217edo: Difference between revisions
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<nowiki>* | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Notation == | == Notation == | ||
=== Sagittal === | === Sagittal === | ||
217edo can be written in Sagittal using almost the entire Athenian extension | 217edo can be written in Sagittal using ''almost'' the entire Athenian extension by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to [[224edo]]'s Sagittal notation, but it uses the 11/7C for the +6/-6 alteration instead of 55C.<ref name=":1">[[George Secor|George D. Secor]] and [[David Keenan|David C. Keenan]], [https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''], p. 11.</ref> | ||
It shares the same exact symbol system as the Athenian notation for just intonation or ''Medium-precision JI notation.''<ref name=":1"/> | It shares the same exact symbol system as the Athenian notation for just intonation or ''Medium-precision JI notation.''<ref name=":1"/> | ||
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|+Sagittal notation | |+Sagittal notation | ||
! colspan="2" | Steps | ! colspan="2" | Steps | ||
! 1 | |||
! 2 | |||
! 3 | |||
! 4 | |||
! 5 | |||
! 6 | |||
! 7 | |||
! 8 | |||
! 9 | |||
! 10 | |||
! 11 | |||
! 12 | |||
! 13 | |||
! 14 | |||
! 15 | |||
! 16 | |||
! 17 | |||
! 18 | |||
! 19 | |||
! 20 | |||
! 21 | |||
|- | |- | ||
! rowspan="2" | Symbol | ! rowspan="2" | Symbol | ||
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| rowspan="2" | <big>{{Sagittal| /|) }}</big> | | rowspan="2" | <big>{{Sagittal| /|) }}</big> | ||
| rowspan="2" | <big>{{Sagittal| /|\ }}</big> | | rowspan="2" | <big>{{Sagittal| /|\ }}</big> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| \!/ }}</small> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| \!) }}</small> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| \\! }}</small> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| (!( }}</small> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| (! }}</small> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| !) }}</small> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| \! }}</small> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| ~!( }}</small> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| )!( }}</small> | ||
| < | | <small>{{Sagittal|#}}{{sagittal| !( }}</small> | ||
| < | | <small>{{Sagittal|#}}</small> | ||
|- | |- | ||
! Revo | ! Revo | ||
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| <big>{{Sagittal| /||\ }}</big> | | <big>{{Sagittal| /||\ }}</big> | ||
|} | |} | ||
Because it uses the entire Athenian system (except for {{sagittal| |\ }} {{sagittal| !/ }} {{sagittal| /|| }} {{sagittal| \!! }} since it tempers out [[1240029/1239040]]), it allows no accidental enharmonic respellings. | |||
=== Ups-and-downs notation === | === Ups-and-downs notation === | ||
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Since {{nowrap| 217 {{=}} 31 × 7 }}, one ''could'' base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth. | Since {{nowrap| 217 {{=}} 31 × 7 }}, one ''could'' base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth. | ||
This could be useful when [[31edo]] is used as a base tuning, where the whole palette of 217edo is only used to provide subtle inflections of the 31edo pitches, similar to how one might use [[159edo]] to provide subtle corrections of [[53edo]] pitches. It also can be written with SZ half-sharps and up/down. The main drawback of this notation is that | This could be useful when [[31edo]] is used as a base tuning, where the whole palette of 217edo is only used to provide subtle inflections of the 31edo pitches, similar to how one might use [[159edo]] to provide subtle corrections of [[53edo]] pitches. It also can be written with SZ half-sharps and up/down. The main drawback of this notation is that 3/2 is no longer P5 but '''^P5''' , so that ~4:5:6 is from C written as C-E-^G, compared to C-^<E-G to the patent val chain-of-fifths ups-and-downs notation. | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
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=== 7edo-based whitewood notation === | === 7edo-based whitewood notation === | ||
Since {{nowrap| 217 {{=}} 7 × 31}}, one ''could'' use the inherited [[Whitewood family|whitewood]] fifth, ditch sharps and flats, and instead use ups and downs to represent pitch deviations from the | Since {{nowrap| 217 {{=}} 7 × 31}}, one ''could'' use the inherited [[Whitewood family|whitewood]] fifth 124\217, ditch sharps and flats, and instead use ups and downs to represent pitch deviations from the 7edo nominals. Since 31=15*2+1, all pitches can be notated with ups, downs, quips and quids. ~4:5:6 becomes C-^^^>E-^^^G. | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||