3-limit: Difference between revisions
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A '''3-limit''' interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as 2^a 3^b, where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. Confining intervals to the 3-limit is known as [[Pythagorean tuning]], and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music. | {{Wikipedia|Pythagorean tuning}} | ||
A '''3-limit''' interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as <math>2^a \cdot 3^b</math>, where ''a'' and ''b'' can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. Confining intervals to the 3-limit is known as [[Pythagorean tuning]], and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music. | |||
[[EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [[wikipedia: Continued fraction|continued fraction]] for the logarithm of 3 base 2. These are 1, 2, 3, [[5edo|5]], [[7edo|7]], [[12edo|12]], [[17edo|17]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[306edo|306]], ... | [[EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [[wikipedia: Continued fraction|continued fraction]] for the logarithm of 3 base 2. These are 1, 2, 3, [[5edo|5]], [[7edo|7]], [[12edo|12]], [[17edo|17]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[306edo|306]], ... | ||
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* [[3-odd-limit]] | * [[3-odd-limit]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
[[Category:3-limit| ]] <!-- main article --> | [[Category:3-limit| ]] <!-- main article --> | ||
[[Category:Prime limit]] | [[Category:Prime limit]] | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
Revision as of 14:46, 16 December 2021
A 3-limit interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as [math]\displaystyle{ 2^a \cdot 3^b }[/math], where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are 3/2, 4/3, 9/8. Confining intervals to the 3-limit is known as Pythagorean tuning, and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music.
EDOs which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the continued fraction for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, ...
Another approach is to find EDOs which have more accurate 3 than all smaller EDOs. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ...
3-limit intervals up to odd-limit 19683:
| Ratio | Monzo | Size (¢) | Color name | Interval category | ||
|---|---|---|---|---|---|---|
| 1/1 | [0⟩ | 0.000 | w1 | wa unison | unison | C |
| 2187/2048 | [-11 7⟩ | 113.685 | Lw1 | lawa 1sn | aug. unison | C# |
| 256/243 | [8 -5⟩ | 90.225 | sw2 | sawa 2nd | minor 2nd | Db |
| 9/8 | [-3 2⟩ | 203.910 | w2 | wa 2nd | major 2nd | D |
| 19683/16384 | [-14 9⟩ | 317.595 | Lw2 | lawa 2nd | aug. 2nd | D# |
| 32/27 | [5 -3⟩ | 294.135 | w3 | wa 3rd | minor 3rd | Eb |
| 81/64 | [-6 4⟩ | 407.820 | Lw3 | lawa 3rd | major 3rd | E |
| 8192/6561 | [13 -8⟩ | 384.360 | sw4 | sawa 4th | dim. fourth | Fb |
| 4/3 | [2 -1⟩ | 498.045 | w4 | wa 4th | fourth | F |
| 729/512 | [-9 6⟩ | 611.730 | Lw4 | lawa 4th | aug. fourth | F# |
| 1024/729 | [10 -6⟩ | 588.270 | sw5 | sawa 5th | dim. fifth | Gb |
| 3/2 | [-1 1⟩ | 701.955 | w5 | wa 5th | fifth | G |
| 6561/4096 | [-12 8⟩ | 815.640 | Lw5 | lawa 5th | aug. fifth | G# |
| 128/81 | [7 -4⟩ | 792.180 | sw6 | sawa 6th | minor 6th | Ab |
| 27/16 | [-4 3⟩ | 905.865 | w6 | wa 6th | major 6th | A |
| 32768/19683 | [15 -9⟩ | 882.405 | sw7 | sawa 7th | dim. 7th | Bbb |
| 16/9 | [4 -2⟩ | 996.090 | w7 | wa 7th | minor 7th | Bb |
| 243/128 | [-7 5⟩ | 1109.775 | Lw7 | lawa 7th | major 7th | B |
| 4096/2187 | [12 -7⟩ | 1086.315 | sw8 | sawa 8ve | dim. octave | Cb |
| 2/1 | [1⟩ | 1200.000 | w8 | wa 8ve | octave | C |