Porcupine extensions

< Porcupine

Porcupine has various extensions to the 13-limit. Adding the 13th harmonic to porcupine is not very simple, because 13 tends to fall in between the simple intervals produced by porcupine's generator. The extensions are:

• Tridecimal porcupine (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;
• Porkpie (15f & 22) – tempering out 55/54, 64/63, 65/63, 100/99;
• Porcupinefish (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;
• Porcup (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.

Tridecimal porcupine maps 13/8 to -2 generator steps and conflates it with 5/3 and 18/11, tempering out 40/39. This is where the generator, representing 10/9, 11/10, and 12/11, goes one step further to stand in for ~13/12. Porkpie maps 13/8 to +5 generator steps and conflates it with 8/5, tempering out 65/64. The generator now represents ~14/13. Without optimization for the 13-limit, tridecimal porcupine sharpens the interval class of 13 by about 30 cents, and porkpie flattens it by about 20.

The other pair of extensions are of higher complexity, but are well rewarded with better intonation. Porcupinefish's mapping of 13 is available at -17 generator steps. This equates the sharply tuned diatonic major third of porcupine with 13/10 along with 9/7, and requires a much more precise tuning of the porcupine generator to 161.5–163.5 cents to tune the 13th harmonic well. Porcup's mapping of 13 is available at +20 generator steps. They unite in 37edo, which can be recommended as a tuning for both.

Prime 17 can be found at +8 generator steps, in which case -14 generator steps represent 18/17. This conflates 16/15 with 17/16, tempering out 256/255, and 15/14 with 18/17, tempering out 85/84. It can also be found at -14 generator steps, in which case +8 generator steps represent 18/17. This conflates 17/16 with 15/14, tempering out 120/119, and 18/17 with 16/15, tempering out 136/135. Both steps tend to be tuned between around 90 and 130 cents.

Interval chain

In the following table, odd harmonics and subharmonics 1–13 are in bold.

#	Cents*	Approximate ratios
		11-limit	13-limit extensions
			Porcupine	Porcupinefish	Porkpie	Porcup
0	0.0	1/1
1	162.8	10/9, 11/10, 12/11	13/12		14/13
2	325.6	6/5, 11/9	13/11, 16/13		26/21
3	488.4	4/3	13/10
4	651.3	16/11, 22/15	13/9
5	814.1	8/5	21/13		13/8
6	976.9	7/4, 16/9	26/15
7	1139.7	48/25, 64/33, 160/81	52/27	25/13	39/20
8	102.5	16/15, 21/20	14/13, 26/25	27/26	13/12
9	265.3	7/6		15/13	13/11
10	428.2	14/11			13/10
11	591.0	7/5		18/13	13/9
12	753.8	14/9		20/13
13	916.6	42/25		22/13	26/15
14	1079.4	28/15		24/13	52/27	13/7
15	42.2	28/27, 49/48		40/39	26/25
16	205.0	28/25
17	367.9	49/40, 56/45		16/13		26/21
18	530.7	49/36
19	693.5	49/33
20	856.3	49/30		21/13		13/8
21	1019.1	49/27
22	1181.9	49/25				39/20

* In 11-limit CWE tuning, octave reduced

Tuning spectrum

Tridecimal porcupine

Tuning spectrum of 13-limit porcupine

Edo generator	Eigenmonzo (Unchanged-interval)	Generator (¢)	Comments
	13/12	138.573
	13/11	144.605
1\8		150.000	8d val, lower bound of 5-odd-limit diamond monotone
	11/6	150.637	Lower bound of 11-odd-limit diamond tradeoff
	13/10	151.405
	5/3	157.821	Lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff
	15/13	158.710
	13/9	159.154
2\15		160.000	Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
	7/4	161.471
	11/7	161.751
	7/5	162.047
5\37		162.162	37ff val
	11/8	162.171	13- and 15-odd-limit minimax
8\59		162.712	59fff val
	5/4	162.737	5- and 7-odd-limit minimax
	15/14	162.897
	7/6	162.986
3\22		163.636	22f val, upper bound of 7-, 9-, 11, and 13-odd-limit diamond monotone
	9/7	163.743	9- and 11-odd-limit minimax
	15/8	163.966
	11/10	165.004
	15/11	165.762
	3/2	166.015	Upper bound of 5- and 7-odd-limit diamond tradeoff
	13/7	166.037
1\7		171.429	Upper bound of 5-odd-limit diamond monotone
	11/9	173.704
	13/8	179.736
	9/5	182.404	Upper bound of 9- and 11-odd-limit diamond tradeoff

Porcupinefish

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	11/6	150.637
	5/3	157.821
2\15		160.000	Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
	13/9	160.307
	15/13	160.860
	7/4	161.471
	13/12	161.531
	11/7	161.751
	7/5	162.047
	13/7	162.100
	13/10	162.149
5\37		162.162	Upper bound of 13-odd-limit diamond monotone
	11/8	162.171
	13/8	162.322
	13/11	162.368	13- and 15-odd-limit minimax
8\59		162.712
	5/4	162.737	5- and 7-odd-limit minimax
	15/14	162.897
	7/6	162.986
3\22		163.636	Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
	9/7	163.743	9- and 11-odd-limit minimax
	15/8	163.966
	11/10	165.004
	15/11	165.762
	3/2	166.015
1\7		171.429	Upper bound of 5-odd-limit diamond monotone
	11/9	173.704
	9/5	182.404