Tuning-Math Digests messages 10828 - 10852

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Message: 10828

Date: Fri, 16 Apr 2004 22:00:03

Subject: Re: On

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > <<0, 0, 12, 0, 19, 28]], on the other hand, seems to be 
functionally 
> > the same as Jon Catler's '12-tone plus' tuning -- except that the 
> > offset is cleverly an eighthtone instead of a sixthtone, making 
for 
> > better 7:5s . . . right?
> 
> One way to look at it is that it is 5-limit 12-equal with 7s tacked
> on, which may as well be pure 7s (the TOP tuning.) TOP 12-et of 
course
> has flat octaves, but this is hardly a requirement. "12 tone plus"
> seems like a good description of this--is this Catler's idea?

Apparently. See the microtones.com website.

> Is
> "catler" a good name for this temperament?

He might object, since his extended 13-limit JI systems are clearly 
dearer to his heart.


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Message: 10829

Date: Fri, 16 Apr 2004 22:43:43

Subject: Re: On

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > <<0, 0, 12, 0, 19, 28]], on the other hand, seems to be 
functionally 
> > the same as Jon Catler's '12-tone plus' tuning -- except that the 
> > offset is cleverly an eighthtone instead of a sixthtone, making 
for 
> > better 7:5s . . . right?
> 
> One way to look at it is that it is 5-limit 12-equal with 7s tacked
> on, which may as well be pure 7s (the TOP tuning.) TOP 12-et of 
course
> has flat octaves, but this is hardly a requirement. "12 tone plus"
> seems like a good description of this--is this Catler's idea? Is
> "catler" a good name for this temperament?

I might use "catler" in my paper, but I'm having trouble with some of 
the other names.

In 5-limit, <3    0   -7] is called "augmented" -- fine.

In 7-limit, we have
<3     0     6    -7     1    14]
and
<3     0    -6    -7   -18   -14]
Why is one of these "augmented" and the other "tripletone"?


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Message: 10830

Date: Sat, 17 Apr 2004 17:06:36

Subject: Re: On

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > In 5-limit, <3    0   -7] is called "augmented" -- fine.
> > 
> > In 7-limit, we have
> > <3     0     6    -7     1    14]
> > and
> > <3     0    -6    -7   -18   -14]
> > Why is one of these "augmented" and the other "tripletone"?
> 
> By my proposal to use TOP to guide us, the second one should be 
called
> "augmented" also, and the name "tripletone" retired.
> 
> TOP generators:
> 
> 128/125 [399.020, 93.145]
> 
> <3 0 6 -7 1 14| [399.992 107.311]
> 
> <3 0 -6 -7 -18 -14| [399.020 92.460]

Close, but no cigar. In my paper, I will only use the same name if 
the tuning is exactly the same, because the names will be references 
to horagrams.

> That leaves us without a name for the other system, which is also an
> important one.

I propose both names be some sort of variants of "augmented".


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Message: 10833

Date: Mon, 19 Apr 2004 15:28:37

Subject: Re: On

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Close, but no cigar. In my paper, I will only use the same name 
if 
> > the tuning is exactly the same, because the names will be 
> references 
> > to horagrams.
> 
> There's no reason not to use the same tuning for both,

I consoled John Chalmers by telling him I'd use only one tuning -- 
the TOP tuning -- for each temperament. Are you suggesting a 
deviation from this strategy? Can you make it quantitative?


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Message: 10834

Date: Mon, 19 Apr 2004 21:55:53

Subject: notation of monzos

From: monz

hello all,


did we ever reach a consensus on the use or
non-use of angle-brakets and/or pipe symbols
in expressing monzos?  i have not been following
tuning-math much lately, and just want to be
sure that the Encyclopaedia entry is up-to-date.


-monz


--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

Yahoo groups: /tuning-math/message/7527 *

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > I admit to being shamefully ignorant of these things.
> > It sounds then like vals could use the angle-brackets.
> 
> Physicists sometimes use bra and ket vectors; if you 
> did that, the monzo for 81/80 would be a ket, [-4 4 -1> 
> and the val for 5-limit 12-et would be a bra, <12 19 28].
> Putting them together would give the bra-ket, angle bracket,
> or inner product: <12 19 28 | -4 4 -1> = 0.
> 
> See:
> 
> Contravariant Vector -- from MathWorld *
> 
> Covariant Vector -- from MathWorld *
> 
> Ket -- from MathWorld *
> 
> Bra -- from MathWorld *
> 
> One-Form -- from MathWorld *
> 
> Angle Bracket -- from MathWorld *


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Message: 10836

Date: Mon, 19 Apr 2004 22:02:39

Subject: Re: TM-reduced bases for ETs

From: monz

hello all,


i've just updated the Encyclopaedia of Tuning
entry for "TM-reduced lattice", to include examples
of TM-reduced bases for ETs that were posted here
by Gene at the beginning of November 2003.

Tonalsoft Encyclopaedia of Tuning - TM-reduced lattice basis, (c) 2004 Tonalsoft Inc. *

feedback appreciated.

(paul, i know that i also need to quote your post
about how certain ETs can have multiple equally-good/bad
val mappings ... i will when i get more time.)



-monz


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Message: 10839

Date: Tue, 20 Apr 2004 19:52:24

Subject: Re: notation of monzos

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > hello all,
> > 
> > 
> > did we ever reach a consensus on the use or
> > non-use of angle-brakets and/or pipe symbols
> > in expressing monzos? 
> 
> Not really. Angle-brakets seem to be well accepted, but so far 
as "|"
> versus "]" goes, some people use one, and some the other.

The monzo would have a "[", not a "]". I'm using these, not pipe 
symbols or even vertical lines, in my paper.


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Message: 10840

Date: Tue, 20 Apr 2004 22:33:12

Subject: Re: TM-reduced bases for ETs

From: monz

hi paul,


--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> > Tonalsoft Encyclopaedia of Tuning - TM-reduced lattice basis, (c) 2004 Tonalsoft Inc. *

> Right off the bat:
> 
> "A method for reducing the bases of a lattice."
> 
> should read
> 
> "A method for reducing the basis of a lattice."
> 
> TM-reduction results in a *single* basis for the lattice, not 
> multiple _bases_.



thanks.  i'd really like to add a little more "regular English"
to the opening part of that definition, describing exactly
what TM-reduction does ... before heading into Gene's
mathematical definition.  can you or anyone else help?

i'm thinking something like this:

"A method for reducing the basis of a lattice to its
most compact representation, with all unison-vectors 
as small as possible in prime-space."



-monz


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Message: 10841

Date: Tue, 20 Apr 2004 22:37:38

Subject: Re: notation of monzos

From: monz

hi paul and Gene,

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> > 
> > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > > hello all,
> > > 
> > > 
> > > did we ever reach a consensus on the use or
> > > non-use of angle-brakets and/or pipe symbols
> > > in expressing monzos? 
> > 
> > Not really. Angle-brakets seem to be well accepted, 
> > but so far as "|" versus "]" goes, some people use one,
> > and some the other.
> 
> The monzo would have a "[", not a "]". I'm using these, 
> not pipe symbols or even vertical lines, in my paper.


it was my understanding that the monzo by itself uses [...>
and the val uses <...] , and that putting them together
one would use the pipe symbol instead of the two square
brackets thus: <...|...> .  has this become established usage?

also, what about the suggestion to use comma punctuation
after the exponents of 3, 11, 19, 31, etc.?  is that
established at all?



-monz


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Message: 10842

Date: Tue, 20 Apr 2004 16:47:51

Subject: Re: TM-reduced bases for ETs

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hello all,
> 
> 
> i've just updated the Encyclopaedia of Tuning
> entry for "TM-reduced lattice", to include examples
> of TM-reduced bases for ETs that were posted here
> by Gene at the beginning of November 2003.
> 
> Tonalsoft Encyclopaedia of Tuning - TM-reduced lattice basis, (c) 2004 Tonalsoft Inc. *
> 
> feedback appreciated.

Right off the bat:

"A method for reducing the bases of a lattice."

should read

"A method for reducing the basis of a lattice."

TM-reduction results in a *single* basis for the lattice, not 
multiple _bases_.



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Message: 10843

Date: Wed, 21 Apr 2004 20:05:42

Subject: Re: 270 equal as the universal temperament

From: Graham Breed

Gene Ward Smith wrote:

> If "four digit frequency ratios" (which from context I take to mean
> superparticular ones) have no audible meaning, it seems like a nifty
> idea to temper them out. In the 7-limit this gives ennealimmal, and in
> the 11-limit hemiennealimmal. If we take Graham's view, which I think
> has something to be said for it, we go up to the 13-limit but no
> farther. In the 13-limit, there are twelve four digit superparticular
> commas; the kernel of all of these taken together is 270-equal. This
> sort of fact I've discussed before; it does seem there is some
> justification for considering 270-et to be a sort of universal
> replacement for just intonation.

I did mean the complexity was too great (and obviously so).   Probably 
harmonic commas can be heard for much smaller intervals -- particularly 
mistuned unisons.  But commas aren't usually hidden within chords, are 
they?  Such small commas should be melodically inaudible, especially if 
shared among a few chords.  So an adaptive tuning scheme with these 
commas should be indistinguishable on a chord-by-chord basis from JI. 
In which case a 270 note system would be a replacement for JI if used 
with adaptive temperament.

13-prime limit is one place to stop.  I think harmony works clearly up 
to the 9-limit.  Then at the 11-limit you get exotic intervals like 
neutral thirds and seconds.  The 13-limit is roughly more of the same, 
but you get 8:10:13 and 8:11:13 chords.  These are theoretically 
important for a number of reasons:

- They start with a power of two, and so may have a strong virtual pitch.

- All intervals, including the implied octave, are larger than 7:8 
(roughly a critical bandwidth).

- Along with 4:5:6, they're the only such chords with the first number 
less than 16.

Such chords may be useful in cadences, I still haven't decided.  And 
slightly mistuned versions may work as well, if each interval is a 
better approximation to an 11-limit interval.  But, anyway, they might 
be a reason for going to the 13-limit, at which point you may as well go 
to 15.  It may even be possible to hear higher limits.  With 16:19:24, 
you get that alternative tuning of a minor triad.


                Graham


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Message: 10844

Date: Wed, 21 Apr 2004 20:45:16

Subject: Re: 270 equal as the universal temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
> 
> > If "four digit frequency ratios" (which from context I take to 
mean
> > superparticular ones) have no audible meaning, it seems like a 
nifty
> > idea to temper them out. In the 7-limit this gives ennealimmal, 
and in
> > the 11-limit hemiennealimmal. If we take Graham's view, which I 
think
> > has something to be said for it, we go up to the 13-limit but no
> > farther. In the 13-limit, there are twelve four digit 
superparticular
> > commas; the kernel of all of these taken together is 270-equal. 
This
> > sort of fact I've discussed before; it does seem there is some
> > justification for considering 270-et to be a sort of universal
> > replacement for just intonation.
> 
> I did mean the complexity was too great (and obviously so).   
Probably 
> harmonic commas can be heard for much smaller intervals -- 
particularly 
> mistuned unisons.  But commas aren't usually hidden within chords, 
are 
> they?

64:63 is famous for being hidden within chords . . .


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Message: 10845

Date: Wed, 21 Apr 2004 20:59:00

Subject: Re: 270 equal as the universal temperament

From: Graham Breed

Paul Erlich wrote:

> 64:63 is famous for being hidden within chords . . .

Yes, but it doesn't run to four digits, or not in each number.  The 
neutral third comma makes it to three -- 243:242.


                       Graham


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Message: 10847

Date: Wed, 21 Apr 2004 20:58:28

Subject: Re: notation of monzos

From: monz

hi paul,


--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 
> > >However I don't use the word "monzo"
> > 
> > Why not?
> > 
> > -Carl
> 
> I want the minimum of jargon, and the maximum sense
> of "I could have thought of this myself and I probably
> did at one point" in the reader.


i guess i'm just being selfish, but i am disappointed,
and wish you'd use "monzo" in your paper ... mainly 
because it's my feeling that its use in a paper by *you* 
would give the term a cachet that it doesn't currently have.

:)

as Gene pointed out when he first named the "vector of
prime-factor exponents" after me, it's useful to have
one word to replace a whole phrase, if you're going to
be referring to it often.





-monz


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Message: 10849

Date: Wed, 21 Apr 2004 21:00:51

Subject: Re: 270 equal as the universal temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
> 
> > If "four digit frequency ratios" (which from context I take to 
mean
> > superparticular ones) have no audible meaning, it seems like a 
nifty
> > idea to temper them out. In the 7-limit this gives ennealimmal, 
and in
> > the 11-limit hemiennealimmal. If we take Graham's view, which I 
think
> > has something to be said for it, we go up to the 13-limit but no
> > farther. In the 13-limit, there are twelve four digit 
superparticular
> > commas; the kernel of all of these taken together is 270-equal. 
This
> > sort of fact I've discussed before; it does seem there is some
> > justification for considering 270-et to be a sort of universal
> > replacement for just intonation.
> 
> I did mean the complexity was too great (and obviously so).   
Probably 
> harmonic commas can be heard for much smaller intervals -- 
particularly 
> mistuned unisons.  But commas aren't usually hidden within chords, 
are 
> they?  Such small commas should be melodically inaudible, 
especially if 
> shared among a few chords.  So an adaptive tuning scheme with these 
> commas should be indistinguishable on a chord-by-chord basis from 
JI. 
> In which case a 270 note system would be a replacement for JI if 
used 
> with adaptive temperament.

Being just like JI (and ~2 cent errors don't bother me) doesn't 
strike me as a sufficient qualification for being a/the "universal 
temperament". In addition, multiple vals/breeds should be supported, 
including some useful non-micro-temperaments.

My favorites among those include the ones that support 
omnitetrachordal scales and have relatively low complexity, such as 
meantone, double-meantone (injera), and pajara.

You'll find my suggested "universal tuning" here:

Tonalsoft Encyclopaedia of Tuning - equal-temperament, (c) 2004 Tonalsoft Inc. *


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