Quarter Comma Meantone Scales

1/4-Comma Meantone scales are related to Pythagorean scales in terms of being built from fifths, but in this meantone, the perfect 2:3 fifth is reduced in size by 1/4 of the Syntonic Comma, which makes a stack of four of these fifths minus two octaves equal to a just 4:5 major third, compared to the Pythagorean 64:81 major third, which is 21 cents sharper. Thus the only two ‘simple’ in-tune intervals here are the octave and the major third. In these examples, the notation is assuming Pythagorean tuning of perfect fourths and fifths when perfect intervals are written. Thus, the additional accidentals shown indicate how much deviation there is for a note compared to Pythagorean tuning. The E and A-flat contain the Syntonic Comma accidental; the F and G contain the 1/4-comma accidental; other accidentals are thus composites of these. (Note that I use these highly flexible Sagittal accidentals in other ways in other contexts.)

These playbacks are realized via Dorico 2.2.1, using a 224EDO custom tonality system. 224EDO turns out to realize this particular meantone fairly easily, because the Syntonic Comma accidental ends up representing 4 steps out of 224.

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