16edo: Difference between revisions

In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though there are exceptions: it has a [[7/4|7/1]] which is only six cents sharp, and a [[5/4|5/1]] which is only eleven cents flat. Most low harmonics are tuned very flat, but some such as [[21/16|21]]:[[11/8|22]]:[[23/16|23]]:[[3/2|24]]:[[25/16|25]]:[[13/8|26]] are well in tune with each other. Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and gamelans, with [[25edt]], [[41ed6]], and [[57ed12]] being good options.

Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), and thus the familiar [[diminished seventh chord]] may be built on any scale step with 4 unique tetrads up to [[octave equivalence]].

16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first defines sharp/flat, major/minor and aug/dim in terms of the native antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules (see [[16edo#Interval_arithmetic_examples]]). Chord names don't follow diatonic nominals because {{dash|C, E, G|med}} is not {{dash|P1, M3, P5|med}}. (But see below in "Chord Names".)

 

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

Mos scales like Mavila[7] (or "inverse/anti-diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale, while maintaining conventional A–G ♯/♭ notation as described above. Alternatively, one can utilize the Mavila[9] mos, for a sort of "hyper-diatonic" scale of 7 large steps and 2 small steps. [[Armodue theory|Armodue notation]] of 16edo "Mavila[9] Staff" does just this, and places the arrangement (222122221) on nine white "natural" keys of the 16edo keyboard. If the 9-note (enneatonic) mos is adopted as a notational basis for 16edo, then we need an entirely different set of interval classes than any of the heptatonic classes described above; perhaps it even makes sense to refer to the octave ([[2/1]]) as the "[[decave]]".

For translations of parts of the Armodue pages see the [[Armodue]] on this wiki

16edo chords can be named using ups and downs. Using diatonic interval names, the names correspond to diatonic names, but they bear little relationship to the sound: a minor chord (spelled {{dash|A, C, E|med}}) sounds like [[4:5:6]], the classical major triad, and a major chord (spelled {{dash|C, E, G|med}}) sounds like [[10:12:15]], a classical minor triad! Instead, using antidiatonic names, the chord names will match the sound—but finding the name from the spelling follows the rules of antidiatonic rather than diatonic interval arithmetic.

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| [[magic]]/[[muggles]]

* Pathological [[magic]] chromatic 11121121112 3L10s (5\16, 1\1)

Because 16edo does not approximate 3/2 well at all, triadic harmony based on heptatonic thirds is not a great option for typical harmonic timbres.

 

However, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16edo approximates 7/4 well enough to use

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