 11-limit extensions of aberschismic, inheriting the chain-of-fifths-based structure and the generic comma step that results from equating the syntonic and septimal commas. It 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 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 quartertone-up augmented second (C–^D♯).

See Aberschismic family #Pele for technical data.

Interval lattice

  • 13-limit pele
    13-limit pele
  • 17-limit pele
    17-limit pele

This lattice shows pele as an extension of parapyth, generated by ~2, ~3/2, and ~7/4.

Chords and harmony

Pele enables essentially tempered chords of werckismic and pentacircle in the 11-odd-limit, in addition to mynucumic, major minthmic, minor minthmic and cuthbert in the 13-odd-limit.

Scales

Tunings

11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 703.2829 ¢, ~5/4 = 386.5647 ¢ CWE: ~3/2 = 703.2804 ¢, ~5/4 = 387.3911 ¢ POTE: ~3/2 = 703.2791 ¢, ~5/4 = 387.7906 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 703.4398 ¢, ~5/4 = 386.8933 ¢ CWE: ~3/2 = 703.4225 ¢, ~5/4 = 387.7761 ¢ POTE: ~3/2 = 703.4143 ¢, ~5/4 = 388.1971 ¢
17-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 703.5544 ¢, ~5/4 = 387.9654 ¢ CWE: ~3/2 = 703.4518 ¢, ~5/4 = 388.4909 ¢ POTE: ~3/2 = 703.4265 ¢, ~5/4 = 388.6202 ¢
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