10edo: Difference between revisions
m cat sorting, links |
added M2, m2 and A1 to the template, moved the primes-error table up to the top |
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| Prime factorization = 2 * 5 | | Prime factorization = 2 * 5 | ||
| Subgroup = 2.7.13.15 | | Subgroup = 2.7.13.15 | ||
| Step size = | | Step size = 120¢ | ||
| Fifth type = [[5edo]] | | Fifth type = 6\10 = 720¢ = [[5edo]] | ||
| Major 2nd = 2\10 = 240¢ | |||
| Minor 2nd = 0\10 = 0¢ | |||
| Augmented 1sn = 2\10 = 240¢ | |||
| Common uses = chromatic [[5edo]], neutral thirds MOS | | Common uses = chromatic [[5edo]], neutral thirds MOS | ||
| Important MOS = neutral thirds ([[beatles]]) 3L4s 2121211 (3\10, 1\1) | | Important MOS = neutral thirds ([[beatles]]) 3L4s 2121211 (3\10, 1\1) | ||
| Line 18: | Line 21: | ||
== Theory == | == Theory == | ||
{| class="wikitable center-all" | |||
! colspan="2" | | |||
! prime 2 | |||
! prime 3 | |||
! prime 5 | |||
! prime 7 | |||
! prime 11 | |||
! prime 13 | |||
! prime 17 | |||
|- | |||
! rowspan="2" |Error | |||
! absolute (¢) | |||
| 0.0 | |||
| +18.0 | |||
| -26.3 | |||
| -8.8 | |||
| +48.7 | |||
| -0.5 | |||
| +12.5 | |||
|- | |||
![[Relative error|relative]] (%) | |||
| 0 | |||
| +15 | |||
| -22 | |||
| -7 | |||
| +41 | |||
| -0.4 | |||
| +15 | |||
|- | |||
! colspan="2" |[[nearest edomapping]] | |||
|10 | |||
|6 | |||
|3 | |||
|8 | |||
|5 | |||
|7 | |||
|1 | |||
|} | |||
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 [[MOS scales|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. | 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 [[MOS scales|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. | ||
| Line 157: | Line 198: | ||
== Just approximation == | == Just approximation == | ||
=== Selected just intervals by error === | === Selected just intervals by error === | ||
==== Selected 13-limit intervals ==== | ==== Selected 13-limit intervals ==== | ||
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||