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 [[Stretched_and_compressed_tuning|flat tendency]], 16et is best tuned with the octave approximately 5{{c}} sharp, slightly improving the accuracy of wide-voiced JI chords and [[rooted]] harmonics.

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. The disadvantage to this approach is that, 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. Chord names are different 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]]

! rowspan="2" | Approximate<br>rtios*

! colspan="2" | Melodic<br>(major wider than minor)

! colspan="2" | Harmonic<br>(major narrower than minor)

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 harmonic (circle-of-fifths) interval names, the names are easy to find, 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 melodic names, the chord names will match the sound&mdash;but finding the name from the spelling is more complicated (see below).

! colspan="3" | Harmonic

! colspan="3" | Melodic

Using melodic names, interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples:

| what chord is C E G♭ B♭?

| C major scale = C + M2 + M3<br>+ P4 + P5 + M6 + M7 + P8

| C minor scale = C + M2 + m3<br>+ P4 + P5 + m6 + m7 + P8

It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it very discordant, although playing this in the context of a larger chord, and with specialized timbres, can make this less noticeable.

The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents [[support]]s Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third.

16edo is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600¢), and a generator of a flat septimal major 2nd, for which 16edo uses 3\16. For this, there are mos scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16edo supports both, but is not a very accurate tuning of either).

From a harmonic series perspective, if we take 13\16 as a 7/4 ratio approximation, sharp by 6.174 cents, and take the 300-cent minor third as an approximation of the harmonic 19th ([[19/16]], approximately 297.5 cents), that can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .

The interval between the 28th &amp; 19th harmonics, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19

{{Uniform map|13|15.5|16.5}}

| [[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.

 

 

Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads".

The ssLsssL mode of Mavila[7] contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025¢. In Mavila[9], hard and soft triads cease to share a triad class, as 975¢ is a major 8th, while 1050¢ is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.

This Layout places mavila[7] on the black keys and mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower, as per the "harmonic notation" above. Simply swap sharps with flats for "melodic notation".

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Please explain this image. {{todo|annotate}}

'''Lumatone mapping'''