9edo: Difference between revisions
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9: [[2/1]] 1200.000 octave | 9: [[2/1]] 1200.000 octave | ||
Here the characterizations are taken from [[Scala]], which also describes the scale itself as "Pelog Nawanada: Sunda". Chords such as 1/1 | Here the characterizations are taken from [[Scala]], which also describes the scale itself as "Pelog Nawanada: Sunda". Chords such as {{dash|1/1, 7/6, 49/36, 12/7|med}} are therefore natural ones for 9edo. The above scale generates the [[just intonation subgroup]] 2.27/25.7/3, which is closely related to 9edo. | ||
9edo has very little to offer in terms of accuracy for harmonics. Most other systems that lack good perfect fifths such as [[6edo]], [[11edo]], [[13edo]] and [[18edo]] at least contain a reasonable approximation of 9/8 (or (3/2)<sup>2</sup>), so they can be viewed as temperaments in subgroups like 2.9.5 and 2.9.7. 9edo, on the other hand, completely misses both 3/2 and 9/8. You could then try to treat 9edo as an entirely no-threes system without any 3/2 or 9/8, in a subgroup like 2.5.7.11, but even here 9edo performs somewhat poorly, because its best [[7/4]] is much closer to 12/7 and is off by 36 cents, while its best [[11/8]] is off by 18 cents. The 13th harmonic is also entirely missed by 9edo. This being said, 9edo does approximate [[47/32]] to within about 1.2 cents. | 9edo has very little to offer in terms of accuracy for harmonics. Most other systems that lack good perfect fifths such as [[6edo]], [[11edo]], [[13edo]] and [[18edo]] at least contain a reasonable approximation of 9/8 (or (3/2)<sup>2</sup>), so they can be viewed as temperaments in subgroups like 2.9.5 and 2.9.7. 9edo, on the other hand, completely misses both 3/2 and 9/8. You could then try to treat 9edo as an entirely no-threes system without any 3/2 or 9/8, in a subgroup like 2.5.7.11, but even here 9edo performs somewhat poorly, because its best [[7/4]] is much closer to 12/7 and is off by 36 cents, while its best [[11/8]] is off by 18 cents. The 13th harmonic is also entirely missed by 9edo. This being said, 9edo does approximate [[47/32]] to within about 1.2 cents. | ||
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9edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Chord names are different because {{dash|C, E, G|med}} isn't {{dash|P1, M3, P5}}. | 9edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Chord names are different because {{dash|C, E, G|med}} isn't {{dash|P1, M3, P5}}. | ||
The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 9edo "on the fly". In this notation, the [[enharmonic unison]] is the augmented 2nd, e.g. | The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 9edo "on the fly". In this notation, the [[enharmonic unison]] is the augmented 2nd, e.g. E♭ to F♯. | ||
{| class="wikitable center-all right-1 right-2" | {| class="wikitable center-all right-1 right-2" | ||
| Line 70: | Line 70: | ||
| 1 | | 1 | ||
| 133.33 | | 133.33 | ||
| [[14/13]] (+5.035), [[13/12]] ( | | [[14/13]] (+5.035), [[13/12]] (−5.239),<br>[[12/11]] (−17.304) | ||
| minor 2nd | | minor 2nd | ||
| E | | E | ||
| Line 79: | Line 79: | ||
| 2 | | 2 | ||
| 266.67 | | 266.67 | ||
| [[7/6]] ( | | [[7/6]] (−0.204) | ||
| major 2nd, minor 3rd | | major 2nd, minor 3rd | ||
| | | E♯, F♭ | ||
| minor 2nd, major 3rd | | minor 2nd, major 3rd | ||
| | | E♭, F♯ | ||
| [[File:0-266,67 major 2nd, minor 3rd (9-EDO).mp3|frameless]] | | [[File:0-266,67 major 2nd, minor 3rd (9-EDO).mp3|frameless]] | ||
|- | |- | ||
| 3 | | 3 | ||
| 400.00 | | 400.00 | ||
| [[5/4]] (+13.686), [[14/11]] ( | | [[5/4]] (+13.686), [[14/11]] (−17.508),<br>[[9/7]] (−35.084) | ||
| major 3rd | | major 3rd | ||
| F | | F | ||
| Line 97: | Line 97: | ||
| 4 | | 4 | ||
| 533.33 | | 533.33 | ||
| [[4/3]] (+35.288), [[11/8]] ( | | [[4/3]] (+35.288), [[11/8]] (−17.985) | ||
| perfect 4th | | perfect 4th | ||
| G | | G | ||
| Line 106: | Line 106: | ||
| 5 | | 5 | ||
| 666.67 | | 666.67 | ||
| [[16/11]] (+17.985), [[3/2]] ( | | [[16/11]] (+17.985), [[3/2]] (−35.288) | ||
| perfect 5th | | perfect 5th | ||
| A | | A | ||
| Line 115: | Line 115: | ||
| 6 | | 6 | ||
| 800.00 | | 800.00 | ||
| [[14/9]] (+35.084) [[11/7]] (+17.508),<br>[[8/5]] ( | | [[14/9]] (+35.084) [[11/7]] (+17.508),<br>[[8/5]] (−13.686) | ||
| minor 6th | | minor 6th | ||
| B | | B | ||
| Line 126: | Line 126: | ||
| [[12/7]] (+0.204) | | [[12/7]] (+0.204) | ||
| major 6th, minor 7th | | major 6th, minor 7th | ||
| | | B♯, C♭ | ||
| minor 6th, major 7th | | minor 6th, major 7th | ||
| | | B♭, C♯ | ||
| [[File:0-933,33 major 6th, minor 7th (9-EDO).mp3|frameless]] | | [[File:0-933,33 major 6th, minor 7th (9-EDO).mp3|frameless]] | ||
|- | |- | ||
| 8 | | 8 | ||
| 1066.67 | | 1066.67 | ||
| [[11/6]] (+17.304) [[13/7]] ( | | [[11/6]] (+17.304) [[13/7]] (−5.035) | ||
| major 7th | | major 7th | ||
| C | | C | ||