10edo: Difference between revisions
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| ja = 10平均律 | | ja = 10平均律 | ||
}}'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[ | }} | ||
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]]. | |||
== Theory == | == Theory == | ||
10edo can be thought of as two circles of [[ | 10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup. | ||
== Intervals == | == Intervals == | ||
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! Additional Ratios <br> of 3, 5 and 9<ref>adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals</ref> | ! Additional Ratios <br> of 3, 5 and 9<ref>adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals</ref> | ||
! Interval Names | ! Interval Names | ||
! colspan="3" | [[Ups and Downs Notation | ! colspan="3" | [[Ups and Downs Notation]] | ||
|- | |- | ||
| 0 | | 0 | ||
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| 1 | | 1 | ||
| 1\10 | | 1\10 | ||
| Messed-up [[ | | Messed-up [[negri]] (or [[miracle]]) | ||
|- | |- | ||
| 1 | | 1 | ||
| 3\10 | | 3\10 | ||
| [[ | | [[Dicot]]/[[beatles]]/neutral thirds scale | ||
|- | |- | ||
| 2 | | 2 | ||
| 1\10 | | 1\10 | ||
| Messed-up [[ | | Messed-up [[pajara]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 2\10 | | 2\10 | ||
| [[ | | [[Decimal]] / messed-up [[lemba]] | ||
|- | |- | ||
| 5 | | 5 | ||
| 1\10 | | 1\10 | ||
| [[ | | [[Blackwood]]/[[blacksmith]] | ||
|} | |} | ||
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]] | [[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]] | ||
| Line 216: | Line 217: | ||
|- | |- | ||
!Prime<br>Limit | !Prime<br>Limit | ||
! [[ | ! [[Ratio]] | ||
! [[Monzo]] | ! [[Monzo]] | ||
! [[Cent|Cents]] | ! [[Cent|Cents]] | ||
| Line 222: | Line 223: | ||
! Name(s) | ! Name(s) | ||
|- | |- | ||
|3 | | 3 | ||
| 256/243 | | 256/243 | ||
| <nowiki> | 8 -5 </nowiki>> | | <nowiki> | 8 -5 </nowiki>> | ||
| Line 229: | Line 230: | ||
| Limma, Pythagorean minor 2nd | | Limma, Pythagorean minor 2nd | ||
|- | |- | ||
|5 | | 5 | ||
| 25/24 | | 25/24 | ||
| <nowiki> | -3 -1 2 </nowiki>> | | <nowiki> | -3 -1 2 </nowiki>> | ||
| Line 236: | Line 237: | ||
| 5-limit large semitone, 5-limit chromatic semitone | | 5-limit large semitone, 5-limit chromatic semitone | ||
|- | |- | ||
|" | | " | ||
| 16875/16384 | | 16875/16384 | ||
| <nowiki> | -14 3 4 </nowiki>> | | <nowiki> | -14 3 4 </nowiki>> | ||
| Line 243: | Line 244: | ||
| Negri comma, double augmentation diesis | | Negri comma, double augmentation diesis | ||
|- | |- | ||
|" | | " | ||
| 9931568/9752117 | | 9931568/9752117 | ||
| <nowiki> | -25 7 6 </nowiki>> | | <nowiki> | -25 7 6 </nowiki>> | ||
| Line 250: | Line 251: | ||
| Ampersand's comma | | Ampersand's comma | ||
|- | |- | ||
|" | | " | ||
| 2048/2025 | | 2048/2025 | ||
| <nowiki> | 11 -4 -2 </nowiki>> | | <nowiki> | 11 -4 -2 </nowiki>> | ||
| Line 257: | Line 258: | ||
| Diaschisma | | Diaschisma | ||
|- | |- | ||
|7 | | 7 | ||
| 525/512 | | 525/512 | ||
| <nowiki> | -9 1 2 1 </nowiki>> | | <nowiki> | -9 1 2 1 </nowiki>> | ||
| Line 264: | Line 265: | ||
| Avicennma, Avicenna's enharmonic diesis | | Avicennma, Avicenna's enharmonic diesis | ||
|- | |- | ||
|" | | " | ||
| 49/48 | | 49/48 | ||
| <nowiki> | -4 -1 0 2 </nowiki>> | | <nowiki> | -4 -1 0 2 </nowiki>> | ||
| Line 271: | Line 272: | ||
| Slendro diesis | | Slendro diesis | ||
|- | |- | ||
|" | | " | ||
| 50/49 | | 50/49 | ||
| <nowiki> | 1 0 2 -2 </nowiki>> | | <nowiki> | 1 0 2 -2 </nowiki>> | ||
| Line 278: | Line 279: | ||
| Tritonic diesis, jubilisma | | Tritonic diesis, jubilisma | ||
|- | |- | ||
|" | | " | ||
| 686/675 | | 686/675 | ||
| <nowiki> | 1 -3 -2 3 </nowiki>> | | <nowiki> | 1 -3 -2 3 </nowiki>> | ||
| Line 285: | Line 286: | ||
| Senga | | Senga | ||
|- | |- | ||
|" | | " | ||
| 64/63 | | 64/63 | ||
| <nowiki> | 6 -2 0 -1 </nowiki>> | | <nowiki> | 6 -2 0 -1 </nowiki>> | ||
| Line 292: | Line 293: | ||
| Septimal comma, Archytas' comma, Leipziger Komma | | Septimal comma, Archytas' comma, Leipziger Komma | ||
|- | |- | ||
|" | | " | ||
| 9859966/9733137 | | 9859966/9733137 | ||
| <nowiki> | -10 7 8 -7 </nowiki>> | | <nowiki> | -10 7 8 -7 </nowiki>> | ||
| Line 299: | Line 300: | ||
| Blackjackisma | | Blackjackisma | ||
|- | |- | ||
|" | | " | ||
| 1029/1024 | | 1029/1024 | ||
| <nowiki> | -10 1 0 3 </nowiki>> | | <nowiki> | -10 1 0 3 </nowiki>> | ||
| Line 306: | Line 307: | ||
| Gamelisma | | Gamelisma | ||
|- | |- | ||
|" | | " | ||
| 225/224 | | 225/224 | ||
| <nowiki> | -5 2 2 -1 </nowiki>> | | <nowiki> | -5 2 2 -1 </nowiki>> | ||
| Line 313: | Line 314: | ||
| Septimal kleisma, marvel comma | | Septimal kleisma, marvel comma | ||
|- | |- | ||
|" | | " | ||
| 16875/16807 | | 16875/16807 | ||
| <nowiki> | 0 3 4 -5 </nowiki>> | | <nowiki> | 0 3 4 -5 </nowiki>> | ||
| Line 320: | Line 321: | ||
| Mirkwai | | Mirkwai | ||
|- | |- | ||
|" | | " | ||
| 6772805/6751042 | | 6772805/6751042 | ||
| <nowiki> | 11 -10 -10 10 </nowiki>> | | <nowiki> | 11 -10 -10 10 </nowiki>> | ||
| Line 327: | Line 328: | ||
| Linus | | Linus | ||
|- | |- | ||
|" | | " | ||
| 2401/2400 | | 2401/2400 | ||
| <nowiki> | -5 -1 -2 4 </nowiki>> | | <nowiki> | -5 -1 -2 4 </nowiki>> | ||
| Line 334: | Line 335: | ||
| Breedsma | | Breedsma | ||
|- | |- | ||
|11 | | 11 | ||
| 243/242 | | 243/242 | ||
| <nowiki> | -1 5 0 0 -2 </nowiki>> | | <nowiki> | -1 5 0 0 -2 </nowiki>> | ||
| Line 341: | Line 342: | ||
| Rastma | | Rastma | ||
|- | |- | ||
|" | | " | ||
| 385/384 | | 385/384 | ||
| <nowiki> | -7 -1 1 1 1 </nowiki>> | | <nowiki> | -7 -1 1 1 1 </nowiki>> | ||
| Line 348: | Line 349: | ||
| Keenanisma | | Keenanisma | ||
|- | |- | ||
|" | | " | ||
| 441/440 | | 441/440 | ||
| <nowiki> | -3 2 -1 2 -1 </nowiki>> | | <nowiki> | -3 2 -1 2 -1 </nowiki>> | ||
| Line 355: | Line 356: | ||
| Werckisma | | Werckisma | ||
|- | |- | ||
|" | | " | ||
| 540/539 | | 540/539 | ||
| <nowiki> | 2 3 1 -2 -1 </nowiki>> | | <nowiki> | 2 3 1 -2 -1 </nowiki>> | ||
| Line 362: | Line 363: | ||
| Swetisma | | Swetisma | ||
|- | |- | ||
|" | | " | ||
| 3025/3024 | | 3025/3024 | ||
| <nowiki> | -4 -3 2 -1 2 </nowiki>> | | <nowiki> | -4 -3 2 -1 2 </nowiki>> | ||
| Line 369: | Line 370: | ||
| Lehmerisma | | Lehmerisma | ||
|- | |- | ||
|13 | | 13 | ||
| 91/90 | | 91/90 | ||
| <nowiki> | -1 -2 -1 1 0 1 </nowiki>> | | <nowiki> | -1 -2 -1 1 0 1 </nowiki>> | ||
| Line 376: | Line 377: | ||
| Superleap | | Superleap | ||
|- | |- | ||
|" | | " | ||
| 676/675 | | 676/675 | ||
| <nowiki> | 2 -3 -2 0 0 2 </nowiki>> | | <nowiki> | 2 -3 -2 0 0 2 </nowiki>> | ||
| Line 394: | Line 395: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]] | |||
|- | |- | ||
| A Decaphonic (10-EDO) Classical Guitar | |||
|} | |} | ||
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]] | [[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]] | ||