Parapyth

Parapyth is generally viewed as a no-5 2.3.7.11.13 rank-3 regular temperament 17 & 41 & 46, sometimes called Parapythic. (In the strict sense, parapyth is Margo Schulter's rank-3 tuning construct inspired by medieval European and Middle Eastern music theory, where Schulter imposes commas that are to be observed as well as commas that are to be tempered out.^{[citation needed]})

Structure

Parapyth has two non-period generators: the fifth (which is tuned in the gentle region, i.e. somewhat sharp of just but flatter than 17edo) and the "spacer" 28/27, which is equated to 33/32. This is equivalent to taking the spacer 64/63, which may be equated with 81/80 (as in 46edo and 87edo) or tempered smaller than 81/80 (mapping 5120/5103 negatively, as in 63edo and 80edo) assuming prime 5 is added.

Intervals

# spacers	-1		0		+1
# fifths	Cents*	JI	Cents*	JI	Cents*	JI
-7	1015.9		1072.1	13/7	1128.4	52/27
-6	519.9		576.1	39/28	632.3	13/9
-5	23.9	64/63	80.1	22/21	136.3	13/12
-4	727.9	32/21	784.1	11/7	840.3	13/8
-3	231.9	8/7	288.1	13/11	344.3	11/9
-2	935.8	12/7	992.0	16/9, 39/22	1048.3	11/6
-1	439.8	9/7	496.0	4/3	552.2	11/8
0	1143.8	27/14, 64/33	0	1/1	56.2	28/27, 33/32
1	647.8	16/11	704.0	3/2	760.2	14/9, 99/64
2	151.7	12/11	208.0	9/8, 44/39	264.2	7/6
3	855.7	18/11	911.9	22/13	968.1	7/4
4	359.7	16/13	415.9	14/11	472.1	21/16
5	1063.7	24/13	1119.9	21/11	1176.1	63/32
6	567.7	18/13	623.9	56/39	680.1
7	71.6	27/26	127.9	14/13	184.1

(* Cent values in 2.3.7.11.13 CE tuning. The 104edo tuning (~3/2 = 703.846c, ~28/27 = 57.692c) is close.)

Scales

Diachrome, Penthesilia[12]

The ternary scale diachrome is defined as one of the following patterns:

• 5sC: LsLsLmsLsLsm (subst 2m(5L5s))
• 5sR: LmsLsLsLmsLs
• 5sL: LsLsLsmLsLsm

Diachrome can be given a Parapyth tempering:

• The L step becomes 12/11;
• The m step becomes 22/21~104/99~256/243;
• The s step becomes 28/27~33/32~1053/1024.

The tempered tuning thus has the mappings

• 3/2 = 3L + m + 3s,
• 7/4 = 4L + m + 5s,
• 11/8 = 2L + m + 3s,
• 13/8 = 3L + 2m + 4s.

If 144/143 is not tempered out, Parapyth distinguishes m + s from L by tuning the former to 13/12 and the latter to 12/11.

The 5sL version of diachrome tempered to parapyth is known by Margo Schulter under the name "Penthesilia[12]".

Scale Workshop links:

• 5sRA Aeolian (46edo 6:3:2 Parapyth chromedye)
• 5sRA Aeolian (87edo 11:6:4 Parapyth chromedye)

Supporting temperaments

Rodan (Parapyth + Slendric, 41 & 46) is arguably the most important rank-2 temperament that supports Parapyth (besides the gentle-fifth-generated 2.3.7.11.13 temperament Leapfrog, which see Gentle tuning). As an aside, having Rodan also enables penslen in a Parapyth tuning.

Extensions

If prime 5 is added to 2.3.7.11.13 Parapyth via Aberschismic tempering equating 64/63 to 81/80, the resulting 2.3.5.7.11.13 temperament is called Pele (41 & 46 & 58). However, note that there is no canonical 5 extension applying broadly to Parapyth, as 17edo lacks a good prime 5.

If the diatonic augmented fourth is equated to 23/16, the resulting 2.3.7.11.13.23 temperament is called Skidoo (17 & 41 & 46). 63edo is a particularly good Skidoo tuning and also adds prime 29.

Prime 31 can be easily be added by equating 28/27~33/32 with 32/31.

Patent vals

Main article: Parapyth/Patent vals