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Prime harmonic series: Difference between revisions

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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.
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{| 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)
|}
|}


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{| 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
|}
|}


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|54.964428
|54.964428
|}
|}
[[Category:harmonic_series]]
 
[[Category:intervals]]
[[Category:Harmonic series]]
[[Category:just]]
[[Category:Lists of intervals]]
[[Category:prime]]
[[Category:Just intonation]]
[[Category:theory]]
[[Category:Prime]]
[[Category:Xenharmonic series]]
Retrieved from "https://en.xen.wiki/w/Prime_harmonic_series"