Sharpness: Difference between revisions

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The sharpness of an edo has implications for the heptatonic fifth-generated notation of that edo. For example, all sharp-1 edos (5, 12, 19, 26...) can be notated conventionally with just 7 letters and #/b. Another example: the half-sharp and half-flat accidentals are applicable to an edo only if its sharpness is an even number.

The '''penta-sharpness''' or <span id="eefness";>'''eefness'''</span> of an edo is the number of steps to which it maps the diatonic semitone aka 3-limit minor 2nd aka limma ([[256/243]]). In other words, it's three octaves minus five of its best approximation of [[3/2]]. If one's notation were pentatonic instead of heptatonic, the concept of sharpness would be applied to the limma not the apotome, hence the first name. The second name derives from the limma being notated as the interval E-F and is used ~~for~~ the [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation|Sagittal limma-fraction notation]].

The '''penta-sharpness''' or <span id="eefness";>'''eefness'''</span> of an edo is the number of steps to which it maps the diatonic semitone aka 3-limit minor 2nd aka limma ([[256/243]]). In other words, it's three octaves minus five of its best approximation of [[3/2]]. If one's notation were pentatonic instead of heptatonic, the concept of sharpness would be applied to the limma not the apotome, hence the first name. The second name derives from the limma being notated as the interval E-F and is used by the [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation|Sagittal limma-fraction notation]].

For example, 12, 17 and 22 are all pentasharp-1 edos, and 19 and 24 are both pentasharp-2 edos. A pentasharp-0 edo (5, 10, 15, etc.) is also known as a "pentatonic edo".

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 The sharpness of an edo has implications for the heptatonic fifth-generated notation of that edo. For example, all sharp-1 edos (5, 12, 19, 26...) can be notated conventionally with just 7 letters and #/b. Another example: the half-sharp and half-flat accidentals are applicable to an edo only if its sharpness is an even number.
-The '''penta-sharpness''' or <span id="eefness";>'''eefness'''</span> of an edo is the number of steps to which it maps the diatonic semitone aka 3-limit minor 2nd aka limma ([[256/243]]). In other words, it's three octaves minus five of its best approximation of [[3/2]]. If one's notation were pentatonic instead of heptatonic, the concept of sharpness would be applied to the limma not the apotome, hence the first name. The second name derives from the limma being notated as the interval E-F and is used for the [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation|Sagittal limma-fraction notation]].
+The '''penta-sharpness''' or <span id="eefness";>'''eefness'''</span> of an edo is the number of steps to which it maps the diatonic semitone aka 3-limit minor 2nd aka limma ([[256/243]]). In other words, it's three octaves minus five of its best approximation of [[3/2]]. If one's notation were pentatonic instead of heptatonic, the concept of sharpness would be applied to the limma not the apotome, hence the first name. The second name derives from the limma being notated as the interval E-F and is used by the [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation|Sagittal limma-fraction notation]].
 For example, 12, 17 and 22 are all pentasharp-1 edos, and 19 and 24 are both pentasharp-2 edos. A pentasharp-0 edo (5, 10, 15, etc.) is also known as a "pentatonic edo".