Kite's thoughts on enharmonic unisons: Difference between revisions

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The notation of every temperament, including every edo, has at least one '''enharmonic unison''', abbreviated as '''EU''' (with one rare exception, see below). An EU is by definition enharmonically equivalent to a perfect unison. (''Enharmonically equivalent'' is used here in the modern sense of "the same exact pitch, merely named differently".) Any note or interval can be respelled by adding or subtracting an EU.

For example, in 12edo, {{nowrap|A4 {{=}} d5}} and {{nowrap|F♯ {{=}} G♭}}. Such equivalences result from adding or subtracting a diminished 2nd, abbreviated as d2. But in 19edo, {{nowrap|A4 {{=}} dd5}} and {{nowrap|F♯ {{=}} G𝄫}}. 19edo's EU is the dd2.

For example, in 12edo, {{nowrap|A4 {{=}} d5}} and {{nowrap|F♯ {{=}} G♭}}. Such equivalences result from adding or subtracting a diminished 2nd, abbreviated as d2. But in 19edo, {{nowrap|A4 {{=}} dd5}} and {{nowrap|F♯ {{=}} G𝄫}}. 19edo's EU is the dd2, a double-diminished 2nd. As these two examples show, an enharmonic unison may actually be not a unison but a second.

EUs are very useful for respelling notes and intervals less awkwardly. For example, in 12edo we can add a d2 to {{nowrap|B♯}} to convert it to C, or we can subtract a d2 from a diminished 4th to get a major 3rd.

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Just as a temperament can be defined by a list of primes (or more generally a JI subgroup) and a list of linearly independent commas, a notation can be defined by a list of accidentals and a list of linearly independent EUs. Furthermore, that notation implies a certain edo or pergen.For example, conventional notation (the usual 7 letters plus sharps and flats) with a dd2 EU must be 19edo, because that EU reduces the infinite chain of 5ths to a closed loop of 19 notes. Conventional notation plus ups and downs with a vvA1 EU must be (P8, P5/2), because P5 = vM3 + ^m3 = vM3 + ^m3 + vvA1 = vM3 + vM3.

Some notations have just one EU, others are multi-EU. A multi-comma temperament can be defined by various equivalent but different comma lists. Likewise, a multi-EU notation can be defined by various EUs. Some notations define a canonical list of EUs.

== Notation-specific observations ==

There's one type of edo notation that does not produce any EUs: giving each note a unique letter. For example, an octave of 7edo is notated C D E F G A B C. The intervals are named 1sn, 2nd, 3rd, 4th, 5th, 6th, 7th, and octave, all perfect. There are no major or minor or augmented or diminished intervals. As long as one refrains from using sharps or flats, there will be one and only one name for each note and each interval. Because there is a finite number of possible note names, this notation is rank-1 not rank-2.

There's one type of edo notation that does not produce any EUs: giving each note a unique letter. For example, an octave of 7edo is notated {{nowrap|C D E F G A B C}}. The intervals are named 1sn, 2nd, 3rd, 4th, 5th, 6th, 7th, and octave, all perfect. There are no major or minor or augmented or diminished intervals. As long as one refrains from using sharps or flats, there will be one and only one name for each note and each interval. Because there is a finite number of possible note names, this notation is rank-1 not rank-2.

Likewise, if an octave of 8edo were notated as J K L M N O P Q J with no sharps or flats, there would be no EUs. Though, this type of notation is obviously only practical for small edos.

Likewise, if an octave of 8edo were notated as {{nowrap|J K L M N O P Q J}} with no sharps or flats, there would be no EUs. Though, this type of notation is obviously only practical for small edos.

The usage of half-sharps and half-flats ({{demisharp2}} and {{demiflat2}}) creates a rather obvious EU: {{demiflat2}}{{demiflat2}}{{nbhsp}}A1.

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 The notation of every temperament, including every edo, has at least one '''enharmonic unison''', abbreviated as '''EU''' (with one rare exception, see below). An EU is by definition enharmonically equivalent to a perfect unison. (''Enharmonically equivalent'' is used here in the modern sense of "the same exact pitch, merely named differently".) Any note or interval can be respelled by adding or subtracting an EU.
-For example, in 12edo, {{nowrap|A4 {{=}} d5}} and {{nowrap|F&#x266F; {{=}} G&#x266D;}}. Such equivalences result from adding or subtracting a diminished 2nd, abbreviated as d2. But in 19edo, {{nowrap|A4 {{=}} dd5}} and {{nowrap|F&#x266F; {{=}} G&#x1D12B;}}. 19edo's EU is the dd2.
+For example, in 12edo, {{nowrap|A4 {{=}} d5}} and {{nowrap|F&#x266F; {{=}} G&#x266D;}}. Such equivalences result from adding or subtracting a diminished 2nd, abbreviated as d2. But in 19edo, {{nowrap|A4 {{=}} dd5}} and {{nowrap|F&#x266F; {{=}} G&#x1D12B;}}. 19edo's EU is the dd2, a double-diminished 2nd. As these two examples show, an enharmonic unison may actually be not a unison but a second.
 EUs are very useful for respelling notes and intervals less awkwardly. For example, in 12edo we can add a d2 to {{nowrap|B&#x266F;}} to convert it to C, or we can subtract a d2 from a diminished 4th to get a major 3rd.
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+Just as a temperament can be defined by a list of primes (or more generally a JI subgroup) and a list of linearly independent commas, a notation can be defined by a list of accidentals and a list of linearly independent EUs. Furthermore, that notation implies a certain edo or pergen.For example, conventional notation (the usual 7 letters plus sharps and flats) with a dd2 EU must be 19edo, because that EU reduces the infinite chain of 5ths to a closed loop of 19 notes. Conventional notation plus ups and downs with a vvA1 EU must be (P8, P5/2), because P5 = vM3 + ^m3 = vM3 + ^m3 + vvA1 = vM3 + vM3.
 Some notations have just one EU, others are multi-EU. A multi-comma temperament can be defined by various equivalent but different comma lists. Likewise, a multi-EU notation can be defined by various EUs. Some notations define a canonical list of EUs.
 == Notation-specific observations ==
-There's one type of edo notation that does not produce any EUs: giving each note a unique letter. For example, an octave of 7edo is notated C D E F G A B C. The intervals are named 1sn, 2nd, 3rd, 4th, 5th, 6th, 7th, and octave, all perfect. There are no major or minor or augmented or diminished intervals. As long as one refrains from using sharps or flats, there will be one and only one name for each note and each interval. Because there is a finite number of possible note names, this notation is rank-1 not rank-2.
+There's one type of edo notation that does not produce any EUs: giving each note a unique letter. For example, an octave of 7edo is notated {{nowrap|C D E F G A B C}}. The intervals are named 1sn, 2nd, 3rd, 4th, 5th, 6th, 7th, and octave, all perfect. There are no major or minor or augmented or diminished intervals. As long as one refrains from using sharps or flats, there will be one and only one name for each note and each interval. Because there is a finite number of possible note names, this notation is rank-1 not rank-2.
-Likewise, if an octave of 8edo were notated as J K L M N O P Q J with no sharps or flats, there would be no EUs. Though, this type of notation is obviously only practical for small edos.
+Likewise, if an octave of 8edo were notated as {{nowrap|J K L M N O P Q J}} with no sharps or flats, there would be no EUs. Though, this type of notation is obviously only practical for small edos.
 The usage of half-sharps and half-flats ({{demisharp2}} and {{demiflat2}}) creates a rather obvious EU: {{demiflat2}}{{demiflat2}}{{nbhsp}}A1.