10edo: Difference between revisions

| Major 2nd = 2\10 = 240¢

| Minor 2nd = 0\10 = 0¢

| Augmented 1sn = 2\10 = 240¢

'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==

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.

===Differences between distributionally-even scales and smaller edos===

!L-Nedo

|3

|80¢

|4

|60¢

|40¢

|7

|68.571¢

|8

|90¢

|30¢

|9

|106.667¢

The following table shows [[TE temperament measures]] (RMS normalized) of 10et.  

! colspan="2" |Octave stretch (¢)

* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively.  

* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}.  

| [[Linus]]

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

| A Decaphonic (10-EDO) Classical Guitar

* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]

* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]