Xenharmonic Wiki

Prime harmonic series: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>FREEZE
No edit summary
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,232 edits
m Added to category "xenharmonic series"
 
(6 intermediate revisions by 4 users not shown)
Line 1: Line 1:
The '''acoustic prime harmonic series''' is similar to the set of [[prime_numbers|prime numbers]], except that it begins with 1, and skips 2 because of [[octave_equivalence|octave equivalence]] : 1, 3, 5, 7, 11, 13 etc.
The '''acoustic prime harmonic series''' is similar to the set of [[prime number]]s, 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.
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.
Line 9: Line 9:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | N (primes)
| N (primes)
| | scale
| scale
|-
|-
| | 1 (1)
| 1 (1)
| | 1/1
| 1/1
|-
|-
| | 2 (1,3)
| 2 (1,3)
| | 1/1, 3/2
| 1/1, 3/2
|-
|-
| | 3 (1,3,5)
| 3 (1,3,5)
| | 1/1, 5/4, 3/2
| 1/1, 5/4, 3/2
|-
|-
| | 4 (1,3,5,7)
| 4 (1,3,5,7)
| | 1/1, 5/4, 3/2, 7/4
| 1/1, 5/4, 3/2, 7/4
|-
|-
| | 5 (1,3,5,7,11)
| 5 (1,3,5,7,11)
| | 1/1, 5/4, 11/8, 3/2, 7/4 (pentatonic)
| 1/1, 5/4, 11/8, 3/2, 7/4 (pentatonic)
|-
|-
| | 6 (1,3,5,7,11,13)
| 6 (1,3,5,7,11,13)
| | 1/1, 5/4, 11/8, 3/2, 13/8, 7/4 (hexatonic)
| 1/1, 5/4, 11/8, 3/2, 13/8, 7/4 (hexatonic)
|-
|-
| | 7 (1,3,5,7,11,13,17)
| 7 (1,3,5,7,11,13,17)
| | 1/1, 17/16, 5/4, 11/8, 3/2, 13/8, 7/4 (heptatonic)
| 1/1, 17/16, 5/4, 11/8, 3/2, 13/8, 7/4 (heptatonic)
|-
|-
| | 8 (1,3,5,7,11,13,17,19)
| 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)
| 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)
| 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)
| 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)
| 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)
| 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)
| 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)
| 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)
| 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)
| 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)
|}
|}


Line 61: Line 61:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | 17/16
| 17/16
| | 20/17
| 20/17
| | 22/20
| 22/20 <br> (11/10)
 
| 24/22 <br> (12/11)
(11/10)
| 26/24 <br> (13/12)
| | 24/22
| 28/26 <br> (14/13)
 
| 32/28 <br> (8/7)
(12/11)
| | 26/24
 
(13/12)
| | 28/26
 
(14/13)
| | 32/28
 
(8/7)
|-
|-
| | 104.96
| 104.96
| | 281.36
| 281.36
| | 165
| 165
| | 150.64
| 150.64
| | 138.57
| 138.57
| | 128.3
| 128.3
| | 231.17
| 231.17
|}
|}


Line 96: Line 86:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | 34/32  
| 34/32 <br> (17/16)
 
| 37/34
(17/16)
| 38/37
| | 37/34
| 40/38 <br> (20/19)
| | 38/37
| 44/40 <br> (11/10)
| | 40/38
| 46/44 <br> (23/22)
 
| 48/46 <br> (24/23)
(20/19)
| 52/48 <br> (13/12)
| | 44/40  
| 56/52 <br> (14/13)
 
| 58/56 <br> (29/28)
(11/10)
| 62/58 <br> (31/29)
| | 46/44  
| 64/62 <br> (32/31)
 
(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
| 104.96
| | 146.39
| 146.39
| | 46.17
| 46.17
| | 88.8
| 88.8
| | 165
| 165
| | 76.96
| 76.96
| | 73.68
| 73.68
| | 138.57
| 138.57
| | 128.3
| 128.3
| | 60.75
| 60.75
| | 115.46
| 115.46
| | 54.97
| 54.97
|}
|}


Line 163: Line 133:
<span style="font-size: 90%;">Because the lowest partial is 17, there are no familiar lower prime limit intervals like octaves (2/1), fifths (3/2), thirds (5/4), etc. Nevertheless, some intervals between higher primes approximate the lower prime limit intervals (e.g. 61/31 is very close to an octave), without of course reproducing them exactly." [http://users.rcn.com/dante.interport//justguitar.html (original page source)]</span>
<span style="font-size: 90%;">Because the lowest partial is 17, there are no familiar lower prime limit intervals like octaves (2/1), fifths (3/2), thirds (5/4), etc. Nevertheless, some intervals between higher primes approximate the lower prime limit intervals (e.g. 61/31 is very close to an octave), without of course reproducing them exactly." [http://users.rcn.com/dante.interport//justguitar.html (original page source)]</span>


<span style="font-size: 90%;">[http://www.youtube.com/watch?v=tP9iafbjlOw "Tarkovsky's Mirror" for prime guitar by Dante Rosati]</span>     [[Category:harmonic_series]]
<span style="font-size: 90%;">[http://www.youtube.com/watch?v=tP9iafbjlOw "Tarkovsky's Mirror" for prime guitar by Dante Rosati]</span>
[[Category:intervals]]
 
[[Category:just]]
{| class="wikitable"
[[Category:prime]]
|+prime harmonics through the 6th octave of the harmonic series
[[Category:theory]]
|'''prime harmonic'''
|'''cents (octave reduced)'''
|'''sorted'''
|'''delta'''
|-
|2
|0
|0
|104.9554095
|-
|3
|701.9550009
|104.9554095
|146.3886293
|-
|5
|386.3137139
|251.3440388
|46.16897738
|-
|7
|968.8259065
|297.5130161
|88.80069773
|-
|11
|551.3179424
|386.3137139
|42.74869168
|-
|13
|840.5276618
|429.0624055
|82.4553001
|-
|17
|104.9554095
|511.5177056
|39.80023672
|-
|19
|297.5130161
|551.3179424
|76.9564049
|-
|23
|628.2743473
|628.2743473
|37.23227474
|-
|29
|1029.577194
|665.506622
|36.44837885
|-
|31
|1145.035572
|701.9550009
|138.5726609
|-
|37
|251.3440388
|840.5276618
|32.97688371
|-
|41
|429.0624055
|873.5045455
|95.32136099
|-
|43
|511.5177056
|968.8259065
|60.75128768
|-
|47
|665.506622
|1029.577194
|29.59446508
|-
|53
|873.5045455
|1059.171659
|57.71314584
|-
|59
|1059.171659
|1116.884805
|28.15076739
|-
|61
|1116.884805
|1145.035572
|54.964428
|}
 
[[Category:Harmonic series]]
[[Category:Lists of intervals]]
[[Category:Just intonation]]
[[Category:Prime]]
[[Category:Xenharmonic series]]
Retrieved from "https://en.xen.wiki/w/Prime_harmonic_series"