Well-temperaments in a micro-equal

We can find the possible patterns of well-temperaments in a given
equal temperament by brute force if it isn't too complex. For instace,
if we look for oombinations (partions) of fifth from 270-et using 156,
157, 158 and 159 steps, we find only four possibilities for a
well-temperament on 12 notes; [0,6,6,0] (a version of Valotti-Young)
[1,4,7,0], [2,2,8,0] and [3,0,9,0] (a version of Stanhope.) 311 would
give [0,7,5,0], [1,5,6,0], [2,3,7,0] and [3,1,8,0] for 180,181,182,
and 183 steps. By the time we get up to 494 the number of
possibilities starts to get much larger; I find 83 for fifths of size
286,287,288,289, and 290 steps.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> We can find the possible patterns of well-temperaments in a given
> equal temperament by brute force if it isn't too complex. For
instace,
> if we look for oombinations (partions) of fifth from 270-et using
156,
> 157, 158 and 159 steps, we find only four possibilities for a
> well-temperament on 12 notes; [0,6,6,0] (a version of Valotti-Young)
> [1,4,7,0], [2,2,8,0] and [3,0,9,0] (a version of Stanhope.) 311
would
> give [0,7,5,0], [1,5,6,0], [2,3,7,0] and [3,1,8,0] for 180,181,182,
> and 183 steps. By the time we get up to 494 the number of
> possibilities starts to get much larger; I find 83 for fifths of
size
> 286,287,288,289, and 290 steps.

I see the pattern but don't understand the significance of the
numbers in [ ]

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> We can find the possible patterns of well-temperaments
> in a given equal temperament by brute force if it isn't
> too complex. For instace, if we look for oombinations
> (partions) of fifth from 270-et using 156, 157, 158 and
> 159 steps, we find only four possibilities for a
> well-temperament on 12 notes; [0,6,6,0] (a version of
> Valotti-Young) [1,4,7,0], [2,2,8,0] and [3,0,9,0]
> (a version of Stanhope.) 311 would give [0,7,5,0],
> [1,5,6,0], [2,3,7,0] and [3,1,8,0] for 180,181,182,
> and 183 steps. By the time we get up to 494 the number
> of possibilities starts to get much larger; I find 83
> for fifths of size 286,287,288,289, and 290 steps.

hmm ... this kind of thing sounds familiar to me ...

have you seen my analysis of Werckmeister III
well-temperament as subsets of 612edo and 200edo?

http://tonalsoft.com/enc/werckmeister.htm

-monz

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:

> I see the pattern but don't understand the significance of the
> numbers in [ ]

The numbers say, for instance, I can produce a well temperament in
270-equal by means of 6 steps of 157 and 6 of 158, by one step of 155,
4 of 157 and 7 of 158, by 2 of 155, 2 of 156 and 8 of 158, or by
3 of 156 and 9 of 158. Since a symmetrical arragement of notes makes
sense this means I could give these without much difficulty.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@m...> wrote:
>
> > I see the pattern but don't understand the significance of the
> > numbers in [ ]
>
> The numbers say, for instance, I can produce a well temperament in
> 270-equal by means of 6 steps of 157 and 6 of 158, by one step of
155,
> 4 of 157 and 7 of 158, by 2 of 155, 2 of 156 and 8 of 158, or by
> 3 of 156 and 9 of 158. Since a symmetrical arragement of notes makes
> sense this means I could give these without much difficulty.

Thanks. It's good to branch out into well-temperaments and other non-
equal temperaments. I suppose you could combine things and include
systems with "tempered" or non-standard octaves and non-tempered
steps, or would that be too troublesome?