Herman Miller's introduction to Regular Temperaments

For some reason I had not seen this, or at least never looked
at it properly. It is easily the best introduction I've seen
on the topic:

http://www.io.com/~hmiller/music/regular-temperaments.html

If I ever retire a young millionaire, I will have this printed
up into pamphlets (with Herman's permission of course) and drop
them from my helicopter onto select college campuses and rock
concert arenas.

-Carl

Some highlights:
--By definition, a temperament is a system of altering (or "tempering") --the size of musical intervals, resulting in a deviation away from the --pure rational intervals of just intonation.
  Again, on the surface this seems to say JI and mean-tone are closely linked.

---The thing that is special about the 5, 7, and 12-note meantone scales ---is that they have exactly two sizes of steps; in fact, two sizes of every ---interval smaller than an octave. Scales with this property are called ---"distributionally even"; related terms include "moments of symmetry" ---(MOS) and "Myhill's property".
  Hmm...this seems to denote a possible pattern that the most consonant ET scales are also MOS scales.    Come to think of it, I could swear I read an article about the best ET tunings' being defined by primes in the fibonacci series IE 5,7,12,19,31...ET (see http://www.bikexprt.com/tunings/fibonaci.htm).  Bizarrely enough, my PHITER scale is also formed form MOS intervals, despite having the tuning it is based on being generated using the irrational ratio PHI.
  I'm not saying making a scale under MOS tuning is the only way to make a scale under a tuning sound good...but, in general, MOS tunings do seem to tend to sound good.
***********************************************************************
---For example, the usual 5-limit interpretation of 12-note equal
---temperament (12-ET), the "nearest prime mapping" as it is called, can
---be represented as <12, 19, 28], where 12 is the number of steps to
---approximate 2:1, 19 is the number of steps to approximate 3:1, and ---28
is the number of steps to approximate 5:1
   So it takes 12 steps in 12TET to get to the octave, 19 to get to the tritave, and 28 to get to the "5/1-tave"...this makes sense.

--The vals <1, 2, 4] and <0, -1, -4] (in the case of this
particular ---mapping of meantone temperament) can be combined by a
---mathematical operation called a "wedge product", then normalized by
--factoring out the greatest common divisor, resulting in a wedgie
(short --for "wedge invariant"). The wedgie formed by wedging these
meantone --vals (or any pair of vals that represents a meantone
temperament map) ---is <<1, 4, 4]].

funny, I get
*******************
1 2 4
vs
0 -1 -4
i = 0
j = 1
(val1[0] * val2[1]) - (val2[0] * val1[1])
wedgiesize[0] = (1*-1) - (0*2) = -1
----------
i = 0
j = 2
(val1[0] * val2[2]) - (val2[0] * val1[2])
wedgiesize[1] = (1*-4) - (0*4) = -4
----------
i = 1
j = 2
(val1[1] * val2[2]) - (val2[1] * val1[2])
wedgiesize[2] = (2*4) - (-1*-4) = 4

IE the result vector -1,-4,4 and not 1,4,4.
***************************************
   These "Monzo vectors" are confusing me...what did I do wrongly in the calculation?

-Michael

--- On Wed, 2/18/09, Carl Lumma <carl@...g> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Herman Miller's introduction to Regular Temperaments
To: tuning@yahoogroups.com
Date: Wednesday, February 18, 2009, 11:58 PM

For some reason I had not seen this, or at least never looked

at it properly. It is easily the best introduction I've seen

on the topic:

http://www.io. com/~hmiller/ music/regular- temperaments. html

If I ever retire a young millionaire, I will have this printed

up into pamphlets (with Herman's permission of course) and drop

them from my helicopter onto select college campuses and rock

concert arenas.

-Carl