15edt: Difference between revisions

== Theory ==

15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1|5]] and [[13/1|13]] closely in tune, but does not do so well for [[11/1|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1|4th harmonic]], and terrible approximation of the [[2/1|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:5:7. It also tempers out the syntonic comma, [[81/80]], in the 3.4.5 subgroup, as the major third is three perfect fourths below a tritave. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping| 1 0 0 0 | 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.  

Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[k*N subgroups|2*15 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].

 

=== Harmonics ===

{{Harmonics in equal|15|3|1|prec=2|columns=12|start=12|collapsed=true|Approximation of harmonics in 15edt (continued)}}

== Intervals ==

{| class="wikitable center-1 center-2 center-3"

! #

! Cents

! Hekts

! Approximate ratios

! [[Polaris]] nonatonic notation

| 0

| 0.0

| 0.0

| [[1/1]]

| H

| 1

| 126.8

| 86.7

| [[14/13]], [[15/14]], [[16/15]], 29/27

| Ib

| 2

| 253.6

| 173.3

| [[15/13]]

| vH#, ^Ib

| 3

| 380.4

| 260.0

| [[5/4]]

| H#

| 4

| 507.2

| 346.7

| [[4/3]]

| I

| 5

| 634.0

| 433.3

| [[13/9]]

| J

| 6

| 760.8

| 520.0

| [[14/9]]

| K

| 7

| 887.6

| 606.7

| [[5/3]]

| L

| 8

| 1014.4

| 793.3

| [[9/5]]

| Mb

| 9

| 1141.2

| 780.0

| [[27/14]]

| vL#, ^Mb

| 10

| 1268.0

| 866.7

| [[27/13]]

| L#

| 11

| 1394.8

| 953.3

| [[9/4]]

| M

| 12

| 1521.6

| 1040.0

| [[12/5]]

| N

| 13

| 1648.4

| 1126.7

| [[13/5]]

| O

| 14

| 1775.2

| 1213.3

| [[14/5]]

| 15

| 1902.0

| 1300.0

| [[3/1]]

15edt contains 4 intervals from [[5edt]] and 2 intervals from [[3edt]], meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16…

 

15edt also contains a [[5L 5s (3/1-equivalent)|5L 5s]] mos similar to Blackwood Decatonic, which I{{who}} call Ebony. This mos has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.

 

15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L 3s mos "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.

Below is a plot of the [[The Riemann zeta function and tuning #Removing primes|no-twos Z function]] in the vicinity of 15edt:

[[File:15edt.png|alt=15edt.png|15edt.png]]

== Music ==

; [[nationalsolipsism]]

* [https://www.youtube.com/watch?v=bC_Pc4jKm2k ''ox-idation''] (2012)