List of superparticular intervals: Difference between revisions
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This '''list of superparticular intervals''' ordered by prime limit. It reaches to the 101-limit and is complete up to the [[19-limit]]. | This '''list of superparticular intervals''' ordered by prime limit. It reaches to the 101-limit and is complete up to the [[19-limit]]. | ||
[[Superparticular]] numbers are ratios of the form (''n'' + 1)/''n'', or 1 + 1/''n'', where ''n'' is a whole number other than 1. They appear frequently in [[just intonation]] and [[ | [[Superparticular]] numbers are ratios of the form (''n'' + 1)/''n'', or 1 + 1/''n'', where ''n'' is a whole number other than 1. They appear frequently in [[just intonation]] and [[harmonic series]] music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[comma]]s are superparticular ratios. | ||
The list below is ordered by [[harmonic limit]], or the largest prime involved in the prime factorization. [[36/35]], for instance, is an interval of the [[7-limit]], as it factors to (2<sup>2</sup>×3<sup>2</sup>)/(5×7), while 37/36 would belong to the 37-limit. | The list below is ordered by [[harmonic limit]], or the largest prime involved in the prime factorization. [[36/35]], for instance, is an interval of the [[7-limit]], as it factors to (2<sup>2</sup>×3<sup>2</sup>)/(5×7), while 37/36 would belong to the 37-limit. | ||
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| 7/(2*3) | | 7/(2*3) | ||
| {{Monzo|-1 -1 0 1 }} | | {{Monzo|-1 -1 0 1 }} | ||
| (septimal) subminor third, septimal minor third | | (septimal) subminor third, septimal minor third | ||
|- | |- | ||
| [[8/7]] | | [[8/7]] | ||
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| 2<sup>3</sup>/7 | | 2<sup>3</sup>/7 | ||
| {{Monzo|3 0 0 -1}} | | {{Monzo|3 0 0 -1}} | ||
| (septimal) supermajor second, septimal whole tone | | (septimal) supermajor second, septimal whole tone, 7th subharmonic | ||
|- | |- | ||
| [[15/14]] | | [[15/14]] | ||
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| (3*7)/(2<sup>2</sup>*5) | | (3*7)/(2<sup>2</sup>*5) | ||
| {{Monzo|-2 1 -1 1}} | | {{Monzo|-2 1 -1 1}} | ||
| minor semitone, large septimal | | minor semitone, large septimal chroma | ||
|- | |- | ||
| [[28/27]] | | [[28/27]] | ||
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| (2<sup>2</sup>*7)/3<sup>3</sup> | | (2<sup>2</sup>*7)/3<sup>3</sup> | ||
| {{Monzo|2 -3 0 1}} | | {{Monzo|2 -3 0 1}} | ||
| septimal chroma, | | septimal third-tone, small septimal chroma, (septimal) subminor second, septimal minor second, trienstonic comma | ||
|- | |- | ||
| [[36/35]] | | [[36/35]] | ||
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| (3*7*13)/(2<sup>4</sup>*17) | | (3*7*13)/(2<sup>4</sup>*17) | ||
| {{Monzo|-4 1 0 1 0 1 -1}} | | {{Monzo|-4 1 0 1 0 1 -1}} | ||
| | | tannisma | ||
|- | |- | ||
| [[289/288]] | | [[289/288]] | ||
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| (3*19)/(2<sup>3</sup>*7) | | (3*19)/(2<sup>3</sup>*7) | ||
| {{Monzo|-3 1 0 -1 0 0 0 1}} | | {{Monzo|-3 1 0 -1 0 0 0 1}} | ||
| | | hendrix comma | ||
|- | |- | ||
| [[76/75]] | | [[76/75]] | ||