User:Vector/Wedgie - Revision history

Vector at 07:51, 14 April 2026

2026-04-14T07:51:52Z

← Older revision		Revision as of 07:51, 14 April 2026
Line 11:		Line 11:
	* the direction of alteration of the imperfect interval, relative to some standard reference (usually the direction of alteration of the imperfect 3/2), conveyed by the sign		* the direction of alteration of the imperfect interval, relative to some standard reference (usually the direction of alteration of the imperfect 3/2), conveyed by the sign

	These two properties are, as it turns out, invariant with MOS size given the same temperament. In fact, this holds when any just interval is considered, not just ratios between primes: this is trivially derivable from the fact that a temperament's base subgroup may be expressed in terms of a set of generators containing any given interval as long as the correct intervals are chosen for the remaining generators. (For example, this holds for 16/15 in meantone - there are always five imperfect 16/15s, because meantone can be thought of as, for instance, a 3/2.5/4.16/15 temperament.) However, it may also be derived from the temperament's generator chain - all but N modes have the note at N generator steps available, invariant of the size of the scale.		These two properties are, as it turns out, invariant with MOS size given the same temperament (and in fact invariant with ''non-MOS scale size'' as long as you define accidentals and degrees with respect to the genchain). In fact, this holds when any just interval is considered, not just ratios between primes: this is trivially derivable from the fact that a temperament's base subgroup may be expressed in terms of a set of generators containing any given interval as long as the correct intervals are chosen for the remaining generators. (For example, this holds for 16/15 in meantone - there are always five imperfect 16/15s, because meantone can be thought of as, for instance, a 3/2.5/4.16/15 temperament.) However, it may also be derived from the temperament's generator chain - all but N modes have the note at N generator steps available, invariant of the size of the scale.

	We may step through a few examples.		We may step through a few examples.

Vector at 07:50, 14 April 2026

2026-04-14T07:50:48Z

← Older revision		Revision as of 07:50, 14 April 2026
Line 11:		Line 11:
	* the direction of alteration of the imperfect interval, relative to some standard reference (usually the direction of alteration of the imperfect 3/2), conveyed by the sign		* the direction of alteration of the imperfect interval, relative to some standard reference (usually the direction of alteration of the imperfect 3/2), conveyed by the sign

	These two properties are, as it turns out, invariant with MOS size given the same temperament. We may step through a few examples.		These two properties are, as it turns out, invariant with MOS size given the same temperament. In fact, this holds when any just interval is considered, not just ratios between primes: this is trivially derivable from the fact that a temperament's base subgroup may be expressed in terms of a set of generators containing any given interval as long as the correct intervals are chosen for the remaining generators. (For example, this holds for 16/15 in meantone - there are always five imperfect 16/15s, because meantone can be thought of as, for instance, a 3/2.5/4.16/15 temperament.) However, it may also be derived from the temperament's generator chain - all but N modes have the note at N generator steps available, invariant of the size of the scale.

			We may step through a few examples.

	=== Meantone ===		=== Meantone ===

Vector at 03:24, 22 January 2026

2026-01-22T03:24:52Z

@@ Line 1: / Line 1: @@
 The '''wedgie''' of a regular temperament is a mathematical object that uniquely characterizes the temperament independently of choice of generator or equave. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w<sub>n</sub>'' ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.
@@ Line 26: / Line 27: @@
 The third entry requires examining tritave-equivalent diaschismic, which has a generator of 600 cents and a period of a tritave. It turns out that in this system there are ''eleven'' imperfect 5/3s, and the imperfect 5/3s alter in the same direction as the 5/4s (flatwards in diaschismic[16], the tritave counterpart of diaschismic[10]), so the third entry is -11.
-WIP
+=== Blackwood ===

Vector: Created page with "The '''wedgie''' of a regular temperament is a mathematical object that uniquely characterizes the temperament independently of choice of generator or equave. A wedgie takes the form ⟨⟨…⟨ ''w''₁ ''w''₂ … ''w_n'' ]]…], with ''n'' entries listed in between multiple val brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Each element conveys information about the structure of a set of primes in the te..."

2026-01-21T19:49:55Z

Created page with "The '''wedgie''' of a regular temperament is a mathematical object that uniquely characterizes the temperament independently of choice of generator or equave. A wedgie takes the form ⟨⟨…⟨ ''w''1 ''w''2 … ''wn'' ]]…], with ''n'' entries listed in between multiple val brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Each element conveys information about the structure of a set of primes in the te..."

New page

The '''wedgie''' of a regular temperament is a mathematical object that uniquely characterizes the temperament independently of choice of generator or equave. A wedgie takes the form ⟨⟨…⟨ ''w''1 ''w''2 … ''wn'' ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.

== Rank-1 temperaments ==
The wedgie of a rank-1 temperament is its [[val]]. As such, wedgies can be thought of as a generalization of vals, called ''multivals.''

== Rank-2 temperaments ==
Each wedgie entry for a rank-2 temperament contains information about a pair of primes. In the rank-2 case, these provide useful information about the MOS scales a regular temperament provides:

* the number of imperfect instances of the ratio between the two primes, conveyed by the absolute value
* the direction of alteration of the imperfect interval, relative to some standard reference (usually the direction of alteration of the imperfect 3/2), conveyed by the sign

These two properties are, as it turns out, invariant with MOS size given the same temperament. We may step through a few examples.

=== Meantone ===
Meantone has one imperfect fifth (ratio 3/2). This is true regardless of whether you take meantone[5], meantone[7], meantone[12] or another meantone MOS. As such, the first entry of meantone's wedgie is ±1 (1 by convention).

Meantone additionally has four imperfect major thirds (ratio 5/2, or equivalently 5/4). Again, this is true regardless of which meantone scale you use. (In diatonic, the "imperfect major third" is the minor third.) As such, the second entry of meantone's wedgie is ±4. Imperfect major thirds are always flat when imperfect fifths are flat, and likewise for when they are sharp. For instance, in diatonic, we see a minor third and a diminished fifth, while in pentic, the perfect fourth is the imperfect major third, and the minor sixth is the imperfect fifth. Thus, the second entry is 4.

For the third entry, we examine 5/3, but tritave-equivalently, so that instead of the octave-equivalent meantone scales we instead use the tritave-equivalent ones. In this context it can be seen that there are four imperfect 5/3s in a tritave-equivalent scale. So the third entry is ±4, but how do we determine whether it is positive or negative? Well, every 2/1-equivalent meantone MOS has a corresponding 3/1-equivalent MOS, and the tritave version of meantone[7] is meantone[11]. In this scale, 5/3's imperfect counterpart is flat of it (and in meantone[8], the tritave counterpart of pentic, 5/3's imperfect counterpart is sharp of it), so the third entry is also 4.

Therefore, meantone's wedgie is ⟨⟨ 1 4 4 ]].

=== Diaschismic ===
Diaschismic has two imperfect fifths, so its first wedgie entry is 2. There are four imperfect major thirds. However, when imperfect fifths are flat, imperfect major thirds are sharp (as in diaschismic[12]), and vice versa for diaschismic[10]. So, the second entry is actually -4.

The third entry requires examining tritave-equivalent diaschismic, which has a generator of 600 cents and a period of a tritave. It turns out that in this system there are ''eleven'' imperfect 5/3s, and the imperfect 5/3s alter in the same direction as the 5/4s (flatwards in diaschismic[16], the tritave counterpart of diaschismic[10]), so the third entry is -11.

WIP