Kite's thoughts on enharmonic unisons: Difference between revisions

Line 5:

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.

~~Just intonation does not have EUs, although as a practical matter just intonation is audibly indistinguishable from certain microtemperaments that do have them.~~

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of microtonal accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. EUs are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank.

Line 53:

Line 51:

Some notations, like ups and downs, notate all but the largest edos with only a single additional pair of accidentals. Other notations, like Sagittal and SKULO, notate edos using various commas such as 81/80, 64/63, and 33/32. Thus they notate an edo interval as a nearby JI interval, indicating the "feel" of the interval. For some edos, these notations use multiple such commas. For example, Sagittal notates [[41edo]] using 81/80 and 33/32, and SKULO notates it using all three commas. Each comma used adds a pair of accidentals, and hence increases the minimum number of EUs needed, complicating the respelling of intervals.

Just intonation notations generally do not have EUs. Each ratio factors uniquely into primes, and each prime number above 3 has its own pair of accidentals, therefore each ratio is uniquely notated. But a JI notation that uses more accidental pairs than that (such as standard Sagittal, as opposed to prime-factor Sagittal) does have EUs. Furthermore, as a practical matter just intonation is audibly indistinguishable from certain microtemperaments that do have EUs.

EUs were first systematically investigated by [[Kite Giedraitis]] in 2018 while developing [[Pergen|pergen notation]].

@@ Line 5: / Line 5: @@
 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.
-Just intonation does not have EUs, although as a practical matter just intonation is audibly indistinguishable from certain microtemperaments that do have them.
 Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of microtonal accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. EUs are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank.
@@ Line 53: / Line 51: @@
 Some notations, like ups and downs, notate all but the largest edos with only a single additional pair of accidentals. Other notations, like Sagittal and SKULO, notate edos using various commas such as 81/80, 64/63, and 33/32. Thus they notate an edo interval as a nearby JI interval, indicating the "feel" of the interval. For some edos, these notations use multiple such commas. For example, Sagittal notates [[41edo]] using 81/80 and 33/32, and SKULO notates it using all three commas. Each comma used adds a pair of accidentals, and hence increases the minimum number of EUs needed, complicating the respelling of intervals.
+Just intonation notations generally do not have EUs. Each ratio factors uniquely into primes, and each prime number above 3 has its own pair of accidentals, therefore each ratio is uniquely notated. But a JI notation that uses more accidental pairs than that (such as standard Sagittal, as opposed to prime-factor Sagittal) does have EUs. Furthermore, as a practical matter just intonation is audibly indistinguishable from certain microtemperaments that do have EUs.
 EUs were first systematically investigated by [[Kite Giedraitis]] in 2018 while developing [[Pergen|pergen notation]].