100edo

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100 EDO is the equal division of the octave into 100 parts of exact 12 cents each. It is closely related to 50 EDO, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 EDO 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like 6 EDO, 35 EDO, 47 EDO and 88 EDO, 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to 12 EDO.

Scales

• Greeley8
• Greeley15

Since 100 EDO has a step of 12 cents, it also allows one to use its MOS scales as circulating temperaments.

Circulating temperaments in 100 EDO

Tones	Pattern	L:s
5	5 EDO	equal
6	4L 2s	17:16
7	2L 5s	15:14
8	4L 4s	13:12
9	1L 8s	12:11
10	10 EDO	equal
11	1L 10s	10:9
12	4L 8s	9:8
13	9L 4s	8:7
14	2L 12s
15	10L 5s	7:6
16	4L 12s
17	15L 2s	6:5
18	10L 8s
19	5L 14s
20	20 EDO	equal
21	16L 5s	5:4
22	12L 10s
23	8L 15s
24	4L 20s
25	25 EDO	equal
26	22L 4s	4:3
27	19L 8s
28	16L 12s
29	13L 16s
30	10L 20s
31	7L 24s
32	4L 28s
33	1L 32s
34	32L 2s	3:2
35	30L 5s
36	28L 8s
37	26L 11s
38	24L 14s
39	22L 17s
40	20L 20s
41	18L 23s
42	16L 26s
43	14L 29s
44	12L 32s
45	10L 35s
46	8L 38s
47	6L 41s
48	4L 44s
49	2L 47s
50	50 EDO	equal
51	49L 2s	2:1
52	48L 4s
53	47L 6s
54	46L 8s
55	45L 10s
56	44L 12s
57	43L 14s
58	42L 16s
59	41L 18s
60	40L 20s
61	39L 22s
62	38L 24s
63	37L 26s
64	36L 28s
65	35L 30s
66	34L 32s
67	33L 34s
68	32L 36s
69	31L 38s
70	30L 40s
71	29L 42s
72	28L 44s
73	27L 46s
74	26L 48s
75	25L 50s
76	24L 52s
77	23L 54s
78	22L 56s
79	21L 58s

100bddd and the 22-note scales

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to 22 EDO for pajara temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range favored by George Secor for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term dog rather than wolf for these intervals. Dog intervals frequently provide closer matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12 Edo.

Steps of 22-note MODMOS	Interval name (decatonic)	Interval name (superpyth diatonic)	Pure interval size [multiplicity] Difference from 22 EDO	Dog interval size [multiplicity] Difference from 22 EDO
1	Diminished 2nd10	Minor second	60¢ [12] 5.4545¢	48¢ [10] -6.5455¢
2	Minor 2nd10	Augmented seventh	108¢ [20] -1.091¢	120¢ [2] 10.909¢
3	Major 2nd10	Augmented unison	168¢ [14] 4.364¢	156¢ [8] -7.636¢
4	Minor 3rd10	Major second	216¢ [18] -2.182¢	228¢ [4] 9.818¢
5	Major 3rd10	Minor third	276¢ [16] 3.273¢	264¢ [6] -8.727¢
6	Minor 4th10	Diminished fourth	324¢ [16] -3.273¢	336¢ [6] 8.727¢
7	Major 4th10	Augmented second	384¢ [18] 2.182¢	372¢ [4] -9.818¢
8	Augmented 4th10 Diminished 5th10	Major third	432¢ [14] -4.364¢	444¢ [8] 7.636¢
9	Perfect 5th10	Perfect fourth	492¢ [20] 1.091¢	480¢ [2] -10.909¢
10	Augmented 5th10 Diminished 6th10	Diminished fifth	540¢ [12] -5.4545¢	552¢ [10] 6.5455¢
11	Perfect 6th10	Augmented third Diminished sixth	600¢ [20]	588¢ [1] -12¢ 612¢ [1] 12¢
12	Augmented 6th10 Diminished 7th10	Augmented fourth	660¢ [12] 6.5455¢	648¢ [10] -5.4545¢
13	Perfect 7th10	Perfect fifth	708¢ [20] -1.091¢	720¢ [2] 10.909¢
14	Augmented 7th10 Diminished 8th10	Minor sixth	768¢ [14] 4.364¢	756¢ [8] -7.636¢
15	Minor 8th10	Diminished seventh	816¢ [18] -2.182¢	828¢ [4] 9.818¢
16	Major 8th10	Augmented fifth	876¢ [16] 3.273¢	864¢ [6] -8.727¢
17	Minor 9th10	Major sixth	924¢ [16] -3.273¢	936¢ [6] 8.727¢
18	Major 9th10	Minor seventh	984¢ [18] 2.182¢	972¢ [4] -9.818¢
19	Minor 10th10	Diminished octave	1032¢ [14] -4.364¢	1044¢ [8] 7.636¢
20	Major 10th10	Diminished second	1092¢ [20] 1.091¢	1080¢ [2] -10.909¢
21	Augmented 10th10 Diminished 11th10	Major seventh	1140¢ [12] -5.4545¢	1152¢ [10] 6.5455¢
22	11th10	Octave	1200¢ [22]	N/A

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th₁₀ is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are tuned in tritones. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow almost all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the Indian sruti system.

Other, "gentle" alternatives to 22 EDO for pajara include 78ddd and 56d. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&34d temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include 58d and 46d. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&34d temperaments up to 58d is also usable.