100edo: Difference between revisions

← 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

VisualWikitext

Latest revision as of 18:44, 8 February 2026

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. It is therefore a strong 2.9.5.7.11.13.17.19 system for its size. 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
Error	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. 200edo, which doubles it, corrects the perfect fifth to near-just quality. 400edo further corrects many harmonics, making for a strong 19-limit system. 1600edo and 2000edo do very well in high prime limits.

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

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 ¢. 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.

Instruments

Lumatone

Lumatone mapping for 100edo

Skip fretting

One way to play 100edo on a 20edo guitar is to tune the strings 13\100 apart, or 156 cents. All examples on this page are for 7-string guitar.

Prime harmonics

1/1: string 2 open

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11

11/8: string 4 fret 4

13/8: string 7 fret 1

17/16: string 5 fret 14

19/16: string 2 fret 5

23/16: string 6 open

29/16: string 4 fret 12

31/16: string 7 fret 6

37/32: string 4 fret 19

41/32: string 4 fret 2

43/32: string 3 fret 6

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

Video

Music

Bryan Deister

100edo (2022)

Iceface H. Wakabayashi (微分音チャンネル)

Microtonal Piano in 100 tone equal temperament (100EDO) (Microtonal Music) (2016) (this is the same as the video linked above, to use in case the embedded video refuses to play)

100edo: Difference between revisions

Latest revision as of 18:44, 8 February 2026

Contents

Theory

Odd harmonics

Subsets and supersets

Intervals

Scales

100bddd and the 22-note scales

Instruments

Lumatone

Skip fretting

Video

Music

Navigation menu

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox ET}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+{{ED intro}}
-: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-06-22 08:18:58 UTC</tt>.<br>
-: The original revision id was <tt>238149773</tt>.<br>
+== Theory ==
-: The revision comment was: <tt></tt><br>
+edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|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 {{nowrap|43 &amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
+Like [[6edo|6-]], [[35edo|35-]], [[47edo|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. It is therefore a strong 2.9.5.7.11.13.17.19 system for its size. 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]].
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Scales=
-[[greeley8]]
+=== Odd harmonics ===
-[[greeley15]]</pre></div>
+{{Harmonics in equal|100}}
-<h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;100edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Scales&lt;/h1&gt;
+=== Subsets and supersets ===
- &lt;a class="wiki_link" href="/greeley8"&gt;greeley8&lt;/a&gt;&lt;br /&gt;
+Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}. [[200edo]], which doubles it, corrects the perfect fifth to near-just quality. [[400edo]] further corrects many harmonics, making for a strong 19-limit system. [[1600edo]] and [[2000edo]] do very well in high prime limits.
-&lt;a class="wiki_link" href="/greeley15"&gt;greeley15&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+== Intervals ==
+{{Interval table}}
+== 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{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf 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]].
+{| class="wikitable"
+|-
+! Steps of 22-note MODMOS
+! Interval name<br />(decatonic)
+! Interval name<br />(superpyth diatonic)
+! Pure interval size [multiplicity]<br />Difference from 22edo
+! Dog interval size [multiplicity]<br />Difference from 22edo
+|-
+| 1
+| Diminished 2nd<sub>10</sub>
+| Minor second
+| 60{{c}} [12] <br />5.4545{{c}}
+| 48{{c}} [10] <br />−6.5455{{c}}
+|-
+| 2
+| Minor 2nd<sub>10</sub>
+| Augmented seventh
+| 108{{c}} [20] <br />−1.091{{c}}
+| 120{{c}} [2] <br />10.909{{c}}
+|-
+| 3
+| Major 2nd<sub>10</sub>
+| Augmented unison
+| ''168{{c}} [14]'' <br />4.364{{c}}
+| ''156{{c}} [8]'' <br />−7.636{{c}}
+|-
+| 4
+| Minor 3rd<sub>10</sub>
+| Major second
+| 216{{c}} [18] <br />−2.182{{c}}
+| 228{{c}} [4] <br />9.818{{c}}
+|-
+| 5
+| Major 3rd<sub>10</sub>
+| Minor third
+| 276{{c}} [16] <br />3.273{{c}}
+| 264{{c}} [6] <br />−8.727{{c}}
+|-
+| 6
+| Minor 4th<sub>10</sub>
+| Diminished fourth
+| ''324{{c}} [16]''<br />−3.273{{c}}
+| ''336{{c}} [6]''<br />8.727{{c}}
+|-
+| 7
+| Major 4th<sub>10</sub>
+| Augmented second
+| 384{{c}} [18] <br />2.182{{c}}
+| 372{{c}} [4] <br />−9.818{{c}}
+|-
+| 8
+| Augmented 4th<sub>10</sub> <br> Diminished 5th<sub>10</sub>
+| Major third
+| 432{{c}} [14] <br />−4.364{{c}}
+| 444{{c}} [8] <br />7.636{{c}}
+|-
+| 9
+| Perfect 5th<sub>10</sub>
+| Perfect fourth
+| 492{{c}} [20] <br />1.091{{c}}
+| 480{{c}} [2] <br />−10.909{{c}}
+|-
+| 10
+| Augmented 5th<sub>10</sub> <br> Diminished 6th<sub>10</sub>
+| Diminished fifth
+| 540{{c}} [12] <br />−5.4545{{c}}
+| 552{{c}} [10] <br />6.5455{{c}}
+|-
+| 11
+| Perfect 6th<sub>10</sub>
+| Augmented third <br> Diminished sixth
+| 600{{c}} [20]
+| 588{{c}} [1] <br>−12{{c}} <br> 612{{c}} [1] <br>12{{c}}
+|-
+| 12
+| Augmented 6th<sub>10</sub> <br> Diminished 7th<sub>10</sub>
+| Augmented fourth
+| 660{{c}} [12] <br />6.5455{{c}}
+| 648{{c}} [10] <br />−5.4545{{c}}
+|-
+| 13
+| Perfect 7th<sub>10</sub>
+| Perfect fifth
+| 708{{c}} [20] <br />−1.091{{c}}
+| 720{{c}} [2] <br />10.909{{c}}
+|-
+| 14
+| Augmented 7th<sub>10</sub>
+Diminished 8th<sub>10</sub>
+| Minor sixth
+| 768{{c}} [14] <br />4.364{{c}}
+| 756{{c}} [8] <br />−7.636{{c}}
+|-
+| 15
+| Minor 8th<sub>10</sub>
+| Diminished seventh
+| 816{{c}} [18] <br />−2.182{{c}}
+| 828{{c}} [4] <br />9.818{{c}}
+|-
+| 16
+| Major 8th<sub>10</sub>
+| Augmented fifth
+| ''876{{c}} [16]''<br />3.273{{c}}
+| ''864{{c}} [6]''<br />−8.727{{c}}
+|-
+| 17
+| Minor 9th<sub>10</sub>
+| Major sixth
+| 924{{c}} [16] <br />−3.273{{c}}
+| 936{{c}} [6] <br />8.727{{c}}
+|-
+| 18
+| Major 9th<sub>10</sub>
+| Minor seventh
+| 984{{c}} [18] <br />2.182{{c}}
+| 972{{c}} [4] <br />−9.818{{c}}
+|-
+| 19
+| Minor 10th<sub>10</sub>
+| Diminished octave
+| ''1032{{c}} [14]''<br />−4.364{{c}}
+| ''1044{{c}} [8]''<br />7.636{{c}}
+|-
+| 20
+| Major 10th<sub>10</sub>
+| Diminished second
+| 1092{{c}} [20] <br />1.091{{c}}
+| 1080{{c}} [2] <br />−10.909{{c}}
+|-
+| 21
+| Augmented 10th<sub>10</sub> <br> Diminished 11th<sub>10</sub>
+| Major seventh
+| 1140{{c}} [12] <br />−5.4545{{c}}
+| 1152{{c}} [10] <br />6.5455{{c}}
+|-
+| 22
+| 11th<sub>10</sub>
+| Octave
+| 1200{{c}} [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 6th<sub>10</sub> is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|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 [[78edo|78ddd]] and [[56edo|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 {{nowrap|22 &amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|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 {{nowrap|12 &amp; [[34edo|34d]]}} temperaments up to 58d is also usable.
+== Instruments ==
+=== Lumatone ===
+[[Lumatone mapping for 100edo]]
+=== Skip fretting ===
+One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.
+; Prime harmonics
+/1: string 2 open
+/1: string 7 fret 7
+/2: string 5 fret 4
+/4: string 1 fret 9
+/4: string 4 fret 11
+/8: string 4 fret 4
+/8: string 7 fret 1
+/16: string 5 fret 14
+/16: string 2 fret 5
+/16: string 6 open
+/16: string 4 fret 12
+/16: string 7 fret 6
+/32: string 4 fret 19
+/32: string 4 fret 2
+/32: string 3 fret 6
+/32: string 4 fret 6
+/32: string 3 fret 12
+/32: string 3 fret 15
+/32: string 3 fret 16
+== Video ==
+<youtube>shcrw2vtmJU</youtube>
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/37bFvBsKXqo ''100edo''] (2022)
+; [[Iceface H. Wakabayashi]] (微分音チャンネル)
+* [https://www.youtube.com/watch?v=shcrw2vtmJU ''Microtonal Piano in 100 tone equal temperament (100EDO) (Microtonal Music)''] (2016) (this is the same as the video linked above, to use in case the embedded video refuses to play)

100edo: Difference between revisions

Latest revision as of 18:44, 8 February 2026

Theory

Odd harmonics

Subsets and supersets

Intervals

Scales

100bddd and the 22-note scales

Instruments

Lumatone

Skip fretting

Video

Music

Navigation menu

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