100edo

Load video

YouTube

YouTube might collect personal data. Privacy Policy

Continue Dismiss

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It 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 50 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 6edo, 35edo, 47edo 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.

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]	Dog interval size [multiplicity]
1	Diminished 2nd10	Minor second	60¢ [12]	48¢ [10]
2	Minor 2nd10	Augmented seventh	108¢ [20]	120¢ [2]
3	Major 2nd10	Augmented unison	168¢ [14]	156¢ [8]
4	Minor 3rd10	Major second	216¢ [18]	228¢ [4]
5	Major 3rd10	Minor third	276¢ [16]	264¢ [6]
6	Minor 4th10	Augmented second	324¢ [16]	336¢ [6]
7	Major 4th10	Diminished fourth	384¢ [18]	372¢ [4]
8	Augmented 4th10 Diminished 5th10	Major third	432¢ [14]	444¢ [8]
9	Perfect 5th10	Perfect fourth	492¢ [20]	480¢ [2]
10	Augmented 5th10 Diminished 6th10	Diminished fifth	540¢ [12]	552¢ [10]
11	Perfect 6th10	Augmented third Diminished sixth	600¢ [20]	588¢ [1] 612¢ [1]
12	Augmented 6th10 Diminished 7th10	Augmented fourth	660¢ [12]	648¢ [10]
13	Perfect 7th10	Perfect fifth	708¢ [20]	720¢ [2]
14	Augmented 7th10 Diminished 8th10	Minor sixth	768¢ [14]	756¢ [8]
15	Minor 8th10	Diminished seventh	816¢ [18]	828¢ [4]
16	Major 8th10	Augmented sixth	876¢ [16]	864¢ [6]
17	Minor 9th10	Major sixth	924¢ [16]	936¢ [6]
18	Major 9th10	Minor seventh	984¢ [18]	972¢ [4]
19	Minor 10th10	Diminished octave	1032¢ [14]	1044¢ [8]
20	Major 10th10	Diminished second	1092¢ [20]	1080¢ [2]
21	Augmented 10th10 Diminished 11th10	Major seventh	1140¢ [12]	1152¢ [10]
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.