100edo: Difference between revisions

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~~'''100~~ EDO~~''' is the [[equal division of the octave]] into~~ 100 ~~parts of exact 12 cents each.~~ It is closely related to [[50 EDO]], but the [[patent val]]s 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 EDO [[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.

==Theory==

It is closely related to [[50 EDO]], but the [[patent val]]s 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 EDO [[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 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO 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 [[12 EDO]].

===Prime harmonics===

== Scales ==

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== 100bddd and the 22-note 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 [[22 EDO]] 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 12 EDO 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 [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

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 {{Infobox ET}}
-'''100 EDO''' is the [[equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to [[50 EDO]], but the [[patent val]]s 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 EDO [[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 intro|100}}
+==Theory==
+It is closely related to [[50 EDO]], but the [[patent val]]s 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 EDO [[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.
 Like [[6 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO 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 [[12 EDO]].
+===Prime harmonics===
 {{harmonics in equal|100|start=2|intervals=prime}}
 == Scales ==
@@ Line 261: / Line 265: @@
 |}
-== 100bddd and the 22-note 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 [[22 EDO]] 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 12 EDO 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 [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.