Sqrt(2187/2048): Difference between revisions
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{{Infobox interval|Name=half-apotome<br>semiaugmented unison|Ratio=\sqrt{2187/2048}|Cents=56.8425}}'''sqrt(2187/2048)''', the '''half-apotome, or semiaugmented unison''', is a [[radical interval]] of about 57 cents, in the [[Hemipyth|sqrt(2).sqrt(3) subgroup]] or [[Ploidacot/Dicot|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 {{Monzo|-11/2 7/2}}. | {{Infobox interval|Name=half-apotome<br>semiaugmented unison|Ratio=\sqrt{2187/2048}|Cents=56.8425}}'''sqrt(2187/2048)''', the '''half-apotome, or semiaugmented unison''', is a [[radical interval]] of about 57 cents, in the [[Hemipyth|sqrt(2).sqrt(3) subgroup]] or [[Ploidacot/Dicot|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 [[Hemififth|hemififths]] octave reduced, with a fractional monzo of {{Monzo|-11/2 7/2}}. | ||
== In temperaments == | == In temperaments == | ||
The half-apotome can be made equal to many rational intervals, most notably, to [[33/32]], tempering out the [[rastma]], or much more accurately to [[1323/1280]], tempering out the [[breedsma]]. | 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 | 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. | ||
Latest revision as of 18:26, 9 June 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. 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.