Overthink: + content (not cleaned up)

2026-03-16T04:25:03Z

+ content (not cleaned up)

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so anyways about 54edo pajara
Even in the 7-odd-limit 6/5 is very sharp
almost 18 cents (slightly over the error of 7/5 and 10/7)
and in the 9-odd-limit 9/5 is just off
sharp by 26.8 cents
and as [HIDDEN] said there's no way to tune 11-odd-limit monotonically
since 10/9 is tuned less than 1/3 of 4/3
even if we don't consider the 11-limit that's still probably unreasonable
and if we only consider the 7-odd-limit...
well 7/4 is about as sharp in 22edo as 5/4 in 12edo
but 6/5 is almost as sharp
and one can argue that keen is the correct extension for tunings flat of 22edo
but no 7-, 9-, and 11-odd-limit intervals of 7 appear in the 10- or 12-note keen MOSes
though the 22-note mos causes issues
There should probably be a consistency metric for mos scales, to determine if it unambiguously uses a single mapping in an odd limit

It would depend on the tuning of the mos, not just the pattern
A good definition would be that the temperament maps every interval in the odd limit to the nearest interval in that MOS
note that if two intervals in the MOS (e.g. A4 and d5 in 12edo diatonic) and an interval is closest to that step (in this case 6\12) it's acceptable to map it to either A4 or d5
There should probably be a term for if a MOS of a rank-2 temperament contain every interval in an odd limit
how about encapsulates

So pajara[10] and pajara[12] are consistent in the 7- and 9-odd-limit in 22edo
in general if an edo is consistent in an odd limit a tuning of any mos of a temperament in that edo supported by its patent val is consistent in that odd limit
if it's the nearest edo step, it must be the nearest mos step
weirdly 54edo pajarous[10] and pajarous[12] seem to be consistent in the 11-odd-limit even though the patent val isn't even monotone
10/9 is very flat but still mapped to the closest mos step, and 11/10 doesn't even appear
though if a tuning fails monotonicity it's arguably too inaccurate to use
the smallest pajara or pajarous MOS that encapsulates the 11-odd-limit is 22 notes
though for both the only consistent tuning is 22edo
which adds quite a bit of further damage on top of the canonical pajara mapping (6/5~11/9, 10/9~11/10~12/11)
cuz porcupine

where the 22-note mos is equalized and IMO isn't a legitimate mos (it's just the edo)
And there should also be a term for if a rank-2 tempermant contains any mos scales that encapsulates and is consistent in an odd limit
how about encapsulable
I define encapsulable to include if that mos is equalized (but not collapsed or with negative steps)
and strictly encapsulable to exclude equalized moses
Unfortunately pajara is not strictly encapsulable in the 11-odd-limit

User:Overthink/Consistency for MOS scales - Revision history

Overthink: + content (not cleaned up)