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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>
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| : This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-11-18 12:20:30 UTC</tt>.<br>
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| : The original revision id was <tt>276984936</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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">=10 equal divisions of the 5th harmonic=
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| Half of [[20ed5]] (obviously). But it has important characteristics of its own:
| | In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to. |
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| Well, as [[17ed5|hyperpyth]] is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.
| | It is especially important as a structural framework for the [[5.7.11.13 subgroup]]. |
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| Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to [[5edo]] and [[4edt]] in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies. This would be the perfect tuning for blues from outer space (perhaps from a gas giant somewhere).
| | == Harmonics == |
| | {{Harmonics in equal |
| | | steps = 10 |
| | | num = 5 |
| | | denom = 1 |
| | }} |
| | {{Harmonics in equal |
| | | steps = 10 |
| | | num = 5 |
| | | denom = 1 |
| | | start = 12 |
| | | collapsed = 1 |
| | }} |
|
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| Adding octaves, strangely enough, relates this tuning to [[53edo]].
| | == Intervals == |
| | {| class="wikitable" |
| | |+ |
| | !Degree |
| | !Cents |
| | !5.7.11.13 intervals |
| | |- |
| | |0 |
| | |0.000 |
| | |1/1 |
| | |- |
| | |1 |
| | |278.631 |
| | |13/11, 55/49 |
| | |- |
| | |2 |
| | |557.263 |
| | |7/5 |
| | |- |
| | |3 |
| | |835.894 |
| | |11/7 |
| | |- |
| | |4 |
| | |1114.525 |
| | |13/7, 25/13 |
| | |- |
| | |5 |
| | |1393.157 |
| | |11/5, 25/11 |
| | |- |
| | |6 |
| | |1671.788 |
| | |13/5, 35/13 |
| | |- |
| | |7 |
| | |1950.420 |
| | |35/11 |
| | |- |
| | |8 |
| | |2229.051 |
| | |49/13 |
| | |- |
| | |9 |
| | |2507.682 |
| | |49/11 |
| | |- |
| | |10 |
| | |2786.314 |
| | |5/1 |
| | |} |
|
| |
|
| 0: 1/1
| | == Subsets and supersets == |
| 1: 278.631 cents 13/11
| | Half of [[20ed5]]. |
| 2: 557.263 cents 7/5
| | |
| 3: 835.894 cents
| | As 5ed5 is the simplest [[hyperpyth]] tuning (analogous to [[5edo]] and [[4edt]] in their own spheres) this, its double, can be compared structurally to [[10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. |
| 4: 1114.525 cents "9/5"
| | |
| 5: 1393.157 cents 11/5
| | Octaves can be added by dividing the step in three to get [[13edo]] with octaves 7 cents sharp. If octaves are instead made just, prime 7 becomes very flat, as well as prime 5 to a lesser extent. Alternatively, the step can be divided in ten to get [[43edo]]. |
| 6: 1671.788 cents 13/5
| | |
| 7: 1950.420 cents
| | == Music == |
| 8: 2229.051 cents "17/5"
| | |
| 9: 2507.682 cents 21/5
| | [http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky |
| 10: 5/1</pre></div>
| | [[Category:5th_harmonic]] |
| <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>10ed5</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x10 equal divisions of the 5th harmonic"></a><!-- ws:end:WikiTextHeadingRule:0 -->10 equal divisions of the 5th harmonic</h1>
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| <br />
| |
| Half of <a class="wiki_link" href="/20ed5">20ed5</a> (obviously). But it has important characteristics of its own:<br /> | |
| <br />
| |
| Well, as <a class="wiki_link" href="/17ed5">hyperpyth</a> is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.<br />
| |
| <br />
| |
| Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to <a class="wiki_link" href="/5edo">5edo</a> and <a class="wiki_link" href="/4edt">4edt</a> in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies. This would be the perfect tuning for blues from outer space (perhaps from a gas giant somewhere).<br />
| |
| <br />
| |
| Adding octaves, strangely enough, relates this tuning to <a class="wiki_link" href="/53edo">53edo</a>.<br />
| |
| <br />
| |
| 0: 1/1<br />
| |
| 1: 278.631 cents 13/11<br />
| |
| 2: 557.263 cents 7/5<br />
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| 3: 835.894 cents<br />
| |
| 4: 1114.525 cents &quot;9/5&quot;<br />
| |
| 5: 1393.157 cents 11/5<br />
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| 6: 1671.788 cents 13/5<br />
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| 7: 1950.420 cents<br />
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| 8: 2229.051 cents &quot;17/5&quot;<br />
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| 9: 2507.682 cents 21/5<br />
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| 10: 5/1</body></html></pre></div>
| |
| Prime factorization
|
2 × 5
|
| Step size
|
278.631 ¢
|
| Octave
|
4\10ed5 (1114.53 ¢) (→ 2\5ed5)
|
| Twelfth
|
7\10ed5 (1950.42 ¢)
(semiconvergent)
|
| Consistency limit
|
3
|
| Distinct consistency limit
|
3
|
In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.
It is especially important as a structural framework for the 5.7.11.13 subgroup.
Harmonics
Approximation of harmonics in 10ed5
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
| Error
|
Absolute (¢)
|
-85
|
+48
|
+108
|
+0
|
-37
|
-25
|
+22
|
+97
|
-85
|
+28
|
-122
|
| Relative (%)
|
-30.7
|
+17.4
|
+38.6
|
+0.0
|
-13.3
|
-9.1
|
+8.0
|
+34.8
|
-30.7
|
+10.1
|
-44.0
|
Steps
(reduced)
|
4
(4)
|
7
(7)
|
9
(9)
|
10
(0)
|
11
(1)
|
12
(2)
|
13
(3)
|
14
(4)
|
14
(4)
|
15
(5)
|
15
(5)
|
Approximation of harmonics in 10ed5
| Harmonic
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
| Error
|
Absolute (¢)
|
+18
|
-111
|
+48
|
-63
|
+110
|
+11
|
-82
|
+108
|
+23
|
-57
|
-134
|
| Relative (%)
|
+6.3
|
-39.7
|
+17.4
|
-22.7
|
+39.6
|
+4.1
|
-29.5
|
+38.6
|
+8.3
|
-20.6
|
-48.2
|
Steps
(reduced)
|
16
(6)
|
16
(6)
|
17
(7)
|
17
(7)
|
18
(8)
|
18
(8)
|
18
(8)
|
19
(9)
|
19
(9)
|
19
(9)
|
19
(9)
|
Intervals
| Degree
|
Cents
|
5.7.11.13 intervals
|
| 0
|
0.000
|
1/1
|
| 1
|
278.631
|
13/11, 55/49
|
| 2
|
557.263
|
7/5
|
| 3
|
835.894
|
11/7
|
| 4
|
1114.525
|
13/7, 25/13
|
| 5
|
1393.157
|
11/5, 25/11
|
| 6
|
1671.788
|
13/5, 35/13
|
| 7
|
1950.420
|
35/11
|
| 8
|
2229.051
|
49/13
|
| 9
|
2507.682
|
49/11
|
| 10
|
2786.314
|
5/1
|
Subsets and supersets
Half of 20ed5.
As 5ed5 is the simplest hyperpyth tuning (analogous to 5edo and 4edt in their own spheres) this, its double, can be compared structurally to 10edo. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.
Octaves can be added by dividing the step in three to get 13edo with octaves 7 cents sharp. If octaves are instead made just, prime 7 becomes very flat, as well as prime 5 to a lesser extent. Alternatively, the step can be divided in ten to get 43edo.
Music
Weird Blues -- Kosmorsky