16edo: Difference between revisions

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16edo represents [[ntiscifer]] temperament.

== Theory ==

16edo is not especially good at representing most low-odd-limit musical intervals, but it has a [[7/4]] which is only six cents sharp, and a [[5/4]] which is only eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12edo, giving it four diminished seventh chords exactly like those of [[12edo]], and a diminished triad on each scale step.

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== Intervals ==

16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G is not P1 - M3 - P5. (But see below in "Chord Names".)

16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works - this shouldn't be surprising as conventional interval arithmetic is designed for meantone/(super)pythagorean systems and 16edo is neither - e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G is not P1 - M3 - P5. (But see below in "Chord Names".)

The second approach is to preserve the *harmonic* meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. ~~While this~~ approach may seem bizarre at first, interval arithmetic and chord names work ~~as usual~~. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".

The second approach is to preserve the *harmonic* meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. This approach may seem bizarre at first. However, it carries over the way interval arithmetic and chord names work from diatonic notation. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".

Alternatively, one can use Armodue nine-nominal notation; see [[Armodue theory]]

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 {{Infobox ET}}
 {{EDO intro|16}}
+edo represents [[ntiscifer]] temperament.
 == Theory ==
 edo is not especially good at representing most low-odd-limit musical intervals, but it has a [[7/4]] which is only six cents sharp, and a [[5/4]] which is only eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12edo, giving it four diminished seventh chords exactly like those of [[12edo]], and a diminished triad on each scale step.
@@ Line 14: / Line 17: @@
 == Intervals ==
-edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G is not P1 - M3 - P5. (But see below in "Chord Names".)
+edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works - this shouldn't be surprising as conventional interval arithmetic is designed for meantone/(super)pythagorean systems and 16edo is neither - e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G is not P1 - M3 - P5. (But see below in "Chord Names".)
-The second approach is to preserve the *harmonic* meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".
+The second approach is to preserve the *harmonic* meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. This approach may seem bizarre at first.  However, it carries over the way interval arithmetic and chord names work from diatonic notation. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".
 Alternatively, one can use Armodue nine-nominal notation; see [[Armodue theory]]