Temperament catalog

I'm uploading a list of notable linear temperaments to
<http://x31eq.com/catalog.htm>. Please report any errors or
omissions. The idea is that next time somebody thinks they have come up
with something new, they can see if it's on the list. Also if they have a
previous reference for anything, they can submit it.

Those 13-limit mappings of Erv Wilson should be added, if anybody can
remember them.

Monz already has a list of equal temperaments. Push the link my way and
I'll add it!

Graham

> From: <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, July 14, 2001 9:51 AM
> Subject: [tuning-math] Temperament catalog
>
>
> I'm uploading a list of notable linear temperaments to
> <http://x31eq.com/catalog.htm>. Please report any errors or
> omissions. The idea is that next time somebody thinks they have come up
> with something new, they can see if it's on the list. Also if they have a
> previous reference for anything, they can submit it.

Good show, Graham!

The first sentence under "Miracle" says "Has its own page
[a link to Graham's], better covered by Monz, I'll link
to that sometime." This should make it easier:
http://www.ixpres.com/interval/dict/miracle.htm

Note that I updated that page so that the mapping is given
the way Graham does it: (period, generator).

Near the end, under "Golden Section Tuning", there's a typo:
"confussed" should be "confused".

> Monz already has a list of equal temperaments. Push the link my way and
> I'll add it!

Sure thing... and I've just updated it again.
http://www.ixpres.com/interval/dict/eqtemp.htm

And, I've added a link to your catalog in my "linear temperament"
definition (which, BTW, could probably be expanded... hint...)
http://www.ixpres.com/interval/dict/lineartemp.htm

-monz
http://www.monz.org
"All roads lead to n^0"

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Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., graham@m... wrote:
> I'm uploading a list of notable linear temperaments to
> <http://x31eq.com/catalog.htm>. Please report any
errors or
> omissions.

The Kleisma is a small interval generated from a series of just minor
thirds. You must be thinking of 225:224 which is the "_septimal_
kleisma".

As for omissions -- How about BP, for one?

In-Reply-To: <9ivqmg+507h@eGroups.com>
Paul wrote:

> The Kleisma is a small interval generated from a series of just minor
> thirds. You must be thinking of 225:224 which is the "_septimal_
> kleisma".

You mean the difference between four 6:5s and an octave?

> As for omissions -- How about BP, for one?

Either a JI scale or an ET, but not a linear temperament, unless you know
better.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9ivqmg+507h@e...>
> Paul wrote:
>
> > The Kleisma is a small interval generated from a series of just
minor
> > thirds. You must be thinking of 225:224 which is the "_septimal_
> > kleisma".
>
> You mean the difference between four 6:5s and an octave?

No. See Scala's intnam.par: 15552:15625 (8.11 c)
or my http://dkeenan.com/Music/ChainOfMinor3rds.htm
"A kleisma is the difference between a just fifth (2:3) and an octave
reduced chain of 6 just minor thirds (5:6)"

Presumably the septimal kleisma was so named because it is close in
size to the kleisma.

The catalog is a brilliant effort Graham. Thanks.

You haven't told us what sort of interval the generator is, for some
of them, and in some you've given a "generator" which is in fact the
period.

You could add to the Miracle entry, the fact that Secor proposed it as
a way of making sense of Partch's 43 note gamut (i.e. not out of a
musical vacuum.)

Some others that have been found interesting by more than one person
(one of those people being me) are:

(-4 3 -2), a wide minor second generator (about 125c), octave period,
19 and 29-EDO.

(-3, -5, 6), a wide neutral second generator (about 163c), octave
period, 15 and 22-EDO.

(7, -3, 8), a subminor third generator (about 272c), octave period, 22
and 31-EDO.

(5, 1), a major third generator (about 380c), octave period, 19 and
60-EDO.

Wilson's 11-limit mapping is:
(-1, 8, 14, 18), perfect fourth generator (about 497 c), octave
period, 41-EDO.

Wilson's 13-limit mapping is:
(-1, 8, 14, -23, -20), perfect fourth generator (about 498 c), octave
period, 41 and 53-EDO.

Your own 13-limit mapping that gives a more compact diamond (68 notes
instead of 75) but with slightly higher errors is:
(1, -2, -8, -12, -15), minor second generator (about 104 c) or
equivalently a perfect fourth generator (about 496c), half-octave
period, 46 and 104-EDO.

-- Dave Keenan

In-Reply-To: <9j36rh+ipou@eGroups.com>
Dave Keenan wrote:

> No. See Scala's intnam.par: 15552:15625 (8.11 c)
> or my http://dkeenan.com/Music/ChainOfMinor3rds.htm
> "A kleisma is the difference between a just fifth (2:3) and an octave
> reduced chain of 6 just minor thirds (5:6)"

Thanks!

> Presumably the septimal kleisma was so named because it is close in
> size to the kleisma.
>
> The catalog is a brilliant effort Graham. Thanks.
>
> You haven't told us what sort of interval the generator is, for some
> of them, and in some you've given a "generator" which is in fact the
> period.

The mapping's enough to define the temperament. I say the period where it
isn't the octave as well, and that should all be correct now.

> You could add to the Miracle entry, the fact that Secor proposed it as
> a way of making sense of Partch's 43 note gamut (i.e. not out of a
> musical vacuum.)

Too much information for a brief list.

<snipped> temperaments added.

> Wilson's 11-limit mapping is:
> (-1, 8, 14, 18), perfect fourth generator (about 497 c), octave
> period, 41-EDO.
>
> Wilson's 13-limit mapping is:
> (-1, 8, 14, -23, -20), perfect fourth generator (about 498 c), octave
> period, 41 and 53-EDO.

I can't see this one. Does his layout contradict his numbers?

> Your own 13-limit mapping that gives a more compact diamond (68 notes
> instead of 75) but with slightly higher errors is:
> (1, -2, -8, -12, -15), minor second generator (about 104 c) or
> equivalently a perfect fourth generator (about 496c), half-octave
> period, 46 and 104-EDO.

I don't want to add *everything* from my automatically generated lists,
because it'd mean duplicating those lists! I'm not sure it is more
compact. I think it should be 70 notes for the diamond, what with it
having a half-octave period. Cassandra(?) is more efficient, but not
unique.

Graham

In-Reply-To: <memo.301982@cix.compulink.co.uk>
I wrote:

> I don't want to add *everything* from my automatically generated lists,
> because it'd mean duplicating those lists! I'm not sure it is more
> compact. I think it should be 70 notes for the diamond, what with it
> having a half-octave period. Cassandra(?) is more efficient, but not
> unique.

Oops! The "I'm not sure" comment is a leftover from before I worked it
out.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9j36rh+ipou@e...>
> Dave Keenan wrote:
> > You haven't told us what sort of interval the generator is, for
some
> > of them ...
>
> The mapping's enough to define the temperament.

I totally agree, but that doesn't mean you shouldn't be kind to those
without math degrees and give at least the interval-class (say from
the Fokker/Miracle naming scheme) of the generator. It's not
simple to extract the generator unless there's a 1 or -1 in the prime
mapping, and many people wouldn't even know to look for that.

> > You could add to the Miracle entry, the fact that Secor proposed
it as
> > a way of making sense of Partch's 43 note gamut (i.e. not out of a
> > musical vacuum.)
>
> Too much information for a brief list.

If you've got room to mention Arabic and Indian etc under Schismic,
Diaschismic and Neutral Thirds, you've got room to mention Partch
under Miracle. All I'm asking for is a phrase inserted, such as
"Discovered by George Secor 1975 as an efficient mapping of Partch's
43 note gamut, rediscovered by ..."

> > Wilson's 13-limit mapping is:
> > (-1, 8, 14, -23, -20), perfect fourth generator (about 498 c),
octave
> > period, 41 and 53-EDO.
>
> I can't see this one. Does his layout contradict his numbers?

Maybe so, or I may be wrongly attributing this temperament to Wilson,
but see page 7 of
http://www.anaphoria.com/tres.pdf
Oops, that's labelled "Casandra". I'll have to have to study this
layout again when I have time.

Either way, (-1, 8, 14, -23, -20) does exist. But it may or may not be
interesting. However it was wrong of me (and Wilson?) to say it was
covered by 41 and 53-EDO. 53-EDO is not 13-limit consistent. It should
be 41 and 94-EDO.

-- Dave Keenan

--- In tuning-math@y..., graham@m... wrote:

> > As for omissions -- How about BP, for one?
>
> Either a JI scale or an ET, but not a linear temperament, unless
you know
> better.

Perhaps no one has viewed it as such yet, but what if you take the
generator of the 9-tone BP diatonic scale, and call that the
generator of the LT?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
>
> > > As for omissions -- How about BP, for one?
> >
> > Either a JI scale or an ET, but not a linear temperament, unless
> you know
> > better.
>
> Perhaps no one has viewed it as such yet, but what if you take the
> generator of the 9-tone BP diatonic scale, and call that the
> generator of the LT?

So what are the generator, period, mapping from primes to generators
and periods, example ED3s. i.e. Please write Grahams catalog entry for
it.

Graham,

I've gone over http://www.anaphoria.com/tres.pdf
pages 6 & 7 again, and can confirm that you've got the name
"Cassandra" on the wrong one in your Catalog, and that Cassandra (p7)
(-1, 8, 14, -23, -20) (497.9c) covers 41 and 94-EDO, but not 53.

The other unnamed one (p6) (-1, 8, 14, 18) (497.4c) is only given as
an 11-limit mapping on that page, although Wilson may have used the
full (-1, 8, 14, 18, 21) (497.2 c) elsewhere.

-- Dave Keenan

In-Reply-To: <9j8mep+v7v2@eGroups.com>
Dave Keenan wrote:

> I've gone over http://www.anaphoria.com/tres.pdf
> pages 6 & 7 again, and can confirm that you've got the name
> "Cassandra" on the wrong one in your Catalog, and that Cassandra (p7)
> (-1, 8, 14, -23, -20) (497.9c) covers 41 and 94-EDO, but not 53.

I said at the top I'm not bothered about consistency. This scale comes
out if you plug 41 and 53 into my temperament algorithm.

> The other unnamed one (p6) (-1, 8, 14, 18) (497.4c) is only given as
> an 11-limit mapping on that page, although Wilson may have used the
> full (-1, 8, 14, 18, 21) (497.2 c) elsewhere.

Ah! Now I see my problem! Page 8 gives alternative mappings for 11 and
13, but I was only looking at the bottom and ignored the dotted lines.

Graham

In-Reply-To: <006a01c110df$bb6cb6a0$f45ed63f@stearns>
Dan Stearns wrote:

> I've posted a bunch on this.

Dan, I know you've got a lot of similar ideas to me, but you don't present
them in a digestible way! If you send a short writeup, I'll add it to the
catalog.

I also know that you and Mats have both used a lot of MOS scales. I'm
going to have to leave them out unless they have some interesting just
approximations. Otherwise I'll end up with the whole of the scale tree in
there!

Graham

--- In tuning-math@y..., graham@m... wrote:
> Ah! Now I see my problem! Page 8 gives alternative mappings for 11
and
> 13, but I was only looking at the bottom and ignored the dotted
lines.

Ah! Now that you mention it. I was wrong too. Pages 9, 10 and 11 make
it very clear that the name "Cassandra" refers to the tuning, and not
to either of the keyboard mappings or their corresponding linear
temperaments.

He simply calls one "41-like" and the other "41- and 53-like". So I
guess you should just call them "29 and 41" and "41 and 53". Your
entries say "p6" and "pp7-8" but they don't say of what.

Regards,
-- Dave Keenan

In-Reply-To: <9j9mb4+j8lb@eGroups.com>
Dave Keenan wrote:

> Ah! Now that you mention it. I was wrong too. Pages 9, 10 and 11 make
> it very clear that the name "Cassandra" refers to the tuning, and not
> to either of the keyboard mappings or their corresponding linear
> temperaments.

I've called them *both* Cassandra now.

> He simply calls one "41-like" and the other "41- and 53-like". So I
> guess you should just call them "29 and 41" and "41 and 53". Your
> entries say "p6" and "pp7-8" but they don't say of what.

Good proof reading! I'd formatted the links e-mail style instead of
HTML-style, and so they weren't showing through at all.

Graham

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > Perhaps no one has viewed it as such yet, but what if you take
the
> > generator of the 9-tone BP diatonic scale, and call that the
> > generator of the LT?
>
> So what are the generator,

5

> period,

3

> example ED3s.

13, 88, 271 . . .

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > > Perhaps no one has viewed it as such yet, but what if you take
> the
> > > generator of the 9-tone BP diatonic scale, and call that the
> > > generator of the LT?
> >
> > So what are the generator,
>
> 5

That can't be right. The generator has to be a supermajor third
(approx 7:9) MA optimum around 440 cents (1/3-BP-comma wide). BP-comma
is 243:245 = 3^5 : 5^1 * 7^2. So the mapping is

Prime No. Generators
----- --------------
5 2
7 -1
and possibly
2 7 (the optimum moves to 442 cents if you include this)

> > period,
>
> 3

Yes. As in approx 1902 cents.

> > example ED3s.
>
> 13, 88, 271 . . .

So these are wrong too.

Only 13-ED3 really makes any sense. Ignoring ratios of two, the next
better one is 160-ED3. If we include ratios of 2 the next better one
is 43-ED3. So any ED3 whose cardinality is of the form 13n+4 where n
is between 4 and 12, may be of interest.

MOS cardinalities for the BP linear-temperament go (not strictly
proper in paren.) 4 (5) (9) 13 (17) (30) ....

-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > > > Perhaps no one has viewed it as such yet, but what if you
take
> > the
> > > > generator of the 9-tone BP diatonic scale, and call that the
> > > > generator of the LT?
> > >
> > > So what are the generator,
> >
> > 5
>
> That can't be right. The generator has to be a supermajor third
> (approx 7:9) MA optimum around 440 cents (1/3-BP-comma wide). BP-
comma
> is 243:245 = 3^5 : 5^1 * 7^2. So the mapping is
>
> Prime No. Generators
> ----- --------------
> 5 2
> 7 -1

I'm glad I got you to find the right answer (my excuse for not
thinking about this myself is that I have had strep bacteria for two
weeks, but was only informed of this today, and just started
penicillin).

> and possibly
> 2 7 (the optimum moves to 442 cents if you include this)

No, you should absolutely not include this. There are no factors of 2
to be approximated by the BP scale as it is understood by any of its
creators.
>
> > > example ED3s.
> >
> > 13, 88, 271 . . .
>
> So these are wrong too.

With respect to the 9-tone diatonic BP scale, I see what you mean.
These are simply ED3s that approximate 2-less JI well.

Good work!

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul Erlich wrote,
>
> <<No, you should absolutely not include this. There are no factors
of
> 2 to be approximated by the BP scale as it is understood by any of
its
> creators.>>
>
> I really don't know why I bother, but whatever... as I mentioned
> before I had a brief off-list correspondence with Heinz Bohlen that
> would seem to contradict the severity of Paul's -- "absolutely
not" --
> view, see:
>
> </tuning/topicId_unknown.html#25735>

Perhaps my view was too severe, but it definitely seems to contradict
the _spirit_ of the near-just approximations to all simple ratios of
odd numbers, since the approximations to simple ratios involving even
numbers are so poor.
>
> And while I'm at it... here's another previous post mentioning the
> linear temperament that 'no one's thought of BP as yet':
>
> </tuning/topicId_20880.html#20880>

You thought of it first! Why didn't you say so?