9edo: Difference between revisions
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| Line 34: | Line 34: | ||
{| class="wikitable" | {| class="wikitable" | ||
! style="text-align:center;" | [[degree]] | |||
! style="text-align:center;" | [[cent]]s | |||
! style="text-align:center;" | Approximate <br>Ratios | |||
! colspan="2" style="text-align:center;" | Major wider <br>than minor | |||
! colspan="2" style="text-align:center;" | Major narrower <br>than minor | |||
|- | |- | ||
| style="text-align:center;" | | | style="text-align:center;" | 0 | ||
| | | style="text-align:center;" | 0 | ||
| style="text-align:center;" | 1/1 | | style="text-align:center;" | 1/1 | ||
| style="text-align:center;" | perfect unison | | style="text-align:center;" | perfect unison | ||
| Line 132: | Line 132: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | | ! | [[Ratios|Ratio]] | ||
! | Monzo | ! | [[Monzo]] | ||
! | | ! | [[Cent]]s | ||
![[Color notation/Temperament Names|Color Name]] | |||
! | Name 1 | ! | Name 1 | ||
! | Name 2 | ! | Name 2 | ||
| Line 142: | Line 143: | ||
| |<nowiki> | 0 3 -2 </nowiki>> | | |<nowiki> | 0 3 -2 </nowiki>> | ||
| style="text-align:right;" | 133.24 | | style="text-align:right;" | 133.24 | ||
| style="text-align:center;" |Gugu | |||
| style="text-align:center;" | Large Limma | | style="text-align:center;" | Large Limma | ||
| style="text-align:center;" | Large Semitone | | style="text-align:center;" | Large Semitone | ||
| Line 149: | Line 151: | ||
| |<nowiki> | -7 3 1 </nowiki>> | | |<nowiki> | -7 3 1 </nowiki>> | ||
| style="text-align:right;" | 92.18 | | style="text-align:right;" | 92.18 | ||
| style="text-align:center;" |Layobi | |||
| style="text-align:center;" | Major Chroma | | style="text-align:center;" | Major Chroma | ||
| style="text-align:center;" | Major Limma | | style="text-align:center;" | Major Limma | ||
| Line 156: | Line 159: | ||
| |<nowiki> | -14 3 4 </nowiki>> | | |<nowiki> | -14 3 4 </nowiki>> | ||
| style="text-align:right;" | 51.12 | | style="text-align:right;" | 51.12 | ||
| style="text-align:center;" |Laquadyo | |||
| style="text-align:center;" | Negri Comma | | style="text-align:center;" | Negri Comma | ||
| style="text-align:center;" | Double Augmentation Diesis | | style="text-align:center;" | Double Augmentation Diesis | ||
| Line 163: | Line 167: | ||
| |<nowiki> | 7 0 -3 </nowiki>> | | |<nowiki> | 7 0 -3 </nowiki>> | ||
| style="text-align:right;" | 41.06 | | style="text-align:right;" | 41.06 | ||
| style="text-align:center;" |Trigu | |||
| style="text-align:center;" | Diesis | | style="text-align:center;" | Diesis | ||
| style="text-align:center;" | Augmented Comma | | style="text-align:center;" | Augmented Comma | ||
| Line 170: | Line 175: | ||
| |<nowiki> | -21 3 7 </nowiki>> | | |<nowiki> | -21 3 7 </nowiki>> | ||
| style="text-align:right;" | 10.06 | | style="text-align:right;" | 10.06 | ||
| style="text-align:center;" |Lasepyo | |||
| style="text-align:center;" | Semicomma | | style="text-align:center;" | Semicomma | ||
| style="text-align:center;" | Fokker Comma | | style="text-align:center;" | Fokker Comma | ||
| Line 177: | Line 183: | ||
| |<nowiki> | 2 2 -1 -1 </nowiki>> | | |<nowiki> | 2 2 -1 -1 </nowiki>> | ||
| style="text-align:right;" | 48.77 | | style="text-align:right;" | 48.77 | ||
| style="text-align:center;" |Rugu | |||
| style="text-align:center;" | Septimal Quarter Tone | | style="text-align:center;" | Septimal Quarter Tone | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 184: | Line 191: | ||
| |<nowiki> | -9 1 2 1 </nowiki>> | | |<nowiki> | -9 1 2 1 </nowiki>> | ||
| style="text-align:right;" | 43.41 | | style="text-align:right;" | 43.41 | ||
| style="text-align:center;" |Lazoyoyo | |||
| style="text-align:center;" | Avicenna | | style="text-align:center;" | Avicenna | ||
| style="text-align:center;" | Avicenna's Enharmonic Diesis | | style="text-align:center;" | Avicenna's Enharmonic Diesis | ||
| Line 191: | Line 199: | ||
| |<nowiki> | -4 -1 0 2 </nowiki>> | | |<nowiki> | -4 -1 0 2 </nowiki>> | ||
| style="text-align:right;" | 35.70 | | style="text-align:right;" | 35.70 | ||
| style="text-align:center;" |Zozo | |||
| style="text-align:center;" | Slendro Diesis | | style="text-align:center;" | Slendro Diesis | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 198: | Line 207: | ||
| |<nowiki> | 1 -3 -2 3 </nowiki>> | | |<nowiki> | 1 -3 -2 3 </nowiki>> | ||
| style="text-align:right;" | 27.99 | | style="text-align:right;" | 27.99 | ||
| style="text-align:center;" |Trizo-agugu | |||
| style="text-align:center;" | Senga | | style="text-align:center;" | Senga | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 205: | Line 215: | ||
| |<nowiki> | 1 5 1 -4 </nowiki>> | | |<nowiki> | 1 5 1 -4 </nowiki>> | ||
| style="text-align:right;" | 20.79 | | style="text-align:right;" | 20.79 | ||
| style="text-align:center;" |Quadru-ayo | |||
| style="text-align:center;" | Nuwell | | style="text-align:center;" | Nuwell | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 212: | Line 223: | ||
| |<nowiki> | 6 3 -1 -3 </nowiki>> | | |<nowiki> | 6 3 -1 -3 </nowiki>> | ||
| style="text-align:right;" | 13.07 | | style="text-align:right;" | 13.07 | ||
| style="text-align:center;" |Triru-agu | |||
| style="text-align:center;" | Orwellisma | | style="text-align:center;" | Orwellisma | ||
| style="text-align:center;" | Orwell Comma | | style="text-align:center;" | Orwell Comma | ||
| Line 219: | Line 231: | ||
| |<nowiki> | -5 2 2 -1 </nowiki>> | | |<nowiki> | -5 2 2 -1 </nowiki>> | ||
| style="text-align:right;" | 7.71 | | style="text-align:right;" | 7.71 | ||
| style="text-align:center;" |Ruyoyo | |||
| style="text-align:center;" | Septimal Kleisma | | style="text-align:center;" | Septimal Kleisma | ||
| style="text-align:center;" | Marvel Comma | | style="text-align:center;" | Marvel Comma | ||
| Line 226: | Line 239: | ||
| |<nowiki> | 11 1 -3 -2 </nowiki>> | | |<nowiki> | 11 1 -3 -2 </nowiki>> | ||
| style="text-align:right;" | 5.36 | | style="text-align:right;" | 5.36 | ||
| style="text-align:center;" |Sarurutrigu | |||
| style="text-align:center;" | Porwell | | style="text-align:center;" | Porwell | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 233: | Line 247: | ||
| |<nowiki> | -16 1 5 1 </nowiki>> | | |<nowiki> | -16 1 5 1 </nowiki>> | ||
| style="text-align:right;" | 2.35 | | style="text-align:right;" | 2.35 | ||
| style="text-align:center;" |Lazoquinyo | |||
| style="text-align:center;" | Horwell | | style="text-align:center;" | Horwell | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 240: | Line 255: | ||
| |<nowiki> | -1 2 0 -2 1 </nowiki>> | | |<nowiki> | -1 2 0 -2 1 </nowiki>> | ||
| style="text-align:right;" | 17.58 | | style="text-align:right;" | 17.58 | ||
| style="text-align:center;" |Loruru | |||
| style="text-align:center;" | Mothwellsma | | style="text-align:center;" | Mothwellsma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 247: | Line 263: | ||
| |<nowiki> | -3 -1 -1 0 2 </nowiki>> | | |<nowiki> | -3 -1 -1 0 2 </nowiki>> | ||
| style="text-align:right;" | 14.37 | | style="text-align:right;" | 14.37 | ||
| style="text-align:center;" |Lologu | |||
| style="text-align:center;" | Biyatisma | | style="text-align:center;" | Biyatisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 254: | Line 271: | ||
| |<nowiki> | 4 0 -2 -1 1 </nowiki>> | | |<nowiki> | 4 0 -2 -1 1 </nowiki>> | ||
| style="text-align:right;" | 9.86 | | style="text-align:right;" | 9.86 | ||
| style="text-align:center;" |Lorugugu | |||
| style="text-align:center;" | Valinorsma | | style="text-align:center;" | Valinorsma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 261: | Line 279: | ||
| |<nowiki> | -7 -1 1 1 1 </nowiki>> | | |<nowiki> | -7 -1 1 1 1 </nowiki>> | ||
| style="text-align:right;" | 4.50 | | style="text-align:right;" | 4.50 | ||
| style="text-align:center;" |Lozoyo | |||
| style="text-align:center;" | Keenanisma | | style="text-align:center;" | Keenanisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 268: | Line 287: | ||
| |<nowiki> | 2 3 1 -2 -1 </nowiki>> | | |<nowiki> | 2 3 1 -2 -1 </nowiki>> | ||
| style="text-align:right;" | 3.21 | | style="text-align:right;" | 3.21 | ||
| style="text-align:center;" |Lururuyo | |||
| style="text-align:center;" | Swetisma | | style="text-align:center;" | Swetisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 275: | Line 295: | ||
| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>> | | |<nowiki> | -1 -2 -1 1 0 1 </nowiki>> | ||
| style="text-align:right;" | 19.13 | | style="text-align:right;" | 19.13 | ||
| style="text-align:center;" |Thozogu | |||
| style="text-align:center;" | Superleap | | style="text-align:center;" | Superleap | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 282: | Line 303: | ||
| |<nowiki> | 2 -3 -2 0 0 2 </nowiki>> | | |<nowiki> | 2 -3 -2 0 0 2 </nowiki>> | ||
| style="text-align:right;" | 2.56 | | style="text-align:right;" | 2.56 | ||
| style="text-align:center;" |Bithogu | |||
| style="text-align:center;" | Parizeksma | | style="text-align:center;" | Parizeksma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
Revision as of 04:56, 13 December 2019
The 9EDO scale, which divides the octave into nine equal parts each of 133+1/3 cents precisely, has the peculiar property of representing certain 7-limit intervals almost exactly. A 7-limit version of 9EDO goes
1: 27/25 133.238 large limma, BP small semitone
2: 7/6 266.871 septimal minor third
3: 63/50 400.108 quasi-equal major third
4: 49/36 533.742 Arabic lute acute fourth
5: 72/49 666.258 Arabic lute grave fifth
6: 100/63 799.892 quasi-equal minor sixth
7: 12/7 933.129 septimal major sixth
8: 50/27 1066.762 grave major seventh
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 - 7/6 - 49/36 - 12/7 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.
Notation
9edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the melodic 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. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
The second approach preserves the harmonic 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".
| degree | cents | Approximate Ratios |
Major wider than minor |
Major narrower than minor | ||
|---|---|---|---|---|---|---|
| 0 | 0 | 1/1 | perfect unison | D | perfect unison | D |
| 1 | 133 | 14/13, 13/12, 12/11 | minor 2nd | E | major 2nd | E |
| 2 | 267 | 7/6 | major 2nd, minor 3rd | E#, Fb | minor 2nd, major 3rd | Eb, F# |
| 3 | 400 | 5/4, 14/11, 9/7 | major 3rd | F | minor 3rd | F |
| 4 | 533 | 4/3, 11/8 | perfect 4th | G | perfect 4th | G |
| 5 | 667 | 16/11, 3/2 | perfect 5th | A | perfect 5th | A |
| 6 | 800 | 14/9, 11/7, 8/5 | minor 6th | B | major 6th | B |
| 7 | 933 | 12/7 | major 6th, minor 7th | B#, Cb | minor 6th, major 7th | Bb, C# |
| 8 | 1067 | 11/6, 13/7 | major 7th | C | minor 7th | C |
| 9 | 1200 | 2/1 | octave | D | octave | D |
9EDO contains a pentatonic MOS scale -- 2L 3s (1 3 1 3 1) -- with a heptatonic extension -- 2L 5s (1 1 2 1 1 2 1, sometimes called "mavila" or "antidiatonic"). Indonesian pelog scales sometimes use five-tone subsets of a seven-tone superset in a similar way, and it has been suggested that Indonesian gamelan music stems from a 9EDO tradition.
Commas
9 EDO tempers out the following commas. (Note: This assumes val < 9 14 21 25 31 33 |.)
| Ratio | Monzo | Cents | Color Name | Name 1 | Name 2 | Name 3 |
|---|---|---|---|---|---|---|
| 27/25 | | 0 3 -2 > | 133.24 | Gugu | Large Limma | Large Semitone | Bug Comma |
| 135/128 | | -7 3 1 > | 92.18 | Layobi | Major Chroma | Major Limma | Pelogic Comma |
| 16875/16384 | | -14 3 4 > | 51.12 | Laquadyo | Negri Comma | Double Augmentation Diesis | |
| 128/125 | | 7 0 -3 > | 41.06 | Trigu | Diesis | Augmented Comma | |
| 2109375/2097152 | | -21 3 7 > | 10.06 | Lasepyo | Semicomma | Fokker Comma | |
| 36/35 | | 2 2 -1 -1 > | 48.77 | Rugu | Septimal Quarter Tone | ||
| 525/512 | | -9 1 2 1 > | 43.41 | Lazoyoyo | Avicenna | Avicenna's Enharmonic Diesis | |
| 49/48 | | -4 -1 0 2 > | 35.70 | Zozo | Slendro Diesis | ||
| 686/675 | | 1 -3 -2 3 > | 27.99 | Trizo-agugu | Senga | ||
| 2430/2401 | | 1 5 1 -4 > | 20.79 | Quadru-ayo | Nuwell | ||
| 1728/1715 | | 6 3 -1 -3 > | 13.07 | Triru-agu | Orwellisma | Orwell Comma | |
| 225/224 | | -5 2 2 -1 > | 7.71 | Ruyoyo | Septimal Kleisma | Marvel Comma | |
| 6144/6125 | | 11 1 -3 -2 > | 5.36 | Sarurutrigu | Porwell | ||
| 65625/65536 | | -16 1 5 1 > | 2.35 | Lazoquinyo | Horwell | ||
| 99/98 | | -1 2 0 -2 1 > | 17.58 | Loruru | Mothwellsma | ||
| 121/120 | | -3 -1 -1 0 2 > | 14.37 | Lologu | Biyatisma | ||
| 176/175 | | 4 0 -2 -1 1 > | 9.86 | Lorugugu | Valinorsma | ||
| 385/384 | | -7 -1 1 1 1 > | 4.50 | Lozoyo | Keenanisma | ||
| 540/539 | | 2 3 1 -2 -1 > | 3.21 | Lururuyo | Swetisma | ||
| 91/90 | | -1 -2 -1 1 0 1 > | 19.13 | Thozogu | Superleap | ||
| 676/675 | | 2 -3 -2 0 0 2 > | 2.56 | Bithogu | Parizeksma |
Images
Compositions
- Tenacious Chorale (only movement I is in 9-EDO) by Stephen Weigel
- In Our Own Lonely Worlds by Stephen Weigel
- Nocturne in 9tet by Daniel Wolf
- Prelude in 9ET by Aaron Andrew Hunt
- Improvisation for Electric Guitar in 9 edo by Chris Vaisvil
- Comets Over Flatland 8 by Randy Winchester
- Nine tones per Octave (9-EDO / 9-TET) by Ivor Darreg
- Gerbils at the Wheel of Government by Chris Vaisvil (in 9 and 18 edo simultaneously)
- New World by Carlo Serafini (blog entry)
- Interdimensional Train Ride by Santiago Cosentino
Ear Training
9 EDO ear-training exercises by Alex Ness available here.
Instruments
Ukulele (MicroUke 1.2) set to 9 EDO with 40 lb. test fishing line (by cenobyte)