Temperamental badness

You can write the quadratic form of Cangwu badness of a
mapping as

(1 + Ek2)G - G[H><H]G/<H]G[H>

where

G is the quadratic form for TE complexity (scalar
complexity) and also the weighting matrix squared, with a
normalization.

Ek2 is the free parameter: the target error squared.

[H> is a column vector with the sizes of the prime
intervals and <H] is its transpose.

The quadratic form for the badness of an interval is

K + [H><H]/<H]G[H>/Ek2

Symbols are as above, and K is the matrix for temperamental
complexity, the inverse of G. This is proportional to the
inverse of the Cangwu badness metric, because multiplying
the two gives the identity matrix times (1+Ek2).

For a temperament, [H> measures the sizes of the
generators. For a mapping <M] (vals as rows):

[H> -> K<M]G[H>
<H]G[H> -> <H]G[M>K<M]G[H>

So it's possible to calculate the badness of a tempered
interval with a free parameter.

This term:

[H><H]/<H]G[H>

measures the square of the size of an interval. The
relative TE error (scalar badness) of a temperament class
is the size of the unison vector it tempers out.
Unfortunately, this doesn't generalize to sets of more than
one unison vector, because the "badness" always comes out
as zero. Something to do with it being a projection.

Anyway, the smaller Ek (the square root of Ek2) gets, the
more weight is given to the size of an interval in the
badness. As Ek tends to infinity, the badness is
proportional to TE complexity. And as it tends to zero, the
badness is proportional to relative error.

This badness can be used to compare intervals of different
sizes. The raw temperamental complexity doesn't do that,
so that 15:1 counts equally with 5:3. This is wrong
because it contradicts Keenan's razor: that intervals
larger than 2 octaves have negligible
consonance/dissonance. Dave stated this in another place
and nobody contradicted him.

Ek seems to be much larger for finding intervals than
finding temperament classes. That makes sense because you
want musical intervals to be larger than unison vectors.
It also ties in with STD error, which is like Cangwu
badness but acting on tunings instead of mappings. The STD
error corrects the RMS error but is obviously wrong because
it's optimized when all intervals are tempered to unisons.
Mixing the standard deviation and RMS would correct
this, hence the free parameter.

As an example, here are the 11-limit meantone intervals with
the lowest temperamental badness for Ek=0.69:

[5,2] T
[8,3] TS
[10,4] TT
[13,5] TTS
[18,7] TTST
[23,9] TTSTT
[28,11] TTSTTT
[31,12] TTSTTTS
[36,14] TTSTTTS T

They're in terms of steps from 31- and 12-equal and also
tones and semitones. By giving extra weight to small
intervals, this almost gives a diatonic scale without
asking for octave equivalence. The exception is that you
have both a major and minor third.

I suggest this badness is the correct thing to take an
octave-orthogonal sublattice of. You may get STD
complexity as a special case.

Graham