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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:Kosmorsky|Kosmorsky]] and made on <tt>2011-10-07 02:41:18 UTC</tt>.<br>
| |
| : The original revision id was <tt>262469926</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=
| |
|
| |
|
| Suppose you go hunting for alternate tonalities, a temperament safari. The octave has been pretty well explored, not that there isn't a lot to be done. The tritave beckons - but Bohlen Pierce and friends have kind of found what you'd be looking for. While there's tremendous ground to cover there, you have the pioneering instinct in the first degree, so you examine the 5th harmonic as an equivalence.
| | == 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. |
|
| |
|
| The first place to look is isoharmonic chords. In diatonic music we take 4:5:6:(7):8 and get the sublime meantone, and assorted other possibilities. In the tritave one finds that 1:2:3 looks back to the octave - although [[11edt]] does it very prettily. But when you crack open that 2/1 shell, you get to the meat of the tritave - [[Bohlen-Pierce]]- based on 3:5:7:9 isoharmony. In the pentave (5/1 and I am perpetuating the technically incorrect terminology, because it works) the first thing you find is 1:2:3:4:5 which is a good lock combination, but not the best pentave-specific chord. But if you pry it open like a clam you'll find a pearl - 5:9:13:17:21:25 a stunning consonance. The full pentad is a little complex, so you might expect to leave out the 21st, just like one does the 7th in diatonicism.
| | But wait, an interesting pattern emerges: |
|
| |
|
| So what does one do with this? Well it took me long enough to stumble on the answer... a year... Anything worth hearing is worth taking the 17th root of. Because bizarrely enough, the pattern known as "superpyth" in an octave context, is the key to tempering together these cosmic harmonies in the pentave! So I dub it "hyperpyth" but if you want to call it "kosmorsky" that's fine by me too. Musical examples forthcoming.
| | [[22ed5|22ED5]] focuses on 9/5 |
|
| |
|
| 17ed5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. Nevertheless it's the simplest equal hyperpyth, and quite consonant, like rolling clouds of colorful interstellar dust. However, in hyperpyth, 21/5 isn't necessarily represented, at least not well - which is the case in 17ed5. Luckily, 27, 29, and 39 do a fair job of it. I wonder if there is an alternate pentave system involving 11 and 7 to correspond to octaves' "[[26edo|Orgone]]"? I bet there is and I'll look for it soon.
| | [[27ed5|27ED5]] focuses on 13/5 |
|
| |
|
| But wait, an interesting pattern emerges:
| | [[29ed5|29ED5]] focuses on 17/5 |
|
| |
|
| 22ed5 focuses on 9/5
| |
| 27ed5 focuses on 13/5
| |
| 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: 1/1 0.000 unison, perfect prime || || ||
| | and behold, of the lot, [[39ed5|39ED5]] offers the best balance between those intervals. |
| || 1: 163.901 cents 163.901 || || ||
| |
| || 2: 327.802 cents 327.802 || || ||
| |
| || 3: 491.702 cents 491.702 || || ||
| |
| || 4: 655.603 cents 655.603 || || ||
| |
| || 5: 819.504 cents 819.504 || || ||
| |
| || 6: 983.405 cents 983.405 || || ||
| |
| || 7: 1147.306 cents 1147.306 || || ||
| |
| || 8: 1311.206 cents 1311.206 || || ||
| |
| || 9: 1475.107 cents 1475.107 || || ||
| |
| || 10: 1639.008 cents 1639.008 || || ||
| |
| || 11: 1802.909 cents 1802.909 || || ||
| |
| || 12: 1966.810 cents 1966.810 || || ||
| |
| || 13: 2130.710 cents 2130.710 || || ||
| |
| || 14: 2294.611 cents 2294.611 || || ||
| |
| || 15: 2458.512 cents 2458.512 || || ||
| |
| || 16: 2622.413 cents 2622.413 || || ||
| |
| || 17: 5/1 2786.314 pentave || || ||</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 />
| |
| Suppose you go hunting for alternate tonalities, a temperament safari. The octave has been pretty well explored, not that there isn't a lot to be done. The tritave beckons - but Bohlen Pierce and friends have kind of found what you'd be looking for. While there's tremendous ground to cover there, you have the pioneering instinct in the first degree, so you examine the 5th harmonic as an equivalence.<br />
| |
| <br />
| |
| The first place to look is isoharmonic chords. In diatonic music we take 4:5:6:(7):8 and get the sublime meantone, and assorted other possibilities. In the tritave one finds that 1:2:3 looks back to the octave - although <a class="wiki_link" href="/11edt">11edt</a> does it very prettily. But when you crack open that 2/1 shell, you get to the meat of the tritave - <a class="wiki_link" href="/Bohlen-Pierce">Bohlen-Pierce</a>- based on 3:5:7:9 isoharmony. In the pentave (5/1 and I am perpetuating the technically incorrect terminology, because it works) the first thing you find is 1:2:3:4:5 which is a good lock combination, but not the best pentave-specific chord. But if you pry it open like a clam you'll find a pearl - 5:9:13:17:21:25 a stunning consonance. The full pentad is a little complex, so you might expect to leave out the 21st, just like one does the 7th in diatonicism.<br />
| |
| <br />
| |
| So what does one do with this? Well it took me long enough to stumble on the answer... a year... Anything worth hearing is worth taking the 17th root of. Because bizarrely enough, the pattern known as &quot;superpyth&quot; in an octave context, is the key to tempering together these cosmic harmonies in the pentave! So I dub it &quot;hyperpyth&quot; but if you want to call it &quot;kosmorsky&quot; that's fine by me too. Musical examples forthcoming.<br />
| |
| <br />
| |
| 17ed5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. Nevertheless it's the simplest equal hyperpyth, and quite consonant, like rolling clouds of colorful interstellar dust. However, in hyperpyth, 21/5 isn't necessarily represented, at least not well - which is the case in 17ed5. Luckily, 27, 29, and 39 do a fair job of it. I wonder if there is an alternate pentave system involving 11 and 7 to correspond to octaves' &quot;<a class="wiki_link" href="/26edo">Orgone</a>&quot;? I bet there is and I'll look for it soon.<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: 1/1 0.000 unison, perfect prime<br />
| | | steps = 17 |
| </td>
| | | num = 5 |
| <td><br />
| | | denom = 1 |
| </td>
| | }} |
| <td><br />
| | {{Harmonics in equal |
| </td>
| | | steps = 17 |
| </tr>
| | | num = 5 |
| <tr>
| | | denom = 1 |
| <td>1: 163.901 cents 163.901<br />
| | | start = 12 |
| </td>
| | | collapsed = 1 |
| <td><br />
| | }} |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2: 327.802 cents 327.802<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3: 491.702 cents 491.702<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4: 655.603 cents 655.603<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5: 819.504 cents 819.504<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6: 983.405 cents 983.405<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7: 1147.306 cents 1147.306<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8: 1311.206 cents 1311.206<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9: 1475.107 cents 1475.107<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10: 1639.008 cents 1639.008<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11: 1802.909 cents 1802.909<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12: 1966.810 cents 1966.810<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13: 2130.710 cents 2130.710<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14: 2294.611 cents 2294.611<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15: 2458.512 cents 2458.512<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16: 2622.413 cents 2622.413<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17: 5/1 2786.314 pentave<br />
| |
| </td>
| |
| <td><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)
|