Stearnsmic clan: Difference between revisions

The '''stearnsmic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[stearnsma]] ({{monzo|legend=1| 1 10 0 -6 }}, [[ratio]]: 118098/117649), a no-5's comma of about 6.59 [[cent]]s.

 

This temperament is generated by a very slightly flat [[~]][[9/7]] major third, three of which minus a [[semi-octave]] period give the [[3/2|perfect fifth]]. Its [[ploidacot]] is diploid alpha-tricot.  

* [[WE]]: ~343/243 = 599.994{{c}}, ~9/7 = 433.884{{c}}

* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.885{{c}}

{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d }}

The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|5]]. Pogo (94 & 130) adds [[32805/32768]], supers (58 & 94) adds [[5120/5103]], echidna (22 & 58) adds [[1728/1715]], and hedgehog (14c & 22) adds [[50/49]]. Those are the [[strong extension]]s.  

Wizard adds [[225/224]]. Harry adds [[2401/2400]]. Those split the generator in two. Septisuperfourth adds [[6144/6125]] and splits the generator in three. Stearnscape adds [[250047/250000]] and splits the period in three. Octoid adds [[4375/4374]] and splits the period in four. Decistearn adds [[3136/3125]] splits the period in five.

 

All of them extend naturally to the 11-limit via tempering out both [[540/539]] and [[4000/3993]], as well as [[9801/9800]], so that ~[[11/10]] and ~[[9/7]] add up to the semi-octave period. For strong extensions, the generator can be taken to be ~11/10, and three ~11/10's then make a ~4/3 as a result of tempering out 4000/3993, making [[10/9]] and [[12/11]] equidistant from 11/10.

 

The rest are considered below.  

The pogo temperament tempers out the [[schisma]], whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It may be described as 94 & 130, and is the temperament that reconciles the difference between [[tertiaschis]] and [[countertertiaschis]], using its semi-octave period.  

* [[WE]]: ~343/243 = 599.9860{{c}}, ~9/7 = 433.8905{{c}}

* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.9010{{c}}

[[Badness]] (Sintel): 2.02

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9125{{c}}

{{Optimal ET sequence|legend=0| 36, 58ce, 94, 130, 224, 354, 578 }}

Badness (Sintel): 1.05

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9112{{c}}

Badness (Sintel): 0.724

* [[WE]]: ~343/243 = 599.8062{{c}}, ~9/7 = 434.0778{{c}}

* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 434.2025{{c}}

{{Optimal ET sequence|legend=1| 36c, 58, 94, 152 }}

[[Badness]] (Sintel): 2.35

Optimal tunings:

* WE: ~99/70 = 599.8051{{c}}, ~9/7 = 434.0755{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 434.2002{{c}}

{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152 }}

Badness (Sintel): 0.934

Optimal tunings:

* WE: ~99/70 = 599.7318{{c}}, ~9/7 = 434.0271{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 434.1985{{c}}

{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152f }}

Badness (Sintel): 0.894

Optimal tunings:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Minimax tuning:

 

 

Badness (Sintel): 0.859

 

 

 

 

* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}

 

 

 

=== 17-limit ===

 

Comma list: 136/135, 176/175, 221/220, 256/255, 540/539

 

Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}

 

* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}

 

 

Badness (Sintel): 1.03

 

 

 

22edo provides an obvious tuning, which happens to be the only [[patent val|patent-val]] edo tuning, but if you are looking for an alternative you could try the {{val| 146 232 338 411 }} (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note [[mos]] gives scope for harmony while stopping well short of 22. A related temperament is [[#Echidna|echidna]], which offers much more accuracy. They merge on 22edo.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}

 

 

 

 

 

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''

 

Wizard tempers out the [[marvel comma]] and may be described as {{nowrap| 22 & 72 }}. It splits the ~9/7 in two parts, which can be treated as ~[[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. The [[ploidacot]] signature of wizard is diploid alpha-hexacot. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.

 

 

 

 

* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}

 

 

 

Subgroup: 2.3.5.7.11

 

 

* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness (Sintel): 0.901

 

 

Comma list: 99/98, 176/175, 1331/1296

 

Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}

 

 

 

Badness (Sintel): 1.91

* [[WE]]: ~2450/2187 = 199.9984{{c}}, ~567/500 = 216.9400{{c}}

* [[CWE]]: ~2450/2187 = 200.0000{{c}}, ~567/500 = 216.9406{{c}}

[[Badness]] (Sintel): 2.29

Optimal tunings:

* WE: ~55/49 = 199.9835{{c}}, ~567/500 = 216.9321{{c}} (~100/99 = 16.9487{{c}})

* CWE: ~55/49 = 200.0000{{c}}, ~567/500 = 216.9381{{c}} (~100/99 = 16.9381{{c}})

Badness (Sintel): 1.06

Optimal tunings:

* WE: ~55/49 = 199.9810{{c}}, ~312/275 = 216.9360{{c}} (~105/104 = 16.9550{{c}})

* CWE: ~55/49 = 200.0000{{c}}, ~312/275 = 216.9474{{c}} (~105/104 = 16.9474{{c}})

Badness (Sintel): 1.06

Optimal tunings:

* WE: ~55/49 = 199.9814{{c}}, ~17/15 = 216.9376{{c}} (~105/104 = 16.9563{{c}})

* CWE: ~55/49 = 200.0000{{c}}, ~17/15 = 216.9487{{c}} (~105/104 = 16.9345{{c}})

Badness (Sintel): 0.782

== Octoid ==

 

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374, the [[ragisma]], and 16875/16807, the [[canopic comma]]. In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].

 

The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].

[[Comma list]]: 4375/4374, 16875/16807

{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}

[[Optimal tuning]]s:  

* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}

* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}

[[Tuning ranges]]:  

* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)

* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)

* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]

* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}

[[Badness]] (Sintel): 1.08

Comma list: 540/539, 1375/1372, 4000/3993

Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}

* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}

* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)

* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

 

==== 13-limit ====

Comma list: 540/539, 625/624, 729/728, 1375/1372

Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}

* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}

{{Optimal ET sequence|legend=0| 72, 152f, 224 }}

Badness (Sintel): 0.631

===== 17-limit =====

Comma list: 375/374, 540/539, 625/624, 715/714, 729/728

Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}

* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}

 

Badness (Sintel): 0.729

Subgroup: 2.3.5.7.11.13.17.19

Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}

 

 

 

A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4.

Subgroup: 2.3.5.7.11.13

Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}

Optimal tunings:  

* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}

Badness (Sintel): 0.896

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 540/539

Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}

* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}

Badness (Sintel): 0.795

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399

Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}

* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}

{{Optimal ET sequence|legend=0| 8d, 72, 152 }}

Badness (Sintel): 0.993

 

==== Hexadecoid ====

 

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}

* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}

* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}

{{Optimal ET sequence|legend=0| 80, 144, 224 }}

Badness (Sintel): 1.27

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}

* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}

* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}

{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}

Badness (Sintel): 1.46

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}

* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}

* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}

{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}

Badness (Sintel): 1.44