81/80: Difference between revisions

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== Relations to other 5-limit intervals ==

[[81/80]] is the difference between a large number of intervals of the 5-limit, so that if tempered, it simplifies the structure of the 5-limit drastically. For some differences in higher limits, see [[#Relations to other superparticular ratios]]. A few important ones are that 81/80 is:

81/80 is the difference between a large number of intervals of the 5-limit, so that if tempered, it simplifies the structure of the 5-limit drastically. For some differences in higher limits, see [[#Relations to other superparticular ratios]]. A few important ones are that 81/80 is:

* The amount by which [[2187/2048]] exceeds [[135/128]].

* The amount by which [[25/24]] exceeds [[250/243]].

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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.

== Sagittal notation ==

== 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| /| }}.

This interval is significant in the [[Functional Just System]] and [[Helmholtz-Ellis notation]] as the classical (5-limit) formal comma which translates a Pythagorean interval to a nearby classical interval.

=== Ben Johnston's notation ===

In [[Ben Johnston's notation]], this interval is denoted with "+" and its reciprocal with "-".

=== 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| /| }}.

== Relations to other superparticular ratios ==

@@ Line 21: / Line 21: @@
 == Relations to other 5-limit intervals ==
-[[81/80]] is the difference between a large number of intervals of the 5-limit, so that if tempered, it simplifies the structure of the 5-limit drastically. For some differences in higher limits, see [[#Relations to other superparticular ratios]]. A few important ones are that 81/80 is:
+/80 is the difference between a large number of intervals of the 5-limit, so that if tempered, it simplifies the structure of the 5-limit drastically. For some differences in higher limits, see [[#Relations to other superparticular ratios]]. A few important ones are that 81/80 is:
 * The amount by which [[2187/2048]] exceeds [[135/128]].
 * The amount by which [[25/24]] exceeds [[250/243]].
@@ Line 32: / Line 32: @@
 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.
-== Sagittal notation ==
+== 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| /| }}.
+This interval is significant in the [[Functional Just System]] and [[Helmholtz-Ellis notation]] as the classical (5-limit) formal comma which translates a Pythagorean interval to a nearby classical interval.
+=== Ben Johnston's notation ===
+In [[Ben Johnston's notation]], this interval is denoted with "+" and its reciprocal with "-".
+=== 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| /| }}.
 == Relations to other superparticular ratios ==