Pele

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
• 
  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

• SNS ((2/1, 3/2)-12, 64/63: 441/440, 896/891)-24
• SNS ((2/1, 3/2)-12, 64/63: 441/440, 896/891)-36
• Dekany pele – a transversal scale
• The compdye scale pattern

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 ¢