Irregular temperaments

Thorin>"I like Michaels suggestion of irregular temparaments. That seems really

progressive to me, offering lots of freedom... but I can also image it's

pretty hard to get it right. For example, will my 'irregular temperament'

allow me to transpose and modulate with all (or another set) of intervals

that I like?"

   Perfect transposition (I'm pretty sure) is not possible outside of EDO scales.  However I've found, in an irregularly tempered scale where all dyads are within about 10 cents of a list of dyads you consider good, you can modulate at will and, though that chords won't sound "the same" when transposed, they will still sound quite stable.
  Making irregular temperaments that are strictly proper is important here...meaning if, say, 5 steps represent a third, they must also represent a third starting from any other root tone.  It gives a sense of consistency for composition...and it follows that very clustered scales are usually not desirable.  MOS scales (though unrelated) are also much based on the idea of avoiding clustering and being strictly proper.

  Far as clusters...I've found you can do things like have two consecutive notes represent a minor and neutral third in one place and a
minor and major third starting from a different root tone (so the neutral substitutes for the major)...or exchanging perfect 5ths for diminished fifths in some places.  However, I've found pushing it much further, I've found, makes things sounds skewed.

   Far as actually making irregular temperament I've found a good way to do this is
A) Find a variety of fifths (or, in some cases, high 4th or low 6th) you like...preferably not just variations on a perfect fifths (doing this often allows you to get intervals you like you can't get otherwise).  Some of my favorites are 22/15, 14/9, 11/7, 10/7, 7/5, and 13/9.

B) Try to chain them together IE if your fifths are 3/2 and 22/15 you can do   3/2 * 22/15 * 22/15 etc.
   Hint...try to keep fifths other than 3/2 a maximum distance apart IE
3/2 * 22/15 * 3/2 * 3/2 * 22/15 * 3/2 is good but
22/15 * 22/15 * 3/2 * 3/2 * 3/2 is likely much worse and
more clustered.
     When you find a combination that hits a power of 2 IE 2^x AKA an octave multiple, or a tri-tave (3^x) multiple if you wish, proceed to C

C) Look at the spacing of the notes you got for B....you must divide all notes in B by 2 repeatedly until you get them to fit between 1 and the 2/1 octave.  If it is fairly even IE there are no areas where a ton of notes cluster together, proceed to D

D) Compare all other dyadic intervals in the scale (every note to every other note within the octave).  Note which ones are furthest off from intervals you like.

E) Try your best to slightly nudge notes up and down to lower the highest error in D.  When you get everything at 10 cents or lower dyadic error...chances are you have a very balanced, easy to play scale.

>"if my dyads sound good, will my triads sound good too?"
 
   Some people are going to hate me for
this opinion...but I'd say "usually".  If all your dyads are good, chances are at least 2 of 3 of the dyads in your triads will work well together and will stabilize anything that does not.  My obvious example of this is the chord C E F in 12TET.  It works out to something like 12:15:16 as a "just" chord...pretty high limit compared to, say, a 4:5:6 major chord...but listening to it, IMVHO, it sounds quite stable because even though the chord is high limit, of the dyads in at (16/12 = 4/3, 15/12 = 5/4, 16/15 = 16/15....only the 16/15 proves high limit).

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--- In MakeMicroMusic@yahoogroups.com, Michael <djtrancendance@...> wrote:

MOS scales (though unrelated) are also much based on the idea of avoiding clustering and being strictly proper.

MOS scales may or may not be strictly proper. If you fire up Scala, go to the pull-down menu item under "files" called "new" and then "linear temperament", put in 7 for the scale size and 699.0 for the for generator, and then run "show data" on the result, it will tell you the scale is strictly proper. Do the same only with 701.0 for the generator, and Scala will tell you it is not proper. MOS scales have a proper range and an improper range, depending on whether the ratio of the largest to the smallest step is greater than or less than two.

However, MOS scales are often a good way to get proper scales. The hobbit construction also often results in proper scales, and is more to the point here as these are in temperaments with more than two generators and hence can be considered "irregular temperaments". If that's not irregular enough for you, it's always possible to convert it into a lesfip scale.