Limmic temperaments: Difference between revisions

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The '''limma family''' tempers out the Pythagorean limma, [[256/243]]. As a consequence, [[3/2]] is always represented by 3\5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it.
This '''limmic temperaments''' page collects various temperaments tempering out the Pythagorean limma, [[256/243]]. As a consequence, [[3/2]] is always represented by 3\5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it.


== Blackwood ==
== Blacksmith ==
=== 5-limit (blackwood) ===
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 16: Line 17:
[[Badness]]: 0.063760
[[Badness]]: 0.063760


=== Blacksmith ===
=== 7-limit ===
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]


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[[Badness]]: 0.025640
[[Badness]]: 0.025640


==== 11-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 48: Line 49:
Badness: 0.024641
Badness: 0.024641


===== 13-limit =====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.020498
Badness: 0.020498


==== Farrier ====
=== Farrier ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 74: Line 75:
Badness: 0.029200
Badness: 0.029200


===== 13-limit =====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 87: Line 88:
Badness: 0.022325
Badness: 0.022325


==== Ferrum ====
=== Ferrum ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Badness: 0.030883
Badness: 0.030883
== Blackweed ==
== Blackweed ==
Blackweed is so named because the 20edo tuning has 4\20 as the period and 420¢ as the generator.
Blackweed is so named because the 20edo tuning has 4\20 as the period and 420¢ as the generator.
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Vals: {{Val list|20, 35b }}
Vals: {{Val list|20, 35b }}
[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament collection]]
[[Category:Limma family]] <!-- main article -->
[[Category:Limmic temperaments]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Blackwood]]
[[Category:Blackwood]]

Revision as of 15:15, 26 September 2021

This limmic temperaments page collects various temperaments tempering out the Pythagorean limma, 256/243. As a consequence, 3/2 is always represented by 3\5, 720 cents assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it.

Blacksmith

5-limit (blackwood)

Subgroup: 2.3.5

Comma list: 256/243

Mapping: [⟨5 8 0], ⟨0 0 1]]

Mapping generators: ~9/8, ~5

POTE generator: ~5/4 = 399.594

Template:Val list

Badness: 0.063760

7-limit

blacksmith10.jpg
Lattice of blacksmith

Subgroup: 2.3.5.7

Comma list: 28/27, 49/48

Mapping: [⟨5 8 0 14], ⟨0 0 1 0]]

Mapping generators: ~7/6, ~5

Wedgie: ⟨⟨ 0 5 0 8 0 -14 ]]

POTE generator: ~5/4 = 392.767

Template:Val list

Badness: 0.025640

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 49/48, 55/54

Mapping: [⟨5 8 0 14 29], ⟨0 0 1 0 -1]]

POTE generator: ~5/4 = 394.948

Vals: Template:Val list

Badness: 0.024641

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 49/48, 55/54

Mapping: [⟨5 8 0 14 29 7], ⟨0 0 1 0 -1 1]]

POTE generator: ~5/4 = 391.037

Vals: Template:Val list

Badness: 0.020498

Farrier

Subgroup: 2.3.5.7.11

Comma list: 28/27, 49/48, 77/75

Mapping: [⟨5 8 0 14 -6], ⟨0 0 1 0 2]]

POTE generator: ~5/4 = 398.070

Vals: Template:Val list

Badness: 0.029200

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 49/48, 66/65

Mapping: [⟨5 8 0 14 -6 7], ⟨0 0 1 0 2 1]]

POTE generator: ~5/4 = 396.812

Vals: Template:Val list

Badness: 0.022325

Ferrum

Subgroup: 2.3.5.7.11

Comma list: 28/27, 35/33, 49/48

Mapping: [⟨5 8 0 14 6], ⟨0 0 1 0 1]]

POTE generator: ~5/4 = 374.763

Vals: Template:Val list

Badness: 0.030883

Blackweed

Blackweed is so named because the 20edo tuning has 4\20 as the period and 420¢ as the generator.

Subgroup: 2.3.14/11

Commas: 256/243

Mapping: [⟨5 8 0], ⟨0 0 1]]

Mapping generators: ~9/8, ~14/11

POTE generator: ~14/11 = 413.7785

Vals: Template:Val list

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