User:Moremajorthanmajor/5L 2s (8/3-equivalent): Difference between revisions

{{interwiki

| en = 5L 2s

| Equave = 8/3

}}

One way of distinguishing the '''diatonic''' scale is by considering it a [[MOS scale|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. Among the most well-known variants of this MOS proper are [[17edXI|17EDXI]]<nowiki/>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 [[12edXI|12EDXI]] diatonic.

When L=3, s=1, you have 17EDXI of standard practice: 3 3 3 1 3 3 1

When L=3, s=2, you have [[19edXI|19EDXI]]: 3 3 3 2 3 3 2

When L=4, s=1, you have [[22edXI|22EDXI]]: 4 4 4 1 4 4 1

When L=4, s=3, you have [[26edXI|26EDXI]]: 4 4 4 3 4 4 3

When L=5, s=1, you have [[27edXI|27EDXI]]: 5 5 5 1 5 5 1

When L=5, s=2, you have [[29edXI|29EDXI]]: 5 5 5 2 5 5 2

When L=5, s=3, you have [[31EDXI]]: 5 5 5 3 5 5 3

When L=5, s=4, you have [[33edXI|33EDXI]]: 5 5 5 4 5 5 4

So you have scales where L and s are nearly equal, which approach [[7edXI|7EDXI]]:

And you have scales where s becomes so small it approaches zero, which would give us [[5edXI|5EDXI]]:

"17/12 Flattone" systems, such as [[26edo|26EDXI]].

"17/12 Meantone" (more properly "septimal meantone") systems, such as [[31edo|31EDXI]].

The near-just part of the region is of interest mainly for those interested in 17/12 [[Pythagorean tuning]] and large, accurate EDO systems based on close-to-Pythagorean fifths, such as [[41edXI|41EDXI]] and [[53edXI|53EDXI]]. This class of tunings is called 17/12 [[schisma|schismic]] temperament; these tunings can approximate 5^17/12-limit harmonies very accurately by [[tempering out]] 17/12 of a small comma called the [[schisma]]. (Technically, 12EDXI tempers out the 17/12 schisma and thus is a 17/12 schismic tuning, but it is nowhere near as accurate as 17/12 schismic tunings can be.)

The sharp-of-just part of this range includes so-called "17/12 [[neogothic]]" or "17/12 parapyth" systems, which tune the 17/12 diatonic major third slightly sharply of [[11/7]] (around [[128/81]]) and the diatonic minor third slightly flatly of [[13/11]] (around [[32/27]]). Good 17/12 neogothic EDXIs include [[29edo|29EDXI]] and [[46edXI|46EDXI]]. [[17edXI|17EDXI]] is often considered the sharper end of the neogothic spectrum; its major third at 800 cents is considerably more discordant than in flatter 17/12 neogothic tunings.

"17/12 Archy" systems such as [[17edo|17EDXI]], [[22edo|22EDXI]], and [[27edo|27EDXI]].  

If 4\7 (four degrees of 7EDXI) is at one extreme and 3\5 (three degrees of 5EDXI) 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 12EDXI.

If we carry this freshman-summing out a little further, new, larger [[EDXI]]<nowiki/>s pop up in our continuum.

*Chroma-positive generator: 960 cents (4\7, normalized) to 1028.5714 cents (3\5, normalized)

*Chroma-negative generator: [[Tel:685.7143|685.7143]] cents (2\5, normalized) to 720 cents (3\7, normalized)

{| class="wikitable center-all"

| || || || || ||41\71|| ||974.{{Overline|2574}}¢||11||8||1.375||

| || || || ||26\45|| || ||975¢||7||5||1.400||17/12 [[Flattone]] is in this region

| || || ||18\31|| || || ||981.{{Overline|81}}¢||5||3||1.667||17/12 [[Meantone]] is in this region

| ||7\12|| || || || || ||988.2353¢||2||1||2.000||Basic 17/12 diatonic <br>(Generators smaller than this are proper)

| || || || || ||41\70|| ||993.{{Overline|93}}¢||12||5||2.400||

|}