Warped diatonic

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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.