100edo: Difference between revisions

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== Theory ==

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 43 & 57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

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. 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]].

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 ===

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=== Subsets and supersets ===

Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

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 ==

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|-

! Steps of 22-note MODMOS

! Interval name (decatonic)

! Interval name<br />(decatonic)

! Interval name (superpyth diatonic)

! Interval name<br />(superpyth diatonic)

! Pure interval size [multiplicity]<br />Difference from 22edo

! Dog interval size [multiplicity]<br />Difference from 22edo

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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 & [[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 & [[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

1/1: 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 ==

<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)

@@ Line 3: / Line 3: @@
 == Theory ==
-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 43 &amp; 57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.
+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.
-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. 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]].
+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 ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.
+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 ==
@@ Line 28: / Line 28: @@
 |-
 ! Steps of 22-note MODMOS
-! Interval name (decatonic)
+! Interval name<br />(decatonic)
-! Interval name (superpyth diatonic)
+! Interval name<br />(superpyth diatonic)
 ! Pure interval size [multiplicity]<br />Difference from 22edo
 ! Dog interval size [multiplicity]<br />Difference from 22edo
@@ Line 171: / Line 171: @@
 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)