5L 4s: Difference between revisions
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=== Hard-of-basic === | === Hard-of-basic === | ||
These tunings satisfy the property that two [[semifourth]] generators make a ''diatonic'' ([[5L 2s]]) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) and 3\14 (257.14¢). | These tunings satisfy the property that two [[semifourth]] generators make a ''diatonic'' ([[5L 2s]]) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) and 3\14 (257.14¢). | ||
==== Hypohard ==== | |||
The sizes of the generator, large step and small step of 5L 4s are as follows in various hypohard (2/1 ≤ L/s ≤ 3/1) tunings. | |||
{| class="wikitable right-2 right-3 right-4 right-5" | |||
|- | |||
! | |||
! [[14edo]] (L/s = 2/1) | |||
! [[47edo]] (L/s = 7/3) | |||
! [[33edo]] (L/s = 5/2) | |||
! [[52edo]] (L/s = 8/3) | |||
! [[19edo]] (L/s = 3/1) | |||
|- | |||
| generator (g) | |||
| 3\14, 257.14 | |||
| 10\47 | |||
| 7\33, 254.54 | |||
| 11\52, 253.85 | |||
| 4\19, 252.63 | |||
|- | |||
| L (octave - 4g) | |||
| 171.43 | |||
| 181.81 | |||
| 184.62 | |||
| 189.47 | |||
|- | |||
| s (5g - octave) | |||
| 85.71 | |||
| 72.73 | |||
| 69.23 | |||
| 63.16 | |||
|} | |||
This range is notable for having many tunings that are close to being "eigentunings" (tunings that tune a certain JI interval exactly): | |||
* 33edo semiquartal has close 7/5 (error -0.69¢), 9/5 (error -0.59¢) and 9/7 (error +1.28¢), thus can be used for the close 5:7:9 in the two Locrian-like modes 1|7 and 0|8 | |||
* 52edo semiquartal has close 22/19 (error +0.04¢) | |||
* 19edo semiquartal has close 6/5 (error +0.15¢) and 28/27 (error +0.20¢) | |||
==== Parahard and ultrahard ==== | |||
One important sub-range is given by stipulating that two semifourth generators must make a ''meantone'' fourth; i.e. that four fifths should approximate a [[5/4]] major third. This can be considered the [[19edo]] (4\19)-to-[[24edo]] (5\24) range, i.e. parahard semiquartal, which also contains [[43edo]] (9\43) and [[62edo]] (13\62). Parahard semiquartal can be given an RTT interpretation known as [[godzilla]]. | |||
The sizes of the generator, large step and small step of 5L 4s are as follows in various | The sizes of the generator, large step and small step of 5L 4s are as follows in various hypohard (2/1 ≤ L/s ≤ 3/1) tunings. | ||
{| class="wikitable right-2 right-3 right-4 right-5" | {| class="wikitable right-2 right-3 right-4 right-5" | ||
|- | |- | ||
! [[19edo]] | ! [[19edo]] | ||
! [[24edo]] | ! [[24edo]] | ||
| Line 40: | Line 74: | ||
| generator (g) | | generator (g) | ||
| 3\14, 257.14 | | 3\14, 257.14 | ||
| 7\33, 254.54 | |||
| 11\52, 253.85 | |||
| 4\19, 252.63 | | 4\19, 252.63 | ||
| 5\24, 250.00 | | 5\24, 250.00 | ||
| Line 46: | Line 82: | ||
| L (octave - 4g) | | L (octave - 4g) | ||
| 171.43 | | 171.43 | ||
| 181.81 | |||
| 184.62 | |||
| 189.47 | | 189.47 | ||
| 200.00 | | 200.00 | ||
| Line 52: | Line 90: | ||
| s (5g - octave) | | s (5g - octave) | ||
| 85.71 | | 85.71 | ||
| 72.73 | |||
| 69.23 | |||
| 63.16 | | 63.16 | ||
| 50.00 | | 50.00 | ||
| 41.38 | | 41.38 | ||
|} | |} | ||
=== Soft-of-basic === | === Soft-of-basic === | ||