Interval size measure: Difference between revisions

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'''Interval size measure''' or '''interval size unit''' means the ''distance'' between pitches. Intervals can be measured [[#logarithmic|logarithmic]] or by frequency [[#ratio|ratios]].

== Logarithmic ==

All logarithmic measures can be combined by adding and subtracting them.

=== Backslash notation ===

A common shorthand in use in the microtonal community is ''k''\''N'', written with a backslash (\) instead of a forwardslash (/), to refer to an interval with a frequency ratio of 2''k''/''N''. ''k''\''N'' is pronounced "''k'' steps of ''N'' [[edo]]", and suffices to define the meaning of "[[step]]s" in the context of edos (unless talking about steps of specific subsets/scales of some edo).

It is logarithmic because it's common to see expressions like 11\19 - 6\19 = 5\19, which is an additive notation, though the frequency ratios referred to in the definition are linear, which is to say:

: ''a''\''N'' + ''b''\''N'' = (''a'' + ''b'')\''N'' expresses the same thing as 2''a''/''N'' × 2''b''/''N'' = 2(''a'' + ''b'')/''N''

Or equivalently, for subtraction/division:

: ''a''\''N'' - ''b''\''N'' = (''a'' - ''b'')\''N'' expresses the same thing as 2''a''/''N'' / 2''b''/''N'' = 2(''a'' - ''b'')/''N''

=== Gross ===

The [[octave]] and the [[decade]] are common coarse units for interval sizes. The {{w|decibel}}, being a relative logarithmic-scale unit for power or root-power quantities, is inappropriate for measuring intervals; the decade is used instead. Similarly, the {{w|neper}} (Np) and the dineper (dNp), like the decibel, should not be used. However, in the absence of a substitute, dinepers have an application in [[logarithmic approximants]].

Intervals are sometimes expressed in the number of scale steps between them. These steps can be of different size, compare for example the names of the major scale in the classic music. An early unit for measuring intervals is the "[[tone]]" which dates back to classic Greece.

In serial music, all intervals were measured by the number of 12edo [[semitone]]s. In analogy, the '''relative interval measure''' is the number of steps between two pitches of an [[equal tuning]], sometimes called "[[degree]]s". These measures can be written using ~~'''~~backslash notation~~'''~~, which looks like a frequency ratio but using a backslash (instead of a forward slash) to indicate a logarithmic ratio. For example, 11\15 means 11 steps of 15edo, 4\9edf means 4 steps of 9edf, and 16\21ed12/7 means 16 steps of 21ed12/7.

In serial music, all intervals were measured by the number of 12edo [[semitone]]s. In analogy, the '''relative interval measure''' is the number of steps between two pitches of an [[equal tuning]], sometimes called "[[degree]]s". These measures can be written using [[#Backslash notation|backslash notation]], which looks like a frequency ratio but using a backslash (instead of a forward slash) to indicate a logarithmic ratio. For example, 11\15 means 11 steps of 15edo, 4\9edf means 4 steps of 9edf, and 16\21ed12/7 means 16 steps of 21ed12/7.

=== Fine ===

@@ Line 1: / Line 1: @@
 '''Interval size measure''' or '''interval size unit''' means the ''distance'' between pitches. Intervals can be measured [[#logarithmic|logarithmic]] or by frequency [[#ratio|ratios]].
 == Logarithmic ==
 All logarithmic measures can be combined by adding and subtracting them.
+=== Backslash notation ===
+A common shorthand in use in the microtonal community is ''k''\''N'', written with a backslash (\) instead of a forwardslash (/), to refer to an interval with a frequency ratio of 2<sup>''k''/''N''</sup>. ''k''\''N'' is pronounced "''k'' steps of ''N'' [[edo]]", and suffices to define the meaning of "[[step]]s" in the context of edos (unless talking about steps of specific subsets/scales of some edo).
+It is logarithmic because it's common to see expressions like 11\19 - 6\19 = 5\19, which is an additive notation, though the frequency ratios referred to in the definition are linear, which is to say:
+: ''a''\''N'' + ''b''\''N'' = (''a'' + ''b'')\''N'' expresses the same thing as 2<sup>''a''/''N''</sup> × 2<sup>''b''/''N''</sup> = 2<sup>(''a'' + ''b'')/''N''</sup>
+Or equivalently, for subtraction/division:
+: ''a''\''N'' - ''b''\''N'' = (''a'' - ''b'')\''N'' expresses the same thing as 2<sup>''a''/''N''</sup> / 2<sup>''b''/''N''</sup> = 2<sup>(''a'' - ''b'')/''N''</sup>
 === Gross ===
 The [[octave]] and the [[decade]] are common coarse units for interval sizes. The {{w|decibel}}, being a relative logarithmic-scale unit for power or root-power quantities, is inappropriate for measuring intervals; the decade is used instead. Similarly, the {{w|neper}} (Np) and the dineper (dNp), like the decibel, should not be used. However, in the absence of a substitute, dinepers have an application in [[logarithmic approximants]].
 Intervals are sometimes expressed in the number of scale steps between them. These steps can be of different size, compare for example the names of the major scale in the classic music. An early unit for measuring intervals is the "[[tone]]" which dates back to classic Greece.
-In serial music, all intervals were measured by the number of 12edo [[semitone]]s. In analogy, the '''relative interval measure''' is the number of steps between two pitches of an [[equal tuning]], sometimes called "[[degree]]s". These measures can be written using '''backslash notation''', which looks like a frequency ratio but using a backslash (instead of a forward slash) to indicate a logarithmic ratio. For example, 11\15 means 11 steps of 15edo, 4\9edf means 4 steps of 9edf, and 16\21ed12/7 means 16 steps of 21ed12/7.
+In serial music, all intervals were measured by the number of 12edo [[semitone]]s. In analogy, the '''relative interval measure''' is the number of steps between two pitches of an [[equal tuning]], sometimes called "[[degree]]s". These measures can be written using [[#Backslash notation|backslash notation]], which looks like a frequency ratio but using a backslash (instead of a forward slash) to indicate a logarithmic ratio. For example, 11\15 means 11 steps of 15edo, 4\9edf means 4 steps of 9edf, and 16\21ed12/7 means 16 steps of 21ed12/7.
 === Fine ===