Blare comma: Difference between revisions

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Akselai (talk | contribs)
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the rank 2 temperament is useless for 2.9.7.11. also write monzos in subgroup instead of in full p-limit. also use dirichlet badness (this temp is actually really really bad).
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== Temperaments ==
== Temperaments ==
=== Loudspeaker ===
Tempering out this interval in the 2.9.7.11 subgroup leads to the '''loudspeaker family''' of microtemperaments.
Tempering out this interval in the 2.9.7.11 subgroup leads to the '''loudspeaker family''' of microtemperaments.


=== Rank-3 ===
==== Rank-3 ====
==== Loudspeaker ====
===== Loudspeaker =====
[[Subgroup]]: 2.9.7.11
[[Subgroup]]: 2.9.7.11


[[Comma list]]: {{monzo| -51 -24 0 12 16 }}
[[Comma list]]: {{monzo| -51 -12 12 16 }}


{{Mapping|legend=2| 4 12 13 12 | 0 1 1 0 | 0 0 -4 3 }}
{{Mapping|legend=2| 4 12 13 12 | 0 1 1 0 | 0 0 -4 3 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~{{monzo| 13 6 0 -3 -4 }} = 1\4 = 300.000, ~9/8 = 203.912, ~{{monzo| 16 8 0 -4 -5 }} = 183.772
* [[CTE]]: ~{{monzo| 13 3 -3 -4 }} = 1\4 = 300.000, ~9/8 = 203.912, ~{{monzo| 16 4 -4 -5 }} = 183.772


{{Optimal ET sequence|legend=1| 20, 32}}[+9], {{EDOs|52, 72, 124, 300, 424, 516, 536, 588, 640, 660, 712, 764, 836, 888, 960, 1012, 1848, 4708, 6556, 11264, 17820, 18708, 21568, 29972, 34680, 41256, 71208, 183652}}[-9], {{EDOs|254860}}[-9]
{{Optimal ET sequence|legend=1| 20, 32}}[+9], {{EDOs|52, 72, 124, 300, 424, 516, 536, 588, 640, 660, 712, 764, 836, 888, 960, 1012, 1848, 4708, 6556, 11264, 17820, 18708, 21568, 29972, 34680, 41256, 71208, 183652}}[-9], {{EDOs|254860}}[-9]


[[Badness]] (Smith): 7 × 10<sup>-6</sup>
[[Badness]] (Dirichlet): 0.78992


=== Rank-2 ===
=== Blazen ===
==== Microphone ====
Tempering out this interval in the no-fives [[11-limit]] (2.3.7.11 subgroup) leads to the '''blazen family'''.
{{see also| 44th-octave temperaments}}


This temperament additionally tempers out the parismina [1 -26 0 2 10⟩, which is even smaller than the blare comma. Surprisingly, this temperament admits an optimal generator very close to [[3/2]].
=== Blare ===
 
Tempering out this interval in the full [[11-limit]] leads to the '''blare family'''.
[[Subgroup]]: 2.9.7.11
 
[[Comma list]]: {{monzo| -52 2 0 10 6 }}, {{monzo| 1 -26 0 2 10 }}
 
{{Mapping|legend=2| 44 88 175 75 | 0 2 -2 3 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~64/63 = 1\44 = 27.273, ~{{monzo| 25 0 0 -5 -3 }} = 701.953
 
{{Optimal ET sequence|legend=1| 176, 484, 660, 836, 1012, 1848, 4708, 6556, 11264, 17820, 29084, 46904, 75988, 122892}}
 
[[Badness]] (Smith): 2.466 × 10<sup>-3</sup>

Revision as of 10:03, 28 April 2024

Interval information
Factorization 2-51 × 3-24 × 712 × 1116
Monzo [-51 -24 0 12 16⟩
Size in cents 0.0779347¢
Name blare comma
Color name Laquadquadlo-aquadtrizo comma
FJS name [math]\displaystyle{ - }[/math]
Special properties reduced
Tenney norm (log2 nd) 178.078
Weil norm (log2 max(n, d)) 178.078
Wilson norm (sopfr(nd)) 434
Comma size unnoticeable
Open this interval in xen-calc

The blare comma is an unnoticeable 11-limit comma in the 2.9.7.11 subgroup, with a size of only 0.078 cents. It is the interval in which twelve 9/7 septimal major thirds plus three octaves fall short by sixteen 11/8 undecimal major fourths.

It is named in analogy with the industrial theme of 2.9.7.11 subgroup temperaments, "blare" being a word used to describe the loud and piercing noise of machines.

Temperaments

Loudspeaker

Tempering out this interval in the 2.9.7.11 subgroup leads to the loudspeaker family of microtemperaments.

Rank-3

Loudspeaker

Subgroup: 2.9.7.11

Comma list: [-51 -12 12 16⟩

Subgroup-val mapping: [⟨4 12 13 12], ⟨0 1 1 0], ⟨0 0 -4 3]]

Optimal tunings:

  • CTE: ~[13 3 -3 -4⟩ = 1\4 = 300.000, ~9/8 = 203.912, ~[16 4 -4 -5⟩ = 183.772

Optimal ET sequence: 20, 32[+9], 52, 72, 124, 300, 424, 516, 536, 588, 640, 660, 712, 764, 836, 888, 960, 1012, 1848, 4708, 6556, 11264, 17820, 18708, 21568, 29972, 34680, 41256, 71208, 183652[-9], 254860[-9]

Badness (Dirichlet): 0.78992

Blazen

Tempering out this interval in the no-fives 11-limit (2.3.7.11 subgroup) leads to the blazen family.

Blare

Tempering out this interval in the full 11-limit leads to the blare family.

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