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List of superparticular intervals: Difference between revisions

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Added some 23-limit intervals.
Complete 23-limit intervals.
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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 [[23-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 [[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.
[[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.
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! colspan="6" | 23-limit (incomplete)
! colspan="6" | 23-limit (complete)
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| [[23/22]]
| [[23/22]]
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|70/69 to 69/68
|70/69 to 69/68
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|5083/5082
|0.34063
|(13*17*23)/(2*3*7*11<sup>2</sup>)
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|7866/7865
|0.22010
|(2*3<sup>2</sup>*19*23)/(5*11<sup>2</sup>*13)
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|8281/8280
|0.20907
|(7<sup>2</sup>*13<sup>2</sup>)/(2<sup>3</sup>*3<sup>2</sup>*5*23)
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|92/91 to 91/90
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|8625/8624
|0.20073
|(3*5<sup>3</sup>*23)/(2<sup>4</sup>*7<sup>2</sup>*11)
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|10626/10625
|0.16293
|(2*3*7*11*23)/(5<sup>4</sup>*17)
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|11271/11270
|0.15361
|(3*13*17<sup>2</sup>)/(2*5*7<sup>2</sup>*23)
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|11662/11661
|0.14846
|(2*7<sup>3</sup>*17)/(3*13<sup>2</sup>*23)
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|12168/12167
|0.14228
|(2<sup>3</sup>*3<sup>2</sup>*13<sup>2</sup>)/(23<sup>3</sup>)
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|16929/16928
|0.10227
|(3<sup>4</sup>*11*19)/(2<sup>5</sup>*23<sup>2</sup>)
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|19551/19550
|0.088552
|(3*7<sup>3</sup>*19)/(2*5<sup>2</sup>*17*23)
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|21505/21504
|0.080506
|(5*11*17*23)/(2<sup>10</sup>*3*7)
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|25921/25920
|0.066790
|(7<sup>2</sup>*23<sup>2</sup>)/(2<sup>6</sup>*3<sup>4</sup>*5)
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|162/161 to 161/160
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|43264/43263
|0.040016
|(2<sup>8</sup>*13<sup>2</sup>)/(3<sup>2</sup>*11*19*23)
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|209/208 to 208/207
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|52326/52325
|0.033086
|(2*3<sup>4</sup>*17*19)/(5<sup>2</sup>*7*13*23)
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|71875/71874
|0.024087
|(5<sup>5</sup>*23)/(2*3<sup>3</sup>*11<sup>3</sup>)
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|75141/75140
|0.023040
|(3<sup>3</sup>*11<sup>2</sup>*23)/(2<sup>2</sup>*5*13*17<sup>2</sup>)
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|76545/76544
|0.022617
|(3<sup>7</sup>*5*7)/(2<sup>8</sup>*13*23)
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|104329/104328
|0.016594
|(17<sup>2</sup>*19<sup>2</sup>)/(2<sup>3</sup>*3<sup>4</sup>*7*23)
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|324/323 to 323/322
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|122452/122451
|0.014138
|(2<sup>2</sup>*11<sup>3</sup>*23)/(3*7<sup>4</sup>*17)
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|126225/126224
|0.013716
|(3<sup>3</sup>*5<sup>2</sup>*11*17)/(2<sup>4</sup>*7<sup>3</sup>*23)
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|152881/152880
|0.011324
|(17<sup>2</sup>*23<sup>2</sup>)/(2<sup>4</sup>*3*5*7<sup>2</sup>*13)
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|392/391 to 391/390
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|202125/202124
|0.0085652
|(3*5<sup>3</sup>*7<sup>2</sup>*11)/(2<sup>2</sup>*13<sup>3</sup>*23)
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|264385/264384
|0.0065482
|(5*11<sup>2</sup>*19*23)/(2<sup>6</sup>*3<sup>5</sup>*17)
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|282625/282624
|0.0061256
|(5<sup>3</sup>*7*17*19)/(2<sup>1</sup><sup>2</sup>*3*23)
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|328510/328509
|0.0052700
|(2*5*7*13*19<sup>2</sup>)/(3<sup>3</sup>*23<sup>3</sup>)
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|2023425/2023424
|0.00085560
|(3<sup>2</sup>*5<sup>2</sup>*17*23<sup>2</sup>)/(2<sup>13</sup>*13*19)
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|4096576/4096575
|0.00042261
|(2<sup>6</sup>*11<sup>2</sup>*23<sup>2</sup>)/(3<sup>4</sup>*5<sup>2</sup>*7*17<sup>2</sup>)
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|2025/2024 to 2024/2023
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|5142501/5142500
|0.00033665
|(3<sup>3</sup>*7<sup>2</sup>*13<sup>2</sup>*23)/(2<sup>2</sup>*5<sup>4</sup>*11<sup>2</sup>*17)
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! colspan="6" | 29-limit (incomplete)
! colspan="6" | 29-limit (incomplete)
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