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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | '''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. |
| : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-09-12 00:42:04 UTC</tt>.<br>
| |
| : The original revision id was <tt>591641538</tt>.<br>
| |
| : The revision comment was: <tt></tt><br>
| |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| |
| <h4>Original Wikitext content:</h4>
| |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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>
| |
|
| |
|
| <span style="background-color: rgba(255,255,255,0);">As the double of [[@9edt]], it is the analog of 14edo insofar 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>
| | == 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 |
| | }} |
| | Note that this tuning is fairly close to [[1ed17/16]], although by [[patent val]] [[mapping]] 18edt maps ~[[17/16]] inconsistently to 2\18edt instead of 1\18edt, while 1ed17/16 maps its own [[equave]] even more inconsistently, to 3\1ed17/16. |
|
| |
|
| <span style="background-color: rgba(255,255,255,0);">0: 1/1</span>
| | == Intervals == |
| <span style="background-color: rgba(255,255,255,0);">1: 105.664 cents 16/15</span>
| | {| class="wikitable" |
| <span style="background-color: rgba(255,255,255,0);">2: 211.328 cents 9/8</span>
| | ! Step |
| 3: 316.993 cents 6/5 | | ! Cents |
| <span style="background-color: rgba(255,255,255,0);">4: 422.657 cents 9/7</span>
| | ! Hekts |
| 5: 528.321 cents 27/20 | | ! Approximated interval |
| <span style="background-color: rgba(255,255,255,0);">6: 633.985 cents 13/9</span>
| | ! [[Electra]] notation (J = 1/1) |
| 7: 739.649 cents 17/13 | | |- |
| <span style="background-color: rgba(255,255,255,0);">8: 845.313 cents 5/3</span>
| | ! colspan="3" | 0 |
| 9: 950.9775 cents 19/11 | | | 1/1 |
| <span style="background-color: rgba(255,255,255,0);">10: 1056.642 cents 9/5</span>
| | | J |
| 11: 1162.306 cents 49/25 | | |- |
| <span style="background-color: rgba(255,255,255,0);">12: 1267.970 cents 27/13</span>
| | | 1 |
| 13: 1373.634 cents 20/9 | | | 105.664 |
| <span style="background-color: rgba(255,255,255,0);">14: 1479.298 cents 7/3</span>
| | |72.222 |
| 15: 1584.963 cents 5/2 | | | 16/15 |
| <span style="background-color: rgba(255,255,255,0);">16: 1690.627 cents 8/3</span>
| | | J#, Kbb |
| 17: 1806.2905 cents 45/16 | | |- |
| <span style="background-color: rgba(255,255,255,0);">18: 3/1</span></pre></div>
| | | 2 |
| <h4>Original HTML content:</h4>
| | | 211.328 |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>18edt</title></head><body>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><br />
| | |144.444 |
| <br />
| | | 9/8 |
| <span style="background-color: rgba(255,255,255,0);">As the double of <a class="wiki_link" href="/9edt" target="_blank">9edt</a>, it is the analog of 14edo insofar as it has a doubled harmonic chain. However, it, like <a class="wiki_link" href="/8edt" target="_blank">8edt</a>, 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><br />
| | | Jx, Kb |
| <br />
| | |- |
| <span style="background-color: rgba(255,255,255,0);">0: 1/1</span><br />
| | | 3 |
| <span style="background-color: rgba(255,255,255,0);">1: 105.664 cents 16/15</span><br />
| | | 316.993 |
| <span style="background-color: rgba(255,255,255,0);">2: 211.328 cents 9/8</span><br />
| | |216.667 |
| 3: 316.993 cents 6/5<br />
| | | 6/5 |
| <span style="background-color: rgba(255,255,255,0);">4: 422.657 cents 9/7</span><br />
| | | K |
| 5: 528.321 cents 27/20<br />
| | |- |
| <span style="background-color: rgba(255,255,255,0);">6: 633.985 cents 13/9</span><br />
| | | 4 |
| 7: 739.649 cents 17/13<br />
| | | 422.657 |
| <span style="background-color: rgba(255,255,255,0);">8: 845.313 cents 5/3</span><br />
| | |288.889 |
| 9: 950.9775 cents 19/11<br />
| | | 9/7 |
| <span style="background-color: rgba(255,255,255,0);">10: 1056.642 cents 9/5</span><br />
| | | K#, Lb |
| 11: 1162.306 cents 49/25<br />
| | |- |
| <span style="background-color: rgba(255,255,255,0);">12: 1267.970 cents 27/13</span><br />
| | | 5 |
| 13: 1373.634 cents 20/9<br />
| | | 528.321 |
| <span style="background-color: rgba(255,255,255,0);">14: 1479.298 cents 7/3</span><br />
| | |361.111 |
| 15: 1584.963 cents 5/2<br />
| | | 27/20 |
| <span style="background-color: rgba(255,255,255,0);">16: 1690.627 cents 8/3</span><br />
| | | L |
| 17: 1806.2905 cents 45/16<br />
| | |- |
| <span style="background-color: rgba(255,255,255,0);">18: 3/1</span></body></html></pre></div>
| | | 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 |
| | |} |
| | |
| | == 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. |
| | |
| | == See also == |
| | [[1ed17/16]] — relative ed17/16 |
| | |
| | [[category:macrotonal]] |
| | [[category:tritave]] |
| 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.
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)
|
Note that this tuning is fairly close to 1ed17/16, although by patent val mapping 18edt maps ~17/16 inconsistently to 2\18edt instead of 1\18edt, while 1ed17/16 maps its own equave even more inconsistently, to 3\1ed17/16.
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
|
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.
See also
1ed17/16 — relative ed17/16