# Harry

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 12 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

## Tuning spectrum

EDO
generator
eigenmonzo
(unchanged-interval)
generator
(¢)
3\44 81.818 Lower bound of 7- and 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
4/3 83.007
14/13 83.019
16/13 83.057
13/12 83.071
9\130 83.077
18/13 83.099 13- and 15-odd-limit minimax
8/7 83.117
16/15 83.119
15/14 83.120
5/4 83.158 5-, 7- and 9-odd-limit minimax
7/5 83.216
6/5 83.240
11/8 83.245 11-odd-limit minimax
7/6 83.282
5\72 83.333
12/11 83.404
14/11 83.502
10/9 83.519
6\86 83.721 Upper bound of 13-odd-limit diamond monotone
11/9 84.197