3-limit: Difference between revisions
m I forgot to state that the two pieces are actually different but used two almost identical titles, and the only difference of the titles is one used Simplified Chinese and the other used Traditional Chinese. |
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== Edo approximation == | == Edo approximation == | ||
[[Edo]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the | [[Edo]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the {{w|continued fraction}} for the logarithm base 2 of 3. The sequence is equivalent to the numbers of notes per octave in the [[mos scale]]s generated by pure 3/2: {{EDOs| 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, 359, 665, … }} ({{OEIS|A206788}}). | ||
Another approach is to find edos which have more accurate approximation to 3 than all smaller edos. This results in {{EDOs|1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, | Another approach is to find edos which have more accurate approximation to 3 than all smaller edos. This results in {{EDOs| '''1''', '''2''', 3, '''5''', 7, '''12''', 29, '''41''', '''53''', 200, 253, '''306''', 359, '''665''', … }} ({{OEIS|A060528}}). Bold ones are records of relative error. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. | ||
A stricter approach is to find edos with an increasingly stronger [[consistent circle]] of 3/2. These are {{EDOs|1, 12, 53, 665, 190537, … }} (with strengths 1, 2, 3, 11, 28, … respectively) | A stricter approach is to find edos with an increasingly stronger [[consistent circle]] of 3/2. These are {{EDOs| 1, 12, 53, 665, 190537, … }} (with strengths 1, 2, 3, 11, 28, … respectively) | ||
== Table of intervals == | == Table of intervals == | ||