John Grieve: SuperParticular #s    
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8 Jun 2010 @ 05:34, by John Grieve

Number Theory/ Music/ Pythagorean

Superparticular number
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Superparticular numbers, also called epimoric ratios, are ratios of the form

Superparticular numbers were written about by Nicomachus in his treatise "Introduction to Arithmetic". They are useful in the study of harmony: many musical intervals can be expressed as a superparticular ratio. In this application, Størmer's theorem can be used to list all possible superparticular numbers for a given limit; that is, all ratios of this type in which both the numerator and denominator are smooth numbers.

In graph theory, superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.) arise as the possible values of the upper density of an infinite graph.

These ratios are also important in visual harmony – most flags of the world's countries have a ratio of 3:2 between their length and width, aspect ratios of 4:3, and 3:2 are common in digital photography, and aspect ratios of 7:6 and 5:4 are used in medium format and large format photography respectively.

The root of the names comes from Latin sesqui- "one and a half" (from semis "a half" -que "and") describing the ratio 3:2.

Examples Ratio Name Related musical interval
2:1 duplex octave
3:2 sesquialterum perfect fifth
4:3 sesquitertium perfect fourth
5:4 sesquiquartum major third
6:5 sesquiquintum minor third
9:8 sesquioctavum major second
10:9 minor tone
16:15 just diatonic semitone
25:24 just chromatic semitone
81:80 syntonic comma

[edit] See also
Mathematics of musical scales
[edit] References
Halsey, G. D.; Hewitt, Edwin (1972). "More on the superparticular ratios in music". American Mathematical Monthly (Mathematical Association of America) 79 (10): 1096–1100. doi:10.2307/2317424. MR0313189. [link]
[edit] External links
An Arithmetical Rubric by Siemen Terpstra, about the application of superparticular numbers to harmony.
Superparticular numbers applied to construct pentatonic scales by David Canright.
De Institutione Arithmetica, liber II by Anicius Manlius Severinus Boethius
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6 Jul 2016 @ 03:29 by king king @ : king  

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