Initial Conditions.

In order to design a temperament whether equal, unequal or mean-tone
one has to start somewhere. Thus the initial conditions. The most
common initial conditions which temperament designers use are the
chords (0-267-702) and (0-435-702) and (0-294-702) and (0-408-702) and
(0-316-702) and (0-386-702) and (0-351-702). A good fit for these seven
chords is 41tet.

robert thomas martin wrote:
> In order to design a temperament whether equal, unequal or mean-tone > one has to start somewhere. Thus the initial conditions. The most > common initial conditions which temperament designers use are the > chords (0-267-702) and (0-435-702) and (0-294-702) and (0-408-702) and > (0-316-702) and (0-386-702) and (0-351-702). A good fit for these seven > chords is 41tet. Which "temperament designers" might those be?

A translation for those who might be more familiar with ratios:

0: 1/1 0.000 unison, perfect prime
1: 7/6 266.871 septimal minor third
2: 32/27 294.135 Pythagorean minor third
3: 6/5 315.641 minor third
4: 49/40 351.338 larger approximation to neutral third
5: 5/4 386.314 major third
6: 81/64 407.820 Pythagorean major third
7: 9/7 435.084 septimal major third, BP third
8: 3/2 701.955 perfect fifth
9: 2/1 1200.000 octave

Seems a bit arbitrary, doesn't it? An 81/64 might come as a result of stacking 3/2 fifths, but it isn't something you'd likely use as an initial condition. If you want a third that's intermediate between 5/4 and 9/7, 14/11 (417.508 cents) is arguably a better choice. And thirds aren't the only intervals you might want to use: what about 7/5 (582.512 cents) for instance? Still, 41-ET is a pretty good choice if you're looking for equal temperaments.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> robert thomas martin wrote:
> > In order to design a temperament whether equal, unequal or mean-
tone
> > one has to start somewhere. Thus the initial conditions. The most
> > common initial conditions which temperament designers use are the
> > chords (0-267-702) and (0-435-702) and (0-294-702) and (0-408-
702) and
> > (0-316-702) and (0-386-702) and (0-351-702). A good fit for these
seven
> > chords is 41tet.
>
> Which "temperament designers" might those be?
>
> A translation for those who might be more familiar with ratios:
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 7/6 266.871 septimal minor third
> 2: 32/27 294.135 Pythagorean minor third
> 3: 6/5 315.641 minor third
> 4: 49/40 351.338 larger approximation to
neutral third
> 5: 5/4 386.314 major third
> 6: 81/64 407.820 Pythagorean major third
> 7: 9/7 435.084 septimal major third, BP
third
> 8: 3/2 701.955 perfect fifth
> 9: 2/1 1200.000 octave
>
> Seems a bit arbitrary, doesn't it? An 81/64 might come as a result
of
> stacking 3/2 fifths, but it isn't something you'd likely use as an
> initial condition. If you want a third that's intermediate between
5/4
> and 9/7, 14/11 (417.508 cents) is arguably a better choice. And
thirds
> aren't the only intervals you might want to use: what about 7/5
(582.512
> cents) for instance? Still, 41-ET is a pretty good choice if you're
> looking for equal temperaments.
>
From Robert. Thankyou for translating my figures to ratios and
their descriptions for those who are more used to this sort of
presentation. Middle eastern, Indian, European and Asian music
traditions have certain expectations about what should be included in
a comprehensive temperament. The seven chords which I listed are
probably the minimum requirements and 41tet is a good choice for a
working system which can act as a level playing field for all the
different musicians around the world. This is not to say that I am
against other microtonal activities. I am quite in favour of having
1200tet as the scientific standard (rounded to the nearest cent).
Every temperament was designed by someone. I am trying to demystify
the process so that some members can feel confident in designing
their own. One way of analysing a temperament is to divide it into
three parts ie. (F-C-G-D) and (A-E-B-F#) and (C#-Ab-Eb-Bb) and
examine the numerical relationships according to certain criteria.
What criteria? Whatever the designer thinks is important.