Cangwu badness

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Cangwu badness is a polynomial function badness measure; the name stems from Cangwu Green Park, Lianyungang, China, where Graham Breed thought it up. It is defined in terms of a matrix determinant as

[math]\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])[/math]

where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.

From this definition, it follows that C(0) is proportional to the square of simple badness, aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to the square of TE complexity. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.

If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a dominates b if Cb(x) - Ca(x) is a positive function for x ≥ 0, which says that the badness of 'a' is always less than the badness of 'b' for every choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is indomitable. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.

Examples of 5-limit indomitable temperaments are:

Father 16/15 |4 -1 -1>

Dicot 25/24 |-3 -1 2>

Meantone 81/80 |-4 4 -1>

Srutal/Diaschismic 2048/2025 |11 -4 -2>

Hanson/Kleismic 15625/15552 |-6 -5 6>

Helmholtz/Schismic 32805/32768 |-15 8 1>

Hemithirds |38 -2 -15>

Ennealimmal |1 -27 18>

Kwazy |-53 10 16>

Monzismic |54 -37 2>

Senior |-17 62 -35>

Pirate |-90 -15 49>

Atomic |161 -84 -12>

Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.

If two temperaments of the same rank are such that neither dominants the other, we may subtract one Cangwu badness polynomial from the other and find the positive root of the result. This gives a value of the parameter 'x' at which the two temperaments are rated equal in badness, which can be applied to rate other temperaments by badness. For example, if 5-limit father and helmholtz are made equally bad, then meantone, augmented, dicot, porcupine, srutal, diminished, magic, hanson and mavila, in that order, rate as better.