Porcupine

Porcupine, [15 & 22] (usually defined in 2.3.5.11 or 2.3.5.7.11), is a temperament that splits 4/3 into three submajor seconds (approximately 11/10), representing 10/9~11/10~12/11. In the 5-limit, it equates 81/80 with 25/24. This makes it an excellent compromise between accuracy and simplicity on the side of simplicity while at the same time not fully exotempering (the intervals it detunes significantly, such as 10/9, can be seen as "connective" intervals rather than distinct harmonic identities, with the notable exception of 11/9).

The simplest Porcupine edo join is [7 & 8], however this results in an inaccurate extension to higher primes than 11. Also note that 29 agrees with 15 & 22 in 2.3.5.11 but not in 2.3.5.7.11, thus [22 & 29] represents a separate extension from [15 & 22] in the full 11-limit.

The Porcupine generator generates 1L6s, 7L1s, and 7L8s.

Interval chain

In the following table, odd harmonics 1–15 and their inverses are in bold. Interpretations in parentheses are only found in the Septimal Porcupine (2.3.5.7.11[15 & 22]) extension.

#	Cents*	Approximate ratios
0	0.0	1/1
1	162.8	10/9, 11/10, 12/11
2	325.6	6/5, 11/9
3	488.4	4/3
4	651.3	16/11
5	814.1	8/5
6	976.9	(7/4), 16/9
7	1139.7	160/81, 48/25
8	102.5	16/15
9	265.3	(7/6)
10	428.2	(14/11), 32/25
11	591.0	(7/5)
12	753.8	(14/9)

* In 2.3.5.7.11 CWE tuning

Scales

Onyx, pine

Onyx (1L6s) is the moment-of-symmetry scale generated by six porcupine generators. It is tuned rather soft in porcupine, resulting in a 7-form structure. Pine adds an additional "blue note" to onyx.

Mode	Brightness	Steps	Interval qualities
Lydian	3	Lssssss	AMMMMP
Ionian	2	sLsssss	PMMMMP
Mixolydian	1	ssLssss	PmMMMP
Dorian	0	sssLsss	PmmMMP
Aeolian	-1	ssssLss	PmmmMP
Phrygian	-2	sssssLs	PmmmmP
Locrian	-3	ssssssL	Pmmmmd

maybe move this to dedicated page for onyx

Zarlino

The 5-limit zarlino scale, in Porcupine temperament, has its two chromas 25/24 and 81/80 equated, making it a MODMOS of onyx. This characteristic defines 5-limit Porcupine.

List of patent vals

EDO	Extension to 7	Generator tuning	25/24 tuning	Fifth tuning
7	[15 & 22]	171.4c	0c	685.7c
29	[22 & 29]	165.5c	41.4c	703.5c
51	[22 & 29]	164.7c	47.1c	705.9c
22	[15 & 22], [22 & 29]	163.6c	54.5c	709.1c
59	[15 & 22]	162.7c	61c	711.9c
37	[15 & 22]	162.2c	64.9c	713.5c
15	[15 & 22]	160c	80c	720c
8	[15 & 22]	150c	150c	750c