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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{interwiki |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | en = 27edt |
| : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-02-24 06:39:09 UTC</tt>.<br>
| | | de = 27-EDT |
| : The original revision id was <tt>204608830</tt>.<br>
| | }} |
| : The revision comment was: <tt></tt><br>
| | {{Infobox ET}} |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | {{ED intro}} |
| <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">=[[#Division of the tritave (3/1) into 12 equal parts]]Division of the tritave (3/1) into 27 equal parts=
| |
|
| |
|
| Dividing the interval of 3/1 into 27 equal parts gives a scale with a basic step of 70.44 cents, which is nearly identical to one step of [[17edo]] (70.59 cents). Hence it has similar melodic and harmonic properties as 17edo, with the difference that 27 is not a prime number.
| | == Theory == |
| | 27edt corresponds to 17.035…edo, which is nearly identical to one step of [[17edo]] (70.59 cents). Hence it has similar melodic and harmonic properties as 17edo, with the difference that 27 is not a [[prime number]]. In fact, the [[prime edo]]s that approximate the 3-limit well often correspond to composite edts: e.g. [[19edo]] → [[30edt]], [[29edo]] → [[46edt]] and [[31edo]] → [[49edt]]. |
|
| |
|
| 27 being the third power of 3, and the base interval being 3/1, 27edt is a tuning where the number 3 prevails. This property seems to predestine 27edt as base tuning for Klingon music (since the tradtional Klingon number system is also based on 3). See, e.g., [[http://launch.dir.groups.yahoo.com/group/tuning/message/86909]] and [[http://www.klingon.org/smboard/index.php?topic=1810.0]].
| | Compared to 17edo, 27edt approximates the [[prime interval|primes]] [[7/1|7]], [[11/1|11]], and [[13/1|13]] better; it approximates prime [[5/1|5]] equally poorly, but maps it to 40 steps rather than 39 in the [[patent val]], corresponding to the 17c [[val]], often considered the better mapping as it equates [[5/4]] and [[6/5]] to major and minor thirds rather than to a neutral third, and 5 has the same sharp tendency as 7 and 11. |
|
| |
|
| ==Intervals==
| | From a purely tritave-based perspective, it [[support]]s the [[minalzidar]] temperament, but otherwise it can be used as a retuning of 17edo with closer-to-just harmonic properties in the no-fives 2.3.7.11.13 subgroup. |
| ||~ degrees of 27edt ||~ cents value ||~ approximation in 17edo ||
| |
| || 0 || 0.00 || 0.00 ||
| |
| || 1 || 70.44 || 70.59 ||
| |
| || 2 || 140.89 || 141.18 ||
| |
| || 3 || 211.33 || 211.76 ||
| |
| || 4 || 281.77 || 282.35 ||
| |
| || 5 || 352.21 || 352.94 ||
| |
| || 6 || 422.66 || 423.53 ||
| |
| || 7 || 493.10 || 494.12 ||
| |
| || 8 || 563.54 || 564.71 ||
| |
| || 9 || 633.99 || 635.29 ||
| |
| || 10 || 704.43 || 705.88 ||
| |
| || 11 || 774.87 || 776.47 ||
| |
| || 12 || 845.31 || 847.06 ||
| |
| || 13 || 915.76 || 917.65 ||
| |
| || 14 || 986.20 || 988.24 ||
| |
| || 15 || 1056.64 || 1058.82 ||
| |
| || 16 || 1127.08 || 1129.41 ||
| |
| || 17 || 1197.53 || 1200.00 ||
| |
| || 18 || 1267.97 || 1270.59 ||
| |
| || 19 || 1338.41 || 1341.18 ||
| |
| || 20 || 1408.86 || 1411.76 ||
| |
| || 21 || 1479.30 || 1482.35 ||
| |
| || 22 || 1549.74 || 1551.94 ||
| |
| || 23 || 1620.18 || 1623.53 ||
| |
| || 24 || 1690.63 || 1694.12 ||
| |
| || 25 || 1761.07 || 1764.71 ||
| |
| || 26 || 1831.51 || 1835.29 ||
| |
| || 27 || 1901.96 || 1905.88 ||</pre></div>
| |
| <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>27edt</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Division of the tritave (3/1) into 27 equal parts"></a><!-- ws:end:WikiTextHeadingRule:0 --><!-- ws:start:WikiTextAnchorRule:4:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Division of the tritave (3/1) into 12 equal parts&quot; title=&quot;Anchor: Division of the tritave (3/1) into 12 equal parts&quot;/&gt; --><a name="Division of the tritave (3/1) into 12 equal parts"></a><!-- ws:end:WikiTextAnchorRule:4 -->Division of the tritave (3/1) into 27 equal parts</h1>
| |
| <br />
| |
| Dividing the interval of 3/1 into 27 equal parts gives a scale with a basic step of 70.44 cents, which is nearly identical to one step of <a class="wiki_link" href="/17edo">17edo</a> (70.59 cents). Hence it has similar melodic and harmonic properties as 17edo, with the difference that 27 is not a prime number.<br />
| |
| <br />
| |
| 27 being the third power of 3, and the base interval being 3/1, 27edt is a tuning where the number 3 prevails. This property seems to predestine 27edt as base tuning for Klingon music (since the tradtional Klingon number system is also based on 3). See, e.g., <a class="wiki_link_ext" href="http://launch.dir.groups.yahoo.com/group/tuning/message/86909" rel="nofollow">http://launch.dir.groups.yahoo.com/group/tuning/message/86909</a> and <a class="wiki_link_ext" href="http://www.klingon.org/smboard/index.php?topic=1810.0" rel="nofollow">http://www.klingon.org/smboard/index.php?topic=1810.0</a>.<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Division of the tritave (3/1) into 27 equal parts-Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h2>
| |
|
| |
|
| |
|
| <table class="wiki_table">
| | === Harmonics === |
| <tr>
| | {{Harmonics in equal|27|3|1|intervals=integer|columns=11}} |
| <th>degrees of 27edt<br />
| | {{Harmonics in equal|27|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 27edt (continued)}} |
| </th>
| |
| <th>cents value<br />
| |
| </th>
| |
| <th>approximation in 17edo<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0.00<br />
| |
| </td>
| |
| <td>0.00<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>70.44<br />
| |
| </td>
| |
| <td>70.59<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>140.89<br />
| |
| </td>
| |
| <td>141.18<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>211.33<br />
| |
| </td>
| |
| <td>211.76<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>281.77<br />
| |
| </td>
| |
| <td>282.35<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>352.21<br />
| |
| </td>
| |
| <td>352.94<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>422.66<br />
| |
| </td>
| |
| <td>423.53<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>493.10<br />
| |
| </td>
| |
| <td>494.12<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>563.54<br />
| |
| </td>
| |
| <td>564.71<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>633.99<br />
| |
| </td>
| |
| <td>635.29<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>704.43<br />
| |
| </td>
| |
| <td>705.88<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>774.87<br />
| |
| </td>
| |
| <td>776.47<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>845.31<br />
| |
| </td>
| |
| <td>847.06<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>915.76<br />
| |
| </td>
| |
| <td>917.65<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>986.20<br />
| |
| </td>
| |
| <td>988.24<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>1056.64<br />
| |
| </td>
| |
| <td>1058.82<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>1127.08<br />
| |
| </td>
| |
| <td>1129.41<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>1197.53<br />
| |
| </td>
| |
| <td>1200.00<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>1267.97<br />
| |
| </td>
| |
| <td>1270.59<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>1338.41<br />
| |
| </td>
| |
| <td>1341.18<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>1408.86<br />
| |
| </td>
| |
| <td>1411.76<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>1479.30<br />
| |
| </td>
| |
| <td>1482.35<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22<br />
| |
| </td>
| |
| <td>1549.74<br />
| |
| </td>
| |
| <td>1551.94<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>1620.18<br />
| |
| </td>
| |
| <td>1623.53<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24<br />
| |
| </td>
| |
| <td>1690.63<br />
| |
| </td>
| |
| <td>1694.12<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>1761.07<br />
| |
| </td>
| |
| <td>1764.71<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26<br />
| |
| </td>
| |
| <td>1831.51<br />
| |
| </td>
| |
| <td>1835.29<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>27<br />
| |
| </td>
| |
| <td>1901.96<br />
| |
| </td>
| |
| <td>1905.88<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div>
| | === Subsets and supersets === |
| | Since 27 factors into primes as 3<sup>3</sup>, 27edt contains [[3edt]] and [[9edt]] as subset edts. |
| | |
| | === Miscellany === |
| | 27 being the third power of 3, and the base interval being 3/1, 27edt is a tuning where the number 3 prevails. This property seems to predestine 27edt as base tuning for {{w|Klingon}} music since the tradtional Klingon number system is also based on 3. The rather harsh harmonic character of 27edt would suit very well, too<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_86909.html Yahoo! Tuning Group | ''the evil 27 equal temp scale from outer space'']</ref><ref>[https://web.archive.org/web/20100624113458/https://www.klingon.org/smboard/index.php?topic=1810.0 Klingon Imperial Forums | ''klingon music theory'']</ref>. |
| | |
| | This being said, such a proposal is rather short-sighted from a general cultural perspective, since any kind of living creature would most likely gravitate towards some form of [[low-complexity JI]], and while 27edt will gain appreciation in base-3 cultures at some point, it may not be the first temperament they discover. That would be like aliens assuming dominant tuning in human music is [[100ed10]] (or 1000ed10 or variation thereof) just because we count in base 10. |
| | |
| | == Intervals == |
| | {{Interval table}} |
| | |
| | == See also == |
| | * [[10edf]] – relative edf |
| | * [[17edo]] – relative edo |
| | * [[44ed6]] – relative ed6 |
| | |
| | == Notes == |
| Prime factorization
|
33
|
| Step size
|
70.4428 ¢
|
| Octave
|
17\27edt (1197.53 ¢)
(semiconvergent)
|
| Consistency limit
|
4
|
| Distinct consistency limit
|
4
|
27 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 27edt or 27ed3), is a nonoctave tuning system that divides the interval of 3/1 into 27 equal parts of about 70.4 ¢ each. Each step represents a frequency ratio of 31/27, or the 27th root of 3.
Theory
27edt corresponds to 17.035…edo, which is nearly identical to one step of 17edo (70.59 cents). Hence it has similar melodic and harmonic properties as 17edo, with the difference that 27 is not a prime number. In fact, the prime edos that approximate the 3-limit well often correspond to composite edts: e.g. 19edo → 30edt, 29edo → 46edt and 31edo → 49edt.
Compared to 17edo, 27edt approximates the primes 7, 11, and 13 better; it approximates prime 5 equally poorly, but maps it to 40 steps rather than 39 in the patent val, corresponding to the 17c val, often considered the better mapping as it equates 5/4 and 6/5 to major and minor thirds rather than to a neutral third, and 5 has the same sharp tendency as 7 and 11.
From a purely tritave-based perspective, it supports the minalzidar temperament, but otherwise it can be used as a retuning of 17edo with closer-to-just harmonic properties in the no-fives 2.3.7.11.13 subgroup.
Harmonics
Approximation of harmonics in 27edt
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
| Error
|
Absolute (¢)
|
-2.5
|
+0.0
|
-4.9
|
+31.4
|
-2.5
|
+12.4
|
-7.4
|
+0.0
|
+28.9
|
+4.8
|
-4.9
|
| Relative (%)
|
-3.5
|
+0.0
|
-7.0
|
+44.6
|
-3.5
|
+17.6
|
-10.5
|
+0.0
|
+41.1
|
+6.8
|
-7.0
|
Steps
(reduced)
|
17
(17)
|
27
(0)
|
34
(7)
|
40
(13)
|
44
(17)
|
48
(21)
|
51
(24)
|
54
(0)
|
57
(3)
|
59
(5)
|
61
(7)
|
Approximation of harmonics in 27edt (continued)
| Harmonic
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
| Error
|
Absolute (¢)
|
-2.6
|
+10.0
|
+31.4
|
-9.9
|
+26.0
|
-2.5
|
-25.6
|
+26.5
|
+12.4
|
+2.3
|
-4.2
|
-7.4
|
| Relative (%)
|
-3.7
|
+14.1
|
+44.6
|
-14.0
|
+37.0
|
-3.5
|
-36.4
|
+37.6
|
+17.6
|
+3.3
|
-5.9
|
-10.5
|
Steps
(reduced)
|
63
(9)
|
65
(11)
|
67
(13)
|
68
(14)
|
70
(16)
|
71
(17)
|
72
(18)
|
74
(20)
|
75
(21)
|
76
(22)
|
77
(23)
|
78
(24)
|
Subsets and supersets
Since 27 factors into primes as 33, 27edt contains 3edt and 9edt as subset edts.
Miscellany
27 being the third power of 3, and the base interval being 3/1, 27edt is a tuning where the number 3 prevails. This property seems to predestine 27edt as base tuning for Klingon music since the tradtional Klingon number system is also based on 3. The rather harsh harmonic character of 27edt would suit very well, too[1][2].
This being said, such a proposal is rather short-sighted from a general cultural perspective, since any kind of living creature would most likely gravitate towards some form of low-complexity JI, and while 27edt will gain appreciation in base-3 cultures at some point, it may not be the first temperament they discover. That would be like aliens assuming dominant tuning in human music is 100ed10 (or 1000ed10 or variation thereof) just because we count in base 10.
Intervals
| Steps
|
Cents
|
Hekts
|
Approximate ratios
|
| 0
|
0
|
0
|
1/1
|
| 1
|
70.4
|
48.1
|
22/21, 23/22, 24/23
|
| 2
|
140.9
|
96.3
|
12/11, 13/12
|
| 3
|
211.3
|
144.4
|
9/8, 17/15, 26/23
|
| 4
|
281.8
|
192.6
|
13/11, 20/17
|
| 5
|
352.2
|
240.7
|
11/9, 16/13
|
| 6
|
422.7
|
288.9
|
14/11, 23/18
|
| 7
|
493.1
|
337
|
4/3
|
| 8
|
563.5
|
385.2
|
18/13
|
| 9
|
634
|
433.3
|
13/9, 23/16
|
| 10
|
704.4
|
481.5
|
3/2
|
| 11
|
774.9
|
529.6
|
11/7, 14/9
|
| 12
|
845.3
|
577.8
|
13/8, 18/11
|
| 13
|
915.8
|
625.9
|
17/10, 22/13
|
| 14
|
986.2
|
674.1
|
16/9, 23/13
|
| 15
|
1056.6
|
722.2
|
11/6, 24/13
|
| 16
|
1127.1
|
770.4
|
21/11, 23/12
|
| 17
|
1197.5
|
818.5
|
2/1
|
| 18
|
1268
|
866.7
|
23/11
|
| 19
|
1338.4
|
914.8
|
13/6
|
| 20
|
1408.9
|
963
|
9/4
|
| 21
|
1479.3
|
1011.1
|
26/11
|
| 22
|
1549.7
|
1059.3
|
22/9
|
| 23
|
1620.2
|
1107.4
|
23/9
|
| 24
|
1690.6
|
1155.6
|
8/3
|
| 25
|
1761.1
|
1203.7
|
11/4
|
| 26
|
1831.5
|
1251.9
|
23/8, 26/9
|
| 27
|
1902
|
1300
|
3/1
|
See also
Notes