18edt: Difference between revisions

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18edt means the division of the tritave into 18 equal parts <span style="background-color: rgba(255,255,255,0);">of size 105.664 cents each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a sixth, it would count as a neutral sixth. The corresponding 5/3 is 845 cents, which is a neutral sixth between 8/5 and 5/3, which is really more of a 13/8, though this is allegedly a no-twos tuning. With octaves added, it also has a minor third and a major tenth which are both excellent as well as a minor thirteenth and major seventeenth which are still decent even though it skips actual octaves (in fact it is the non-octave semitone scale of 34edo). On the 3.7.13 subgroup it tempers out 351/343 and 2197/2187.</span>[[category:macrotonal]]
{{Infobox ET}}
'''18edt''' is the division of the tritave into 18 equal parts of size 105.664 cents each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a sixth, it would count as a neutral sixth. The corresponding 5/3 is 845 cents, which is a neutral sixth between 8/5 and 5/3, which is really more of a 13/8, though this is allegedly a no-twos tuning. With octaves added, it also has a minor third and a major tenth which are both excellent as well as a minor thirteenth and major seventeenth which are still decent even though it skips actual octaves (in fact it is the non-octave semitone scale of 34edo). On the 3.7.13 subgroup it tempers out 351/343 and 2197/2187.


<span style="background-color: rgba(255,255,255,0);">As the double of [[9edt|9edt]], it is the analog of 14edo insofar as it has a doubled harmonic chain. However, it, like [[8edt|8edt]], is not schismatic because 3:5:7 is a redundant chord. As a multiple of 9edt, it is the widest variety of 'White-Extraterrestrial-Tree' temperament.</span>
== Temperaments ==
As the double of [[9edt]], it is the analog of 14edo insofar if treating as it has a doubled harmonic chain. However, it, like [[8edt]], is not schismatic because 3:5:7 is a redundant chord. As a multiple of 9edt, it is the widest variety of 'White-Extraterrestrial-Tree' temperament.


<span style="background-color: rgba(255,255,255,0);">0: 1/1</span>
18edt can also be used for the [[Electra]] temperament based on [[15/11]], although in this case its approximation to [[13/11]] is very sharp.
== Intervals ==
{| class="wikitable"
! Step
! Cents
! Hekts
! Approximated interval
! [[Electra]] notation (J = 1/1)
|-
! colspan="3" | 0
| 1/1
| J
|-
| 1
| 105.664
|72.222
| 16/15
| J#, Kbb
|-
| 2
| 211.328
|144.444
| 9/8
| Jx, Kb
|-
| 3
| 316.993
|216.667
| 6/5
| K
|-
| 4
| 422.657
|288.889
| 9/7
| K#, Lb
|-
| 5
| 528.321
|361.111
| 27/20
| L
|-
| 6
| 633.985
|433.333
| 13/9
| L#, Mbb
|-
| 7
| 739.649
|505.556
| 17/13
| Lx, Mb
|-
| 8
| 845.313
|577.778
| 5/3
| M
|-
| 9
| 950.978
|650
| 19/11
| M#, Nbb
|-
| 10
| 1056.642
|722.222
| 9/5
| Mx, Nb
|-
| 11
| 1162.306
|794.444
| 49/25
| N
|-
| 12
| 1267.97
|866.667
| 27/13
| N#, Ob
|-
| 13
| 1373.634
|938.889
| 20/9
| O
|-
| 14
| 1479.298
|1011.111
| 7/3
| O#, Pbb
|-
| 15
| 1584.963
|1083.333
| 5/2
| Ox, Pb
|-
| 16
| 1690.627
|1155.556
| 8/3
| P
|-
| 17
| 1806.291
|1227.778
| 45/16
| P#, Jb
|-
| 18
| 1901.955
|1300
| 3/1
| J
|}


<span style="background-color: rgba(255,255,255,0);">1: 105.664 cents 16/15</span>
== Harmonics ==
{{Harmonics in equal
| steps = 18
| num = 3
| denom = 1
| intervals = integer
}}
{{Harmonics in equal
| steps = 18
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}


<span style="background-color: rgba(255,255,255,0);">2: 211.328 cents 9/8</span>
[[category:macrotonal]]
 
[[category:tritave]]
3: 316.993 cents 6/5
 
<span style="background-color: rgba(255,255,255,0);">4: 422.657 cents 9/7</span>
 
5: 528.321 cents 27/20
 
<span style="background-color: rgba(255,255,255,0);">6: 633.985 cents 13/9</span>
 
7: 739.649 cents 17/13
 
<span style="background-color: rgba(255,255,255,0);">8: 845.313 cents 5/3</span>
 
9: 950.9775 cents 19/11
 
<span style="background-color: rgba(255,255,255,0);">10: 1056.642 cents 9/5</span>
 
11: 1162.306 cents 49/25
 
<span style="background-color: rgba(255,255,255,0);">12: 1267.970 cents 27/13</span>
 
13: 1373.634 cents 20/9
 
<span style="background-color: rgba(255,255,255,0);">14: 1479.298 cents 7/3</span>
 
15: 1584.963 cents 5/2
 
<span style="background-color: rgba(255,255,255,0);">16: 1690.627 cents 8/3</span>
 
17: 1806.2905 cents 45/16
 
<span style="background-color: rgba(255,255,255,0);">18: 3/1</span>

Latest revision as of 15:31, 31 July 2025

← 17edt 18edt 19edt →
Prime factorization 2 × 32
Step size 105.664 ¢ 
Octave 11\18edt (1162.31 ¢)
Consistency limit 3
Distinct consistency limit 3

18edt is the division of the tritave into 18 equal parts of size 105.664 cents each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a sixth, it would count as a neutral sixth. The corresponding 5/3 is 845 cents, which is a neutral sixth between 8/5 and 5/3, which is really more of a 13/8, though this is allegedly a no-twos tuning. With octaves added, it also has a minor third and a major tenth which are both excellent as well as a minor thirteenth and major seventeenth which are still decent even though it skips actual octaves (in fact it is the non-octave semitone scale of 34edo). On the 3.7.13 subgroup it tempers out 351/343 and 2197/2187.

Temperaments

As the double of 9edt, it is the analog of 14edo insofar if treating as it has a doubled harmonic chain. However, it, like 8edt, is not schismatic because 3:5:7 is a redundant chord. As a multiple of 9edt, it is the widest variety of 'White-Extraterrestrial-Tree' temperament.

18edt can also be used for the Electra temperament based on 15/11, although in this case its approximation to 13/11 is very sharp.

Intervals

Step Cents Hekts Approximated interval Electra notation (J = 1/1)
0 1/1 J
1 105.664 72.222 16/15 J#, Kbb
2 211.328 144.444 9/8 Jx, Kb
3 316.993 216.667 6/5 K
4 422.657 288.889 9/7 K#, Lb
5 528.321 361.111 27/20 L
6 633.985 433.333 13/9 L#, Mbb
7 739.649 505.556 17/13 Lx, Mb
8 845.313 577.778 5/3 M
9 950.978 650 19/11 M#, Nbb
10 1056.642 722.222 9/5 Mx, Nb
11 1162.306 794.444 49/25 N
12 1267.97 866.667 27/13 N#, Ob
13 1373.634 938.889 20/9 O
14 1479.298 1011.111 7/3 O#, Pbb
15 1584.963 1083.333 5/2 Ox, Pb
16 1690.627 1155.556 8/3 P
17 1806.291 1227.778 45/16 P#, Jb
18 1901.955 1300 3/1 J

Harmonics

Approximation of harmonics in 18edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -37.7 +0.0 +30.3 -39.0 -37.7 +12.4 -7.4 +0.0 +28.9 -30.4 +30.3
Relative (%) -35.7 +0.0 +28.7 -37.0 -35.7 +11.8 -7.0 +0.0 +27.4 -28.8 +28.7
Steps
(reduced) 		11
(11) 		18
(0) 		23
(5) 		26
(8) 		29
(11) 		32
(14) 		34
(16) 		36
(0) 		38
(2) 		39
(3) 		41
(5)
Approximation of harmonics in 18edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -2.6 -25.3 -39.0 -45.1 -44.4 -37.7 -25.6 -8.8 +12.4 +37.6 -39.4
Relative (%) -2.5 -23.9 -37.0 -42.7 -42.0 -35.7 -24.3 -8.3 +11.8 +35.5 -37.3
Steps
(reduced) 		42
(6) 		43
(7) 		44
(8) 		45
(9) 		46
(10) 		47
(11) 		48
(12) 		49
(13) 		50
(14) 		51
(15) 		51
(15)
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