User:TallKite/Midpoints

There are two types of midpoints, neutral and interordinal.

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Background

Any interval roughly midway between a M2 and a m3 is an interordinal interval. As is one between a M3 and a P4, and so forth. While it's easy to label or categorize most intervals as 2nds, 3rds, etc., interordinals are difficult to label. Kite Giedraitis proposes labeling them "plus-second", "plus-third", etc., written +2nd or +2. See Kite's Rationales for terminology below.

"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.

MID IS ~, NEED A SYMBOL FOR OUTSIDE
+2nd includes interordinal plus-2nds like 15/13 as well
o+2nd or o2nd or ~+2nd?
or 15/13 is "a" +2nd, but 5\19 is "the" +2nd?

MIDPOINTS
+1sn = m2/2
~2nd = m3/2
+2nd = P4/2
~3rd = P5/2
+3rd = M6/2
~4th = M7/2
+4th = P8/2 = half-octave = 600¢ if the 8ve is pure
~5th = m9/2 = m2/2 + 600c
+5th = m10/2 = P8/2 + m3/2
~6th = P11/2
+6th = P12/2 = P8/2 + P5/2 = P8/2 + mid 3rd
~7th = M13/2
+7th = M14/2 = P8 - m2/2 = P8 - +1sn

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

Calling an outside +4th a tritone is a little problematic.

• A tritone is literally three M2's, which is an aug 4th.
• A large odd edo like 41 has two tritones, each of which is an interordinal but not an outsider.

One could label various interordinals as perfect, major, minor, etc. But outsiders aren't major or minor, they're just outside. Analogous to how mids are just mid. IOW mid and outside are themselves qualities. Neutral is also a quality, although a vague one, whereas mid is exact. Major, minor, etc. are in theory exact, but in practice vague. If they were in practice exact, we would describe e.g. 5/4 as "majorish".

Size in cents

The exact cents of the midpoints are defined by the size of the 5th. As well as the octave, if it's stretched or compressed. Assume unstretched 8ves and a 5th of 700+c cents. Thus c = one twelfth of a (possibly tempered) pyth comma.

+1sn = 50 - 2.5c = quartertone

~2nd = 150 - 1.5c

+2nd = 250 - c/2

~3rd = 350 + c/2

+3rd = 450 + 1.5c

~4th = 550 + 2.5c

+4th = 600

~5th = 650 - 2.5c

+5th = 750 - 1.5c

~6th = 850 - c/2

+6th = 950 + c/2

~7th 1050 + 1.5c

+7th 1150 + 2.5c

Except for the +4th, a plus interval is always half a m2 above the major or perfect version of that interval.

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.

Except in the region of the half-octave, every midpoint is midway between the two midpoints nearest to it.

Occurences

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.

What about (P8/2, P5)? it has vvd2 and a +4.

Any other pergens? ask Praveen

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.

Interval arithmetic

When you add two plus-intervals, the sum is plus-one from what it would be without the plusses: the sum of a 2nd and a 3rd is a 4th the sum of a +2nd and a +3rd is a 5th

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

The logic is: flatten the +something to a major-something, sharpen the mid-something to a major-something, and add a pyth comma.

m3rd = 300 - 3c ~3rd = 350 + c/2 M3rd = 400 + 4c +3rd = 450 + 1.5c P4th = 500 - c

mid to major = 50 + 3.5c = A1/2 major to plus = 50 - 2.5c = m2/2 = A1/2 - pyth/2

Generalizations

All the above generalizes beyond heptatonic notation.

duodecatonic: Midway between a m2 and an A1 (both duodeca-2nds) lies the mid duodeca-2nd = 100¢ + c.. Gamuts are tricky to define because the duodeca-2nd 's gamut is either A1-m2 or m2-A1.

pentatonic: A mid penta-2nd is exactly the same as a heptatonic outside plus-2nd, and a plus penta-2nd is the same as a heptatonic mid 3rd.

Midpoints only require a MOS scale. Going to the next larger or smaller MOS exchanges mids with outsiders and vice versa. (As first described and proven by ________.)

If one generalizes the "outsider" concept so that mids become "insiders", this theorem can be called the inside/outside theorem.

Kite's rationales

Terminology

Calling interordinals 2.5, 3.5, etc. doesn't work well in conversation. "Second-point-five"? Yuck!

While semi-4th works for 2.5, "semi-12th" and "semi-14th" are awkward.

Supermajor and subminor are already used for approximate cents ranges. And 9/7 is often called supermajor but rarely called interordinal.

"Second-third" etc. is too long, and conflicts with "the second third in Kum Ba Yah's melody is minor".

"Surd", "thorth", "forfth" etc. are too cringe.

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.

Gamuts

The gamut of imperfect degrees runs from minor to major. Extending that to include aug and dim wouldn't change the location of the +2nd or +6th, and would mess up (i.e. give non-intuitive results for) the +1sn and +7th.

The 4th's gamut runs perfect-aug and the 5th's runs dim-perfect. Limiting each to perfect only, or expanding each to dim-aug doesn't change the location of the +4th, and the latter messes up the +3rd and +5th.

The unison's gamut is very small, just P1, to avoid messing up the +1sn. Likewise the octave's gamut is P8, to avoid messing up the +7th.

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.