# Porcupine

Porcupine is a linear temperament in the porcupine family that tempers out 250/243, the porcupine comma, and whose generator is somewhere around 160-165 cents. It can be thought of as a 5-limit, 7-limit, or 11-limit temperament, or a 2.3.5.11 subgroup temperament. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.

The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)^2 equivalent to (6/5)^3. In perhaps more familiar musical terms, this means two "perfect fourths" equals three "minor thirds". As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. This is obviously in stark contrast to 12edo, and to meantone, in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.

Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them. (Tuning in 22edo)

## Interval chain

Main article: Porcupine intervals

Generators Cents Ratios Ups and Downs

notation

Generators 2/1 inverse Ratios Ups and Downs

notation

0 0.00 1/1 P1 0 1200.00 2/1 P8
1 162.75 12/11~11/10~10/9 vM2 = ^^m2 -1 1037.25 9/5~20/11~11/6 ^m7 = vvM7
2 325.50 6/5~11/9 ^m3 = vvM3 -2 874.50 18/11~5/3 vM6 = ^^m6
3 488.25 4/3 P4 -3 711.75 3/2 P5
4 651.00 16/11~22/15 v5 = ^^d5 -4 549.00 15/11~11/8 ^4 = vvA4
5 813.75 8/5 ^m6 = vvM6 -5 386.25 5/4 vM3 = ^^m3
6 976.50 7/4~16/9 m7 -6 223.50 9/8~8/7 M2
7 1139.25 48/25~160/81 v8 = ^^d8 -7 60.75 81/80~25/24 ^1 = vvA1
8 102.00 16/15~21/20 ^m2 = vvM2 -8 1098.00 40/21~15/8 vM7 = ^^m7
9 264.75 7/6 m3 -9 935.25 12/7 M6
10 427.50 14/11 v4 = ^^d4 -10 772.50 11/7 ^5 = vvA5
11 590.25 7/5 ^d5 = vv5 -11 609.75 10/7 vA4 = ^^4
12 753.00 14/9 m6 -12 447.00 9/7 M3

The specific tuning shown is the full 11-limit POTE tuning, but of course there is a range of acceptible porcupine tunings that includes generators as small as 160 cents (15edo) and as large as 165.5 cents (29edo). (However, the 29edo patent val does not support 11-limit porcupine proper, not annihilating 64/63.)

12/11, 11/10, and 10/9 are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find.

 https://en.xen.wiki/w/File:OtonalPentad_JI.mp3 https://en.xen.wiki/w/File:OtonalPentad_22edo.mp3 https://en.xen.wiki/w/File:OtonalPentad_29edo.mp3 8:9:10:11:12 chord, in just intonation. All intervals are slightly different. Porcupine-tempered 8:9:10:11:12 chord, in 22edo. Except the first, the intervals are the same. Porcupine-tempered 8:9:10:11:12 chord, in 29edo. Except the first, the intervals are the same.

The 11/9 interval, usually considered a "neutral third", is in porcupine identical to the 6/5 "minor third". This means that the 27/20 "acute fourth" of the JI diatonic scale is equivalent to 11/8 (rather than becoming 4/3 as in meantone).

The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from <50 to 80 cents in general, represents both 25/24 and 81/80.

## Spectrum of Porcupine Tunings by Eigenmonzos

Eigenmonzo Neutral Second
13/12 138.573
13/11 144.605
12/11 150.637
13/10 151.405
6/5 157.821
15/13 158.710
18/13 159.154
2\15 160.000
8/7 161.471
14/11 161.751
7/5 162.047
5\37 162.162
11/8 162.171 13- and 15-limit minimax
8\59 162.712
5/4 162.737 5-limit minimax
15/14 162.897
7/6 162.986
3\22 163.636
9/7 163.743 7- 9- and 11-limit minimax
16/15 163.966
7\51 164.706
11/10 165.004
4\29 165.517
15/11 165.762
4/3 166.015
14/13 166.037
11/9 173.704
16/13 179.736
10/9 182.404

[8/5 12/7] eigenmonzos: porcupinewoo15 porcupinewoo22

### Spectrum of Porcupinefish Tunings

 12/11 150.637 6/5 157.821 2\15 160.000 18/13 160.307 15/13 160.860 8/7 161.471 13/12 161.531 14/11 161.751 7/5 162.047 14/13 162.100 13/10 162.149 5\37 162.162 11/8 162.171 16/13 162.322 13/11 162.368 13- and 15-limit minimax 8\59 162.712 5/4 162.737 15/14 162.897 7/6 162.986 3\22 163.636 9/7 163.743 16/15 163.966 7\51 164.706 11/10 165.004 4\29 165.517 15/11 165.762 4/3 166.015 11/9 173.704 10/9 182.404

## History

Porcupine temperament/scales were discovered by Dave Keenan, but didn't have a name until Herman Miller mentioned that his Mizarian Porcupine Overture in 15-tET had a section that pumps the 250:243 comma. Although this music did not use a Porcupine MOS or MODMOS (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of DE scales was fully evident. It was clear that even though Herman's piece was in 15, 22 was a porcupine tuning par excellence, and that was an interesting development in itself.