Compton family: Difference between revisions

The '''compton family''', otherwise known as the '''aristoxenean family''', of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[Pythagorean comma]] ([[ratio]]: 531441/524288, {{monzo|legend=1| -19 12 }}, and hence the fifths form a closed 12-note [[circle of fifths]], identical to [[12edo]]. While the tuning of the fifth will be that of 12edo, two [[cent]]s flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.

5-limit compton is also known as ''aristoxenean''. It tempers out the Pythagorean comma and has a period of 1\12, so it is the 12edo circle of fifths with an independent dimension for the harmonic 5. Equivalent generators are [[5/4]], [[6/5]], [[10/9]], [[16/15]] (the [[secor]]), [[45/32]], [[135/128]] and most importantly, [[81/80]]. In terms of [[equal temperament]]s, it is the {{nowrap| 12 & 72 }} temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings.  

* [[CTE]]: ~256/243 = 100.000{{c}}, ~5/4 = 386.314{{c}} (~81/80 = 13.686{{c}})

: [[error map]]: {{val| 0.000 -1.955 0.000 }}

* [[POTE]]: ~256/243 = 100.000{{c}}, ~5/4 = 384.884{{c}} (~81/80 = 15.116{{c}})

: error map: {{val| 0.000 -1.955 -1.431 }}

[[Badness]] (Smith): 0.094494

Septimal compton is also known as ''waage''. In terms of the normal list, compton adds 413343/409600 ({{monzo| -14 10 -2 1 }}) to the Pythagorean comma; however, it can also be characterized by saying it adds [[225/224]].  

In either the 5- or 7-limit, 240edo is an excellent tuning, with 81/80 coming in at 15 cents exactly. In the 12edo, the major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.

In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds [[441/440]]. For this 72edo can be recommended as a tuning. In 11-limit compton, intervals of 5 are off by one generator, intervals of 7 are off by two generators, and intervals of 11 are off by 3 generators.  

* [[CTE]]: ~200/189 = 100.000{{c}}, ~5/4 = 384.922{{c}} (~126/125 = 15.078{{c}})

: [[error map]]: {{val| 0.000 -1.955 -1.392 -1.017 }}

* [[POTE]]: ~200/189 = 100.000{{c}}, ~5/4 = 383.775{{c}} (~126/125 = 16.225{{c}})

: error map: {{val| 0.000 -1.955 -2.538 -1.275 }}

{{Optimal ET sequence|legend=1| 12, 48d, 60, 72, 228, 300c, 372bc, 444bc }}

[[Badness]] (Smith): 0.035686

* CTE: ~35/33 = 100.000{{c}}, ~5/4 = 384.324{{c}} (~100/99 = 15.676{{c}})

* POTE: ~35/33 = 100.000{{c}}, ~5/4 = 383.266{{c}} (~100/99 = 16.734{{c}})

{{Optimal ET sequence|legend=0| 12, 48dee, 60e, 72 }}

Badness (Smith): 0.022235

* CTE: ~35/33 = 100.000{{c}}, ~5/4 = 384.685{{c}} (~105/104 = 15.315{{c}})

* POTE: ~35/33 = 100.000{{c}}, ~5/4 = 383.963{{c}} (~105/104 = 16.037{{c}})

{{Optimal ET sequence|legend=0| 12f, 48deefff, 60eff, 72, 228f }}

Badness (Smith): 0.021852

* CTE: ~18/17 = 100.000{{c}}, ~5/4 = 384.685{{c}} (~105/104 = 15.315{{c}})

* POTE: ~18/17 = 100.000{{c}}, ~5/4 = 383.750{{c}} (~105/104 = 16.250{{c}})

Badness (Smith): 0.017131

* CTE: ~35/33 = 100.000{{c}}, ~5/4 = 383.552{{c}} (~100/99 = 16.448{{c}})

* POTE: ~35/33 = 100.000{{c}}, ~5/4 = 382.612{{c}} (~100/99 = 17.388{{c}})

Badness (Smith): 0.025144

* CTE: ~18/17 = 100.000{{c}}, ~5/4 = 383.552{{c}} (~100/99 = 16.448{{c}})

* POTE: ~18/17 = 100.000{{c}}, ~5/4 = 382.597{{c}} (~100/99 = 17.403{{c}})

Badness (Smith): 0.016361

In terms of the normal comma list, catler is characterized by the addition of the [[schisma]], 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, [[128/125]] or [[648/625]]. In any event, the 5-limit is exactly the same as the 5-limit of 12edo. Catler can also be characterized as the {{nowrap| 12 & 24 }} temperament. [[36edo]] or [[48edo]] are possible tunings. Possible generators are [[36/35]], [[21/20]], [[15/14]], [[8/7]], [[7/6]], [[9/7]], [[7/5]], and most importantly, [[64/63]].   

* [[CTE]]: ~16/15 = 100.000{{c}}, ~7/4 = 968.826{{c}} (~64/63 = 31.174{{c}})

: [[error map]]: {{val| 0.000 -1.955 +13.686 0.000 }}

* [[POTE]]: ~16/15 = 100.000{{c}}, ~7/4 = 973.210{{c}} (~64/63 = 26.790{{c}})

: error map: {{val| 0.000 -1.955 +13.686 +4.384 }}

[[Badness]] (Smith): 0.050297

* CTE: ~16/15 = 100.000{{c}}, ~7/4 = 973.779{{c}} (~64/63 = 26.221{{c}})

* POTE: ~16/15 = 100.000{{c}}, ~7/4 = 977.277{{c}} (~64/63 = 22.723{{c}})

Badness (Smith): 0.058213

* CTE: ~16/15 = 100.000{{c}}, ~7/4 = 972.823{{c}} (~64/63 = 27.177{{c}})

* POTE: ~16/15 = 100.000{{c}}, ~7/4 = 972.136{{c}} (~64/63 = 27.864{{c}})

Badness (Smith): 0.081909

* CTE: ~16/15 = 100.000{{c}}, ~7/4 = 961.874{{c}} (~64/63 = 38.126{{c}})

* POTE: ~16/15 = 100.000{{c}}, ~7/4 = 967.224{{c}} (~64/63 = 32.776{{c}})

{{Optimal ET sequence|legend=0| 12, 24, 36, 72ce }}

Badness (Smith): 0.034478

* CTE: ~16/15 = 100.000{{c}}, ~7/4 = 956.375{{c}} (~40/39 = 43.625{{c}})

* POTE: ~16/15 = 100.000{{c}}, ~7/4 = 962.778{{c}} (~40/39 = 37.232{{c}})

Badness (Smith): 0.028363

* CTE: ~18/17 = 100.000{{c}}, ~7/4 = 956.375{{c}} (~40/39 = 43.625{{c}})

* POTE: ~18/17 = 100.000{{c}}, ~7/4 = 960.223{{c}} (~40/39 = 39.777{{c}})

Badness (Smith): 0.023246

* CTE: ~18/17 = 100.000{{c}}, ~7/4 = 956.375{{c}} (~40/39 = 43.625{{c}})

* POTE: ~18/17 = 100.000{{c}}, ~7/4 = 959.835{{c}} (~40/39 = 40.165{{c}})

Badness (Smith): 0.018985

* CTE: ~16/15 = 100.000{{c}}, ~7/4 = 961.255{{c}} (~64/63 = 38.745{{c}})

* POTE: ~16/15 = 100.000{{c}}, ~7/4 = 962.312{{c}} (~64/63 = 37.688{{c}})

Badness (Smith): 0.038307

* CTE: ~18/17 = 100.000{{c}}, ~7/4 = 961.255{{c}} (~64/63 = 38.745{{c}})

* POTE: ~18/17 = 100.000{{c}}, ~7/4 = 961.903{{c}} (~64/63 = 38.097{{c}})

Badness (Smith): 0.027487

* CTE: ~18/17 = 100.000{{c}}, ~7/4 = 961.255{{c}} (~64/63 = 38.745{{c}})

* POTE: ~18/17 = 100.000{{c}}, ~7/4 = 961.920{{c}} (~64/63 = 38.080{{c}})

Badness (Smith): 0.020939

* [[CTE]]: ~16/15 = 100.000{{c}}, ~11/8 = 551.318{{c}} (~33/32 = 48.682{{c}})

: [[error map]]: {{val| 0.000 -1.955 +13.686 +31.174 0.000 }}

* [[POTE]]: ~16/15 = 100.000{{c}}, ~11/8 = 565.023{{c}} (~55/54 = 34.977{{c}})

: error map: {{val| 0.000 -1.955 +13.686 +31.174 +13.705 }}

[[Badness]] (Smith): 0.030536

The hours temperament has a period of 1/24 octave and tempers out the [[cataharry comma]] (19683/19600) and the mirwomo comma (33075/32768). The name ''hours'' was named for the reason that the period is 1/24 octave and there are 24 hours per day.

* [[CTE]]: ~36/35 = 50.000{{c}}, ~5/4 = 384.226{{c}} (~81/80 = 15.774{{c}})

: [[error map]]: {{val| 0.000 -1.955 -2.088 -3.052 }}

* [[POTE]]: ~36/35 = 50.000{{c}}, ~5/4 = 384.033{{c}} (~81/80 = 15.967{{c}})

: error map: {{val| 0.000 -1.955 -2.280 -2.859 }}

[[Badness]] (Smith): 0.116091

* CTE: ~36/35 = 50.000{{c}}, ~5/4 = 384.226{{c}} (~121/120 = 15.774{{c}})

* POTE: ~36/35 = 50.000{{c}}, ~5/4 = 384.054{{c}} (~121/120 = 15.946{{c}})

Badness (Smith): 0.036248

* CTE: ~36/35 = 50.000{{c}}, ~5/4 = 385.420{{c}} (~121/120 = 14.580{{c}})

* POTE: ~36/35 = 50.000{{c}}, ~5/4 = 384.652{{c}} (~121/120 = 15.348{{c}})

Badness (Smith): 0.026931

The gamelstearn temperament has a period of 1/36 octave and tempers out the [[gamelisma]] (1029/1024) and the [[stearnsma]] (118098/117649).  

It used to be named "decades".

* [[CTE]]: ~49/48 = 33.333{{c}}, ~5/4 = 386.314{{c}} (~81/80 = 13.686{{c}})

: [[error map]]: {{val| 0.000 -1.955 0.000 -2.159 }}

* [[POTE]]: ~49/48 = 33.333{{c}}, ~5/4 = 384.764{{c}} (~81/80 = 15.236{{c}})

: error map: {{val| 0.000 -1.955 -1.549 -2.159 }}

[[Badness]] (Smith): 0.108016

* CTE: ~49/48 = 33.333{{c}}, ~5/4 = 385.797{{c}} (~81/80 = 14.203{{c}})

* POTE: ~49/48 = 33.333{{c}}, ~5/4 = 385.150{{c}} (~81/80 = 14.850{{c}})

{{Optimal ET sequence|legend=0| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}

Badness (Smith): 0.043088

* [[CTE]]: ~100/99 = 16.667{{c}}, ~13/8 = 840.528{{c}} (~325/324 = 7.194{{c}})

: [[error map]]: {{val| 0.000 -1.955 -2.980 -2.159 -1.318 0.000 }}

* [[POTE]]: ~100/99 = 16.667{{c}}, ~13/8 = 837.814{{c}} (~364/363 = 4.481{{c}})

: error map: {{val| 0.000 -1.955 -2.980 -2.159 -1.318 -2.713 }}

[[Badness]] (Smith): 0.029956