Superparticular ratio: Difference between revisions

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:<math>\frac{n + 1}{n} = 1 + \frac{1}{n}</math> where <math>n</math> is a [[Wikipedia:Positive integer|positive integer]].

Superparticular ratios appear frequently in [[just intonation]] and [[harmonic series]] music. Consecutive [[harmonic]]s are separated by superparticular ~~intervals~~: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the harmonics 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 ratios appear frequently in [[just intonation]] and [[harmonic series]] music. Consecutive [[harmonic]]s are separated by superparticular [[interval]]s: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the harmonics 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.

A ratio greater than 1 which is not superparticular is a [[superpartient ratio]].

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Superparticular ratios have some peculiar properties:

* The [[Wikipedia:Difference tone|difference tone]] of the ~~dyad~~ is also the [[Wikipedia:Missing fundamental|virtual fundamental]].

* The [[Wikipedia:Difference tone|difference tone]] of the interval is also the [[Wikipedia:Missing fundamental|virtual fundamental]].

* The first 6 such ratios ([[3/2]], [[4/3]], [[5/4]], [[6/5]], [[7/6]], [[8/7]]) are notable [[Harmonic Entropy|harmonic entropy]] minima.

* The logarithmic difference (i.e. quotient) between two successive superparticular ratios is always a superparticular ratio.

@@ Line 5: / Line 5: @@
 :<math>\frac{n + 1}{n} = 1 + \frac{1}{n}</math>  where <math>n</math> is a [[Wikipedia:Positive integer|positive integer]].
-Superparticular ratios appear frequently in [[just intonation]] and [[harmonic series]] music. Consecutive [[harmonic]]s are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the harmonics 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 ratios appear frequently in [[just intonation]] and [[harmonic series]] music. Consecutive [[harmonic]]s are separated by superparticular [[interval]]s: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the harmonics 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.
 A ratio greater than 1 which is not superparticular is a [[superpartient ratio]].
@@ Line 22: / Line 22: @@
 Superparticular ratios have some peculiar properties:
-* The [[Wikipedia:Difference tone|difference tone]] of the dyad is also the [[Wikipedia:Missing fundamental|virtual fundamental]].
+* The [[Wikipedia:Difference tone|difference tone]] of the interval is also the [[Wikipedia:Missing fundamental|virtual fundamental]].
 * The first 6 such ratios ([[3/2]], [[4/3]], [[5/4]], [[6/5]], [[7/6]], [[8/7]]) are notable [[Harmonic Entropy|harmonic entropy]] minima.
 * The logarithmic difference (i.e. quotient) between two successive superparticular ratios is always a superparticular ratio.