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__FORCETOC__
'''15EDT''' is the [[Edt|equal division of the third harmonic]] into 15 parts of 126.7970 [[cent|cents]] each, corresponding to 9.4639 [[edo]].
=<span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span>=


=Properties=
Lookalikes: [[19ed4]]
The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo, or 18.928 ed4. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15&gt; in the 5-limit, which is tempered out by [[19edo|19edo]] but has an [[Optimal_patent_val|optimal patent val]] of [[303edo|303edo]]. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&amp;123, which has[[category:macrotonal]] a mapping [&lt;1 0 0 0|, &lt;0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15edt.


=Intervals of 15edt=
==Properties==
15EDT has harmonics 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15&gt; in the 5-limit, which is tempered out by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. 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). 15EDT is related to the 2.3.5.13 subgroup temperament 19&amp;123, which has[[category:macrotonal]] a mapping [&lt;1 0 0 0|, &lt;0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15EDT.
 
With the patent 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.
 
==Intervals==


{| class="wikitable"
{| class="wikitable"
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| | 0
| | 0
| | 0
| | 0
| | <span style="color: #660000;">[[1/1|1/1]]</span>
| | <span style="color: #660000;">[[1/1]]</span>
|-
|-
| | 1
| | 1
| | 126.797
| | 126.797
| | [[14/13|14/13]], [[15/14|15/14]], [[16/15|16/15]], 29/27
| | [[14/13]], [[15/14]], [[16/15]], 29/27
|-
|-
| | 2
| | 2
| | 253.594
| | 253.594
| | [[15/13|15/13]]
| | [[15/13]]
|-
|-
| | 3
| | 3
| | 380.391
| | 380.391
| | <span style="color: #660000;">[[5/4|5/4]]</span>
| | <span style="color: #660000;">[[5/4]]</span>
|-
|-
| | 4
| | 4
| | 507.188
| | 507.188
| | [[4/3|4/3]]
| | [[4/3]]
|-
|-
| | 5
| | 5
| | 633.985
| | 633.985
| | [[13/9|13/9]]
| | [[13/9]]
|-
|-
| | 6
| | 6
| | 760.782
| | 760.782
| | <span style="color: #660000;">[[14/9|14/9]]</span>
| | <span style="color: #660000;">[[14/9]]</span>
|-
|-
| | 7
| | 7
| | 887.579
| | 887.579
| | [[5/3|5/3]]
| | [[5/3]]
|-
|-
| | 8
| | 8
| | 1014.376
| | 1014.376
| | [[9/5|9/5]]
| | [[9/5]]
|-
|-
| | 9
| | 9
| | 1141.173
| | 1141.173
| | <span style="color: #660000;">[[27/14|27/14]]</span>
| | <span style="color: #660000;">[[27/14]]</span>
|-
|-
| | 10
| | 10
| | 1267.970
| | 1267.970
| | [[27/13|27/13]]
| | [[27/26|27/13]]
|-
|-
| | 11
| | 11
| | 1394.767
| | 1394.767
| | [[9/4|9/4]] ([[9/8|9/8]] plus an octave)
| | [[9/4]] ([[9/8]] plus an octave)
|-
|-
| | 12
| | 12
| | 1521.564
| | 1521.564
| | [[12/5|12/5]] (<span style="color: #660000;">[[6/5|6/5]]</span> plus an octave)
| | [[12/5]] (<span style="color: #660000;">[[6/5]]</span> plus an octave)
|-
|-
| | 13
| | 13
| | 1648.361
| | 1648.361
| | [[13/5|13/5]] ([[13/10|13/10]] plus an octave)
| | [[13/5]] ([[13/10]] plus an octave)
|-
|-
| | 14
| | 14
| | 1775.158
| | 1775.158
| | [[14/5|14/5]] ([[7/5|7/5]] plus an octave)
| | [[14/5]] ([[7/5]] plus an octave)
|-
|-
| | 15
| | 15
| | 1901.955
| | 1901.955
| | [[3/1|3/1]]
| | [[3/1]]
|}
|}


15edt contains 4 intervals from [[5edt|5edt]] and 2 intervals from [[3edt|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 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 5L5s MOS similar to Blackwood Decatonic, which I 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 also contains a 5L5s MOS similar to Blackwood Decatonic, which I 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.
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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.
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.


=Z function=
==Z function==
Below is a plot of the [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos Z function]] in the vicinity of 15edt:
Below is a plot of the [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos Z function]] in the vicinity of 15edt:


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[http://www.youtube.com/watch?v=bC_Pc4jKm2k http://www.youtube.com/watch?v=bC_Pc4jKm2k]
[http://www.youtube.com/watch?v=bC_Pc4jKm2k http://www.youtube.com/watch?v=bC_Pc4jKm2k]
[[Category:Edt]]
[[Category:Edonoi]]
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