67edo: Difference between revisions
67edo doesn't support septimal meantone, so the disqualification of 26edo for being flattone is an unfair comparison |
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== Theory == | == Theory == | ||
67edo [[tempering out|tempers out]] [[81/80]], [[support]]ing [[meantone]], with a tuning which is slightly sharp of [[1/6-comma meantone|1/6-comma]] (the tuning favored by {{w|Wolfgang Amadeus Mozart|Mozart}} and contemporaries, though they suggested the flatter | 67edo [[tempering out|tempers out]] [[81/80]], [[support]]ing [[meantone]], with a tuning which is slightly sharp of [[1/6-comma meantone|1/6-comma]] (the tuning favored by {{w|Wolfgang Amadeus Mozart|Mozart}} and contemporaries, though they suggested the flatter and composite [[55edo]] as an approximation). It is indistinguishable from {{nowrap|{{frac|4|25}} {{=}} 0.16-comma}} meantone. In the 7-limit the [[patent val]] tempers out [[1029/1024]] and [[1728/1715]], so that it supports [[mothra]]. In the 11-limit it tempers out [[176/175]] and [[540/539]], supporting [[mosura]], an alternative 11-limit mothra. In the 13-limit it tempers out [[144/143]] and [[196/195]], supporting 13-limit mosura. It tempers out the [[orgonisma]], and on the 2.7.11 subgroup it supports the [[orgone]] temperament. | ||
It is a promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the second edo after [[26edo]] to have both meantone and an orgone temperament. It has relatively good approximations of the [[3/1|3rd]], [[7/1|7th]], [[11/1|11th]], [[13/1|13th]], [[15/1|15th]], [[17/1|17th]] [[harmonic]]s, although the [[5/1|5th]], [[9/1|9th]], and [[19/1|19th]] as well as certain higher ones are workable as well. {{nowrap|33 + 34}} can be used to construct this temperament explaining some of its properties. It does well on the 2.3.7.11.13.17.23.31.37.41 [[subgroup]]. | It is a promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the second edo after [[26edo]] to have both meantone and an orgone temperament. It has relatively good approximations of the [[3/1|3rd]], [[7/1|7th]], [[11/1|11th]], [[13/1|13th]], [[15/1|15th]], [[17/1|17th]] [[harmonic]]s, although the [[5/1|5th]], [[9/1|9th]], and [[19/1|19th]] as well as certain higher ones are workable as well. {{nowrap|33 + 34}} can be used to construct this temperament explaining some of its properties. It does well on the 2.3.7.11.13.17.23.31.37.41 [[subgroup]]. | ||
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== Notation == | == Notation == | ||
===Ups and downs notation=== | === Ups and downs notation === | ||
67edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down). | 67edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down). | ||
{{Sharpness-sharp5a}} | {{Sharpness-sharp5a}} | ||
Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]: | Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]: | ||
{{Sharpness-sharp5}} | {{Sharpness-sharp5}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
==== Evo flavor ==== | ==== Evo flavor ==== | ||
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* Meantone[7]: 11 11 6 11 11 11 6 | * Meantone[7]: 11 11 6 11 11 11 6 | ||
* Barbados[5], Bustling Docks (original/default tuning): 14 14 11 14 14 | * Barbados[5], Bustling Docks (original/default tuning): 14 14 11 14 14 | ||
* Barbados[9]: 11 3 11 3 11 3 11 3 11 | |||
=== Modmos scales === | === Modmos scales === | ||