[Joe Monzo]

Wedge-product calculations are useful in tuning theory in two ways:

- calculating the wedge-product of a set of monzos (or vectors) gives a multimonzo (multivector), which shows a temperament's prime-mapping.
- calculating the wedge-product of a set of vals (or breeds) gives a wedgie (multival, multibreed), which shows the vanishing unison-vector which any set of temperaments shares in common.

Both multimonzos and wedgies show common traits shared by different tunings belonging to the same family, and thus are a great aid in tuning taxonomy.

. . . . . . . . .

[Dave Keenan, Yahoo tuning-math, message 8009 (Wed Nov 26, 2003 9:28 am)]

[how to calculate a wedge-product:]

Lets first take the simplest case worth considering. The wedge product of two 3-limit (2D) vectors.

[a1 a2> ^ [b1 b2>

The procedure is to first list every product of a coefficient from A with a coefficient from B, i.e. their ordinary scalar products. So with 2 coefficients in each there will be 2x2 = 4 products to consider,

a1*b1, a1*b2, a2*b1, a2*b2.

As you calculate each product, combine the indices of the two coefficients to make a compound index for it. It is important to keep the indices in their original order at this stage. So we have

product index a1*b1 11 a1*b2 12 a2*b1 21 a2*b2 22

There are certain rules about what to to with each product now, depending on its compound index. There are 3 possibilities:

- If the indexes have a digit in common then ignore it. Just throw the product away. So we throw away a1*b1 and a2*b2.
- Otherwise if the digits in the compound index are already in alphabetical order, do nothing. So a1*b2 is just fine as it is.
- Otherwise if they are not in alphabetical order, then put them in alphabetical order. But first, look at each digit of the compound index in turn, and count how many larger digits are to the left of it. Add up all these left-and-larger counts as you go, or just keep counting so their counts accumulate. If the result is an odd number then negate the product, otherwise leave it as it was. Consider the index 21. There are zero larger digits to the left of the 2 (because there are _no_ digits to the left of it), and there is one larger digit to the left of the 1, namely the 2. So the total of the left-and-larger counts is 1, an odd number. So a2*b1 becomes -a2*b1. We now have

product index a1*b2 12 -a2*b1 12

Now find any products that have the same index and add them together. So we have only

-- product -- index a1*b2 - a2*b1 12.

Now list all these sums in alphabetical order of their indices, inside as many brackets as the sum of the number of brackets in the two arguments, and pointing in the same direction. The wedge product is only defined for values having their brackets pointing the same way.

So our answer is

[[a1*b2-a2*b1>>

Now lets try something more messy. A 7-limit (4D) vector wedged with a 7-limit bivector. This might represent combining a third comma with two that have already been combined, as an intermediate result on the way to finding the ET mapping where these all vanish.

[a1 a2 a3 a4> ^ [[b12 b13 b14 b23 b24 b34>>

We first make the list of products of all pairs, with their compound indices.

product index a1*b12 112 a1*b13 113 a1*b14 114 a1*b23 123 a1*b24 124 a1*b34 134 a2*b12 212 a2*b13 213 a2*b14 214 a2*b23 223 a2*b24 224 a2*b34 234 a3*b12 312 a3*b13 313 a3*b14 314 a3*b23 323 a3*b24 324 a3*b34 334 a4*b12 412 a4*b13 413 a4*b14 414 a4*b23 423 a4*b24 424 a4*b34 434

Now we get rid of all those with two digits the same. Of course once you've got the idea, you wouldn't even bother writing them down in the first place. This leaves.

product index left-and-larger count a1*b23 123 a1*b24 124 a1*b34 134 a2*b13 213 1 a2*b14 214 1 a2*b34 234 a3*b12 312 2 a3*b14 314 1 a3*b24 324 1 a4*b12 412 2 a4*b13 413 2 a4*b23 423 2

And we do the left-and-larger counts on the indices that aren't already in alphabetical order (shown above), and negate the product if this is odd. And we end up with:

product index a1*b23 123 a1*b24 124 a1*b34 134 -a2*b13 123 -a2*b14 124 a2*b34 234 a3*b12 123 -a3*b14 134 -a3*b24 234 a4*b12 124 a4*b13 134 a4*b23 234

Now we sum the products having the same index.

------- product -------- index a1*b23 + a3*b12 - a2*b13 123 a1*b24 - a2*b14 + a4*b12 124 a1*b34 - a3*b14 + a4*b13 134 a2*b34 - a3*b24 + a4*b23 234

Now we list them in alphabetical (also numerical) order of index inside the correct number of brackets.

[[[a1*b23+a3*b12-a2*b13 a1*b24-a2*b14+a4*b12 a1*b34-a3*b14+a4*b13 a2*b34-a3*b24+a4*b23>>>

Voila!

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

tredek (unit of interval measurement)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)