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=Properties=
'''16EDT''' is [[EDT|equal division of the third harmonic]] into 16 parts of 118.872 cents each (corresponding to 10.0949 [[EDO]]).
As the double of [[8edt|8edt]], this division of the tritave is harmonically fraternal to [[10edo|10edo]]. Its unit step is ~1.128 cents flat of 1\10edo. Unlike 10edo, it does not really have a 7 or 13 because it is not using its approximation of 2 as equivalent though the accumulated flatness of a stack of its unit step leads to an excellent [[21/13|13:21]] and a decent [[13/7|7:13]]. When twos are admitted, it turns into a tritave-repeating version of Blackwood temperament.
[[category:macrotonal]]


=Intervals=
== Properties ==
As the double of [[8edt|8EDT]], this division of the tritave is harmonically fraternal to [[10edo|10EDO]]. Its unit step is ~1.128 cents flat of 1\10EDO. Unlike 10EDO, it does not really have a 7 or 13 because it is not using its approximation of 2 as equivalent though the accumulated flatness of a stack of its unit step leads to an excellent [[21/13|13:21]] and a decent [[13/7|7:13]]. When twos are admitted, it turns into a tritave-repeating version of Blackwood temperament.
 
== Intervals ==


{| class="wikitable"
{| class="wikitable"
Line 11: Line 12:
|-
|-
![[cent|Cents]]
![[cent|Cents]]
!Hekts
! Hekts
|-
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 1
| style="text-align:right;" | 118.87219
| style="text-align:right;" | 118.87219
|81.25
| 81.25
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 2
| style="text-align:right;" | 237.74438
| style="text-align:right;" | 237.74438
|162.5
| 162.5
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 3
| style="text-align:right;" | 356.61656
| style="text-align:right;" | 356.61656
|243.75
| 243.75
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 4
| style="text-align:right;" | 475.48875
| style="text-align:right;" | 475.48875
|325
| 325
|-
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 5
| style="text-align:right;" | 594.36094
| style="text-align:right;" | 594.36094
|406.25
| 406.25
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 6
| style="text-align:right;" | 713.23312
| style="text-align:right;" | 713.23312
|487.5
| 487.5
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 7
| style="text-align:right;" | 832.10531
| style="text-align:right;" | 832.10531
|568.75
| 568.75
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 8
| style="text-align:right;" | 950.9775
| style="text-align:right;" | 950.9775
|650
| 650
|-
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 9
| style="text-align:right;" | 1069.84969
| style="text-align:right;" | 1069.84969
|731.25
| 731.25
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 10
| style="text-align:right;" | 1188.72188
| style="text-align:right;" | 1188.72188
|812.5
| 812.5
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 11
| style="text-align:right;" | 1307.59406
| style="text-align:right;" | 1307.59406
|893.75
| 893.75
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 12
| style="text-align:right;" | 1426.46625
| style="text-align:right;" | 1426.46625
|975
| 975
|-
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 13
| style="text-align:right;" | 1545.33844
| style="text-align:right;" | 1545.33844
|1056.25
| 1056.25
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 14
| style="text-align:right;" | 1664.21063
| style="text-align:right;" | 1664.21063
|1137.5
| 1137.5
|-
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 15
| style="text-align:right;" | 1783.08281
| style="text-align:right;" | 1783.08281
|1218.75
| 1218.75
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 16
| style="text-align:right;" | 1901.955
| style="text-align:right;" | 1901.955
|1300
| 1300
|}
|}


=Music=
== Related temperament ==
16EDT is also be thought of as a generator of the ''subsedia'' temperament, which is a [[cluster temperament]] with 10 clusters of notes in an octave.
 
; [[Mirkwai clan #Subsedia|Subsedia]] (10 & 111)
'''7-limit'''<br>
Comma list: 16875/16807, 65536/64827<br>
Mapping: [{{val|1 0 5 4}}, {{val|0 16 -27 -12}}]<br>
POTE generator: ~15/14 = 118.965<br>
Vals: 10, 101, 111, 121, 232d<br>
Badness: 0.157658<br><br>
'''11-limit'''<br>
Comma list: 540/539, 1375/1372, 65536/64827<br>
Mapping: [{{val|1 0 5 4 -1}}, {{val|0 16 -27 -12 45}}]<br>
POTE generator: ~15/14 = 118.968<br>
Vals: 10, 101, 111, 121, 232d<br>
Badness: 0.066838<br><br>
'''13-limit'''<br>
Comma list: 352/351, 540/539, 676/675, 1375/1372<br>
Mapping: [{{val|1 0 5 4 -1 4}}, {{val|0 16 -27 -12 45 -3}}]<br>
POTE generator: ~15/14 = 118.968<br>
Vals: 10, 101, 111, 121, 232d<br>
Badness: 0.031635<br><br>
'''17-limit'''<br>
Comma list: 256/255, 352/351, 442/441, 540/539, 715/714<br>
Mapping: [{{val|1 0 5 4 -1 4 3}}, {{val|0 16 -27 -12 45 -3 11}}]<br>
POTE generator: ~15/14 = 118.968<br>
Vals: 10, 101, 111, 121, 232dg<br>
Badness: 0.019707<br><br>
 
== Music ==
[http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/16-edt.mp3 A Short Tune in 16EDT] by [[Peter_'Rush'_Kosmorsky|Peter 'Rush' Kosmorsky]]
[http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/16-edt.mp3 A Short Tune in 16EDT] by [[Peter_'Rush'_Kosmorsky|Peter 'Rush' Kosmorsky]]
[[Category:Edt]]
[[Category:Macrotonal]]

Revision as of 12:14, 11 September 2021

16EDT is equal division of the third harmonic into 16 parts of 118.872 cents each (corresponding to 10.0949 EDO).

Properties

As the double of 8EDT, this division of the tritave is harmonically fraternal to 10EDO. Its unit step is ~1.128 cents flat of 1\10EDO. Unlike 10EDO, it does not really have a 7 or 13 because it is not using its approximation of 2 as equivalent though the accumulated flatness of a stack of its unit step leads to an excellent 13:21 and a decent 7:13. When twos are admitted, it turns into a tritave-repeating version of Blackwood temperament.

Intervals

Degree Size in
Cents Hekts
1 118.87219 81.25
2 237.74438 162.5
3 356.61656 243.75
4 475.48875 325
5 594.36094 406.25
6 713.23312 487.5
7 832.10531 568.75
8 950.9775 650
9 1069.84969 731.25
10 1188.72188 812.5
11 1307.59406 893.75
12 1426.46625 975
13 1545.33844 1056.25
14 1664.21063 1137.5
15 1783.08281 1218.75
16 1901.955 1300

Related temperament

16EDT is also be thought of as a generator of the subsedia temperament, which is a cluster temperament with 10 clusters of notes in an octave.

Subsedia (10 & 111)

7-limit
Comma list: 16875/16807, 65536/64827
Mapping: [⟨1 0 5 4], ⟨0 16 -27 -12]]
POTE generator: ~15/14 = 118.965
Vals: 10, 101, 111, 121, 232d
Badness: 0.157658

11-limit
Comma list: 540/539, 1375/1372, 65536/64827
Mapping: [⟨1 0 5 4 -1], ⟨0 16 -27 -12 45]]
POTE generator: ~15/14 = 118.968
Vals: 10, 101, 111, 121, 232d
Badness: 0.066838

13-limit
Comma list: 352/351, 540/539, 676/675, 1375/1372
Mapping: [⟨1 0 5 4 -1 4], ⟨0 16 -27 -12 45 -3]]
POTE generator: ~15/14 = 118.968
Vals: 10, 101, 111, 121, 232d
Badness: 0.031635

17-limit
Comma list: 256/255, 352/351, 442/441, 540/539, 715/714
Mapping: [⟨1 0 5 4 -1 4 3], ⟨0 16 -27 -12 45 -3 11]]
POTE generator: ~15/14 = 118.968
Vals: 10, 101, 111, 121, 232dg
Badness: 0.019707

Music

A Short Tune in 16EDT by Peter 'Rush' Kosmorsky

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