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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>
| | {{ED intro}} |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04-09 22:29:36 UTC</tt>.<br>
| |
| : The original revision id was <tt>318940230</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">=**Intervals**=
| |
| || 1 || 43.872 cents ||
| |
| || 2 || 87.744 ||
| |
| || 3 || 131.61656 ||
| |
| || 4 || 175.48875 ||
| |
| || 5 || 219.3609 ||
| |
| || 6 || 263.233 ||
| |
| || 7 || 307.105 ||
| |
| || 8 || 350.9775 ||
| |
| || 9 || 394.84969 ||
| |
| || 10 || 438.72188 ||
| |
| || 11 || 482.594 ||
| |
| || 12 || 526.466 ||
| |
| || 13 || 570.338 ||
| |
| || 14 || 614.2106 ||
| |
| || 15 || 658.0828 ||
| |
| || 16 || 701.955 ||
| |
| || 17 || 745.827 ||
| |
| || 18 || 789.699 ||
| |
| || 19 || 833.57156 ||
| |
| || 20 || 877.44375 ||
| |
| || 21 || 921.3159 ||
| |
| || 22 || 965.188 ||
| |
| || 23 || 1009.0603 ||
| |
| || 24 || 1052.932 ||
| |
| || 25 || 1096.80469 ||
| |
| || 26 || 1140.67688 ||
| |
| || 27 || 1184.549 ||
| |
|
| |
|
| =Compositions= | | == Theory == |
| | 16edf corresponds to 27.3522…[[edo]]. It is similar to every third step of [[82edo]] but not quite similar to [[27edo]]; the octave is compressed by 15.45{{c}}, a small but significant deviation. It contains good approximations of the [[7/1|7th]] and [[13/1|13th]] [[harmonic]]s. |
|
| |
|
| [[http://www.youtube.com/watch?v=-RUeO6hJLBY|schizophrenic lullaby fugue]]</pre></div> | | It serves as a good approximation to [[halftone]] temperament, containing the [[~]][[7/5]] generator at 13 steps. |
| <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>16edf</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 --><strong>Intervals</strong></h1>
| |
|
| |
|
| |
|
| <table class="wiki_table">
| | === Harmonics === |
| <tr>
| | {{Harmonics in equal|16|3|2}} |
| <td>1<br />
| | {{Harmonics in equal|16|3|2|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 16edf (continued)}} |
| </td>
| |
| <td>43.872 cents<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>87.744<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>131.61656<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>175.48875<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>219.3609<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>263.233<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>307.105<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>350.9775<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>394.84969<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>438.72188<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>482.594<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>526.466<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>570.338<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>614.2106<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>658.0828<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>701.955<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>745.827<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>789.699<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>833.57156<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>877.44375<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>921.3159<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22<br />
| |
| </td>
| |
| <td>965.188<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>1009.0603<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24<br />
| |
| </td>
| |
| <td>1052.932<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>1096.80469<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26<br />
| |
| </td>
| |
| <td>1140.67688<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>27<br />
| |
| </td>
| |
| <td>1184.549<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | === Subsets and supersets === |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:2 -->Compositions</h1>
| | Since 16 factors into primes as 2<sup>4</sup>, 16edf contains subset edfs {{EDs|equave=f| 2, 4, and 8 }}. |
| <br />
| | |
| <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=-RUeO6hJLBY" rel="nofollow">schizophrenic lullaby fugue</a></body></html></pre></div>
| | == Intervals == |
| | {| class="wikitable center-1 right-2 mw-collapsible" |
| | |+ Intervals of 16edf |
| | |- |
| | ! # |
| | ! Cents |
| | ! Approximate ratios |
| | ! Halftone[6] notation<br>(using [[ups and downs notation|ups and downs]]) |
| | ! Comments |
| | |- |
| | | 0 |
| | | 0.0 |
| | | [[1/1]] |
| | | C |
| | | |
| | |- |
| | | 1 |
| | | 43.9 |
| | | 40/39, 39/38 |
| | | ^C |
| | | |
| | |- |
| | | 2 |
| | | 87.7 |
| | | [[20/19]] |
| | | Db |
| | | |
| | |- |
| | | 3 |
| | | 131.6 |
| | | 55/51, ([[27/25]]) |
| | | vD |
| | | |
| | |- |
| | | 4 |
| | | 175.5 |
| | | ([[21/19]]) |
| | | D |
| | | |
| | |- |
| | | 5 |
| | | 219.4 |
| | | |
| | | vE |
| | | |
| | |- |
| | | 6 |
| | | 263.2 |
| | | ([[7/6]]) |
| | | E |
| | | |
| | |- |
| | | 7 |
| | | 307.1 |
| | | |
| | | Fb |
| | | |
| | |- |
| | | 8 |
| | | 351.0 |
| | | 60/49, 49/40 |
| | | vF |
| | | |
| | |- |
| | | 9 |
| | | 394.8 |
| | | (44/35) |
| | | F |
| | | |
| | |- |
| | | 10 |
| | | 438.7 |
| | | ([[9/7]]) |
| | | Ab |
| | | |
| | |- |
| | | 11 |
| | | 482.6 |
| | | |
| | | vA |
| | | |
| | |- |
| | | 12 |
| | | 526.5 |
| | | ([[19/14]]) |
| | | A |
| | | |
| | |- |
| | | 13 |
| | | 570.3 |
| | | ([[25/18]]), 153/110, 112/81 |
| | | B |
| | | |
| | |- |
| | | 14 |
| | | 614.2 |
| | | ([[10/7]]) |
| | | Cb |
| | | |
| | |- |
| | | 15 |
| | | 658.1 |
| | | [[19/13]] |
| | | vC |
| | | |
| | |- |
| | | 16 |
| | | 702.0 |
| | | [[3/2]] |
| | | C |
| | | Just perfect fifth |
| | |- |
| | | 17 |
| | | 745.8 |
| | | [[20/13]] |
| | | |
| | | |
| | |- |
| | | 18 |
| | | 789.7 |
| | | [[30/19]] |
| | | |
| | | |
| | |- |
| | | 19 |
| | | 833.6 |
| | | 55/34 |
| | | |
| | | |
| | |- |
| | | 20 |
| | | 877.4 |
| | | |
| | | |
| | | |
| | |- |
| | | 21 |
| | | 921.3 |
| | | |
| | | |
| | | |
| | |- |
| | | 22 |
| | | 965.2 |
| | | [[7/4]] |
| | | |
| | | |
| | |- |
| | | 23 |
| | | 1009.0 |
| | | |
| | | |
| | | |
| | |- |
| | | 24 |
| | | 1052.9 |
| | | 90/49, ([[11/6]]) |
| | | |
| | | |
| | |- |
| | | 25 |
| | | 1096.8 |
| | | (66/35) |
| | | |
| | | |
| | |- |
| | | 26 |
| | | 1140.7 |
| | | |
| | | |
| | | |
| | |- |
| | | 27 |
| | | 1184.5 |
| | | |
| | | |
| | | |
| | |- |
| | | 28 |
| | | 1228.4 |
| | | 128/63 |
| | | |
| | | |
| | |- |
| | | 29 |
| | | 1272.3 |
| | | 25/12 |
| | | |
| | | |
| | |- |
| | | 30 |
| | | 1316.2 |
| | | 15/7 |
| | | |
| | | |
| | |- |
| | | 31 |
| | | 1360.0 |
| | | 57/26 |
| | | |
| | | |
| | |- |
| | | 32 |
| | | 1403.9 |
| | | [[9/4]] |
| | | |
| | | Pythagorean major ninth |
| | |} |
| | |
| | == Music == |
| | ; [[Nae Ayy]] |
| | * [https://www.youtube.com/watch?v=8YegsoiO1Co ''Neptune''] (2021) |
| | |
| | ; [[nationalsolipsism]] |
| | * [https://www.youtube.com/watch?v=-RUeO6hJLBY ''schizophrenic lullaby fugue''] (2011) |
| | |
| | == See also == |
| | * [[27edo]] – relative edo |
| | * [[43edt]] – relative edt |
| | * [[70ed6]] – relative ed6 |
| | * [[90ed10]] – relative ed10 |
| | * [[97ed12]] – relative ed12 |
| | |
| | {{Todo|expand}} |
| | |
| | [[Category:27edo]] |
| Prime factorization
|
24
|
| Step size
|
43.8722 ¢
|
| Octave
|
27\16edf (1184.55 ¢)
|
| Twelfth
|
43\16edf (1886.5 ¢)
|
| Consistency limit
|
3
|
| Distinct consistency limit
|
3
|
16 equal divisions of the perfect fifth (abbreviated 16edf or 16ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 16 equal parts of about 43.9 ¢ each. Each step represents a frequency ratio of (3/2)1/16, or the 16th root of 3/2.
Theory
16edf corresponds to 27.3522…edo. It is similar to every third step of 82edo but not quite similar to 27edo; the octave is compressed by 15.45 ¢, a small but significant deviation. It contains good approximations of the 7th and 13th harmonics.
It serves as a good approximation to halftone temperament, containing the ~7/5 generator at 13 steps.
Harmonics
Approximation of harmonics in 16edf
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
| Error
|
Absolute (¢)
|
-15.5
|
-15.5
|
+13.0
|
+21.5
|
+13.0
|
+9.3
|
-2.5
|
+13.0
|
+6.1
|
+16.5
|
-2.5
|
| Relative (%)
|
-35.2
|
-35.2
|
+29.6
|
+49.0
|
+29.6
|
+21.3
|
-5.7
|
+29.6
|
+13.8
|
+37.7
|
-5.7
|
Steps
(reduced)
|
27
(11)
|
43
(11)
|
55
(7)
|
64
(0)
|
71
(7)
|
77
(13)
|
82
(2)
|
87
(7)
|
91
(11)
|
95
(15)
|
98
(2)
|
Approximation of harmonics in 16edf (continued)
| Harmonic
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
| Error
|
Absolute (¢)
|
-9.4
|
-6.1
|
+6.1
|
-17.9
|
+8.7
|
-2.5
|
-8.3
|
-9.4
|
-6.1
|
+1.1
|
+11.9
|
-17.9
|
| Relative (%)
|
-21.5
|
-13.9
|
+13.8
|
-40.9
|
+19.9
|
-5.7
|
-19.0
|
-21.4
|
-13.9
|
+2.5
|
+27.1
|
-40.9
|
Steps
(reduced)
|
101
(5)
|
104
(8)
|
107
(11)
|
109
(13)
|
112
(0)
|
114
(2)
|
116
(4)
|
118
(6)
|
120
(8)
|
122
(10)
|
124
(12)
|
125
(13)
|
Subsets and supersets
Since 16 factors into primes as 24, 16edf contains subset edfs 2, 4, and 8.
Intervals
Intervals of 16edf
| #
|
Cents
|
Approximate ratios
|
Halftone[6] notation
(using ups and downs)
|
Comments
|
| 0
|
0.0
|
1/1
|
C
|
|
| 1
|
43.9
|
40/39, 39/38
|
^C
|
|
| 2
|
87.7
|
20/19
|
Db
|
|
| 3
|
131.6
|
55/51, (27/25)
|
vD
|
|
| 4
|
175.5
|
(21/19)
|
D
|
|
| 5
|
219.4
|
|
vE
|
|
| 6
|
263.2
|
(7/6)
|
E
|
|
| 7
|
307.1
|
|
Fb
|
|
| 8
|
351.0
|
60/49, 49/40
|
vF
|
|
| 9
|
394.8
|
(44/35)
|
F
|
|
| 10
|
438.7
|
(9/7)
|
Ab
|
|
| 11
|
482.6
|
|
vA
|
|
| 12
|
526.5
|
(19/14)
|
A
|
|
| 13
|
570.3
|
(25/18), 153/110, 112/81
|
B
|
|
| 14
|
614.2
|
(10/7)
|
Cb
|
|
| 15
|
658.1
|
19/13
|
vC
|
|
| 16
|
702.0
|
3/2
|
C
|
Just perfect fifth
|
| 17
|
745.8
|
20/13
|
|
|
| 18
|
789.7
|
30/19
|
|
|
| 19
|
833.6
|
55/34
|
|
|
| 20
|
877.4
|
|
|
|
| 21
|
921.3
|
|
|
|
| 22
|
965.2
|
7/4
|
|
|
| 23
|
1009.0
|
|
|
|
| 24
|
1052.9
|
90/49, (11/6)
|
|
|
| 25
|
1096.8
|
(66/35)
|
|
|
| 26
|
1140.7
|
|
|
|
| 27
|
1184.5
|
|
|
|
| 28
|
1228.4
|
128/63
|
|
|
| 29
|
1272.3
|
25/12
|
|
|
| 30
|
1316.2
|
15/7
|
|
|
| 31
|
1360.0
|
57/26
|
|
|
| 32
|
1403.9
|
9/4
|
|
Pythagorean major ninth
|
Music
- Nae Ayy
- nationalsolipsism
See also