3-limit: Difference between revisions
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== Table of intervals == | == Table of intervals == | ||
3-limit intervals up to [[odd-limit]] 19683: | 3-limit intervals up to [[odd-limit]] 19683: | ||
{| class="wikitable center-1 right-3" | {| class="wikitable center-1 right-3 center-6 center-7" | ||
|- | |- | ||
! [[Ratio]] | ! [[Ratio]] | ||
| Line 17: | Line 17: | ||
! Size ([[Cent|¢]]) | ! Size ([[Cent|¢]]) | ||
! colspan="2" | [[Kite's color notation|Color Name]] | ! colspan="2" | [[Kite's color notation|Color Name]] | ||
! colspan="2" | | ! colspan="2" | Diatonic Category | ||
|- | |- | ||
| [[1/1]] | | [[1/1]] | ||
| Line 24: | Line 24: | ||
| w1 | | w1 | ||
| wa unison | | wa unison | ||
| | | P1 | ||
| C | | C | ||
|- | |- | ||
| Line 32: | Line 32: | ||
| Lw1 | | Lw1 | ||
| lawa 1sn | | lawa 1sn | ||
| | | A1 | ||
| C# | | C# | ||
|- | |- | ||
| Line 40: | Line 40: | ||
| sw2 | | sw2 | ||
| sawa 2nd | | sawa 2nd | ||
| | | m2 | ||
| Db | | Db | ||
|- | |- | ||
| Line 48: | Line 48: | ||
| w2 | | w2 | ||
| wa 2nd | | wa 2nd | ||
| | | M2 | ||
| D | | D | ||
|- | |- | ||
| Line 56: | Line 56: | ||
| Lw2 | | Lw2 | ||
| lawa 2nd | | lawa 2nd | ||
| | | A2 | ||
| D# | | D# | ||
|- | |- | ||
| Line 64: | Line 64: | ||
| w3 | | w3 | ||
| wa 3rd | | wa 3rd | ||
| | | m3 | ||
| Eb | | Eb | ||
|- | |- | ||
| Line 72: | Line 72: | ||
| Lw3 | | Lw3 | ||
| lawa 3rd | | lawa 3rd | ||
| | | M3 | ||
| E | | E | ||
|- | |- | ||
| Line 80: | Line 80: | ||
| sw4 | | sw4 | ||
| sawa 4th | | sawa 4th | ||
| | | d4 | ||
| Fb | | Fb | ||
|- | |- | ||
| Line 88: | Line 88: | ||
| w4 | | w4 | ||
| wa 4th | | wa 4th | ||
| | | P4 | ||
| F | | F | ||
|- | |- | ||
| Line 96: | Line 96: | ||
| Lw4 | | Lw4 | ||
| lawa 4th | | lawa 4th | ||
| | | A4 | ||
| F# | | F# | ||
|- | |- | ||
| Line 104: | Line 104: | ||
| sw5 | | sw5 | ||
| sawa 5th | | sawa 5th | ||
| | | d5 | ||
| Gb | | Gb | ||
|- | |- | ||
| Line 112: | Line 112: | ||
| w5 | | w5 | ||
| wa 5th | | wa 5th | ||
| | | P5 | ||
| G | | G | ||
|- | |- | ||
| Line 120: | Line 120: | ||
| Lw5 | | Lw5 | ||
| lawa 5th | | lawa 5th | ||
| | | A5 | ||
| G# | | G# | ||
|- | |- | ||
| Line 128: | Line 128: | ||
| sw6 | | sw6 | ||
| sawa 6th | | sawa 6th | ||
| | | m6 | ||
| Ab | | Ab | ||
|- | |- | ||
| Line 136: | Line 136: | ||
| w6 | | w6 | ||
| wa 6th | | wa 6th | ||
| | | M6 | ||
| A | | A | ||
|- | |- | ||
| Line 144: | Line 144: | ||
| sw7 | | sw7 | ||
| sawa 7th | | sawa 7th | ||
| | | d7 | ||
| Bbb | | Bbb | ||
|- | |- | ||
| Line 152: | Line 152: | ||
| w7 | | w7 | ||
| wa 7th | | wa 7th | ||
| | | m7 | ||
| Bb | | Bb | ||
|- | |- | ||
| Line 160: | Line 160: | ||
| Lw7 | | Lw7 | ||
| lawa 7th | | lawa 7th | ||
| | | M7 | ||
| B | | B | ||
|- | |- | ||
| Line 168: | Line 168: | ||
| sw8 | | sw8 | ||
| sawa 8ve | | sawa 8ve | ||
| | | d8 | ||
| Cb | | Cb | ||
|- | |- | ||
| Line 176: | Line 176: | ||
| w8 | | w8 | ||
| wa 8ve | | wa 8ve | ||
| | | P8 | ||
| C | | C | ||
|} | |} | ||
Revision as of 12:29, 9 February 2023
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, as well as genres derived from it.
Edo approximation
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, ...
Table of intervals
3-limit intervals up to odd-limit 19683:
| Ratio | Monzo | Size (¢) | Color Name | Diatonic Category | ||
|---|---|---|---|---|---|---|
| 1/1 | [0⟩ | 0.000 | w1 | wa unison | P1 | C |
| 2187/2048 | [-11 7⟩ | 113.685 | Lw1 | lawa 1sn | A1 | C# |
| 256/243 | [8 -5⟩ | 90.225 | sw2 | sawa 2nd | m2 | Db |
| 9/8 | [-3 2⟩ | 203.910 | w2 | wa 2nd | M2 | D |
| 19683/16384 | [-14 9⟩ | 317.595 | Lw2 | lawa 2nd | A2 | D# |
| 32/27 | [5 -3⟩ | 294.135 | w3 | wa 3rd | m3 | Eb |
| 81/64 | [-6 4⟩ | 407.820 | Lw3 | lawa 3rd | M3 | E |
| 8192/6561 | [13 -8⟩ | 384.360 | sw4 | sawa 4th | d4 | Fb |
| 4/3 | [2 -1⟩ | 498.045 | w4 | wa 4th | P4 | F |
| 729/512 | [-9 6⟩ | 611.730 | Lw4 | lawa 4th | A4 | F# |
| 1024/729 | [10 -6⟩ | 588.270 | sw5 | sawa 5th | d5 | Gb |
| 3/2 | [-1 1⟩ | 701.955 | w5 | wa 5th | P5 | G |
| 6561/4096 | [-12 8⟩ | 815.640 | Lw5 | lawa 5th | A5 | G# |
| 128/81 | [7 -4⟩ | 792.180 | sw6 | sawa 6th | m6 | Ab |
| 27/16 | [-4 3⟩ | 905.865 | w6 | wa 6th | M6 | A |
| 32768/19683 | [15 -9⟩ | 882.405 | sw7 | sawa 7th | d7 | Bbb |
| 16/9 | [4 -2⟩ | 996.090 | w7 | wa 7th | m7 | Bb |
| 243/128 | [-7 5⟩ | 1109.775 | Lw7 | lawa 7th | M7 | B |
| 4096/2187 | [12 -7⟩ | 1086.315 | sw8 | sawa 8ve | d8 | Cb |
| 2/1 | [1⟩ | 1200.000 | w8 | wa 8ve | P8 | C |