Alphatricot family: Difference between revisions

The '''alphatricot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[alphatricot comma]] ({{monzo|legend=1| 39 -29 3 }}, [[ratio]]: 68 719 476 736 000 / 68 630 377 364 883).  

Strong 7-limit extensions of this temperament include alphatrimot ({{nowrap| 53 & 70 }}), alphatrident ({{nowrap| 53 & 229 }}) and alphatrillium ({{nowrap| 53 & 441 }}). Tempering out [[5120/5103|hemifamity comma]] (5120/5103) leads to alphatrimot, [[6144/6125|porwell comma]] (6144/6125) leads to alphatrident, and [[4375/4374|ragisma]] (4375/4374) leads to alphatrillium.

== Alphatricot ==

Alphatricot is a [[microtemperament]] whose generator is the real cube root of the [[3/1|3rd]] [[harmonic]], 3^1/3, tuned between [[63/44]] and [[13/9]] and representing the acute augmented fourth of 59049/40960, that is, a [[729/512|Pythagorean augmented fourth]] plus a [[81/80|syntonic comma]]. Its [[ploidacot]] is alpha-tricot. It is a member of the [[schismic–Mercator equivalence continuum]] with {{nowrap|''n'' {{=}} 3 }}, so unless 53edo is used as a tuning, the [[schisma]] is always observed.  

The temperament was named by [[Paul Erlich]] in 2002 as ''tricot''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5041.html Yahoo! Tuning Group | ''Paul's new names'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5080.html#5113 Yahoo! Tuning Group | ''Ultimate 5-limit comma list'']</ref>, but renamed in 2025 following the specifications of ploidacot.  

[[Comma list]]: {{monzo| 39 -29 3 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.9762{{c}}, ~59049/40960 = 633.9998{{c}}

{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258, 15365bbc, 17632bbc }}

[[Badness]] (Sintel): 1.08

=== Alphatrimot (2.3.5.13 subgroup) ===

This extension identifies the generator with [[13/9]] by tempering out the threedie, [[2197/2187]], providing a relatively low-complexity mapping for 13.

Comma list: 2197/2187, 41067/40960

 

 

 

 

However, alphatricot in the 5-limit is far more accurate than threedic. Alphatrillium interprets the generator as ~[[75/52]] instead of 13/9, making the tempering of [[140625/140608]], the catasma, instead of the threedie. It also tempers out [[256000/255879]], the phaotisma.

Subgroup: 2.3.5.13

Comma list: 140628/140625, 256000/255879

Mapping: {{mapping| 1 0 -13 -28 | 0 3 29 60 }}

* WE: ~2 = 1199.9796{{c}}, ~75/52 = 634.0000{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0103{{c}}

{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935, 6051f, 6986f, …, 10726bff }}

== Alphatrillium ==

Alphatrillium, named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament, tempering out the [[ragisma]] aside from the alphatricot comma. [[441edo]] is a good tuning for this temperament, with generator 233\441. The harmonic 7 is found at -95 generator steps, so that the smallest [[mos scale]] that contains it is the 123-note one, though otonal and utonal tetrads don't occur until the 176-note mos due to 7/5 being mapped to -124 generators. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].

 

It can be extended to the 11-limit by tempering out [[131072/130977]], and to the 13-limit by tempering out [[2080/2079]], [[4096/4095]] and [[4225/4224]]. The optimal tunings in the 11- and 13-limit lean towards [[494edo]]; [[935edo]] and especially [[1429edo]] are recommendable tunings.  

[[Comma list]]: 4375/4374, {{monzo| 40 -22 -1 -1 }}

 

 

 

 

 

 

 

 

* CWE: ~2 = 1200.0000{{c}}, ~3888/2695 = 634.0094{{c}}

 

 

 

Mapping: {{mapping| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0094{{c}}

{{Optimal ET sequence|legend=0| 53, 388e, 441, 494, 935, 1429, 1923e, 3352de }}

Badness (Sintel): 0.801

=== Pseudotrillium ===

Comma list: 4375/4374, 5632/5625, 4108797/4096000

Mapping: {{mapping| 1 0 -13 53 -61 | 0 3 29 -95 122 }}

Optimal tunings:

* WE: ~2 = 1200.0692{{c}}, ~231/160 = 634.0556{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0191{{c}}

{{Optimal ET sequence|legend=0| 53, 335, 388 }}

Badness (Sintel): 3.70

==== 13-limit ====

Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374

Mapping: {{mapping| 1 0 -13 53 -61 -28 | 0 3 29 -95 122 60 }}

Optimal tunings:

* WE: ~2 = 1200.0351{{c}}, ~75/52 = 634.0366{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0181{{c}}

{{Optimal ET sequence|legend=0| 53, 335, 388 }}

Badness (Sintel): 2.27

== Alphatrident ==

Alphatrident, also named by [[Xenllium]] in 2021 as ''trident'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 229 }} temperament. It tempers out the [[garischisma]], 33554432/33480783 ({{monzo| 25 -14 0 1 }}), and finds the harmonic 7 at -14 fifths or {{nowrap| (-14) × 3 {{=}} -42 }} generator steps, so that the smallest mos scale that includes it is the 53-note one.  

* [[WE]]: ~2 = 1199.7509{{c}}, ~4096/2835 = 633.9164{{c}}

: [[error map]]: {{val| -0.249 -0.206 +0.500 +0.458 }}

: error map: {{val| 0.000 +0.189 +1.081 +1.155 }}

{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511, 793cd }}

[[Badness]] (Sintel): 2.57

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0662{{c}}

{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}

Badness (Sintel): 2.46

Optimal tunings:

* WE: ~2 = 1199.9675{{c}}, ~13/9 = 634.0480{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0651{{c}}

{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}

Badness (Sintel): 1.93

== Alphatrimot ==

Alphatrimot, named by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref> but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 70 }} temperament. It finds prime 7 at only 11 generators up so that the generator is interpreted as a flat ~[[81/56]], but is more of a full 13-limit system in its own right. [[123edo]] in the 123de val is a great tuning for it. Mos scales of 5, 7, 9, 11, 13, 15, 17, 19, 36 or 53 notes are available.  

[[Comma list]]: 2430/2401, 5120/5103

{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}

* [[WE]]: ~2 = 1199.4448{{c}}, ~81/56 = 633.7326{{c}}

: [[error map]]: {{val| -0.555 -0.757 +0.851 +3.898 }}

: error map: {{val| 0.000 +0.066 -0.108 +5.252 }}

{{Optimal ET sequence|legend=1| 17c, 36c, 53, 229dd, 282dd }}

[[Badness]] (Sintel): 2.53

Comma list: 99/98, 121/120, 5120/5103

Mapping: {{mapping| 1 0 -13 -3 -5 | 0 3 29 11 16 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~63/44 = 634.0253{{c}}

{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}

Badness (Sintel): 1.86

=== 13-limit ===

Comma list: 99/98, 121/120, 169/168, 352/351

Mapping: {{mapping| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}

Optimal tunings:

* WE: ~2 = 1200.1213{{c}}, ~13/9 = 634.0757{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0154{{c}}

{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}

Badness (Sintel): 1.33

[[Comma list]]: 250047/250000, {{monzo| 35 -23 -3 3 }}

{{Mapping|legend=1| 3 0 -39 -74 | 0 3 29 52 }}

[[Optimal tuning]]s:

* [[WE]]: ~63/50 = 399.9887{{c}}, ~59049/40960 = 633.7326{{c}} (~100352/91125 = 165.9790{{c}})

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~59049/40960 = 634.0155{{c}} (~100352/91125 = 165.9845{{c}})

{{Optimal ET sequence|legend=1| 159, 282, 441, 1605, 2046, 2487, 2928 }}

[[Badness]] (Sintel): 2.18

Mapping: {{mapping| 3 0 -39 -74 -34 | 0 3 29 52 28 }}

Optimal tunings:

* WE: ~63/50 = 399.9686{{c}}, ~3969/2750 = 633.9667{{c}} (~11/10 = 165.9705{{c}})

* CWE: ~63/50 = 400.0000{{c}}, ~3969/2750 = 634.0142{{c}} (~11/10 = 165.9858{{c}})

{{Optimal ET sequence|legend=0| 159, 282, 441 }}

Badness (Sintel): 2.45

Mapping: {{mapping| 3 0 -39 -74 -34 -84 | 0 3 29 52 28 60 }}

Optimal tunings:

* WE: ~63/50 = 399.9692{{c}}, ~75/52 = 633.9669{{c}} (~11/10 = 165.9714{{c}})

* CWE: ~63/50 = 400.0000{{c}}, ~75/52 = 634.0137{{c}} (~11/10 = 165.9863{{c}})

{{Optimal ET sequence|legend=0| 159, 282, 441 }}

Badness (Sintel): 1.47

Mapping: {{mapping| 3 0 -39 -74 -34 -84 -2 | 0 3 29 52 28 60 9 }}

Optimal tunings:

* WE: ~34/27 = 399.9491{{c}}, ~75/52 = 633.9389{{c}} (~11/10 = 165.9594{{c}})

* CWE: ~34/27 = 400.0000{{c}}, ~75/52 = 634.0166{{c}} (~11/10 = 165.9834{{c}})

{{Optimal ET sequence|legend=0| 159, 282, 441, 723efg, 1164eefgg }}

Badness (Sintel): 1.32

Noletaland is described as {{nowrap| 282 & 1323 }}, and it combines the smallest consistent edo in the 29-odd-limit with the smallest uniquely consistent. It reaches 4/3 in nine generators ([[noleta]]-…) and tempers out the landscape comma (…-land). Noletaland reaches [[13/11]] in 2 generators, and [[29/19]] in 5. Then there is [[44/25]] in 4, and [[152/115]] in also 4.

Optimal tunings:

* WE: ~63/50 = 399.9895{{c}}, ~1936/1875 = 55.3269{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3286{{c}}

{{Optimal ET sequence|legend=0| 282, 759de, 1041, 1323, 4251e }}

Badness (Sintel): 5.23

Optimal tunings:

* WE: ~63/50 = 399.9896{{c}}, ~1936/1875 = 55.3273{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3289{{c}}

{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}

Badness (Sintel): 2.99

Optimal tunings:

* WE: ~63/50 = 399.9876{{c}}, ~351/340 = 55.3270{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3290{{c}}

{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}

Badness (Sintel): 1.93

Optimal tunings:

* WE: ~63/50 = 399.9914{{c}}, ~351/340 = 55.3277{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3291{{c}}

{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}

Badness (Sintel): 1.64

Optimal tunings:

* WE: ~63/50 = 399.9899{{c}}, ~351/340 = 55.3276{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3291{{c}}

{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}

Badness (Sintel): 1.39

Optimal tunings:

* WE: ~63/50 = 399.9940{{c}}, ~351/340 = 55.3283{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3293{{c}}

{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}

Badness (Sintel): 1.40

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