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Two conditions on temperaments

🔗genewardsmith <genewardsmith@juno.com>

12/19/2001 3:59:45 PM

It seems to me there are two conditions it would make sense to put on something intended as an odd-limit temperament:

(1) Weak condition--no element of the tonality diamond is allowed to be a unison

(2) Strong condition--all elements of the tonality diamond are distinct

Clearly (2)==>(1); in the 5-limit case, (1) excludes 6/5, 5/4, 4/3 and
3/2; and (2) also excludes 25/24 and 16/15, leaving 27/25 to be king of funkiness.

🔗genewardsmith <genewardsmith@juno.com>

12/19/2001 4:01:30 PM

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> (1) Weak condition--no element of the tonality diamond is allowed to be a unison (exluding 1/1)
>
> (2) Strong condition--all elements of the tonality diamond are distinct (including 1/1)

🔗paulerlich <paul@stretch-music.com>

12/19/2001 7:00:15 PM

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> It seems to me there are two conditions it would make sense to put
on something intended as an odd-limit temperament:
>
> (1) Weak condition--no element of the tonality diamond is allowed
to be a unison

Fine. Then we won't need a lower bound on g anymore, will we?
>
> (2) Strong condition--all elements of the tonality diamond are
distinct

That would make me cry, as it excludes twintone in the 7-limit.

🔗genewardsmith <genewardsmith@juno.com>

12/19/2001 7:02:56 PM

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> That would make me cry, as it excludes twintone in the 7-limit.

That occurred to me also; I think the strong condition is too strong, but I see no problem with the weak condition.

🔗dkeenanuqnetau <d.keenan@uq.net.au>

12/19/2001 7:33:18 PM

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > That would make me cry, as it excludes twintone in the 7-limit.
>
> That occurred to me also; I think the strong condition is too
strong, but I see no problem with the weak condition.

I agree. but we'd like to know when the strong condition applies to a
temperament. This is shown as "unique" in Graham Breed's output, as in
all elements of the diamond are represented by a unique interval in
the temperament.