Extended meantone notation: Difference between revisions

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... {{dash|F𝄫, C𝄫, G𝄫, D𝄫, A𝄫, E𝄫, B𝄫, F♭, C♭, G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, E♯, B♯, F𝄪, C𝄪, G𝄪, D𝄪, A𝄪, E𝄪, B𝄪|s=thin|d=long}} ...

In some tunings, such as [[24-tone equal temperament]] (quarter tones) and [[31-tone equal temperament]] (extended [[quarter-comma meantone]]), sharps can be split in half. Thus, some notes can be notated using semisharps and semiflats, or with [[ups and downs notation|ups and downs]].

In a general meantone tuning, a sharp is split into 2 different parts, the diesis and the kleisma.

~~For example, in 31edo, the chromatic scale becomes:~~

~~{{dash|C, D𝄫, C♯, D♭, C𝄪, D, E𝄫, D♯, E♭, D𝄪, E, F♭, E♯, F, G𝄫, F♯, G♭, F𝄪, G, A𝄫, G♯, A♭, G𝄪, A, B𝄫, A♯, B♭, A𝄪, B, C♭, B♯, C|s=thin|d=long}}~~

~~Note that the base note letters alternate.~~

~~Using semisharps and semiflats, this can be re-written as:~~

{{dash|C, C{{demisharp2}}, C♯, D♭, D{{demiflat2}}, D, D{{demisharp2}}, D♯, E♭, E{{demiflat2}}, E, E{{demisharp2}}, F{{demiflat2}}, F, F{{demisharp2}}, F♯, G♭, G{{demiflat2}}, G, G{{demisharp2}}, G♯, A♭, A{{demiflat2}}, A, A{{demisharp2}}, A♯, B♭, B{{demiflat2}}, B, B{{demisharp2}}, C{{demiflat2}}, C|s=thin|d=long}}

~~The generalized chain of fifths, however, does not have a single semisharp or semiflat.~~ In a general meantone tuning, a sharp is split into 2 different parts, the diesis and the kleisma.

== Generalizing accidentals ==

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{| class="wikitable center-all"

! rowspan="2" style="width: 50px;" | Notes per octave

! rowspan="2" style="width: ~~75px~~;" | [[81/80|~~Syntonic~~ comma]] fraction

! rowspan="2" style="width: 150px;" | Approximate [[81/80|syntonic comma]] fraction

! colspan="4" | Steps

! rowspan="2" style="width: 275px;" | Explanation

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== True half-sharps and half-flats ==

If sharps raise by an even number of steps, such as [[24-tone equal temperament]] (quarter tones) and [[31-tone equal temperament]] (extended [[quarter-comma meantone]]), they (along with flats) can be split in half. Thus, some notes can be notated using semisharps and semiflats, or with [[ups and downs notation|ups and downs]].

For example, in 31 equal, the chromatic scale becomes:

{{dash|C, D𝄫, C♯, D♭, C𝄪, D, E𝄫, D♯, E♭, D𝄪, E, F♭, E♯, F, G𝄫, F♯, G♭, F𝄪, G, A𝄫, G♯, A♭, G𝄪, A, B𝄫, A♯, B♭, A𝄪, B, C♭, B♯, C|s=thin|d=long}}

Note that the base note letters alternate.

Using semisharps and semiflats, this can be re-written as:

{{dash|C, C{{demisharp2}}, C♯, D♭, D{{demiflat2}}, D, D{{demisharp2}}, D♯, E♭, E{{demiflat2}}, E, E{{demisharp2}}, F{{demiflat2}}, F, F{{demisharp2}}, F♯, G♭, G{{demiflat2}}, G, G{{demisharp2}}, G♯, A♭, A{{demiflat2}}, A, A{{demisharp2}}, A♯, B♭, B{{demiflat2}}, B, B{{demisharp2}}, C{{demiflat2}}, C|s=thin|d=long}}

<!--

If true half-sharps and true half-flats are desired, which exactly bisect the chromatic semitone, the meantone fifth is split in half. This creates a new tuning system consisting of a two-dimensional lattice generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class.

While real-world Arabic and Persian music often involve many very fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and are very difficult to notate exactly as a result, they are commonly notated using half-sharps and half-flats. If we take these to be exactly equal to one-half of a chromatic semitone, then mathematically, this notation system results in the aforementioned two-dimensional lattice. If ~~adjacent sharps and flats~~, such as C♯ and D♭, are made enharmonically equivalent, this lattice degenerates further into 24edo, which is often suggested as a simplified framework and tuning system for notating and playing Arabic and Persian music. But the usual written notation typically lets musicians and composers treat adjacent sharps and flats as two distinct entities if it is decided that they should be different.

While real-world Arabic and Persian music often involve many very fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and are very difficult to notate exactly as a result, they are commonly notated using half-sharps and half-flats. If we take these to be exactly equal to one-half of a chromatic semitone, then mathematically, this notation system results in the aforementioned two-dimensional lattice. If notes separated by a diminished second, such as C♯ and D♭, are made enharmonically equivalent, this lattice degenerates further into 24edo, which is often suggested as a simplified framework and tuning system for notating and playing Arabic and Persian music. But the usual written notation typically lets musicians and composers treat adjacent sharps and flats as two distinct entities if it is decided that they should be different.

The chain-of-neutral-thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any just interval mapping to it in the 7-limit. But, if we bring in the 11th harmonic, and decide that there should only be a single neutral second (resulting in 11/10 and 12/11, the greater and lesser neutral seconds, both being mapped to a single equally-tempered interval), we obtain [[mohajira]], a very accurate 11-limit temperament. The neutral third approximates 11/9, and two of them make a perfect fifth, resulting in [[243/242]] being tempered out. Furthermore, flattening a minor third by a half-flat results in an approximation of 7/6, while sharpening a major third by a half-sharp gives an approximation of 9/7. Mohajira is supported very well by 24edo and 31edo.

(This section needs considerable re-wording, I'm commenting it out for now) - ArrowHead294

-->

== Notes ==

@@ Line 3: / Line 3: @@
 ... {{dash|F𝄫, C𝄫, G𝄫, D𝄫, A𝄫, E𝄫, B𝄫, F♭, C♭, G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, E♯, B♯, F𝄪, C𝄪, G𝄪, D𝄪, A𝄪, E𝄪, B𝄪|s=thin|d=long}} ...
-In some tunings, such as [[24-tone equal temperament]] (quarter tones) and [[31-tone equal temperament]] (extended [[quarter-comma meantone]]), sharps can be split in half. Thus, some notes can be notated using semisharps and semiflats, or with [[ups and downs notation|ups and downs]].
+In a general meantone tuning, a sharp is split into 2 different parts, the diesis and the kleisma.
-For example, in 31edo, the chromatic scale becomes:
-{{dash|C, D𝄫, C♯, D♭, C𝄪, D, E𝄫, D♯, E♭, D𝄪, E, F♭, E♯, F, G𝄫, F♯, G♭, F𝄪, G, A𝄫, G♯, A♭, G𝄪, A, B𝄫, A♯, B♭, A𝄪, B, C♭, B♯, C|s=thin|d=long}}
-Note that the base note letters alternate.
-Using semisharps and semiflats, this can be re-written as:
-{{dash|C, C{{demisharp2}}, C♯, D♭, D{{demiflat2}}, D, D{{demisharp2}}, D♯, E♭, E{{demiflat2}}, E, E{{demisharp2}}, F{{demiflat2}}, F, F{{demisharp2}}, F♯, G♭, G{{demiflat2}}, G, G{{demisharp2}}, G♯, A♭, A{{demiflat2}}, A, A{{demisharp2}}, A♯, B♭, B{{demiflat2}}, B, B{{demisharp2}}, C{{demiflat2}}, C|s=thin|d=long}}
-The generalized chain of fifths, however, does not have a single semisharp or semiflat. In a general meantone tuning, a sharp is split into 2 different parts, the diesis and the kleisma.
 == Generalizing accidentals ==
@@ Line 52: / Line 40: @@
 {| class="wikitable center-all"
 ! rowspan="2" style="width: 50px;" | Notes per octave
-! rowspan="2" style="width: 75px;" | [[81/80|Syntonic comma]] fraction
+! rowspan="2" style="width: 150px;" | Approximate [[81/80|syntonic comma]] fraction
 ! colspan="4" | Steps
 ! rowspan="2" style="width: 275px;" | Explanation
@@ Line 185: / Line 173: @@
 == True half-sharps and half-flats ==
+If sharps raise by an even number of steps, such as [[24-tone equal temperament]] (quarter tones) and [[31-tone equal temperament]] (extended [[quarter-comma meantone]]), they (along with flats) can be split in half. Thus, some notes can be notated using semisharps and semiflats, or with [[ups and downs notation|ups and downs]].
+For example, in 31 equal, the chromatic scale becomes:
+{{dash|C, D𝄫, C♯, D♭, C𝄪, D, E𝄫, D♯, E♭, D𝄪, E, F♭, E♯, F, G𝄫, F♯, G♭, F𝄪, G, A𝄫, G♯, A♭, G𝄪, A, B𝄫, A♯, B♭, A𝄪, B, C♭, B♯, C|s=thin|d=long}}
+Note that the base note letters alternate.
+Using semisharps and semiflats, this can be re-written as:
+{{dash|C, C{{demisharp2}}, C♯, D♭, D{{demiflat2}}, D, D{{demisharp2}}, D♯, E♭, E{{demiflat2}}, E, E{{demisharp2}}, F{{demiflat2}}, F, F{{demisharp2}}, F♯, G♭, G{{demiflat2}}, G, G{{demisharp2}}, G♯, A♭, A{{demiflat2}}, A, A{{demisharp2}}, A♯, B♭, B{{demiflat2}}, B, B{{demisharp2}}, C{{demiflat2}}, C|s=thin|d=long}}
+<!--
 If true half-sharps and true half-flats are desired, which exactly bisect the chromatic semitone, the meantone fifth is split in half. This creates a new tuning system consisting of a two-dimensional lattice generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class.
-While real-world Arabic and Persian music often involve many very fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and are very difficult to notate exactly as a result, they are commonly notated using half-sharps and half-flats. If we take these to be exactly equal to one-half of a chromatic semitone, then mathematically, this notation system results in the aforementioned two-dimensional lattice. If adjacent sharps and flats, such as C♯ and D♭, are made enharmonically equivalent, this lattice degenerates further into 24edo, which is often suggested as a simplified framework and tuning system for notating and playing Arabic and Persian music. But the usual written notation typically lets musicians and composers treat adjacent sharps and flats as two distinct entities if it is decided that they should be different.
+While real-world Arabic and Persian music often involve many very fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and are very difficult to notate exactly as a result, they are commonly notated using half-sharps and half-flats. If we take these to be exactly equal to one-half of a chromatic semitone, then mathematically, this notation system results in the aforementioned two-dimensional lattice. If notes separated by a diminished second, such as C♯ and D♭, are made enharmonically equivalent, this lattice degenerates further into 24edo, which is often suggested as a simplified framework and tuning system for notating and playing Arabic and Persian music. But the usual written notation typically lets musicians and composers treat adjacent sharps and flats as two distinct entities if it is decided that they should be different.
 The chain-of-neutral-thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any just interval mapping to it in the 7-limit. But, if we bring in the 11th harmonic, and decide that there should only be a single neutral second (resulting in 11/10 and 12/11, the greater and lesser neutral seconds, both being mapped to a single equally-tempered interval), we obtain [[mohajira]], a very accurate 11-limit temperament. The neutral third approximates 11/9, and two of them make a perfect fifth, resulting in [[243/242]] being tempered out. Furthermore, flattening a minor third by a half-flat results in an approximation of 7/6, while sharpening a major third by a half-sharp gives an approximation of 9/7. Mohajira is supported very well by 24edo and 31edo.
+(This section needs considerable re-wording, I'm commenting it out for now) - ArrowHead294
+-->
 == Notes ==