Harry

Harry

Subgroups	2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis	2401/2400, 19683/19600 (7-limit); 243/242, 441/440, 4000/3993 (11-limit); 243/242, 351/350, 364/363, 441/440 (13-limit)
Reduced mapping	⟨2; -6 -17 -10 -15 -26]
ET join	58 & 72
Generators (CWE)	~21/20 = 83.1 ¢
MOS scales	2L 12s, 14L 2s, 14L 16s, 14L 30s
Ploidacot	diploid delta-hexacot
Minimax error	9-odd-limit: 1.81 ¢; 13-limit 21-odd-limit: 3.30 ¢
Target scale size	9-odd-limit: 44 notes; 13-limit 21-odd-limit: 58 notes

Harry is the rank-2 temperament with a period of half an octave and a generator somewhere between 22/21 and 21/20 (which are tempered together in harry), or around 83 cents. Two generators are thus equal to 11/10 (which is made a third of 4/3) and three of which made equal to 15/13 (which is made a half of 4/3). This means that harry splits 4/3 into 6 equal parts, a highly composite number, and splitting 2/1 into two equal parts (representing 24/17~99/70) means it also splits 3/2 into two equal parts (representing 11/9~49/40). Alternatively, it can be viewed as a cluster temperament with 14 clusters and a chroma that represents many important intervals including 81/80, 99/98, 100/99, and 121/120. In any case the first important mos of harry has the shape 2L 12s.

Harry was named after Harry Partch, which is ironic given that Harry Partch was adamantly opposed to the very idea of tempering. This is perhaps not so insulting to Harry when you consider that these mathematical structures can also be used to arrange JI intervals into patterns (constant structures) and create JI detemperings of the temperament.

This particular rank-2 temperament might be called "harry" because the lowest edo in which Harry Partch's 43-tone scale is represented distinctly is 58edo, and harry is one of the best temperaments supported by 58edo (it is 58 & 72). Alternatively, if you look at the tempered image of the 43-tone JI scale in this temperament, it is relatively compact and never "backtracks" from one of the 14 clusters to the previous one. In fact, the entire temperament can be derived from knowing that the fragment [12/11, 11/10, 10/9, 9/8] is supposed to be equidistant, and [14/11, 9/7] also has that same separation. The steps of those scale fragments are 121/120, 100/99, 81/80, and 99/98. Tempering these together means that 4000/3993, 243/242, and 9801/9800 are all tempered out, and harry is the unique 11-limit rank-2 temperament tempering those out.

See Gravity family #Harry for more technical data.

Interval chain

#	Period 0		Period 1
	Cents	Approx. Ratios	Cents	Approx. Ratios
0	0.00	1/1	600.00	99/70, 140/99
1	83.12	21/20, 22/21	683.12	40/27
2	166.23	11/10	766.23	14/9
3	249.35	15/13	849.35	18/11, 44/27
4	332.46	40/33	932.46	12/7
5	415.58	14/11	1015.58	9/5
6	498.70	4/3	1098.70	66/35
7	581.81	7/5	1181.81	160/81
8	664.92	22/15	64.92	26/25, 27/26, 28/27
9	748.04	54/35, 20/13	148.04	12/11
10	831.16	21/13	231.16	8/7
11	914.28	22/13	314.28	6/5
12	997.39	16/9	397.39	44/35, 63/50
13	1080.51	28/15	480.51	33/25
14	1163.62	49/25, 88/45, 108/55	563.62	18/13
15	46.74	36/35	646.74	16/11

As a detemperament of 14et

Harry is naturally considered a detemperament of 14 equal temperament, thus containing both diatonic and interordinal interval catgories. The small step at 1/2 octave minus seven ~21/20 generators serves as a spacer between intervals in the same category, representing 81/80~91/90~99/98~100/99~105/104~121/120.

Chords and harmony

See also: Chords of harry

Scales

• Harry58

Tunings

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~21/20 = 83.1249 ¢	CWE: ~21/20 = 83.1427 ¢	POTE: ~21/20 = 83.1560 ¢

11-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~21/20 = 83.1477 ¢	CWE: ~21/20 = 83.1589 ¢	POTE: ~21/20 = 83.1670 ¢

13-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~21/20 = 83.1175 ¢	CWE: ~21/20 = 83.1169 ¢	POTE: ~21/20 = 83.1164 ¢

Tuning spectrum

Edo generator	Eigenmonzo (unchanged interval)	Generator (¢)	Comments
3\44		81.818	44ceff val, lower bound of 7- to 11-odd-limit diamond monotone
	9/7	82.458
	11/10	82.502
	15/13	82.580
4\58		82.759	Lower bound of 13-odd-limit diamond monotone
	13/11	82.799
	13/10	82.865
	15/11	82.881
	3/2	83.007
	13/7	83.019
	13/8	83.057
	13/12	83.071
9\130		83.077
	13/9	83.099	13- and 15-odd-limit minimax
	7/4	83.117
	15/8	83.119
	15/14	83.120
	5/4	83.158	5-, 7- and 9-odd-limit minimax
	7/5	83.216
	5/3	83.240
	11/8	83.245	11-odd-limit minimax
	7/6	83.282
5\72		83.333	Upper bound of 13-odd-limit diamond monotone
	11/6	83.404
	11/7	83.502
	9/5	83.519
6\86		83.721	86ceff val
	11/9	84.197

See also

• Lumatone mapping for harry