Tuning-Math Digests messages 9854 - 9878

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Message: 9854

Date: Fri, 06 Feb 2004 08:10:20

Subject: Re: 126 7-limit linears

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> 1 [0, 0, 2, 0, 3, 5] 662.236987 77.285947 2.153690
> 2 [1, 1, 0, -1, -3, -3] 806.955502 64.326132 2.467788
> 3 [0, 0, 3, 0, 5, 7] 829.171704 30.152577 3.266201
...

Thanks for the list. I can certainly get it into a spreadsheet and
plot it easily, but I have no idea what I'm plotting. I assume the
last two columns are error and complexity but I have no idea which is
which.

Also, I'm not yet up to speed on reading wedgies directly so I have no
idea of the identity of the temperaments. Can we please have
generators or mappings or comma pairs, if not names (where they exist)?

I suppose you figure it was difficult to generate so it should be
difficult to interpret as well. ;-)


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Message: 9864

Date: Sun, 08 Feb 2004 20:04:48

Subject: Re: Some warped egresses

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> I think I've found a couple of good JI approximations for retuning 
> _Egress_. The first one is a nice symmetrical looking one with lots 
of 
> consonances, which looks like it'd work nicely with any of the 
> pelog-type approximations, and would also work as 14 consecutive 
steps 
> of meantone, Fb-B.
> 
> 404 Not Found * Search for http://www.io.com/~hmiller/midi/egress/egress-jimajor.mid in Wayback Machine
> 
>   0  0  0  0  0  ! Bb  5/4  -2  0  1  0
>   1 -2  1 -1  1  ! B
>   2  4 -1 -1  0  ! Cb
>   3  2  0 -2  1  ! C   7/5   0  0 -1  1       Ab    Eb    Bb
>   4  1  1 -1  0  ! Db  3/2  -1  1  0  0
>   5  3 -2 -1  1  ! D                          D     A     E     B
>   6  2 -1  0  0  ! Eb                      Fb    Cb    Gb    Db
>   7  0  0 -1  1  ! E   7/4  -2  0  0  1
>   8  6 -2 -1  0  ! Fb                            F     C     G
>   9  4 -1 -2  1  ! F
> 10  3  0 -1  0  ! Gb  1/1   1  0  0  0
> 11  1  1 -2  1  ! G
> 12  4 -2  0  0  ! Ab
> 13  2 -1 -1  1  ! A   7/6  -1 -1  0  1
> 14  1  0  0  0  ! Bb  5/4  -2  0  1  0
> 
> Does anyone know of any better 14-note block of JI that might be 
useful 
> for this purpose?

Maybe not, but I did post one or two 14-note blocks once -- they're 
in Scala. I would search the tuning list for "What's your favorite 
number?", but it seems the yahoogroups search engine has suddenly 
become near-useless as it only searches a very small number of posts 
at a time.


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Message: 9865

Date: Sun, 08 Feb 2004 08:50:19

Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > Since there's a huge empty gap between complexity ~25+ and ~31, I 
was 
> > forced to look for a lower-complexity moat (probably a good thing 
> > anyway). I'll upload a graph showing the temperaments indicated 
by 
> > their ranking according to error/8.125 + complexity/25, since I 
saw a 
> > reasonable linear moat where this measure equals 1. Twenty 
> > temperaments make it in:
> 
> Given that we normally relate error and complexity multiplicitively,

normally . . .

> I
> think using log(err) and log(complexity) makes far more sense.

I don't think they make more sense practically. 

> Can you
> justify using them additively?

Yes, or else some small power of them. Dave and I discussed this in 
depth. He initially proposed a*error^2 + b*complexity^2, partly 
because the local minima of harmonic entropy are parabolic.


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Message: 9866

Date: Sun, 08 Feb 2004 12:11:14

Subject: Re: Some warped egresses

From: Carl Lumma

>Maybe not, but I did post one or two 14-note blocks once -- they're 
>in Scala. I would search the tuning list for "What's your favorite 
>number?", but it seems the yahoogroups search engine has suddenly 
>become near-useless as it only searches a very small number of posts 
>at a time.

Also I remember it not enforcing "".

I don't find any tuning stuff with this on google.

-C.


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Message: 9867

Date: Sun, 08 Feb 2004 08:52:11

Subject: Re: Comma reduction?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" 
<perlich@a...> 
> > > wrote:
> > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> > > > <paul.hjelmstad@u...> wrote:
> > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> > > > <gwsmith@s...> 
> > > > > wrote:
> > > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
> > > > > > <paul.hjelmstad@u...> wrote:
> > > > > > 
> > > > > > > Thanks. Are they called 2-val and 2-monzo because they 
> > > > > are "linear"
> > > > > > > or is there some other reason?
> > > > > > 
> > > > > > 2-vals are two vals wedged, 2-monzos are two monzos 
wedged. 
> > The 
> > > > > former
> > > > > > is linear unless it reduces to the zero wedgie, the 
latter 
> is 
> > > > linear
> > > > > > only in the 7-limit.
> > > > > 
> > > > > Thanks! So the latter is linear in the 7-limit because the 
7-
> > > limit 
> > > > is 
> > > > > formed from two commas...I see.
> > > > 
> > > > The 7-limit is 4-dimensional, so if you temper out 2 commas 
> > you're 
> > > > left with a 2-dimensional system, which is what we usually 
> refer 
> > to 
> > > > as "linear". Is that what you meant?
> > > 
> > > Yes, I guess so. Why does tempering out two commas in a 4-
> > dimensional
> > > system leave a 2-dimensional system?
> > 
> > Roughly: the two commas in addition to two other basis vectors 
will 
> > span the 4-dimensional system (only if the four vectors are 
> linearly 
> > independent). If you temper out the two commas, the remaining two 
> > basis vectors will form a basis for the entire resulting system 
of 
> > pitches, which we therefore regard as two-dimensional.
> 
> Got it. How does one find the "remaining two basis vectors?" Is it 
> with Graham's matrix method?

I suppose, or with Gene's algorithm, in which he uses Hermite 
reduction.


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Message: 9870

Date: Sun, 08 Feb 2004 08:55:36

Subject: Re: Basis change for monzos, vals and wedgies

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> For whatever insight it may bring, here is an example. Suppose 
instead
> of 2,3,5,7 as a basis for 7-limit, we use 27/25, 21/20, 2401/2400 
and
> 4375/4374.

Paul Hj., this would interest you.

 Then the corresponding basis for vals is 
> <19 13 19 24|, <0 2 3 2|, <-1 -2 -3 -3| and <4 6 9 11|. The
> definitions for bimonzo, bival and compliment

Why, thank you :)

> are the same, giving a
> new basis there as well. We have
> 
> 441-et: <49 31 0 0|
> 612-et: <68 43 0 0|
> 
> ennealimmal: <<1 0 0 0 0 0||
> 
> This can aslo be computed from
> 
> 2401/2400: |0 0 1 0>
> 4375/4374: |0 0 0 1>
> 
> For miracle, we have
> 
> 225/224: |-5 8 -2 -1>
> 1029/1024: |-5 8 -1 -1>
> 
> miracle: <<1 0 8 0 5 0||
> 
> We could also have used, for instance
> 
> 72-et: <8 5 0 0|
> 175-et: <19 12 0 1|
> 
> I'm fond of 12-et in this system: 
> 
> 12-et: <1 1 1 1|



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Message: 9872

Date: Sun, 08 Feb 2004 20:30:18

Subject: Re: Jamesbond in 14-et

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> The jamesbond temperament, from the 007 in the wedgie <0 0 7 0 11 
16|,
> has TM basis {25/24, 81/80}. If you look at the TOP tuning of its
> generator pair, you find one generator is almost exacly twice 
another;
> this strongly suggests we may effectively identify jamesbond with an
> et--but what et? 

Did you see the horagram I posted?


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Message: 9874

Date: Sun, 08 Feb 2004 20:33:28

Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > > I
> > > think using log(err) and log(complexity) makes far more sense.
> > 
> > I don't think they make more sense practically. 
> 
> I think they probably will make more sense both practically and
> theoretically,

As I see it, no way. Example: when you look at the graph with log
(err) as one of the axes, the indication is that JI is infinitely far 
away. This is ridiculous. The JI line should be right there, with 
some temperaments many times more distant from it than others. 
Otherwise, you're operating in the realm of hopelessly impractical 
abstraction.

> but you've been ignoring this issue. Are you going to
> think about it, at least?

Countless hours already spent thinking about it, and discussing it 
here.


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