Sqrt(2187/2048): Difference between revisions
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The half-apotome can be made equal to many rational intervals, most notably, to [[33/32]], tempering out the [[rastma|rastma (243/242)]], or much more accurately to [[1323/1280]], tempering out the [[breedsma|breedsma (2401/2400)]]. | The half-apotome can be made equal to many rational intervals, most notably, to [[33/32]], tempering out the [[rastma|rastma (243/242)]], or much more accurately to [[1323/1280]], tempering out the [[breedsma|breedsma (2401/2400)]]. | ||
It is most often written in scores by a half-sharp, {{Sagittal|t}}. However, this does not mean C{{Sagittal|t}} = D{{Sagittal|d}} always; D{{Sagittal|d}} is a lower pitch than C{{Sagittal|t}}. [[Pythagorean comma#Importance in diatonic|This antiintuitive notion]] is due to the fact that the apotome - an augmented unison - is wider than a minor second in [[Pythagorean tuning]], and [[JI]] more generally. The equality holds true if and only if the pythagorean comma is tempered out. | It is most often written in scores by a half-sharp, {{Sagittal|t}}. However, this does not mean C{{Sagittal|t}} = D{{Sagittal|d}} always; D{{Sagittal|d}} is a lower pitch than C{{Sagittal|t}} in most cases. [[Pythagorean comma#Importance in diatonic|This antiintuitive notion]] is due to the fact that the apotome - an augmented unison - is wider than a minor second in [[Pythagorean tuning]], and [[JI]] more generally. The equality holds true if and only if the pythagorean comma is tempered out. | ||
Revision as of 16:52, 11 July 2026
| Interval information |
semiaugmented unison
sqrt(2187/2048), the half-apotome, or semiaugmented unison, is a radical interval of about 57 cents, in the sqrt(2).sqrt(3) subgroup or dicot (ploidacot) more precisely. It is a quartertone-like interval; differing from the smaller quartertone sqrt(256/243) by half a pythagorean comma. It is reached by +7 hemififths octave reduced, with a fractional monzo of [-11/2 7/2⟩.
In temperaments
The half-apotome can be made equal to many rational intervals, most notably, to 33/32, tempering out the rastma (243/242), or much more accurately to 1323/1280, tempering out the breedsma (2401/2400).
It is most often written in scores by a half-sharp, . However, this does not mean C = D always; D is a lower pitch than C in most cases. This antiintuitive notion is due to the fact that the apotome - an augmented unison - is wider than a minor second in Pythagorean tuning, and JI more generally. The equality holds true if and only if the pythagorean comma is tempered out.