100edo: Difference between revisions

100edo 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 & 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

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]].

=== Odd harmonics ===

{{Harmonics in equal|100}}

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.

== Intervals ==

== 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]].

! 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 2nd₁₀

| Minor second

| 60{{c}} [12] <br />5.4545{{c}}

| 48{{c}} [10] <br />−6.5455{{c}}

| 2

| Minor 2nd₁₀

| Augmented seventh

| 108{{c}} [20] <br />−1.091{{c}}

| 120{{c}} [2] <br />10.909{{c}}

| 3

| Major 2nd₁₀

| Augmented unison

| ''168{{c}} [14]'' <br />4.364{{c}}

| ''156{{c}} [8]'' <br />−7.636{{c}}

| 4

| Minor 3rd₁₀

| Major second

| 216{{c}} [18] <br />−2.182{{c}}

| 228{{c}} [4] <br />9.818{{c}}

| 5

| Major 3rd₁₀

| Minor third

| 276{{c}} [16] <br />3.273{{c}}

| 264{{c}} [6] <br />−8.727{{c}}

| 6

| Minor 4th₁₀

| Diminished fourth

| ''324{{c}} [16]''<br />−3.273{{c}}

| ''336{{c}} [6]''<br />8.727{{c}}

| 7

| Major 4th₁₀

| Augmented second

| 384{{c}} [18] <br />2.182{{c}}

| 372{{c}} [4] <br />−9.818{{c}}

| 8

| Augmented 4th₁₀ <br> Diminished 5th₁₀

| Major third

| 432{{c}} [14] <br />−4.364{{c}}

| 444{{c}} [8] <br />7.636{{c}}

| 9

| Perfect 5th₁₀

| Perfect fourth

| 492{{c}} [20] <br />1.091{{c}}

| 480{{c}} [2] <br />−10.909{{c}}

| 10

| Augmented 5th₁₀ <br> Diminished 6th₁₀

| Diminished fifth

| 540{{c}} [12] <br />−5.4545{{c}}

| 552{{c}} [10] <br />6.5455{{c}}

| 11

| Perfect 6th₁₀

| Augmented third <br> Diminished sixth

| 600{{c}} [20]

| 588{{c}} [1] <br>−12{{c}} <br> 612{{c}} [1] <br>12{{c}}

| 12

| Augmented 6th₁₀ <br> Diminished 7th₁₀

| Augmented fourth

| 660{{c}} [12] <br />6.5455{{c}}

| 648{{c}} [10] <br />−5.4545{{c}}

| 13

| Perfect 7th₁₀

| Perfect fifth

| 708{{c}} [20] <br />−1.091{{c}}

| 720{{c}} [2] <br />10.909{{c}}

| 14

| Augmented 7th₁₀

Diminished 8th₁₀

| Minor sixth

| 768{{c}} [14] <br />4.364{{c}}

| 756{{c}} [8] <br />−7.636{{c}}

| 15

| Minor 8th₁₀

| Diminished seventh

| 816{{c}} [18] <br />−2.182{{c}}

| 828{{c}} [4] <br />9.818{{c}}

| 16

| Major 8th₁₀

| Augmented fifth

| ''876{{c}} [16]''<br />3.273{{c}}

| ''864{{c}} [6]''<br />−8.727{{c}}

| 17

| Minor 9th₁₀

| Major sixth

| 924{{c}} [16] <br />−3.273{{c}}

| 936{{c}} [6] <br />8.727{{c}}

| 18

| Major 9th₁₀

| Minor seventh

| 984{{c}} [18] <br />2.182{{c}}

| 972{{c}} [4] <br />−9.818{{c}}

| 19

| Minor 10th₁₀

| Diminished octave

| ''1032{{c}} [14]''<br />−4.364{{c}}

| ''1044{{c}} [8]''<br />7.636{{c}}

| 20

| Major 10th₁₀

| Diminished second

| 1092{{c}} [20] <br />1.091{{c}}

| 1080{{c}} [2] <br />−10.909{{c}}

| 21

| Augmented 10th₁₀ <br> Diminished 11th₁₀

| Major seventh

| 1140{{c}} [12] <br />−5.4545{{c}}

| 1152{{c}} [10] <br />6.5455{{c}}

| 22

| 11th₁₀

| 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 6th₁₀ 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

; [[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)