Kite Giedraitis's Categorizations of 41edo Scales: Difference between revisions

Kite's personal thoughts on 41-edo scales as they relate to [[the Kite Guitar]]. See also [[Kite Guitar Scales]].

== A review of 12-edo scales ==

There are three broad categories of 12-edo scales: pentatonic, diatonic and chromatic:

Imperfect degrees in 12-edo have two qualities, major and minor, and each one implies two [[Color notation|colors]].  

12-edo accurately represents only primes 2, 3 and 5 (as well as 17 and 19, and various other higher primes). 41-edo accurately represents primes 7, 11 and 13 as well. There are 7 qualities:

In [[color notation]], these subgroups are named wa = 2.3 = 3-limit, ya = 2.3.5 = 5-limit, za = 2.3.7, and ila = 2.3.11. The subgroups can combine, e.g. yaza = 2.3.5.7. Note that 7-limit includes both za and yaza. 41-edo doesn't distinguish between the ila subgroup and the tha subgroup 2.3.13, so tha is lumped in with ila.  

== 41-edo scales ==

41-edo has an enormous variety of scales. There are many thousands of unconventional scales, but we will focus on the ones that map compactly to the JI lattice. These are scales that contain numerous perfect 5ths. Two notes a perfect fifth apart generally have the same quality. So compact scales use only a few qualities, and thus a small prime subgroup.   

In practice, 41-edo scales tend to be "fuzzy", meaning that one or two scale notes may sometimes shift by an edostep. For example, a major scale may contain both a M2 and a vM2, and use whichever one is required by the harmony at the moment.

There are five broad categories of 41-edo scales: pentatonic, diatonic, semitonal, chromatic and microtonal.

A scale needn't have every single step size on the list in order to be in the category, just most of them. In practice, a non-wa pentatonic scale will often lack a m3 step, as in the examples. But a fuzzy pentatonic scale often will have a m3, e.g. C D vE G vA/A C. Ya and za scales generally contain a wolf 5th (either an ^5 or a v5), and would often become fuzzy to avoid the wolf.  

In addition to these broad categories, every 41-edo scale has a unique name that uses ups and downs. The 4 pentatonic examples above are major pentatonic, downmajor pentatonic, downminor pentatonic and double-upminor pentatonic.  

=== Semitonal, chromatic and microtonal scales ===

Most 41-edo intervals suggest a specific ratio, but those only a few edosteps wide don't. Thus the remaining categories don't imply any prime subgroups. On the Kite guitar, playing a run of notes one fret apart inherits the term "chromatic" from 12-edo. This term is especially appropriate if using color notation to describe 41-edo intervals, since chromatic means "colorful". Traditional 12-edo chromaticism, which translates to runs played on every other fret, is called semitonal, a conventional term referring to the 12-edo semitone. Microtonal scales differ from fuzzy scales in having many sequential ^1 intervals, and no steps larger than a vm2. Thus fuzzy means partly but not fully microtonal, and a fuzzy diatonic scale could be called a diatonic/microtonal scale.  

! colspan="2" |chromatic

From this we can deduce that chromatic scales are often 19 tones, and microtonal ones are often 22. We can also deduce that a semitonal scale of 12 notes usually has two vm2's. If there are more vm2's, the scale is semitonal/chromatic. Scales of 1, 2 and 3 edosteps are chromatic/microtonal.   

In Western music, harmonies often require notes that the melody doesn't. For example, "Auld Lang Syne" has a pentatonic melody but diatonic harmonies. Often the melody is diatonic but the harmonies are at least somewhat chromatic. The score will have accidentals in the piano part but not the vocal part. The scale used by the melody is the melodic scale, and the one used to construct chords is the harmonic scale. 41-edo yaza harmonic scales are usually semitonal or chromatic.   

In 12-edo, a song is generally in a major or minor key, and uses a major or minor scale. A ya piece in 41-edo often is as well. But unlike 12-edo, 41-edo allows the use of yaza chords such as 4:5:6:7. If this is one's tonic chord, both major and minor are used simultaneously. A simple Iv7 - IVv7 progression has both a downmajor 3rd and a downminor 3rd. Clearly the major/minor duality no longer applies. Instead, there is an up/down duality.     

For ya or za scales, one chooses a 7-note subset of the 12 notes, and lets the imperfect degrees be either major or minor, or some combination. For yaza scales, choose a 12-note subset, and let all but the tonic, 4th and 5th be either upped or downed. (The M2 and m7 may also be plain.) Up is utonal and down is otonal. Combining upped and downed intervals in a 41-edo scale creates double-up and double-down intervals, i.e. mid intervals. This increases the odd limit and/or the prime limit, so scales tend not to mix up and down.

==41-edo MOS scales==

Mathematically, 41edo has 20 edo-distinct temperaments, and each one has infinite MOS scales. This table only lists musically useful ones. MOS scales listed are those with:

|+Table of 41edo Temperaments by generator