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family name: mutt
'''Mutt''' is the [[regular temperament|temperament]] [[tempering out]] the [[horwell comma]] and the [[landscape comma]] in the 7-limit. [[Gene Ward Smith]] noted two remarkable properties of this temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12028.html#12037 Yahoo! Tuning Group | ''mutt microtemperament'']</ref>. 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 {{monzo| ''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 {{monzo| ''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 {{monzo| ''a'' ''b'' ''c'' ''d'' }}; where both ''b'' and ''d'' are in the range from -1 to 1, so that |''b''| ≤ 1 and |''d''| ≤ 1.


period: 1/3 octave
The other remarkable property explains its name: it is supported by the standard val for [[768edo]]. 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", as was named by [[Gene Ward Smith]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_53634.html#53850 Yahoo! Tuning Group | ''Retuning using midi to produce beatless music - You gotta be kidding!'']</ref>.


generator: 5/4
The fact that the smallest proper 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 768edo 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]].


5-limit
See [[Horwell temperaments]] for technical data.


name: mutt
== 5-limit ==
Comma: {{monzo| -44 -3 21 }} (the [[mutt comma]])


comma: |-44 -3 21&gt;, the mutt comma
Mapping: [&lt;3 5 7|, &lt;0 -7 -1|]


mapping: [&lt;3 5 7|, &lt;0 -7 -1|]
Poptimal generator: 9\771


poptimal generator: 9/771
TOP period: ~98304/78125 = 400.0227


TOP period: 400.023
TOP generator: ~5/4 = 386.0017 or 14.0210


TOP generator: 386.016 or 14.007
MOS: 84, 87, 171, 429, 600, 771, 942, 1113, 1284, 1455


MOS: 84, 87, 171, 429, 600, 771
== 7-limit ==
Commas: 65625/65536, 250047/250000


7-limit
Mapping: [&lt;3 5 7 8|, &lt;0 -7 -1 12|]


name: mutt
7-limit poptimal generator: 21\1794


wedgie: &lt;&lt;21 3 -36 -44 -116 -92||
9-limit poptimal generator: 2\171


mapping: [&lt;3 5 7 8|, &lt;0 -7 -1 12|]
TOP period: ~63/50 = 400.0352


7-limit poptimal generator: 21/1794
TOP generator: ~5/4 = 385.9987 or ~126/125 = 14.0365
 
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
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&gt;, 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&gt;, 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&gt; where both b and d are in the range from -1 to 1, so that |b|&lt;=1 and |d|&lt;=1.
== Notes ==
 
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|12edo]].


From a [http://tech.groups.yahoo.com/group/tuning-math/message/12037 posting] on tuning-math.
[[Category:Mutt| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Horwell temperaments]]
[[Category:Landscape microtemperaments]]
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