User:TallKite/Midpoints: Difference between revisions

"Neutral" refers to a range of interval sizes. Anything in roughly the 340-360c range can be called a neutral 3rd. But mid refers to a single specific interval midway between major and minor. A mid interval is neutral, but neutral intervals are very often not mid.

Likewise, these "plus-something" intervals could be defined either vaguely as a cents range (e.g. +2nd = 240-260¢) or precisely as a specific interval. While the former are interordinals, Kite calls the latter "outsiders". So-called because a +2nd falls *outside* of the usual gamut of 2nds and the usual gamut of 3rds. An outsider is interordinal, but interordinals are very often not outside.

The gamuts are P1, m2-M2, m3-M3, P4-A4, d5-P5, m6-M6, m7-M7, P8. (See Kite's Rationales for gamuts below.) The endpoints of these gamuts define both outsiders and mids, which collectively Kite calls midpoints. For example, a mid 2nd is the midpoint of m2 and M2, which is (m2 + M2)/2 = m3/2. And a plus 2nd is (M2 + m3)/2 = P4/2.

Mid and outside are qualities like perfect, major and minor. Neutral and interordinal are also qualities, although slightly vague ones. Perfect, major and minor are in theory precise qualities but in practice they are slightly vague. e.g. 5/4. They can be diverge from their exact by adding super- and sub- serve to . Likewise supermid, subneutral and superoutside, but subinterordinal is going too far. -ish serves to  

Mid can be generalized to "middish" to include all neutral intervals. Likewise "outside-ish" for all interordinals.

 

 

or 15/13 is "a" +2nd, but 5\19 is "the" +2nd?<br>

!quality

!degree

!fraction

|outside

| +1sn

|mid

|2nd

|outside

| +2nd

|mid

|3rd

|outside

| +3rd

|mid

|4th

|outside

| +4th

|mid

|5th

|P8/2 + o+1sn

|outside

| +5th

|P8/2 + o+2nd

|mid

|6th

|outside

| +6th

|mid

|7th

|outside

| +7th

|P8 - o+1sn

There is symmetry not only within the octave but also within the half-octave.

There is no mid 1sn or mid 8ve. The +8ve is as expected, a +1sn plus an 8ve.

Note that a half-aug 1sn is 50 + 3.5c, half a pyth comma wider than a +1sn. And a half-dim 2nd is 50 - 8.5c, half a pyth comma narrower. This is because mid is A1/2 below major, so half-aug is A1/2 above major, but plus is m2/2 above major

Instead of c, we can use fractions of a comma. If we make quarter-comma meantone 7-limit by adding the zozo comma, it gains outsiders. The outside +2nd is half a just 4th plus one-eighth of a comma.

While neutrals and interordinals exist in JI, just intervals are never midpoints, because of the unique prime-factorization theorem. Mid intervals only exist in edos of even sharpness and rank-2 or higher temps that have an EI of vvA1 or v⁴A1 or v⁶A1, etc. And interordinals only exist in edos of even penta-sharpness and rank-2 or higher temps that have an EI of vvm2 or v⁴m2 or v⁶m2, etc.

But we can still talk about a just mid 3rd. it's half of a just P5. And the just outside +2nd is half a just P4. Thus a just midpoint is simply half a 3-limit interval, and is really a 3-limit midpoint.

In edos like 24, 34 and 72, an outsider and a mid add up nicely:

+2nd + ~2nd = P4/2 + m3/2 = m6/2 = M3 - pyth/2

+2nd + ~3rd = P4/2 + P5/2 = P8/2 = aug4 - pyth/2

+3rd + ~2nd = M6/2 + m3/2 = P8/2

+3rd + ~3rd

m3rd = 300 - 3c

~3rd = 350 + c/2

M3rd = 400 + 4c  

+3rd = 450 + 1.5c

Calling 1.5 a minus 2nd conflicts with calling 50/49 or the pyth comma a negative 2nd, since both are written -2nd or -2.  

The 4th's gamut runs perfect-aug and the 5th's runs dim-perfect. Limiting each gamut to perfect only would mess up the ~4th and the ~5th. Expanding each to dim-aug additionally messes up the +3rd and +5th. Note that often A4 > d5, and the outside +4th isn't actually outside of the nearest gamuts.  

Each gamut is a m2 wide, except for the 1sn and the 8ve. Each gap between gamuts is a m2, except for at the half-8ve. The gaps plus the widths run m2 A1 m2 A1 m2 A1 pyth A1 m2 A1 m2 A1 m2.