Unlisted equal temperaments

Obviously, one must stop listing at some point, so this isn't a
criticism. I'm taking note of notable, for one reason or another,
equal temperaments which Monzo does not list.

38

This has been discussed on tuning-math for situations where
introducing a period of 1/2 or splitting the fifth into two over
19-equal might be useful. 2^(11/38) is almost precisely an 11/9.

65

Mentioned by Barbour as a 5-limit system. It is simultaneously a
schismatic system and a semisixths (78732/78125 comma) system.

87

Mentioned by Barbour as a 5-limit system; it supports kleismic tempering.

99

The first really important division left off the list, which I think
should be on it. It is significant as a 7-limit system, having commas
of 2401/2400, 3136/3125 and 4375/4374. Its errors, well under two
cents, are by some people's ears just enough to be pleasing. If music
needs to be written in something to make it worth talking about, I've
done that.

130

Twice 65, and important in the 7, 13 (and 15) limits in particular.

140

Mentioned in a manner of speaking by Edward Charles Titchmarsh in his
book "The Theory of the Riemann Zeta Function", where he discusses a
high value of the Riemann zeta function which corresponds to the 140
division. It is an important 7-limit division.

224

Important in the 13-limit in particular.

311

This remarkable division is important in the 13 through 41 limits, in
every one of those odd limits. As a generic way of representing what
some might maintain is anything anyone could reasonably want to
represent it is of interest.

441

A very strong 5 or 7 limit system, along with 612 a good way to tune
ennealimmal.

494

Strong 11-15 limit system, and still good as a 17-limit system.

559

A 5-limit system mentioned by Barbour.

More possibilities:

80

A strong 19-limit system. Chains of 80-equal fifths have been proposed
for neo-Gothic or Arabic inspired music.

111

Strong 13-21 limit system.

3125

A strong 7 or 9 limit system, but I mention it here because it is 5^5,
which might be useful for something.

46032

The "Diophantine clarity" division, useful for discussing 5-limit
tempering. If one step is a "flu", then a Pythagorean comma is 900
flus and a Didymus comma 825 flus, and therefore a schisma is 75 flus.
The flu system is plenty accurate enough while tempering the atom out
of the discussion. I recommed it as a replacement for Tuning Units.

hi Gene,

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

> Obviously, one must stop listing at some point, so this
> isn't a criticism. I'm taking note of notable, for one
> reason or another, equal temperaments which Monzo does
> not list.
>
> <Gene's list snipped>

i'll be happy to add these to the new version of the
Equal Temperament page (... which won't be uploaded until
the whole Encyclopedia is converted to the new format).

should i put "2005 Gene Ward Smith" as an advocate for
all of them? (please correct the date if necessary)

are there any other dates/names from tuning-math which
should be included?

-monz

hi Gene (and all),

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

> 46032
>
> The "Diophantine clarity" division, useful for discussing
> 5-limit tempering. If one step is a "flu", then a Pythagorean
> comma is 900 flus and a Didymus comma 825 flus, and therefore
> a schisma is 75 flus.
> The flu system is plenty accurate enough while tempering the
> atom out of the discussion. I recommed it as a replacement
> for Tuning Units.

please note that "tuning unit" is an obsolete term ... it
was actually an error, and that unit of measurement is
properly called a "temperament unit", and was so named by
its inventer John Brombaugh.

(i thought i had fixed this in the Encyclopedia when i
found out about it a couple of months ago, but apparently
not ... please just bear with the error for now, until
the Encyclopedia conversion is complete.)

PS -- i've made a nice new page for "flu" ... but
unfortunately it won't appear online until the conversion
is finished.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> should i put "2005 Gene Ward Smith" as an advocate for
> all of them? (please correct the date if necessary)

I think Paul may have been the main person interested in 38, but let's
see if someone responds to that. As for the date, I'm not clear on
what the date is supposed to date.

hi Gene,

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > should i put "2005 Gene Ward Smith" as an advocate for
> > all of them? (please correct the date if necessary)
>
> I think Paul may have been the main person interested in 38,
> but let's see if someone responds to that. As for the date,
> I'm not clear on what the date is supposed to date.

it's supposed to be the earliest published reference
by someone who advocated or described an EDO.

if you have written about any of these before, then please
give me that date instead of 2005. i'm assuming that you
are an advocate of all of them, since you are the person
who suggested that they be added to my list. :)

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> it's supposed to be the earliest published reference
> by someone who advocated or described an EDO.

That raises some questions, such as what "published" means and whether
mentioning something, or listing it, is the same as advocating it.

> if you have written about any of these before, then please
> give me that date instead of 2005.

I've written music tuned to 99 and 441 equal, which strikes me as a
stronger form of advocacy than most. Both 311 and 494 were being
tossed around during discussions of notation systems, but have also
been mentioned prior to that. I was maintaining last year that 99, 130
and 140 were in a good range so far as a balance between deviation
from JI and usefulness of commas goes, which should count as advocacy,
I suppose. By the way, do the Klingons deserve mention wrt 60 equal?