Meantone family: Difference between revisions

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! class="unsortable"| Approximate ratios<ref>2.3.5, odd limit ≤ 27</ref>

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<div style="line-height:1.6;">Technical data</div>

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Comma: 81/80

Mapping generator: ~3

{{EDOs|legend=1| 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb }}

[[Badness]]: 0.00736

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! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>

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<div style="line-height:1.6;">Technical data</div>

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[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]

7 and [[9-odd-limit|9-limit]] minimax

strict range: [694.786, 700.000]

Algebraic generator: Cybozem, the real root of 15''x''³ - 10''x''² - 18, which comes to 503.4257 cents. The recurrence converges quickly.

[[Vals]]: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb

[[POTE generator]]: ~3/2 = 696.967

[[Algebraic generator]]: Traverse, the positive real root of ''x''⁴ + 2''x'' - 13, or 696.9529 cents.

[[Map]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]

{{EDOs|legend=1| 12, 31, 43, 74, 105, 198be }}

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! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>

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<div style="line-height:1.6;">Technical data</div>

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[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]

[[7-odd-limit|7-limit]] minimax

strict range: [692.353, 694.737]

Algebraic generator: Squarto, the positive root of 8''x''² - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}

{{EDOs|legend=1| 7, 19, 26, 45 }}

== 13-limit ==

valid range: [692.308, 694.737] (26 to 19)

<span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>

Godzilla tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo]] is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.

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! Cents <ref>octave-reduced</ref>

! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>

<div style="line-height:1.6;">Technical data</div>

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[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 2}}, {{val|0 2 8 1}}]

valid range: [240.000, 257.143] (5 to 14c)

strict range: [240.000, 257.143]

[[Wedgie]]: &lt;&lt;2 8 1 8 -4 -20||

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! class="unsortable"| Approximate ratios<ref>2.3.5.7.11, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>

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<div style="line-height:1.6;">Technical data</div>

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Commas: 81/80, 121/120, 176/175

[[11-odd-limit|11-limit]] minimax 1/4 comma

Eigenmonzos: 2, 5

{{EDOs|legend=1| 7, 24, 31, 38, 55 }}