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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>
| | '''[[Ed5|Division of the 5th harmonic]] into 17 equal parts''' (17ED5) is a good [[hyperpyth]] tuning. The step size is about 163.9008 cents, corresponding to 7.3215 [[EDO]]. |
| : This revision was by author [[User:guest|guest]] and made on <tt>2011-12-31 21:40:59 UTC</tt>.<br>
| |
| : The original revision id was <tt>288945041</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">=Division of the 5/1 into 17 tones=
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| A hyperpyth tuning, 17ed5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. However, in hyperpyth, 21/5 isn't necessarily represented, at least not as well. In 17ed5, the 21/5 is represented about as well as the 9/5 is, although that's not too good. Luckily, 27, 29, and 39 do a fair job of it. Nevertheless it's the simplest equal hyperpyth after 5ed5, and quite consonant. I imagine it to be the traditional tonality of the tiny creatures living on subatomic particles. | | == Division of the 5/1 into 17 tones == |
| | A hyperpyth tuning, 17ED5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. However, in hyperpyth, 21/5 isn't necessarily represented, at least not as well. In 17ED5, the 21/5 is represented about as well as the 9/5 is, although that's not too good. Luckily, 27, 29, and 39 do a fair job of it. Nevertheless it's the simplest equal hyperpyth after 5ED5, and quite consonant. I imagine it to be the traditional tonality of the tiny creatures living on subatomic particles. |
|
| |
|
| But wait, an interesting pattern emerges: | | But wait, an interesting pattern emerges: |
|
| |
|
| 22ed5 focuses on 9/5 | | [[22ed5|22ED5]] focuses on 9/5 |
| 27ed5 focuses on 13/5 | | |
| 29ed5 focuses on 17/5 | | [[27ed5|27ED5]] focuses on 13/5 |
| | |
| | [[29ed5|29ED5]] focuses on 17/5 |
| | |
| (and 34=17*2) | | (and 34=17*2) |
|
| |
|
| so: 22+27+29=78=39*2 | | so: 22+27+29=78=39*2 |
| and behold, of the lot, 39ed5 offers the best balance between those intervals.
| |
|
| |
|
| || 0: 0.000 cents || 1/1 || ||
| | and behold, of the lot, [[39ed5|39ED5]] offers the best balance between those intervals. |
| || 1: 163.901 || || ||
| |
| || 2: 327.802 || || ||
| |
| || 3: 491.702 || || ||
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| || 4: 655.603 || || ||
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| || 5: 819.504 || || ||
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| || 6: 983.405 || 9/5, 16/9, 7/4 || 1017 ||
| |
| || 7: 1147.306 || || ||
| |
| || 8: 1311.206 || || ||
| |
| || 9: 1475.107 || || ||
| |
| || 10: 1639.008 || 13/5 || 1654 ||
| |
| || 11: 1802.909 || || ||
| |
| || 12: 1966.810 || || ||
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| || 13: 2130.710 || 17/5 || 2118 ||
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| || 14: 2294.611 || || ||
| |
| || 15: 2458.512 || (21/5) || 2486 ||
| |
| || 16: 2622.413 || || ||
| |
| || 17: 2786.314 || 5/1 || ||</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>17ed5</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Division of the 5/1 into 17 tones"></a><!-- ws:end:WikiTextHeadingRule:0 -->Division of the 5/1 into 17 tones</h1>
| |
| <br />
| |
| A hyperpyth tuning, 17ed5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. However, in hyperpyth, 21/5 isn't necessarily represented, at least not as well. In 17ed5, the 21/5 is represented about as well as the 9/5 is, although that's not too good. Luckily, 27, 29, and 39 do a fair job of it. Nevertheless it's the simplest equal hyperpyth after 5ed5, and quite consonant. I imagine it to be the traditional tonality of the tiny creatures living on subatomic particles.<br />
| |
| <br />
| |
| But wait, an interesting pattern emerges:<br />
| |
| <br />
| |
| 22ed5 focuses on 9/5<br />
| |
| 27ed5 focuses on 13/5<br />
| |
| 29ed5 focuses on 17/5<br />
| |
| (and 34=17*2)<br />
| |
| <br />
| |
| so: 22+27+29=78=39*2<br />
| |
| and behold, of the lot, 39ed5 offers the best balance between those intervals.<br /> | |
| <br />
| |
|
| |
|
| | {| class="wikitable" |
| | |- |
| | ! | degree |
| | ! | cents value |
| | ! | corresponding <br>JI intervals |
| | ! | comments |
| | |- |
| | | | 0 |
| | | | 0.000 |
| | | | '''exact [[1/1]]''' |
| | | | |
| | |- |
| | | | 1 |
| | | | 163.901 |
| | | | [[11/10]] |
| | | | |
| | |- |
| | | | 2 |
| | | | 327.802 |
| | | | [[6/5]] |
| | | | |
| | |- |
| | | | 3 |
| | | | 491.702 |
| | | | [[4/3]] |
| | | | |
| | |- |
| | | | 4 |
| | | | 655.603 |
| | | | [[16/11]], [[19/13]], <br>[[22/15]] |
| | | | |
| | |- |
| | | | 5 |
| | | | 819.504 |
| | | | [[8/5]] |
| | | | |
| | |- |
| | | | 6 |
| | | | 983.405 |
| | | | [[7/4]], [[9/5]], [[16/9]] |
| | | | |
| | |- |
| | | | 7 |
| | | | 1147.306 |
| | | | [[25/13]], [[27/14]], <br>[[35/18]], [[64/33]] |
| | | | |
| | |- |
| | | | 8 |
| | | | 1311.206 |
| | | | [[16/15|32/15]] |
| | | | |
| | |- |
| | | | 9 |
| | | | 1475.107 |
| | | | [[75/64|75/32]] |
| | | | |
| | |- |
| | | | 10 |
| | | | 1639.008 |
| | | | [[13/5]], [[9/7|18/7]] |
| | | | |
| | |- |
| | | | 11 |
| | | | 1802.909 |
| | | | [[17/12|17/6]] |
| | | | |
| | |- |
| | | | 12 |
| | | | 1966.810 |
| | | | [[14/9|28/9]] |
| | | | |
| | |- |
| | | | 13 |
| | | | 2130.710 |
| | | | [[17/10|17/5]], [[12/7|24/7]] |
| | | | |
| | |- |
| | | | 14 |
| | | | 2294.611 |
| | | | [[19/10|19/5]], [[32/17|64/17]] |
| | | | |
| | |- |
| | | | 15 |
| | | | 2458.512 |
| | | | [[21/20|21/5]], [[25/24|25/6]], <br>[[33/32|33/8]] |
| | | | |
| | |- |
| | | | 16 |
| | | | 2622.413 |
| | | | [[17/15|68/15]] |
| | | | |
| | |- |
| | | | 17 |
| | | | 2786.314 |
| | | | '''exact [[5/1]]''' |
| | | | just major third plus two octaves |
| | |} |
|
| |
|
| <table class="wiki_table">
| | == Harmonics == |
| <tr>
| | {{Harmonics in equal |
| <td>0: 0.000 cents<br />
| | | steps = 17 |
| </td>
| | | num = 5 |
| <td>1/1<br />
| | | denom = 1 |
| </td>
| | }} |
| <td><br />
| | {{Harmonics in equal |
| </td>
| | | steps = 17 |
| </tr>
| | | num = 5 |
| <tr>
| | | denom = 1 |
| <td>1: 163.901<br />
| | | start = 12 |
| </td>
| | | collapsed = 1 |
| <td><br />
| | }} |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2: 327.802<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3: 491.702<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4: 655.603<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5: 819.504<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6: 983.405<br />
| |
| </td>
| |
| <td>9/5, 16/9, 7/4<br />
| |
| </td>
| |
| <td>1017<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7: 1147.306<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8: 1311.206<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9: 1475.107<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10: 1639.008<br />
| |
| </td>
| |
| <td>13/5<br />
| |
| </td>
| |
| <td>1654<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11: 1802.909<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12: 1966.810<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13: 2130.710<br />
| |
| </td>
| |
| <td>17/5<br />
| |
| </td>
| |
| <td>2118<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14: 2294.611<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15: 2458.512<br />
| |
| </td>
| |
| <td>(21/5)<br />
| |
| </td>
| |
| <td>2486<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16: 2622.413<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17: 2786.314<br />
| |
| </td>
| |
| <td>5/1<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div>
| | [[Category:Hyperpyth]] |
| | [[Category:Todo:add sound example]] |
| Prime factorization
|
17 (prime)
|
| Step size
|
163.901 ¢
|
| Octave
|
7\17ed5 (1147.31 ¢)
|
| Twelfth
|
12\17ed5 (1966.81 ¢)
|
| Consistency limit
|
2
|
| Distinct consistency limit
|
2
|
Division of the 5th harmonic into 17 equal parts (17ED5) is a good hyperpyth tuning. The step size is about 163.9008 cents, corresponding to 7.3215 EDO.
Division of the 5/1 into 17 tones
A hyperpyth tuning, 17ED5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. However, in hyperpyth, 21/5 isn't necessarily represented, at least not as well. In 17ED5, the 21/5 is represented about as well as the 9/5 is, although that's not too good. Luckily, 27, 29, and 39 do a fair job of it. Nevertheless it's the simplest equal hyperpyth after 5ED5, and quite consonant. I imagine it to be the traditional tonality of the tiny creatures living on subatomic particles.
But wait, an interesting pattern emerges:
22ED5 focuses on 9/5
27ED5 focuses on 13/5
29ED5 focuses on 17/5
(and 34=17*2)
so: 22+27+29=78=39*2
and behold, of the lot, 39ED5 offers the best balance between those intervals.
| degree
|
cents value
|
corresponding
JI intervals
|
comments
|
| 0
|
0.000
|
exact 1/1
|
|
| 1
|
163.901
|
11/10
|
|
| 2
|
327.802
|
6/5
|
|
| 3
|
491.702
|
4/3
|
|
| 4
|
655.603
|
16/11, 19/13,
22/15
|
|
| 5
|
819.504
|
8/5
|
|
| 6
|
983.405
|
7/4, 9/5, 16/9
|
|
| 7
|
1147.306
|
25/13, 27/14,
35/18, 64/33
|
|
| 8
|
1311.206
|
32/15
|
|
| 9
|
1475.107
|
75/32
|
|
| 10
|
1639.008
|
13/5, 18/7
|
|
| 11
|
1802.909
|
17/6
|
|
| 12
|
1966.810
|
28/9
|
|
| 13
|
2130.710
|
17/5, 24/7
|
|
| 14
|
2294.611
|
19/5, 64/17
|
|
| 15
|
2458.512
|
21/5, 25/6,
33/8
|
|
| 16
|
2622.413
|
68/15
|
|
| 17
|
2786.314
|
exact 5/1
|
just major third plus two octaves
|
Harmonics
Approximation of harmonics in 17ed5
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
| Error
|
Absolute (¢)
|
-52.7
|
+64.9
|
+58.5
|
+0.0
|
+12.2
|
+73.1
|
+5.8
|
-34.2
|
-52.7
|
-53.8
|
-40.5
|
| Relative (%)
|
-32.2
|
+39.6
|
+35.7
|
+0.0
|
+7.4
|
+44.6
|
+3.5
|
-20.9
|
-32.2
|
-32.8
|
-24.7
|
Steps
(reduced)
|
7
(7)
|
12
(12)
|
15
(15)
|
17
(0)
|
19
(2)
|
21
(4)
|
22
(5)
|
23
(6)
|
24
(7)
|
25
(8)
|
26
(9)
|
Approximation of harmonics in 17ed5
| Harmonic
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
| Error
|
Absolute (¢)
|
-15.2
|
+20.4
|
+64.9
|
-46.9
|
+12.1
|
+77.0
|
-16.6
|
+58.5
|
-26.0
|
+57.4
|
-19.5
|
| Relative (%)
|
-9.3
|
+12.4
|
+39.6
|
-28.6
|
+7.4
|
+47.0
|
-10.1
|
+35.7
|
-15.8
|
+35.0
|
-11.9
|
Steps
(reduced)
|
27
(10)
|
28
(11)
|
29
(12)
|
29
(12)
|
30
(13)
|
31
(14)
|
31
(14)
|
32
(15)
|
32
(15)
|
33
(16)
|
33
(16)
|