Beep and ennealimmal

There's a curious connection between beep/bug, the 7-limit 4&5
temperament (wedgie <<2 3 1 0 -4 -6||) and ennealimmal, the 7-limit
171&270 temperament (wedgie <<18 27 18 1 -22 -34||.) If we find the TM
basis for beep, we get {27/25, 21/20}, which happens to be a
period/generator pair for ennealimmal.

The 4-et and 5-et, which give the beep generator of 8/7 as 300 and 240
cents respectively, define its extreme range, but more importantly
4-et is the et of the tetrad, since 1-5/4-3/2-7/4 is an epimorphic
scale with <4 6 9 11| as its val, and 5-et is the et of the quintad,
1-9/8-5/4-3/2-7/4 being epimorphic with <5 8 12 14| as its val. This
means we can use beep to lift to 9 odd limit JI, in something like the
way I've used 4 and 5. In terms of the 8/7 generator, we have

-4: 9/8
-3: 5/4, 9/7
-2: 10/7, 3/2
-1: 5/3, 12/7, 7/4, 9/5

The major quintad is [0,-4,-3,-2,-1], the major and supermajor tetrads
are [0,-3,-2,-1], the minor and subminor tetrads [0,1,-2,-1]; these
are just contiguous sets of generators if we ignore the ordering.
Adding octaves, of course, gives us a note of beep.

The mapping for the generators 27/25 and 21/20 of ennealimma is
[<9 13 19 24|, <0 2 3 2|]. If we add mappings for the 4 and 5 vals,
we get a unimodular matrix which we can invert to get a matrix which
in terms of monzos is

[|0 3 -2 0>, |-2 1 -1 1>, |-6 -8 2 5>, |5 1 2 -4>]

In fractions, this is [27/25, 21/20, 420175/419994, 2400/2401]. The
perhaps unfamiliar comma is 420175/419994 = (2401/2400)(4375/4375).
This defines a "notation" for 7-limit JI; a note of ennealimmal plus a
note of beep giving the interval, where roughly speaking the note of
ennealimmal tells you the tuning, and the note of beep the chord position.

By the way, is it beep or bug, and why?

Gene Ward Smith wrote:
> There's a curious connection between beep/bug, the 7-limit 4&5
> temperament (wedgie <<2 3 1 0 -4 -6||) and ennealimmal, the 7-limit
> 171&270 temperament (wedgie <<18 27 18 1 -22 -34||.) If we find the TM
> basis for beep, we get {27/25, 21/20}, which happens to be a
> period/generator pair for ennealimmal. > By the way, is it beep or bug, and why?

"Beep" was supposed to bring to mind "BP" (as in "Bohlen-Pierce"); 27/25 is a frequent step size of the untempered BP scale. But it's as if we were to call father "diatonic" because it tempers out the diatonic semitone. "Bug" appears to be the older name, and I don't know why it was dropped, unless the name was just forgotten. So I vote for "bug".

>"Beep" was supposed to bring to mind "BP" (as in
>"Bohlen-Pierce"); 27/25 is a frequent step size of
>the untempered BP scale. But it's as if we were to
>call father "diatonic" because it tempers out the
>diatonic semitone. "Bug" appears to be the older
>name, and I don't know why it was dropped, unless
>the name was just forgotten. So I vote for "bug".

I too prefer bug, but Joseph Pehrson has done a piece
called Beepy, so I think we should stick with that name.

-Carl

>X-eGroups-Return: perlich@aya.yale.edu
>Date: Thu, 15 Jul 2004 03:20:28 -0000
>From: "Paul Erlich" <perlich@aya.yale.edu>
>To: Carl Lumma <ekin@lumma.org>
>Subject: Re: Beep and ennealimmal
>User-Agent: eGroups-EW/0.82
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>
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>> >"Beep" was supposed to bring to mind "BP" (as in
>> >"Bohlen-Pierce"); 27/25 is a frequent step size of
>> >the untempered BP scale. But it's as if we were to
>> >call father "diatonic" because it tempers out the
>> >diatonic semitone. "Bug" appears to be the older
>> >name, and I don't know why it was dropped, unless
>> >the name was just forgotten. So I vote for "bug".
>>
>> I too prefer bug, but Joseph Pehrson has done a piece
>> called Beepy, so I think we should stick with that name.
>>
>> -Carl
>
>Joseph's piece wasn't in beep/bug; it was in BP!

Eek- sorry!

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >"Beep" was supposed to bring to mind "BP" (as in
> >"Bohlen-Pierce"); 27/25 is a frequent step size of
> >the untempered BP scale. But it's as if we were to
> >call father "diatonic" because it tempers out the
> >diatonic semitone. "Bug" appears to be the older
> >name, and I don't know why it was dropped, unless
> >the name was just forgotten. So I vote for "bug".
>
> I too prefer bug, but Joseph Pehrson has done a piece
> called Beepy, so I think we should stick with that name.

I think Paul suggested the change, if so he should know why. I kind of
like bug too, though. Of course for my suggested use (to me this idea
is a gold mine) 4&5 really says it all. We had this argument about
whether beepbug was a real temperament, but surely we can regard it as
a rough draft temperament for something we propose to refine. I'm
going to use it, certainly. It should be possible, in bug, to get a
draft that actually sounds almost like music, and it is not so complex
that I can't manage to deal with it. Obviously, by changing the 14-et
tune I'll be forced to change my tune about 14-et--I now think it's
wonderful!

Carl Lumma wrote:

>>"Beep" was supposed to bring to mind "BP" (as in
>>"Bohlen-Pierce"); 27/25 is a frequent step size of
>>the untempered BP scale. But it's as if we were to
>>call father "diatonic" because it tempers out the
>>diatonic semitone. "Bug" appears to be the older
>>name, and I don't know why it was dropped, unless
>>the name was just forgotten. So I vote for "bug".
> > > I too prefer bug, but Joseph Pehrson has done a piece
> called Beepy, so I think we should stick with that name.

hmm... but wasn't Beepy in the actual BP scale? I don't know if he used the JI or ET version, but the JI version features 27/25 as a step, not a vanishing comma. As a linear temperament, it could be mapped [<x, 1, 1, 2|, <x, 0, 2, -1|] or [<x, 1, 3, 3|, <x, 0, -5, -4|] (no octaves), which isn't anything like [<1, 2, 3, 3|, <0, -2, -3, -1|]. The first one (5&19) results from tempering out the minor BP diesis 245;243, and the second one (4&12) from tempering out the major BP diesis 3125;3087. You could also temper out any combination of these intervals. Equal tempered BP is 41&49, [<1, 0, 0, 0|, <0, 13, 19, 23|].