g] d]
e b f]
c[ g d a e]
b[ f c
d[ a[
This lattice diagram illustrates a possible notation for the 15-note porcupine scale. The letters a-g (in lowercase to avoid potential confusion with traditional notation) represent a sequence of 7 notes separated by the interval of a porcupine generator (approximating 12/11, 11/10, or 10/9, which are all equivalent in porcupine temperament). The accidentals "[" and "]" represent a lowering or raising of pitch by one note of 15-ET or 22-ET, two notes of 37-ET, or three notes of 59-ET. This represents an approximate ratio of 25/24, 33/32, or 36/35. This notation can easily be extended to larger scales in porcupine temperament.
In Dave Keenan's ET notation, the basic porcupine notes would be notated A Bv C^ D Ev F^ G. In porcupine temperament, the syntonic comma 80;81 (notated v, ^) and the undecimal diesis 32;33 (notated [, ]) happen to be the same interval, so you can convert between porcupine notation and Keenan notation by canceling out one with the other. So for instance, e] is equivalent to Ev], or just plain E.
This porcupine notation can be extended on the lattice in all directions. Because the comma 250;243 is tempered out, the pattern repeats when extended to the left and right. More notes can be added on the upper right and lower left by using double accidentals.
g] d] a] e]] b]]
b f] c] g]] d]] a]]
g d a e] b] f]] c]]
b[ f c g] d] a] e]]
g[ d[ a[ e b f] c] g]] d]]
b[[ f[ c[ g d a e] b] f]]
g[[ d[[ a[[ e[ b[ f c g] d] a]
f[[ c[[ g[ d[ a[ e b f]
e[[ b[[ f[ c[ g d a
Deviation from the harmonic series (in fractions of a step)
| 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 |
| +0.23 | +0.17 | -0.11 | +0.45 | +0.11 | +0.49 | +0.40 | -0.31 | +0.28 | +0.12 | +0.15 | +0.34 | -0.32 | +0.13 | -0.31 |
22-ET is a versatile scale with a number of interesting properties, explored in detail by Paul Erlich. In addition to being a porcupine temperament, it also belongs to the categories of diaschismic [11 -4 -2], MAGIC [-10 -1 5], and Orwell [-21 3 7] temperaments.
Deviation from the harmonic series (in fractions of a step)
| 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 |
| +0.13 | -0.08 | +0.24 | +0.26 | -0.11 | -0.41 | +0.05 | +0.08 | -0.45 | +0.37 | +0.48 | -0.16 | +0.39 | +0.12 | +0.01 |
29-ET is a schismic temperament which shares the [1 -5 3] (250;243) vector with the porcupine scales. However, it doesn't make a very good porcupine temperament, so it's better treated as a regular schismic temperament.
37-ET is one of the more interesting porcupine scales. Its nearly perfect approximations of the 11th and 13th harmonics contrast with an 11.6-cent sharp fifth. 37-ET also has good approximations of the 17th and 19th harmonics.
Deviation from the harmonic series (in fractions of a step)
| 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 |
| +0.36 | +0.09 | +0.13 | -0.29 | +0.00 | +0.08 | +0.45 | -0.24 | -0.17 | +0.48 | -0.37 | +0.18 | +0.07 | +0.25 | -0.31 |
51-ET is inconsistent in the 5-limit, but it fits on the porcupine line between 22 and 29, and can be interpreted as sharing the [1 -5 3] (250;243) unison vector with porcupine temperament. In other respects it appears to be of little interest.
59-ET is the largest porcupine ET, with an exceptionally good major third. But its other properties don't seem to be good enough to justify such a large scale. Still, 15 or 22-note subsets of 59 might have some use.
Deviation from the harmonic series (in fractions of a step)
| 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 |
| +0.49 | +0.01 | +0.37 | -0.03 | -0.11 | -0.33 | +0.49 | -0.16 | +0.37 | -0.15 | +0.11 | +0.01 | +0.46 | +0.38 | -0.30 |
tuning generator 15-TET (30): 160.0 37-TET: 162.2 59-TET: 162.7 22-TET (44, 66): 163.6 51-TET: 164.7 29-TET: 165.5Note that this progression doesn't work in other tunings. Hear how the 12-TET version starts in Eb, like the other versions, but ends up in F#! Here's a 5-limit JI version for comparison.