User:Moremajorthanmajor/5L 2s (5/3-equivalent): Difference between revisions

|Tuning=5L 2s<5/3>}}

{{MOS intro|Scale Signature=5L 2s<5/3>}}Among the most well-known variants of this '''3/4''' '''diatonic''' MOS proper are 12ED5/3's diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.

Insert 2 for L and 1 for s and you'll get the 12ed5/3 diatonic of standard practice.

When L=3, s=1, you have 17ED5/3: 3 3 1 3 3 3 1

When L=3, s=2, you have 19ED5/3: 3 3 2 3 3 3 2

When L=4, s=1, you have 22ED5/3: 4 4 1 4 4 4 1

When L=4, s=3, you have 26ED5/3: 4 4 3 4 4 4 3

When L=5, s=1, you have 27ED5/3: 5 5 1 5 5 5 1

When L=5, s=2, you have 29ED5/3: 5 5 2 5 5 5 2

When L=5, s=3, you have 31ED5/3: 5 5 3 5 5 5 3

When L=5, s=4, you have 33ED5/3: 5 5 4 5 5 5 4

So you have scales where L and s are nearly equal, which approach 7ED5/3:

And you have scales where s becomes so small it approaches zero, which would give us 5ED5/3:

"3/4 Flattone" systems, such as 26ED5/3.

"3/4 Meantone" (more properly "3/4 septimal meantone") systems, such as 31ED5/3.

The near-just part of the region is of interest mainly for those interested in “3/4” [[Pythagorean tuning]] and large, accurate eds systems based on close-to-Pythagorean fifths, such as 41ED5/3 and 53ED5/3. This class of tunings is called [[schisma|trischismic]] temperament; these tunings can approximate 5^3/4-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12ED5/3 tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)<!--(see [[5L 2s/Temperaments#Schismic]])-->.

The sharp-of-just part of this range includes so-called “3/4 [[neogothic]]" or "3/4 parapyth" systems, which tune the diatonic major third slightly flatly of [[6/5]] and the diatonic minor third slightly sharply of [[12/11]]. Good 3/4 neogothic EDSs include 29ED5/3 and 46ED5/3. 17ED5/3 is often considered the sharper end of the 3/4 neogothic spectrum; its major third at 312 cents (416 śata) is considerably more concordant than in flatter neogothic tunings.

"3/4 Archy" systems such as 17ED5/3, 22ED5/3, and 27ED5/3.

If 4\7 (four degrees of 7ED5/3) is at one extreme and 3\5 (three degrees of 5ED5/3) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12ED5/3:

|}If we carry this freshman-summing out a little further, new, larger ED5/3s pop up in our continuum.

{{MOS tuning spectrum|Scale Signature=5L 2s<5/3>}}Tunings above 7\12 on this chart are called "positive tunings" (as they greaten the size of the fifth) and include 3/4 meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

5L 2s contains the pentatonic MOS [[2L_3s (major sixth equivalent) |2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L_5s (major sixth equivalent) |7L 5s]] or [[5L_7s (major sixth equivalent) |5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (675¢, 700$).