Pajara: Difference between revisions
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→Tunings: Added the minimax tunings according to the power limit method proposed by Dave & Doug and independently by myself. |
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== Interval chains == | == Interval chains == | ||
There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but | There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo, with the optimum at around 707-708 cents. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo. | ||
In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''. | In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''. | ||
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== Chords and harmony == | == Chords and harmony == | ||
{{See also| Chords of pajara }} | {{See also| Chords of pajara }} | ||
In pajara, a decatonic system of interval classification based on the [[2L 8s]] (jaric) [[mos scale]] is preferred over the [[diatonic]] interval classification system traditionally used in western music, which is used in [[meantone]]. If we count scale degrees similarly to diatonic, then [[2/1]] is a "hendecave" (11ve), as there are 10 scale degrees, and we repeat at 2/1 at the 11th. In this system, [[3/2]] is a perfect 7th, and [[4/3]] is a perfect 5th. The intervals [[5/4]] and [[6/5]] are major and minor decatonic 4ths respectively, rather than being major and minor 3rds by diatonic interval classification in meantone. Importantly, [[7/4]] is now a major decatonic 9th, with [[12/7]] being its minor counterpart. This is in contrast to diatonic, where 7/4 is considered a subminor 7th, and 12/7 a supermajor 6th. | |||
By decatonic interval classification, the [[4:5:6:7]] tetrad is written as P1–M4–P7–M9. It can be considered the ''major tetrad'', since the non-perfect intervals, those being the decatonic 4th and 9th, are both major intervals. If we instead use a minor interval for the 4th and 9th; that is, a P1–m4–P7–m9 chord, then we get a tetrad approximating [[70:84:105:120|1/(12:10:8:7)]], which can be considered the ''minor tetrad''. | |||
{{Todo|complete section}} | |||
== Scales == | == Scales == | ||
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=== Scala files === | === Scala files === | ||
* [[Pajara12]] | |||
* [[12-22h]] | * [[12-22h]] | ||
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| POTE: ~3/2 = 707.0477{{c}} | | POTE: ~3/2 = 707.0477{{c}} | ||
|} | |} | ||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings | |+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings | ||
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| CWE: ~3/2 = 707.1826{{c}} | | CWE: ~3/2 = 707.1826{{c}} | ||
| POTE: ~3/2 = 706.8851{{c}} | | POTE: ~3/2 = 706.8851{{c}} | ||
|} | |||
=== Target tunings === | |||
{| class="wikitable center-all mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings | |||
|- | |||
! rowspan="2" | Target | |||
! colspan="2" | Minimax | |||
|- | |||
! Generator | |||
! Eigenmonzo* | |||
|- | |||
| 7-odd-limit | |||
| ~3/2 = 709.363{{c}} | |||
| 35/24 | |||
|- | |||
| 9-odd-limit | |||
| ~3/2 = 708.128{{c}} | |||
| 35/18 | |||
|- | |||
| 11-odd-limit | |||
| ~3/2 = 708.128{{c}} | |||
| 35/18 | |||
|} | |} | ||
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| 706.843 | | 706.843 | ||
| 7- and 11-limit POTT | | 7- and 11-limit POTT | ||
|- | |- | ||
| 33\56 | | 33\56 | ||
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| 707.143 | | 707.143 | ||
| 56d val | | 56d val | ||
|- | |- | ||
| | | | ||
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| 707.234 | | 707.234 | ||
| | | | ||
|- | |- | ||
| | | | ||
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| 11/8 | | 11/8 | ||
| 708.114 | | 708.114 | ||
| | | 11- and 15-odd-limit minimax | ||
|- | |||
| | |||
|36/35 | |||
|708.128 | |||
|9-odd-limit minimax | |||
|- | |- | ||
| | | | ||
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| 708.771 | | 708.771 | ||
| | | | ||
|- | |- | ||
| 13\22 | | 13\22 | ||
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| 709.091 | | 709.091 | ||
| Upper bound of 11-odd-limit diamond monotone | | Upper bound of 11-odd-limit diamond monotone | ||
|- | |||
| | |||
|48/35 | |||
|709.363 | |||
|7-odd-limit minimax | |||
|- | |- | ||
| | | | ||