mutt microtemperament

hi Gene,

in the new version of the Encyclopedia, i've just
created a separate page for "mutt". but the data you
gave (which is currently at the bottom of the "hexamu"
page) looks a bit sketchy.

can you please fill out the temperament family template
for this one? thanks.

-monz

family name: mutt
period: 1/3 octave
generator: 5/4

5-limit

name: mutt
comma: |-44 -3 21>, the mutt comma
mapping: [<3 5 7|, <0 -7 -1|]
poptimal generator: 9/771
TOP period: 400.023
TOP generator: 386.016 or 14.007
MOS: 84, 87, 171, 429, 600, 771

7-limit

name: mutt
wedgie: <<21 3 -36 -44 -116 -92||
mapping: [<3 5 7 8|, <0 -7 -1 12|]
7-limit poptimal generator: 21/1794
9-limit poptimal generator: 2/171
TOP period: 400.025
generator: 385.990 or 14.035
TM basis: {65625/65536, 250047/250000}
MOS: 84, 87, 171

The mutt temperament has two remarkable properties. In the 5-limit,
the mutt comma reduces the lattice of pitch classes to three parallel
strips of major thirds. The strips are three fifths (or three minor
thirds, if you prefer) wide. In other words, tempering via mutt
reduces the 5-limit to monzos of the form |a b c>, where b is -1, 0 or
1. In the 7-limit, the landscape comma 250047/250000 reduces the
entire 7-limit to three layers of the 5-limit; everything in the
7-limit can be written |a b c d>, where d is -1, 0, or +1. Putting
these facts together, we discover that mutt reduces the 7-limit to
nine infinite chains of major thirds. In mutt, everything in the
7-limit can be written |a b c d> where both b and d are in the range
from -1 to 1, so that |b|<=1 and |d|<=1.

The other remarkable property explains its name: it is supported by
the standard val for 768 equal. Since dividing the octave into 768 =
12*64 parts is what some systems use for defining pitch (using the
coarse, but not the fine, conceptual "pitch wheel" of midi) mutt is a
temperament which accords to this kind of midi unit, hence the acronym
Midi Unit Tempered Tuning, or "mutt".

The fact that the smallest MOS is 84 and the generator is about the 14
cent difference between the 400 cent third of equal temperament and a
just third of 386 cents limits the applicability of mutt. If we tune
84 notes in 768 equal to mutt, we divide 400 cents by a step of 9
repeated 27 times, followed by a step of 13. If we now use this to
tune seven rows, each of which divides the octave into twelve parts,
we have rows with the pattern [63 63 63 67 63 63 63 67 63 63 63 67], a
modified version of 12EDO.