Warped diatonic

A warped diatonic scale is a scale (excluding the diatonic scale itself) that contains relatively long substrings of the 5L 2s diatonic scale ("LLsLLLsLLsLLLs...") in its sequence of large and small steps, and such that the sizes of those steps are similar to those of the diatonic scale (namely, in the ballpark of 200 and 100 cents).

Such scales may mislead a diatonic-conditioned listener into assigning the intervals to diatonic scale categories, but the categorization will be violated when either (1) the part of the scale that doesn't agree with 5L2s is reached, or (2) the harmonic nature of the intervals is drastically different than what's expected from the diatonic scale.

Combinatorically, most distributionally even scales with more L steps than s steps do have significantly long substrings of the 5L2s diatonic scale in them. So, when searching for distributionally even warped diatonics, we can use the simple figure of merit that xLys is a good warped diatonic when x > y and 2x+y is in the vicinity of 12. If 2x+y is much less than 12, the steps will be too large to be recognized as the diatonic scale; conversely if 2x+y is much more than 12, the steps will be too small.

The scales at the top and bottom of this table are questionable as "warped diatonics", but the ones near 2x+y=12 are good examples.

2x+y	Formula	Temperaments	5L2s substrings	Comments
4	(2edo)		(SS, SSS)
5	2L 1s		LLs
6	(3edo)		(SS, SSS)
7	3L 1s		LLLs
8	(4edo)		(SS, SSS)
8	3L 2s	Sensi, Squares, Petrtri, A-Team, Father	LsLLsL
9	4L 1s	Bug, superpelog	LLLsLL, LLsLLL
9	3L 3s	Augmented	Ls
10	4L 2s	Decimal, lemba	LLsLLsLL	"Remove one L"
10	(5edo)		(SS, SSS)	"Remove two s’s"
11	4L 3s	Orgone, keemun, sixix	LsLLsL	"Replace one L with s"
11	5L 1s	Machine, gorgo	LLLsLL, LLsLLL	"Remove one s"
12	5L 2s	Ordinary diatonic	(infinitely long)
13	5L 3s	Father, A-Team, Petrtri	LsLLsLL, LLsLLsL	"Add one s"
13	6L 1s	Glacial, leantone, tetracot	LLLsLL, LLsLLL	"Replace one s with L"
14	5L 4s	Superpelog, godzilla	LsLLsL	"Add two s's"
14	6L 2s	Hedgehog	LLsLLLsLL	"Add one L"
15	6L 3s	Triforce	LLsLLsLL
15	7L 1s	Porcupine	LLLsLL, LLsLLL
16	6L 4s	Lemba, antikythera	LsLLsL
16	7L 2s	Mavila	LLsLLLsLL
17	7L 3s	Dicot	LLsLLLsLLs
17	8L 1s	Bleu, Tsaharuk, Quanharuk, bohpier	LLLsLL, LLsLLL
18	8L 2s	Octokaidecal	LLLsLL, LLsLLL
19	8L 3s	Sensi	LLLsLLsLLLs
19	9L 1s	Negri, Twothirdtonic	LLLsLL, LLsLLL
20	9L 2s	Casablanca	LLLsLL, LLsLLL

Rank 3

3x+2y+z	Formula	Temperaments/names	Max variety
13	2L 3m 2s		3
14	3L 1m 3s		3
15	2L 4m 1s		4
16	4L 1m 2s		4
17	4L 2m 1s		4
17	2L 4m 3s		4
18	5L 1m 1s		3
18	4L 1s 3m		4
18	3L 3m 3s		3
19	5L 1m 2s		4
19	4L 2m 3s		4
19	2L 5m 3s		4

Warped antidiatonic

The scales at the top and bottom of this table are questionable as "warped antidiatonics", but the ones near 2x+y=9 are good examples.

2x+y	Formula	Temperaments/names	2L5s substrings	Comments
5	5edo		(ss, sss)
6	1L 4s	Slendric, Gorgo, Machine	sLsss
7	1L 5s	Glacial, leantone, tetracot	ssLsss	"Remove one L"
7	2L 3s	Ordinary pentatonic	LssLs	"Remove two s’s"
8	1L 6s	Porcupine		"Replace one L with s"
8	2L 4s	Hedgehog	LssLss	"Remove one s"
9	2L 5s	Ordinary antidiatonic	(infinitely long)
10	2L 6s	Twothirdtonic, srutal/Pajara, shrutar	ssLsssLs	"Add one s"
10	3L 4s	Ordinary neutral diatonic		"Replace one s with L"
11	2L 7s	Score	ssLsssLss	"Add two s's"
11	3L 5s	Sensi	LsssLssL	"Add one L"
12	3L 6s	August, Augene	LssLss
12	4L 4s	Diminished	Ls
13	3L 7s	Magic	LssLsssLss
13	4L 5s	Orwell

Example: Godzilla[9]

Godzilla[9] in 19edo is 3 3 1 3 1 3 1 3 1. That's LLsLsLsLs where L is 189.5 cents and s is 63.2 cents. These step sizes are well within the range where they are "recognized" as diatonic scale steps. If you play 0 3 6 7 10 in 19edo (with no drone or harmony, just the melody) it will sound like "do re mi fa sol", leading you to believe that "sol" is a 3/2 interval above "do", but in fact it's not - it's closer to 10/7. The "real" 3/2 is another small step above that, in the melodic position of a flatted "la".

Here's a summary of all the contradicted expectations in godzilla[9], of which there are many:

• 2L+s, which is expected to be 4/3, is really more like 9/7.
• 2L+2s, which is expected to be a "dissonant tritone" like 10/7, is actually the "real" 4/3. In "ti do re mi fa", the ti-fa interval is a tempered 4/3.
• 3L+s, which is expected to be 3/2, is really more like 10/7.
• 3L+2s, which is expected to be an 8/5, is actually the "real" 3/2. In the "minor" scale "la ti do re mi fa", it's the "fa" rather than the "mi" that forms a 3/2 with the root "la".
• 4L+2s, which is expected to be a "minor seventh" (16/9~9/5), is actually 5/3. So in the "natural minor" scale "la ti do re mi fa sol", the outer la-sol interval is the easily recognizable consonance 5/3, but that's the "wrong" consonance for a diatonically-conditioned listener.
• L+2s, which is not found in the ordinary diatonic scale but is familiar from the melodic and harmonic minor scales, is expected to be something close to a "major third" (in 12edo it's 400 cents, in meantone it's a tempered 9/7). In godzilla[9] it's the just minor third 6/5.
• Finally, although there are 7 or even 8 consecutive notes of the scale that sound melodically familiar, the way the scale closes at the 2/1 is very unexpected and jarring for a diatonically-conditioned listener. When you get to 5L+3s that sounds melodically like it ought to be at least an "octave", probably something larger like an "augmented octave". But the real 2/1 is actually one small step beyond that, at 5L+4s.