# Mutt temperament

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.

From a posting on tuning-math.

## 5-limit

Comma: [-44 -3 21⟩ (the mutt comma)

Mapping: [<3 5 7|, <0 -7 -1|]

Poptimal generator: 9\771

TOP period: ~98304/78125 = 400.0227

TOP generator: ~5/4 = 386.0017 or 14.0210

MOS: 84, 87, 171, 429, 600, 771, 942, 1113, 1284, 1455

## 7-limit

Commas: 65625/65536, 250047/250000

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: ~63/50 = 400.0352

TOP generator: ~5/4 = 385.9987 or ~126/125 = 14.0365

MOS: 84, 87, 171