Prime harmonic series
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author danterosati and made on 2010-11-06 00:07:05 UTC.
- The original revision id was 176990617.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
The **acoustic prime harmonic series** is similar to the set of prime numbers, except that it begins with 1, and skips 2 because of octave equivalence : 1,3,5,7,11,13…etc. If “new” pitch classes in the harmonic series are always odd numbers (even numbers are always octave duplications), the question is whether there is a useful acoustic/musical distinction between odd composites and primes. The test case is 9, which is the first odd numbered partial that is composite. Suppose one is producing notes with harmonic spectra whose fundamentals are members of an harmonic series. Each note that is played also exhibits its own series of partials as its timbre. Some of these partials may coincide with member of the original harmonic series. For example, a note whose fundamental is the 3rd member of an harmonic series will have its own third partial coincident with the 9th partial of the original series. (9 = 3 x 3). It can be shown that all further odd composite members of the original series will show up in the spectra of notes built on lower partials (e.g. 15 = 5 x 3 = 3 x 5). On the other hand, prime numbered partials of the original series do not occur in notes built on lower partials. So, do composites exhibit a kind of redundancy? Does this result in some kind of qualitative/musical difference between primes and composites, and the notes built upon them? Is the derivative nature of the composite members audible? These are questions which may lie in the realm of subjectivity, but one thing is certain: 7 and 11 sound more unusual to our ears than 9 does. If odd limit as a measure of consonance was all that mattered, wouldn’t there be a continuum of some kind from 7 to 9 to 11? Instead, 9 sounds like it belongs with its 3-limit brothers 3/2 and 4/3 far more than it does with 7 or 11. Fortunately, it is not necessary to wait upon the answers to these admittedly very interesting questions before finding interesting uses for the prime harmonic series as material for scale generation. For example, through the magic of octave reduction, the first n members of the prime harmonic series can produce an n-atonic scale: || N (primes) || scale || || 1 (1) || 1/1 || || 2 (1,3) || 1/1, 3/2 || || 3 (1,3,5) || 1/1, 5/4, 3/2 || || 4 (1,3,5,7) || 1/1, 5/4, 3/2, 7/4 || || 5 (1,3,5,7,11) || 1/1, 5/4, 11/8, 3/2, 7/4 (pentatonic) || || 6 (1,3,5,7,11,13) || 1/1, 5/4, 11/8, 3/2, 13/8, 7/4 (hexatonic) || || 7 (1,3,5,7,11,13,17) || 1/1, 17/16, 5/4, 11/8, 3/2, 13/8, 7/4 (heptatonic) || || 8 (1,3,5,7,11,13,17,19) || 1/1, 17/16, 19/16, 5/4, 11/8, 3/2, 13/8, 7/4 (octatonic) || || 9 (1,3,5,7,11,13,17,19,23) || 1/1, 17/16, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4 (nonotonic) || || 10 (1,3,5,7,11,13,17,19,23,29) || 1/1, 17/16, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4, 29/16 (decatonic) || || 11 (1,3,5,7,11,13,17,19,23,29,31) || 1/1, 17/16, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4, 29/16, 31/16 (hendecatonic) || || 12 (1,3,5,7,11,13,17,19,23,29,31,37) || 1/1, 17/16, 37/32, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4, 29/16, 31/16 (dodecatonic) || Of course, as in scales derived from the full series, there is nothing that says one must start at the beginning of the series or include consecutive members only. The possibilities are endless. ---- The prime heptatonic scale can be notated like this: 16:17:20:22:24:26:28:(32) giving the following step sizes: || 17/16 || 20/17 || 22/20 (11/10) || 24/22 (12/11) || 26/24 (13/12) || 28/26 (14/13) || 32/28 (8/7) || || 104.96 || 281.36 || 165 || 150.64 || 138.57 || 128.3 || 231.17 || The prime dodecatonic scale can be notated: 32:34:37:38:40:44:46:48:52:56:58:62:(64) giving these step sizes: || 34/32 (17/16) || 37/34 || 38/37 || 40/38 (20/19) || 44/40 (11/10) || 46/44 (23/22) || 48/46 (24/23) || 52/48 (13/12) || 56/52 (14/13) || 58/56 (29/28) || 62/58 (31/29) || 64/62 (32/31) || || 104.96 || 146.39 || 46.17 || 88.8 || 165 || 76.96 || 73.68 || 138.57 || 128.3 || 60.75 || 115.46 || 54.97 ||
Original HTML content:
<html><head><title>The Prime Harmonic Series</title></head><body>The <strong>acoustic prime harmonic series</strong> is similar to the set of prime numbers, except that it begins with 1, and skips 2 because of octave equivalence : 1,3,5,7,11,13…etc.<br />
If “new” pitch classes in the harmonic series are always odd numbers (even numbers are always octave duplications), the question is whether there is a useful acoustic/musical distinction between odd composites and primes. The test case is 9, which is the first odd numbered partial that is composite.<br />
<br />
Suppose one is producing notes with harmonic spectra whose fundamentals are members of an harmonic series. Each note that is played also exhibits its own series of partials as its timbre. Some of these partials may coincide with member of the original harmonic series. For example, a note whose fundamental is the 3rd member of an harmonic series will have its own third partial coincident with the 9th partial of the original series. (9 = 3 x 3). It can be shown that all further odd composite members of the original series will show up in the spectra of notes built on lower partials (e.g. 15 = 5 x 3 = 3 x 5). On the other hand, prime numbered partials of the original series do not occur in notes built on lower partials. So, do composites exhibit a kind of redundancy? Does this result in some kind of qualitative/musical difference between primes and composites, and the notes built upon them? Is the derivative nature of the composite members audible? These are questions which may lie in the realm of subjectivity, but one thing is certain: 7 and 11 sound more unusual to our ears than 9 does. If odd limit as a measure of consonance was all that mattered, wouldn’t there be a continuum of some kind from 7 to 9 to 11? Instead, 9 sounds like it belongs with its 3-limit brothers 3/2 and 4/3 far more than it does with 7 or 11.<br />
<br />
Fortunately, it is not necessary to wait upon the answers to these admittedly very interesting questions before finding interesting uses for the prime harmonic series as material for scale generation. For example, through the magic of octave reduction, the first n members of the prime harmonic series can produce an n-atonic scale:<br />
<br />
<table class="wiki_table">
<tr>
<td>N (primes)<br />
</td>
<td>scale<br />
</td>
</tr>
<tr>
<td>1 (1)<br />
</td>
<td>1/1<br />
</td>
</tr>
<tr>
<td>2 (1,3)<br />
</td>
<td>1/1, 3/2<br />
</td>
</tr>
<tr>
<td>3 (1,3,5)<br />
</td>
<td>1/1, 5/4, 3/2<br />
</td>
</tr>
<tr>
<td>4 (1,3,5,7)<br />
</td>
<td>1/1, 5/4, 3/2, 7/4<br />
</td>
</tr>
<tr>
<td>5 (1,3,5,7,11)<br />
</td>
<td>1/1, 5/4, 11/8, 3/2, 7/4 (pentatonic)<br />
</td>
</tr>
<tr>
<td>6 (1,3,5,7,11,13)<br />
</td>
<td>1/1, 5/4, 11/8, 3/2, 13/8, 7/4 (hexatonic)<br />
</td>
</tr>
<tr>
<td>7 (1,3,5,7,11,13,17)<br />
</td>
<td>1/1, 17/16, 5/4, 11/8, 3/2, 13/8, 7/4 (heptatonic)<br />
</td>
</tr>
<tr>
<td>8 (1,3,5,7,11,13,17,19)<br />
</td>
<td>1/1, 17/16, 19/16, 5/4, 11/8, 3/2, 13/8, 7/4 (octatonic)<br />
</td>
</tr>
<tr>
<td>9 (1,3,5,7,11,13,17,19,23)<br />
</td>
<td>1/1, 17/16, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4 (nonotonic)<br />
</td>
</tr>
<tr>
<td>10 (1,3,5,7,11,13,17,19,23,29)<br />
</td>
<td>1/1, 17/16, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4, 29/16 (decatonic)<br />
</td>
</tr>
<tr>
<td>11 (1,3,5,7,11,13,17,19,23,29,31)<br />
</td>
<td>1/1, 17/16, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4, 29/16, 31/16 (hendecatonic)<br />
</td>
</tr>
<tr>
<td>12 (1,3,5,7,11,13,17,19,23,29,31,37)<br />
</td>
<td>1/1, 17/16, 37/32, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4, 29/16, 31/16 (dodecatonic)<br />
</td>
</tr>
</table>
<br />
Of course, as in scales derived from the full series, there is nothing that says one must start at the beginning of the series or include consecutive members only. The possibilities are endless.<br />
<br />
<br />
<hr />
<br />
The prime heptatonic scale can be notated like this:<br />
<br />
16:17:20:22:24:26:28:(32)<br />
<br />
giving the following step sizes:<br />
<br />
<table class="wiki_table">
<tr>
<td>17/16<br />
</td>
<td>20/17<br />
</td>
<td>22/20<br />
(11/10)<br />
</td>
<td>24/22<br />
(12/11)<br />
</td>
<td>26/24<br />
(13/12)<br />
</td>
<td>28/26<br />
(14/13)<br />
</td>
<td>32/28<br />
(8/7)<br />
</td>
</tr>
<tr>
<td>104.96<br />
</td>
<td>281.36<br />
</td>
<td>165<br />
</td>
<td>150.64<br />
</td>
<td>138.57<br />
</td>
<td>128.3<br />
</td>
<td>231.17<br />
</td>
</tr>
</table>
<br />
The prime dodecatonic scale can be notated:<br />
<br />
32:34:37:38:40:44:46:48:52:56:58:62:(64)<br />
<br />
giving these step sizes:<br />
<br />
<table class="wiki_table">
<tr>
<td>34/32 <br />
(17/16)<br />
</td>
<td>37/34<br />
</td>
<td>38/37<br />
</td>
<td>40/38<br />
(20/19)<br />
</td>
<td>44/40 <br />
(11/10)<br />
</td>
<td>46/44 <br />
(23/22)<br />
</td>
<td>48/46 <br />
(24/23)<br />
</td>
<td>52/48 <br />
(13/12)<br />
</td>
<td>56/52 <br />
(14/13)<br />
</td>
<td>58/56 <br />
(29/28)<br />
</td>
<td>62/58 <br />
(31/29)<br />
</td>
<td>64/62 <br />
(32/31)<br />
</td>
</tr>
<tr>
<td>104.96<br />
</td>
<td>146.39<br />
</td>
<td>46.17<br />
</td>
<td>88.8<br />
</td>
<td>165<br />
</td>
<td>76.96<br />
</td>
<td>73.68<br />
</td>
<td>138.57<br />
</td>
<td>128.3<br />
</td>
<td>60.75<br />
</td>
<td>115.46<br />
</td>
<td>54.97<br />
</td>
</tr>
</table>
</body></html>