100edo

← 99edo

100edo

101edo →

Prime factorization	22 × 52
Step size	12¢
Fifth	58\100 (696¢) (→29\50)
Semitones (A1:m2)	6:10 (72¢ : 120¢)
Dual sharp fifth	59\100 (708¢)
Dual flat fifth	58\100 (696¢) (→29\50)
Dual major 2nd	17\100 (204¢) (semiconvergent)
Consistency limit	5
Distinct consistency limit	5

100 equal divisions of the octave (abbreviated 100edo or 100ed2), also called 100-tone equal temperament (100tet) or 100 equal temperament (100et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 100 equal parts of exactly 12 ¢ each. Each step represents a frequency ratio of 2^1/100, or the 100th root of 2.

Theory

100edo is closely related to 50edo, 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 50edo 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-, 35-, 47- and 88edo, 100edo 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 12edo.

Odd harmonics

Approximation of odd harmonics in 100edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-5.96	-2.31	+3.17	+0.09	+0.68	-0.53	+3.73	+3.04	+2.49	-2.78	-4.27
	Relative (%)	-49.6	-19.3	+26.5	+0.7	+5.7	-4.4	+31.1	+25.4	+20.7	-23.2	-35.6
Steps (reduced)		158 (58)	232 (32)	281 (81)	317 (17)	346 (46)	370 (70)	391 (91)	409 (9)	425 (25)	439 (39)	452 (52)

Subsets and supersets

Since 100 factors into 2² × 5², 100edo has subset edos 2, 4, 5, 10, 20, 25, and 50.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation (Dual flat fifth 58\100)	Ups and downs notation (Dual sharp fifth 59\100)
0	0	1/1	D	D
1	12		^D, v3E♭♭	^D, v4E♭
2	24		^^D, vvE♭♭	^^D, v3E♭
3	36		^3D, vE♭♭	^3D, vvE♭
4	48	35/34, 38/37	vvD♯, E♭♭	^4D, vE♭
5	60	29/28	vD♯, ^E♭♭	^5D, E♭
6	72	24/23, 25/24	D♯, ^^E♭♭	^6D, ^E♭
7	84	21/20, 43/41	^D♯, v3E♭	v6D♯, ^^E♭
8	96	37/35	^^D♯, vvE♭	v5D♯, ^3E♭
9	108	33/31	^3D♯, vE♭	v4D♯, ^4E♭
10	120	44/41	vvD𝄪, E♭	v3D♯, ^5E♭
11	132	41/38	vD𝄪, ^E♭	vvD♯, ^6E♭
12	144	25/23, 37/34, 38/35	D𝄪, ^^E♭	vD♯, v6E
13	156	23/21, 35/32	^D𝄪, v3E	D♯, v5E
14	168	32/29	^^D𝄪, vvE	^D♯, v4E
15	180	41/37	^3D𝄪, vE	^^D♯, v3E
16	192	19/17	E	^3D♯, vvE
17	204		^E, v3F♭	^4D♯, vE
18	216	43/38	^^E, vvF♭	E
19	228		^3E, vF♭	^E, v4F
20	240	23/20	vvE♯, F♭	^^E, v3F
21	252	22/19, 37/32	vE♯, ^F♭	^3E, vvF
22	264		E♯, ^^F♭	^4E, vF
23	276	34/29, 41/35	^E♯, v3F	F
24	288	13/11	^^E♯, vvF	^F, v4G♭
25	300	25/21, 44/37	^3E♯, vF	^^F, v3G♭
26	312		F	^3F, vvG♭
27	324	35/29, 41/34	^F, v3G♭♭	^4F, vG♭
28	336	17/14	^^F, vvG♭♭	^5F, G♭
29	348		^3F, vG♭♭	^6F, ^G♭
30	360	16/13	vvF♯, G♭♭	v6F♯, ^^G♭
31	372	26/21, 31/25	vF♯, ^G♭♭	v5F♯, ^3G♭
32	384	5/4	F♯, ^^G♭♭	v4F♯, ^4G♭
33	396	39/31, 44/35	^F♯, v3G♭	v3F♯, ^5G♭
34	408	43/34	^^F♯, vvG♭	vvF♯, ^6G♭
35	420	37/29	^3F♯, vG♭	vF♯, v6G
36	432		vvF𝄪, G♭	F♯, v5G
37	444	22/17, 31/24	vF𝄪, ^G♭	^F♯, v4G
38	456	13/10	F𝄪, ^^G♭	^^F♯, v3G
39	468	38/29	^F𝄪, v3G	^3F♯, vvG
40	480	29/22, 33/25	^^F𝄪, vvG	^4F♯, vG
41	492		^3F𝄪, vG	G
42	504		G	^G, v4A♭
43	516	31/23, 35/26	^G, v3A♭♭	^^G, v3A♭
44	528	19/14, 42/31	^^G, vvA♭♭	^3G, vvA♭
45	540		^3G, vA♭♭	^4G, vA♭
46	552	11/8	vvG♯, A♭♭	^5G, A♭
47	564		vG♯, ^A♭♭	^6G, ^A♭
48	576		G♯, ^^A♭♭	v6G♯, ^^A♭
49	588		^G♯, v3A♭	v5G♯, ^3A♭
50	600	41/29	^^G♯, vvA♭	v4G♯, ^4A♭
51	612	37/26	^3G♯, vA♭	v3G♯, ^5A♭
52	624	33/23	vvG𝄪, A♭	vvG♯, ^6A♭
53	636		vG𝄪, ^A♭	vG♯, v6A
54	648	16/11	G𝄪, ^^A♭	G♯, v5A
55	660	41/28	^G𝄪, v3A	^G♯, v4A
56	672	28/19, 31/21	^^G𝄪, vvA	^^G♯, v3A
57	684	43/29	^3G𝄪, vA	^3G♯, vvA
58	696		A	^4G♯, vA
59	708		^A, v3B♭♭	A
60	720	44/29	^^A, vvB♭♭	^A, v4B♭
61	732	29/19	^3A, vB♭♭	^^A, v3B♭
62	744	20/13, 43/28	vvA♯, B♭♭	^3A, vvB♭
63	756	17/11	vA♯, ^B♭♭	^4A, vB♭
64	768	39/25	A♯, ^^B♭♭	^5A, B♭
65	780		^A♯, v3B♭	^6A, ^B♭
66	792		^^A♯, vvB♭	v6A♯, ^^B♭
67	804	35/22	^3A♯, vB♭	v5A♯, ^3B♭
68	816	8/5	vvA𝄪, B♭	v4A♯, ^4B♭
69	828	21/13	vA𝄪, ^B♭	v3A♯, ^5B♭
70	840	13/8	A𝄪, ^^B♭	vvA♯, ^6B♭
71	852		^A𝄪, v3B	vA♯, v6B
72	864	28/17	^^A𝄪, vvB	A♯, v5B
73	876		^3A𝄪, vB	^A♯, v4B
74	888		B	^^A♯, v3B
75	900	37/22, 42/25	^B, v3C♭	^3A♯, vvB
76	912	22/13, 39/23	^^B, vvC♭	^4A♯, vB
77	924	29/17	^3B, vC♭	B
78	936		vvB♯, C♭	^B, v4C
79	948	19/11	vB♯, ^C♭	^^B, v3C
80	960	40/23	B♯, ^^C♭	^3B, vvC
81	972		^B♯, v3C	^4B, vC
82	984		^^B♯, vvC	C
83	996		^3B♯, vC	^C, v4D♭
84	1008	34/19	C	^^C, v3D♭
85	1020		^C, v3D♭♭	^3C, vvD♭
86	1032	29/16	^^C, vvD♭♭	^4C, vD♭
87	1044	42/23	^3C, vD♭♭	^5C, D♭
88	1056	35/19	vvC♯, D♭♭	^6C, ^D♭
89	1068		vC♯, ^D♭♭	v6C♯, ^^D♭
90	1080	41/22	C♯, ^^D♭♭	v5C♯, ^3D♭
91	1092		^C♯, v3D♭	v4C♯, ^4D♭
92	1104		^^C♯, vvD♭	v3C♯, ^5D♭
93	1116	40/21	^3C♯, vD♭	vvC♯, ^6D♭
94	1128	23/12	vvC𝄪, D♭	vC♯, v6D
95	1140		vC𝄪, ^D♭	C♯, v5D
96	1152	37/19	C𝄪, ^^D♭	^C♯, v4D
97	1164		^C𝄪, v3D	^^C♯, v3D
98	1176		^^C𝄪, vvD	^3C♯, vvD
99	1188		^3C𝄪, vD	^4C♯, vD
100	1200	2/1	D	D

Scales

• Greeley8
• Greeley15

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 22edo 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 12edo 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 12edo.

Steps of 22-note Modmos	Interval Name (Decatonic)	Interval Name (Superpyth diatonic)	Pure Interval Size [Multiplicity] Difference from 22edo	Dog Interval Size [Multiplicity] Difference from 22edo
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 22edo 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 22edo 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.