Mavila: Difference between revisions
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{{Infobox regtemp | {{Infobox regtemp | ||
| Title = Mavila | | Title = Mavila | ||
| Subgroups = 2.3.5 | | Subgroups = 2.3.5, 2.3.5.11 | ||
| Comma basis = [[135/128]] | | Comma basis = [[135/128]] (2.3.5)<br>[[33/32]], [[45/44]] (2.3.5.11) | ||
| Mapping = 1; 1 -3 | | Mapping = 1; 1 -3 -1 | ||
| Edo join 1 = 7 | Edo join 2 = 9 | | Edo join 1 = 7 | Edo join 2 = 9 | ||
| | | Generators = 3/2 | ||
| | | Generators tuning = 679.0 | ||
| Optimization method = CWE | | Optimization method = CWE | ||
| Pergen = (P8, P5) | | Pergen = (P8, P5) | ||
| Color name = Layobiti | | Color name = Layobiti | ||
| MOS scales = [[2L 3s]], [[2L 5s]], [[7L 2s]] | | MOS scales = [[2L 3s]], [[2L 5s]], [[7L 2s]] | ||
| Odd limit 1 = 5 | Mistuning 1 = 23.0 | Complexity 1 = | | Odd limit 1 = 5 | Mistuning 1 = 23.0 | Complexity 1 = 5 | ||
| Odd limit 2 = | | Odd limit 2 = 2.3.5.11 11 | Mistuning 2 = 36.9 | Complexity 2 = 7 | ||
}} | }} | ||
'''Mavila''' is a [[regular temperament|temperament]] where the major chroma, [[135/128]], is [[tempering out|tempered out]]. Like [[meantone]], mavila is based on the [[chain of fifths]], but as a result of tempering out 135/128 rather than [[81/80]], the fifths are supposedly very flat ({{nowrap|~{{dash|670, 680}}}}{{c}} or so), flatter than even that of [[7edo]] (4\7). Consequently, stacking 7 of these fifths gives you an [[2L 5s|antidiatonic]] [[mos scale]], where in a certain sense, major and minor intervals get reversed. For example, stacking four fifths and octave-reducing now gets you a [[6/5]] ''minor'' third, whereas stacking three fourths and octave-reducing now gets you a [[5/4]] ''major'' third. Note that since we have a heptatonic scale, terms like ''fifths'', ''thirds'', etc. make perfect sense and really are the fifth, third, etc. steps in the antidiatonic scale. | '''Mavila''' is a [[regular temperament|temperament]] where the major chroma, [[135/128]], is [[tempering out|tempered out]]. Like [[meantone]], mavila is based on the [[chain of fifths]], but as a result of tempering out 135/128 rather than [[81/80]], the fifths are supposedly very flat ({{nowrap|~{{dash|670, 680}}}}{{c}} or so), flatter than even that of [[7edo]] (4\7). Consequently, stacking 7 of these fifths gives you an [[2L 5s|antidiatonic]] [[mos scale]], where in a certain sense, major and minor intervals get reversed. For example, stacking four fifths and octave-reducing now gets you a [[6/5]] ''minor'' third, whereas stacking three fourths and octave-reducing now gets you a [[5/4]] ''major'' third. Note that since we have a heptatonic scale, terms like ''fifths'', ''thirds'', etc. make perfect sense and really are the fifth, third, etc. steps in the antidiatonic scale. | ||
Mavila tunings range from [[9edo]] to 7edo, with [[16edo]], [[23edo]], and [[25edo]] being typical. These tunings detune 5/4 and 3/2 by significant amounts; it is thus reasonable to call mavila an [[exotemperament]], though it is certainly more accurate than the archetypal exotemperaments such as [[father]]. | Mavila tunings range from [[9edo]] to 7edo, with [[16edo]], [[23edo]], and [[25edo]] being typical. These tunings detune 5/4 and 3/2 by significant amounts; it is thus reasonable to call mavila an [[exotemperament]], though it is certainly more accurate than the archetypal exotemperaments such as [[father]]. | ||
| Line 78: | Line 71: | ||
<nowiki/>* In 2.3.5.11-subgroup CWE tuning, octave reduced | <nowiki/>* In 2.3.5.11-subgroup CWE tuning, octave reduced | ||
== | == Chords and harmony == | ||
{{ | {{See also| Mavila temperament modal harmony }} | ||
Mavila's tuning has some very strange implications for music. The mavila antidiatonic scale is similar to the normal [[5L 2s|diatonic]] scale, except interval classes are flipped. Wherever there was a major third, you will find a minor third, and vice versa. Half steps become whole steps and whole steps become half steps (closer to neutral second range, however). When you sharpen the leading tone in minor, you end up sharpening it down instead, meaning you flatten it. Also, minor is now major – instead of a diminished triad in the major scale, there is now an augmented triad. | |||
As an example, the anti-Ionian scale has steps of ssLsssL, which looks like the regular Ionian scale except the "L" intervals are now "s" and vice versa. | |||
Because of the structure of this unique tuning, every existing piece of common practice music has, effectively, a shadow version in antidiatonic. That is, with {{w|Ludwig van Beethoven|Beethoven}}'s {{w|Für Elise}}, there are actually two compositions – the one that you know, and the antidiatonic equivalent that has never been heard before until now. Examples of this are provided in the [[#Music]] section. | |||
== Scales == | == Scales == | ||
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25edo also supports mavila. The tuning is 672{{c}} and hence very flat, even flatter than 16edo, but not as flat as 9edo. This is 25edo's second-best 3/2; the alternate fifth generates 5edo. | 25edo also supports mavila. The tuning is 672{{c}} and hence very flat, even flatter than 16edo, but not as flat as 9edo. This is 25edo's second-best 3/2; the alternate fifth generates 5edo. | ||
=== Norm-based tunings === | |||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings | |+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings | ||
| Line 139: | Line 139: | ||
| POTE: ~3/2 = 679.788{{c}} | | POTE: ~3/2 = 679.788{{c}} | ||
|} | |} | ||
=== Other tunings === | |||
* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 675.456{{c}} | |||
=== Tuning spectrum === | === Tuning spectrum === | ||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
! Edo<br | ! Edo<br>generator | ||
! [[Eigenmonzo|Eigenmonzo<br | ! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]* | ||
! Generator (¢) | ! Generator (¢) | ||
! Comments | ! Comments | ||
| Line 152: | Line 155: | ||
| 600.000 | | 600.000 | ||
| Lower bound of 5-odd-limit diamond monotone | | Lower bound of 5-odd-limit diamond monotone | ||
|- | |||
| | |||
| 11/8 | |||
| 648.682 | |||
| | |||
|- | |||
| 6\11 | |||
| | |||
| 654.545 | |||
| | |||
|- | |- | ||
| | | | ||
| 15/8 | | 15/8 | ||
| 655.866 | | 655.866 | ||
| 1/2 comma | |||
|- | |||
| | |||
| 15/11 | |||
| 663.049 | |||
| | | | ||
|- | |- | ||
| Line 166: | Line 184: | ||
| 5/4 | | 5/4 | ||
| 671.229 | | 671.229 | ||
| | | 1/3 comma | ||
|- | |- | ||
| 9\16 | | 9\16 | ||
| | | | ||
| 675.000 | | 675.000 | ||
| | |||
|- | |||
| | |||
| 11/6 | |||
| 675.319 | |||
| | | | ||
|- | |- | ||
| Line 176: | Line 199: | ||
| 25/24 | | 25/24 | ||
| 675.618 | | 675.618 | ||
| | | 2/7 comma | ||
|- | |- | ||
| | | | ||
| | | ''f''<sup>4</sup> + ''f''<sup>3</sup> - 8 = 0 | ||
| 676.337 | | 676.337 | ||
| | | 1–3–5 equal-beating tuning, Erv Wilson's meta-mavila | ||
|- | |- | ||
| 13\23 | | 13\23 | ||
| Line 191: | Line 214: | ||
| 5/3 | | 5/3 | ||
| 678.910 | | 678.910 | ||
| 5-odd-limit minimax | | 1/4 comma, 5-odd-limit minimax | ||
|- | |||
| | |||
| 11/10 | |||
| 682.502 | |||
| | |||
|- | |- | ||
| | | | ||
| 9/5 | | 9/5 | ||
| 683.519 | | 683.519 | ||
| 5-limit 9-odd-limit minimax | | 1/5 comma, 5-limit 9-odd-limit minimax | ||
|- | |||
| | |||
| 11/9 | |||
| 684.197 | |||
| | |||
|- | |- | ||
| 4\7 | | 4\7 | ||
| | | | ||
| 685.714 | | 685.714 | ||
| Upper bound of 5-odd-limit diamond monotone<br | | Upper bound of 5-odd-limit diamond monotone<br>5-limit 9-odd-limit and 2.3.5.11-subgroup 11-odd-limit diamond monotone (singleton) | ||
|- | |- | ||
| | | | ||
| Line 208: | Line 241: | ||
| Pythagorean tuning | | Pythagorean tuning | ||
|} | |} | ||
<nowiki />* Besides the octave | <nowiki/>* Besides the octave | ||
== Music == | == Music == | ||