81/80: Difference between revisions
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* The amount by which [[27/25]] exceeds [[16/15]]. | * The amount by which [[27/25]] exceeds [[16/15]]. | ||
* The amount by which [[16/15]] exceeds [[256/243]]. | * The amount by which [[16/15]] exceeds [[256/243]]. | ||
== Notation == | == Notation == | ||
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=== Sagittal notation === | === Sagittal notation === | ||
In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \! }} and is called the '''5 comma''', or '''5C''' for short, because the simplest interval it notates is 5/1 (equiv. 5/4), as for example in C–E{{nbhsp}}{{sagittal | \! }}. The upward version is called '''1/5C''' or '''5C up''' and is represented by {{sagittal| /| }}. | In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \! }} and is called the '''5 comma''', or '''5C''' for short, because the simplest interval it notates is 5/1 (equiv. 5/4), as for example in C–E{{nbhsp}}{{sagittal | \! }}. The upward version is called '''1/5C''' or '''5C up''' and is represented by {{sagittal| /| }}. | ||
== Approximation == | |||
If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w|appoggiatura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave, thus tempering out the [[barium comma]] ({{monzo| -225 224 -56 }}). | |||
== Relations to other superparticular ratios == | == Relations to other superparticular ratios == | ||
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Names in brackets refer to 7-limit [[meantone family|meantone extensions]], or 11-limit rank-3 temperaments from the [[didymus rank-3 family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.) | Names in brackets refer to 7-limit [[meantone family|meantone extensions]], or 11-limit rank-3 temperaments from the [[didymus rank-3 family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.) | ||
{| class="wikitable" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ Relations between 81/80 and other superparticular ratios | |||
|- | |- | ||
! Limit | ! Limit | ||
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* [[40/27]] – its [[fifth complement]] | * [[40/27]] – its [[fifth complement]] | ||
* [[1ed81/80]] – its equal multiplication | * [[1ed81/80]] – its equal multiplication | ||
* [[Pythagorean comma]] | * [[Pythagorean comma]] | ||
* [[64/63]] – the septimal comma or Archytas' comma | |||
* [[Small comma]] | * [[Small comma]] | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||