All of these scales are formatted for use in Scala, but I keep them in bare text form in order to display the contents of the scales directly. If you have Scala, copy-paste the text into a text editor and save it as a .scl file.
You can, alternatively, copy specifically the interval data into the very useful Scale Workshop and screw with it from there.
Tuning theory jargon ahead!!
A 12-step system made entirely with septimal (seven-based) frequency ratios. Makes for an interesting playing experience when mapped to a keyboard. You could try replacing the 49/36 interval with a 11/8 interval if you have no particular use for the tritone.
! custom-septimal.scl ! Custom septimal 12edo approximation 12 ! 28/27 8/7 7/6 9/7 4/3 49/36 3/2 14/9 12/7 7/4 27/14 2/1
A Bohlen-Pierce scale constructed entirely with 9/7 ratios, analagous to Pythagorean.
! bp-pyth.scl ! BP made with pure 9/7 ratios, analagous to Pythagorean 13 ! 2401/2187 19683/16807 9/7 343/243 177147/117649 81/49 49/27 117649/59049 729/343 7/3 16807/6561 6561/2401 3/1
A Bohlen-Pierce scale with the 9/7 thirds tempered so as to make the fifth a pure 5/3.
! bp-temp1.scl ! BP tempered to pure 5/3 13 ! 27/25 308.94178 442.17936 575.41693 125/81 5/3 9/5 243/125 1326.53807 1459.77564 1593.01322 25/9 3/1
A Bohlen-Pierce scale with the 9/7 third tempered so as to make the flat-seventh a pure 2/1. (This also, by nature of the BP scale, tempers the sharp-fourth into a pure 3/2.)
! bp-temp2.scl ! BP tempered to pure 2/1 13 ! 166.01500 267.97000 433.98500 600.00000 3/2 867.97000 1033.98500 2/1 1301.95500 1467.97000 1633.98500 1735.94000 3/1
The Drewnian system consists of a mix of two scales, each built on a septimal tetrachord: one built on a 8/7, 8/7, 49/48 tetrachord generating all of the “major” intervals, and the other built on a 49/48, 8/7, 8/7 tetrachord generating all of the “minor” intervals. This leads to huge major seconds and tiny minor seconds, or a septimal whole tone and a slendro diesis respectively in Scala's terms. For completeness, however, I added a 512/343 interval, theoretically an “augmented fourth”. Perhaps this system is what Western tonality would look like if we looked for a good 7/4 interval instead of a good 5/4.
This scale is compatible with conventional music notation, as is any meantone system like Kirnberger III: the nominals C through B can be mapped to the major scale (the one built from the 8/7, 8/7, 49/48 tetrachord), and the sharp and flat signs can be considered a deviation of 384/343 up or down. Thus the scale below would technically be notated as C, D♭, D, E♭, E, F, F♯, G, A♭, A, B♭, B, C.
The caveat to all of this, of course, is that not all fifths are created equal because the major second isn't based on pure fifths. The G↔D and A↔E fifths are actually 32/21 intervals.
The wolf interval here, in the same sense as usual meantone temperaments, is between F♯ (512/343) and D♭ (49/24), a 24576/16807 interval. The F♯ can be replaced with other intervals though:
! drewnian.scl ! Drewnian scale system 12 ! 49/48 8/7 7/6 64/49 4/3 512/343 3/2 49/32 12/7 7/4 96/49 2/1
The Drewnian temperaments solve the problem of the unequal fifths in the just system. In essence, these are just cycles of relatively sharp fifths that push the minor seventh near the pure 7/4 interval. The comma here is the 64/63 interval, the difference between the harmonic seventh 7/4 and the Pythagorean minor seventh 16/9. Thus, the fifth would need to be sharpened by half of this comma to push the minor seventh to 7/4.
The 1/4-comma temperament here, as you can see, makes the minor sixth perfectly hit the 14/9 interval. The fifth is 708.77 cents large, pretty close to the fifth of 22-EDO. This puts the minor seventh at 982.46 cents, 13.63 cents sharper than a 7/4.
! drewnian-temp2.scl ! Drewnian temperament, 1/4-comma "meantone" with 64/63 comma 12 ! 56.14488 217.54205 273.68693 9/7 491.22898 652.62614 708.77102 14/9 926.31307 982.45795 1143.85512 2/1
The Drewnian temperaments solve the problem of the unequal fifths in the just system. In essence, these are just cycles of relatively sharp fifths that push the minor seventh near the pure 7/4 interval. The comma here is the 64/63 interval, the difference between the harmonic seventh 7/4 and the Pythagorean minor seventh 16/9. Thus, the fifth would need to be sharpened by half of this comma to push the minor seventh to 7/4.
The 1/3-comma temperament here, as you can see, makes the major sixth perfectly hit the 12/7 interval. The fifth is 711.04 cents large, pretty close to the fifth of 27-EDO. This puts the minor seventh at 977.91 cents, 9.09 cents sharper than a 7/4.
! drewnian-temp1.scl ! Drewnian temperament, 1/3-comma "meantone" with 64/63 comma 12 ! 44.78484 222.08606 7/6 444.17213 488.95697 72/49 711.04303 755.82787 12/7 977.91394 1155.21516 2/1
The Drewnian temperaments solve the problem of the unequal fifths in the just system. In essence, these are just cycles of relatively sharp fifths that push the minor seventh near the pure 7/4 interval. The comma here is the 64/63 interval, the difference between the harmonic seventh 7/4 and the Pythagorean minor seventh 16/9. Thus, the fifth would need to be sharpened by half of this comma to push the minor seventh to 7/4.
The 1/2-comma temperament here, as you can see, makes the minor seventh perfectly hit the 7/4 interval. The fifth is 715.59 cents large, pretty close to the fifth of 42-EDO. While the seventh is perfect, the fifth is sharp to the point of dissonance; it's 13.63 cents sharper than pure.
! drewnian-temp3.scl ! Drewnian temperament, 1/2-comma "meantone" with 64/63 comma 12 ! 22.06477 8/7 253.23886 64/49 484.41295 512/343 715.58705 49/32 946.76114 7/4 1177.93523 2/1