9edo: Difference between revisions

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Rank-2 temperaments: use backslashes for edo steps
 
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
9edo contains a pentatonic [[mos scale]] produced by stacking 4/9 of [[2L 3s]] (1 3 1 3 1), which has a heptatonic extension, [[2L 5s]] (1 1 2 1 1 2 1, sometimes called "mavila" or "antidiatonic").  
9edo contains a pentatonic [[mos scale]] produced by stacking 4\9 of [[2L 3s]] (1 3 1 3 1), which has a heptatonic extension, [[2L 5s]] (1 1 2 1 1 2 1, sometimes called "mavila" or "antidiatonic").  


You can also use the 2/9, which generates mos scales of [[1L 3s]] (3 2 2 2) and [[4L 1s]] (2 2 2 2 1) and can be interpreted as either an extremely sharp [[bug]] scale or an extremely flat [[orwell]] one.
You can also use 2\9, which generates mos scales of [[1L 3s]] (3 2 2 2) and [[4L 1s]] (2 2 2 2 1) and can be interpreted as either an extremely sharp [[bug]] scale or an extremely flat [[orwell]] one.


== Historical (and other) relevance ==
== Historical (and other) relevance ==
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9edo's [[prime]]s 3, 7, 11 and 13 are all tuned flat, so it can benefit from [[octave stretching]].  
9edo's [[prime]]s 3, 7, 11 and 13 are all tuned flat, so it can benefit from [[octave stretching]].  


; 9edo
* Step size: 133.333{{c}}, octave size: 1200.0{{c}}
Pure-octaves 9edo makes a decent 2.5.11 tuning, approximating all those three primes within 18{{c}}.
Pure-octaves 9edo makes a decent 2.5.11 tuning, approximating all those three primes within 18{{c}}.
{{Harmonics in equal|9|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 9edo}}
{{Harmonics in equal|9|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 9edo (continued)}}


; [[zpi|22zpi]]
* Step size: 134.078{{c}}, octave size: 1206.7{{c}}
9edo with octaves stretched about 5{{c}}, as in [[zpi|22zpi]], makes a decent 2.7.11.13 tuning, approximating all those four primes within 17{{c}}.
9edo with octaves stretched about 5{{c}}, as in [[zpi|22zpi]], makes a decent 2.7.11.13 tuning, approximating all those four primes within 17{{c}}.
{{Harmonics in cet|134.078|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22zpi}}
{{Harmonics in cet|134.078|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22zpi (continued)}}


; [[32ed12]]
* Step size: 134.436{{c}}, octave size: 1209.9{{c}}
9edo with octaves stretched about 10{{c}}, as in [[ed12|32ed12]], makes a decent 2.3.7.11.13 tuning, approximating all those five primes within 20{{c}}.
9edo with octaves stretched about 10{{c}}, as in [[ed12|32ed12]], makes a decent 2.3.7.11.13 tuning, approximating all those five primes within 20{{c}}.
{{Harmonics in equal|32|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32ed12}}
{{Harmonics in equal|32|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32ed12 (continued)}}


== Diagrams ==
== Diagrams ==