Tuning-Math Digests messages 1200 - 1224

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Message: 1200

Date: Thu, 2 Aug 2001 13:47:39

Subject: Ives stretched-8ve scales (was: Another BP linear temperament?)

From: monz

> From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, August 01, 2001 7:55 PM
> Subject: [tuning-math] Re: Another BP linear temperament?
>
>
> Since the "orthodox" BP consonances are ratios of {3,5,7}, 
> and the triple-BP consonances are ratios of {3,5,7,11,13}, and 
> their tritave-equivalents, and these have been referred to as 
> "7-limit" and "13-limit", respectively, I think this could 
> lead to confusion . . . let's abandon the term "limit" when 
> leaving the octave-equivalent world and adopt more explicit 
> mathematical terminology.


Hi Paul,


This caught my attention and reminded me of the webpage I made
examining Charles Ives's stretched-8ve scales.
A stretched-'octave' scale of Charles Ives, *

On that webpage, I propose 5 different tunings for the three
types described by Ives, with equivalence-intervals of 15.00,
16.00, 15.50, 14.50, and ~14.94 Semitones.

Can you guys (Paul, Graham, Dave) explain some more detailed
mathematical analyses of these tunings?




love / peace / harmony ...

-monz
Yahoo! GeoCities *
"All roads lead to n^0"


 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


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Message: 1201

Date: Thu, 02 Aug 2001 02:55:24

Subject: Re: Another BP linear temperament?

From: Paul Erlich

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

> I get 442.6 cents as the 7-limit optimum. I notice you put "7 limit" 
> in scare quotes, and rightly so. I'd call what you've given "7-limit 
> without 2s".
> 
> By analogy: 
> For an octave repeating scale, "7 limit" means ratios of all (whole 
> number) non-multiples of 2 up to 7, i.e {1,3,5,7} (and their octave 
> equivalents). 
> 
> So for a tritave repeating scale I think "7 limit" should mean ratios 
> of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.

Since the "orthodox" BP 
consonances are ratios of {3,5,7}, 
and the triple-BP consonances 
are ratios of {3,5,7,11,13}, and 
their tritave-equivalents, and 
these have been referred to as 
"7-limit" and "13-limit", 
respectively, I think this could 
lead to confusion . . . let's 
abandon the term "limit" when 
leaving the octave-equivalent 
world and adopt more explicit 
mathematical terminology.

(Remember, Pierce's idea for the 
scale that became known as BP 
was that if the even partials are 
absent, the 3:1 would take on the 
role of the octave, and ratios 
involving even numbers would no 
longer be "consonances" . . . 
whether or not the idea holds up 
in practice, it's conceptually neat 
enough that it makes sense to at 
least honor it on equal footing 
with other conceptions . . . 
Graham, or anyone else who can 
"hear" 3:1 equivalence . . . does it 
help to eliminate even partials 
from the timbres?)


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Message: 1203

Date: Thu, 2 Aug 2001 09:47 +01

Subject: Re: Another BP linear temperament?

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9kafes+8qo4@xxxxxxx.xxx>
Paul wrote:

> Graham, or anyone else who can 
> "hear" 3:1 equivalence . . . does it 
> help to eliminate even partials 
> from the timbres?)

It's all in the timbre to me.  Tritaves with odd-number partials have a 
stronger tritave equivalence than octaves with sawtooth timbres.  The same 
is true of phi with a phi timbre.  I'm guessing it would also be true of 
octaves with only even partials, but haven't tried yet.


                   Graham


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Message: 1204

Date: Thu, 2 Aug 2001 09:47 +01

Subject: Re: Another BP linear temperament?

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9kabno+v09n@xxxxxxx.xxx>
Dave Keenan wrote:

> > Here's the printout:
> > 
> > 7/30, 279.0 cent generator
> 
> Oops! Your generator should be 442.1 cents. You've multiplied by cents 
> per octave when it should have been cents per tritave.

Yes, I mentioned this in a later post.

> I get 442.6 cents as the 7-limit optimum. I notice you put "7 limit" 
> in scare quotes, and rightly so. I'd call what you've given "7-limit 
> without 2s".

Oh, it does include the 2s, but not enough of them.  It uses a 2-5-7 
basis, but the consonance limit is the octave-equivalent 7-limit matrix.  
So it won't include 7:4 or 5:4.

> By analogy: 
> For an octave repeating scale, "7 limit" means ratios of all (whole 
> number) non-multiples of 2 up to 7, i.e {1,3,5,7} (and their octave 
> equivalents). 
> 
> So for a tritave repeating scale I think "7 limit" should mean ratios 
> of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.

Sounds good, but it's a case of typing in the matrices.  This was a quick 
experiment.


                         Graham


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Message: 1205

Date: Thu, 2 Aug 2001 14:13:03

Subject: Re: BP linear temperament

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Paul wrote:
>Your least squares optimization uses what mapping from generators to
>primes? And it considers the errors for intervals 3:5, 3:7, 3:11,
>3:13, 5:7, 5:11, 5:13, 7:11, 7:13, 11:13 . . . right?

Not exactly, it was an approximation to all pitches of the "11-limit"
scale given. A bit more work, but can do that too. Then the result
is a virtually equal scale with a LS generator of 780.2702 cents.
Differences are
3:5    6.647 /  5.906
3:7    3.772 /  4.513
3:11   5.993 /  6.734
3:13   2.880 /  2.139
5:7   -2.876 / -2.135
5:11  -0.654 /  0.087
5:13  -3.768 / -4.509
7:11   2.222 /  1.481
7:13  -0.892 / -1.633
11:13 -3.114 / -3.855
The first one is the optimised value, the second one is the one
occurring fewer times in the scale.
The minimax optimal generator is 780.3520 cents giving a slightly
less equal scale with differences of
3:5    6.157 /  8.604
3:7    4.017 /  1.570
3:11   6.157 /  3.710
3:13   1.817 /  4.264
5:7   -2.140 / -4.587
5:11   0.0   / -2.447
5:13  -4.340 / -1.893
7:11   2.140 /  4.587
7:13   0.247 / -2.200
11:13 -4.340 / -1.893

Manuel


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Message: 1206

Date: Thu, 2 Aug 2001 22:47:07

Subject: Re: Ives stretched-8ve scales (was: Another BP linear temperament?)

From: monz

> From: Dave Keenan <D.KEENAN@xx.xxx.xx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Thursday, August 02, 2001 8:10 PM
> Subject: [tuning-math] Re: Ives stretched-8ve scales (was: Another BP
linear temperament?)
>
>
> BTW, do you suppose you are the only person on the planet that
> routinely specifies intervals in semitones with two decimal places,
> rather than cents? :-)


Yep.  It sure is taking a long time for everyone else to catch on!  ;-P

I mean, really... if +/- 5 cents is pretty much an accepted margin
of error in practice, then do we really need anything more accurate
than 240-EDO?, let alone 1200-EDO (which is exactly what my "Semitones"
specify).

And if we *do* need more accuracy, why not use another widely accepted
standard with finer resolution, such as "cawapus", or even "midipus",
rather than having to deal with more decimal points after the cents?


I really like Semitones (with two decimal places) because isolating
the semitone component makes it very easy for me to see at a glance
how a pitch or interval relates to what I'm already familiar with
as a long-time practicing 12-EDO musician.



love / peace / harmony ...

-monz
Yahoo! GeoCities *
"All roads lead to n^0"






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Message: 1207

Date: Fri, 03 Aug 2001 22:16:53

Subject: Re: Another BP linear temperament?

From: Paul Erlich

--- 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:
> > > So for a tritave repeating scale I think "7 limit" should mean 
> ratios 
> > > of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.
> > 
> > Since the "orthodox" BP 
> > consonances are ratios of {3,5,7},
> 
> So isn't 7:9 included, or do you really mean ratios of {1,5,7} and 
> their tritave equivalents?

The latter.
>  
> > and the triple-BP consonances 
> > are ratios of {3,5,7,11,13}, and 
> > their tritave-equivalents,
> 
> You mean {1,5,7,11,13} and their tritave-equivalents?

Same thing -- right?


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Message: 1208

Date: Fri, 03 Aug 2001 22:55:19

Subject: Re: Another BP linear temperament?

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > and the triple-BP consonances 
> > > are ratios of {3,5,7,11,13}, and 
> > > their tritave-equivalents,
> > 
> > You mean {1,5,7,11,13} and their tritave-equivalents?
> 
> Same thing -- right?

Yes, but not in lowest terms, so could be confusing.

You wouldn't write "ratios of {2,3,5,7,9} and their octave 
equivalents", would you? And there will be times when folks will not 
want to always write "and their tritave equivalents" because they feel 
it is given by the context. In this case "3" or "9" in place of "1" 
would definitely suggest tritave-specific. 

Just being my usual pedantic self. Sorry.

-- Dave Keenan


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Message: 1209

Date: Fri, 03 Aug 2001 23:10:09

Subject: Re: Another BP linear temperament?

From: Paul Erlich

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

> You wouldn't write "ratios of {2,3,5,7,9} and their octave 
> equivalents", would you?

Why not? I might even use 4 instead of 2.


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Message: 1210

Date: Fri, 03 Aug 2001 02:54:40

Subject: Re: Another BP linear temperament?

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave 
> Keenan" <D.KEENAN@U...> wrote:
> > So for a tritave repeating scale I think "7 limit" should mean 
ratios 
> > of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.
> 
> Since the "orthodox" BP 
> consonances are ratios of {3,5,7},

So isn't 7:9 included, or do you really mean ratios of {1,5,7} and 
their tritave equivalents?
 
> and the triple-BP consonances 
> are ratios of {3,5,7,11,13}, and 
> their tritave-equivalents,

You mean {1,5,7,11,13} and their tritave-equivalents?

> these have been referred to as 
> "7-limit" and "13-limit", 
> respectively, I think this could 
> lead to confusion . . . let's 
> abandon the term "limit" when 
> leaving the octave-equivalent 
> world and adopt more explicit 
> mathematical terminology.

Yes please!


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Message: 1211

Date: Fri, 03 Aug 2001 03:02:59

Subject: Re: Another BP linear temperament?

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9kabno+v09n@e...>
> Dave Keenan wrote:
> > So for a tritave repeating scale I think "7 limit" should mean 
ratios 
> > of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.
> 
> Sounds good, but it's a case of typing in the matrices.  This was a 
quick 
> experiment.

Duh! I included 6 above (which is of course tritave equivalent to 2 
and therefore redundant). This stuff is tricky. It should have been:

So for a tritave repeating scale I think "7 limit" should mean 
ratios of all non-multiples of 3 up to 7, i.e. {1,2,4,5,7} (and 
of course their tritave equivalents). 

But I agree with Paul, just spell it out, like after the "i.e." above.

Regards,
-- Dave Keenan


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Message: 1212

Date: Fri, 03 Aug 2001 03:10:47

Subject: Re: Ives stretched-8ve scales (was: Another BP linear temperament?)

From: Dave Keenan

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> On that webpage, I propose 5 different tunings for the three
> types described by Ives, with equivalence-intervals of 15.00,
> 16.00, 15.50, 14.50, and ~14.94 Semitones.
> 
> Can you guys (Paul, Graham, Dave) explain some more detailed
> mathematical analyses of these tunings?

Sorry Monz, I've got other things I should be doing at the moment.

BTW, do you suppose you are the only person on the planet that 
routinely specifies intervals in semitones with two decimal places, 
rather than cents? :-)

-- Dave Keenan


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Message: 1213

Date: Fri, 03 Aug 2001 05:59:17

Subject: Re: Magic lattices

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> > Now Dave Keenan's found an alternative simplified Miracle lattice, 
> > let's see if he can make anything of this.
> 
> I came up with something overnight:
[ lattice deleted ]
> 
> The template is
> 
>    5
> 1-----3-----9
>        \   /
>         \ /
>          7
> 
> Like Dave's new septimal-kleismic lattice, but unlike a normal 
7-limit 
> lattice, pitch increases left-right for a 4:5:6:7:9 chord.

Mine doesn't strictly-increase from left to right, but yours does. 


> That'd make it 
> good as a mapping for a hexagonal keyboard.

Yes indeed. In fact it is _THE_ mapping for such a keyboard. Try 
putting the period, generator and number of notes into my keyboard 
mapper spreadsheet (I think you have to wind the aspect-ratio up to 
max to get it to look hexagonal).
http://uq.net.au/~zzdkeena/Music/KeyboardMapper.xls - Ok *


> I also found this:
> 
> 
> G#--B+--D#--F#+-A#
>    / \     / \
> G+/ B \ D+/ F#\ A+
>  /     \ /     \
> G---Bt--D---F+--A
>  \     / \     /
> Gt\ Bb/ Dt\ Ft/ At
>    \ /     \ /
> Gb--Bbt-Db--Fbt-Ab
> 
> With the template
> 
>     5
> 
> 1-------3-------9---11
>          \     /
>           \   /
>            \ /
>             7
> 
> 
...
> And a unified neutral-second/neutral-third lattice
> 
> 
> B--D+-F#-A+-C#
> |\    |\    |
> | \   | \   |
> A  C+ E  G+ B
> |   \ |   \ |
> |    \|    \|
> G--Bt-D--F+-A
> |\    |\    |
> | \   | \   |
> F  At C  Et G
> |   \ |   \ |
> |    \|    \|
> Eb-Gt-Bb-Dt-F
> 
> 
> It's like Dave Keenan's new Miracle lattice, but with extra rows.  
> Template
> 
>             7
>             |
>             |
>             |
>             |
>             |
> 5           |
> |           |
> |           |
> | 11        |
> |           |
> |           |
> 1-----3-----9--11
> 
> 
> I don't know if there's a simpler position for the 7 ...

Nor do I.

All these tempered lattices coming out of the woodwork! After trying 
the full colour version of the 2D tempered lattice for Blackjack, I'm 
thinking that 3D may be easier to deal with in some ways.

It's also useful to note that the keyboard mapping can always be 
treated as a 2D tempered lattice.

I'm thinking now that one way to find the best 3D lattice for an 
11-limit scale may be to put several periods of it into a spreadsheet 
as a 5D triangular lattice and look at a 3D projection of it in the 
same way I did with the dekany. Then rotate it until it is as thin as 
possible in the into-the-screen dimension, followed by minor 
adjustments so no note is obscured. But I don't have time.

-- Dave Keenan


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Message: 1214

Date: Fri, 3 Aug 2001 15:42:05

Subject: Re: BP linear temperament

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>> Not exactly, it was an approximation to all pitches of the "11-
>> limit" scale given.

>Now why would anyone want that? It's the consonant intervals, not the
>pitches, that we care about approximating well. (At least that's my
>philosophy.)

But the former doesn't exclude the latter. Approximating the
intervals everywhere in the scale smoothes out a just scale more than
somebody might want.
The posted "11-limit" periodicity block contains 13 chords 3:5:7:11.
The "approximation to pitches" scale contains 30 with a maximum
deviation of 7.072 cents and the "approximation to consonant
intervals" scale 39 with a 6.647 cents maximum deviation. The
L/s stepsize ratio of the former is 3:2 and 1.0153:1 of the latter.
So I would say it has merit if you're interested in a scale that is
more equal than just but not as equal as ET. By the way the Scala
command I used, BISTEP, approximates a scale by one with two step
sizes, but it doesn't require the step pattern to be Myhill.

>You mean virtually the 39th root of 3? That's more what I would have
>expected.

Yes.

>> The first one is the optimised value, the second one is the one
>> occurring fewer times in the scale.

>Can you explain what the latter means?

Because the scale is almost equal, both interval sizes of each
interval class are close. These are the deviations of the intervals
5/3 7/3 11/3 13/3 (the chord 3:5:7:11:13) in the minimax best scale,
so you can see it:
Locations of 5/3 7/3 11/3 13/3:
0 - 18 - 30 - 46 - 52  diff.  6.157, 1.570, 3.710, 1.817
1 - 19 - 31 - 47 - 53  diff.  6.157, 4.017, 6.157, 1.817
2 - 20 - 32 - 48 - 54  diff.  6.157, 4.017, 6.157, 1.817
3 - 21 - 33 - 49 - 55  diff.  6.157, 4.017, 6.157, 4.264
4 - 22 - 34 - 50 - 56  diff.  6.157, 4.017, 6.157, 1.817
5 - 23 - 35 - 51 - 57  diff.  6.157, 4.017, 6.157, 4.264
6 - 24 - 36 - 52 - 58  diff.  6.157, 4.017, 6.157, 1.817
7 - 25 - 37 - 53 - 59  diff.  8.604, 4.017, 6.157, 4.264
8 - 26 - 38 - 54 - 60  diff.  6.157, 4.017, 6.157, 1.817
9 - 27 - 39 - 55 - 61  diff.  6.157, 4.017, 6.157, 1.817
10 - 28 - 40 - 56 - 62  diff.  6.157, 4.017, 6.157, 1.817
11 - 29 - 41 - 57 - 63  diff.  6.157, 4.017, 6.157, 1.817
12 - 30 - 42 - 58 - 64  diff.  6.157, 4.017, 6.157, 4.264
13 - 31 - 43 - 59 - 65  diff.  6.157, 4.017, 6.157, 1.817
14 - 32 - 44 - 60 - 66  diff.  8.604, 4.017, 6.157, 4.264
15 - 33 - 45 - 61 - 67  diff.  6.157, 4.017, 6.157, 1.817
16 - 34 - 46 - 62 - 68  diff.  6.157, 1.570, 3.710, 1.817
17 - 35 - 47 - 63 - 69  diff.  6.157, 4.017, 6.157, 1.817
18 - 36 - 48 - 64 - 70  diff.  6.157, 4.017, 6.157, 1.817
19 - 37 - 49 - 65 - 71  diff.  6.157, 4.017, 6.157, 4.264
20 - 38 - 50 - 66 - 72  diff.  6.157, 4.017, 6.157, 1.817
21 - 39 - 51 - 67 - 73  diff.  8.604, 4.017, 6.157, 4.264
22 - 40 - 52 - 68 - 74  diff.  6.157, 4.017, 6.157, 1.817
23 - 41 - 53 - 69 - 75  diff.  8.604, 4.017, 6.157, 4.264
24 - 42 - 54 - 70 - 76  diff.  6.157, 4.017, 6.157, 1.817
25 - 43 - 55 - 71 - 77  diff.  6.157, 4.017, 6.157, 1.817
26 - 44 - 56 - 72 - 78  diff.  6.157, 4.017, 6.157, 4.264
27 - 45 - 57 - 73 - 79  diff.  6.157, 4.017, 6.157, 1.817
28 - 46 - 58 - 74 - 80  diff.  6.157, 4.017, 6.157, 4.264
29 - 47 - 59 - 75 - 81  diff.  6.157, 4.017, 6.157, 1.817
30 - 48 - 60 - 76 - 82  diff.  8.604, 4.017, 6.157, 4.264
31 - 49 - 61 - 77 - 83  diff.  6.157, 4.017, 6.157, 1.817
32 - 50 - 62 - 78 - 84  diff.  6.157, 1.570, 6.157, 1.817
33 - 51 - 63 - 79 - 85  diff.  6.157, 4.017, 6.157, 1.817
34 - 52 - 64 - 80 - 86  diff.  6.157, 4.017, 6.157, 1.817
35 - 53 - 65 - 81 - 87  diff.  6.157, 4.017, 6.157, 4.264
36 - 54 - 66 - 82 - 88  diff.  6.157, 4.017, 6.157, 1.817
37 - 55 - 67 - 83 - 89  diff.  8.604, 4.017, 6.157, 4.264
38 - 56 - 68 - 84 - 90  diff.  6.157, 4.017, 6.157, 1.817

Manuel


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Message: 1216

Date: Sat, 04 Aug 2001 18:50:13

Subject: HyperMOS

From: carl@xxxxx.xxx

Finally, I'll take the time to say something on this
fascinating thread...

() Does "HyperMOS" refer to the generalized higher-D
MOS's of Paul's hypothesis, or are they another type?

() Paul, you said that 'as far as we can tell, CS
and PB are the same thing'.  But aren't there many
irrational CS scales that don't have any sensible
PB interp?

() Paul's (hi, Paul!) hypothesis post may be the most
feverishly dense and impressive post I've seen since
his original harmonic entropy post in '97.  I don't
think a notion of HyperMOS is nearly as important as
harmonic entropy, but who knows....  I'll quote the
pertinent parts here, and maybe say something more
later when (a) I understand more of it and (b) I don't
have an apartment full of boxes to unpack.  For now,
please accept my once-over comments:

>Take an n-dimensional lattice, and pick n independent unison
>vectors.  Use these to divide the lattice into parallelograms or
>parallelepipeds or hyperparallelepipeds, Fokker style. Each one
>contains an identical copy of a single scale (the PB) with N notes.

Paul, did you have to use the letter "n" twice?  You seem to want
to use case to distinguish them... I'll hold you to that.

>Any vector in the lattice now corresponds to a single generic
>interval in this scale no matter where the vector is placed (if
>the PB is CS, which it normally should be).

Check.

>Now suppose all but one of the unison vectors are tempered out.
>The "wolves" now divide the lattice into parallel strips, or
>layers, or hyper-layers.  The "width" of each of these, along the
>direction of the chromatic unison vector (the one that remains
>untempered), is equal to the length of exactly one of this
>chromatic unison vector.

Check.

>Now let's go back to "any vector in the lattice". This vector,
>added to itself over and over, will land one back at a pitch in the
>same equivalence class as the pitch one started with, after N
>iterations (and more often if the vector represents a generic
>interval whose cardinality is not relatively prime with N).

Okay.

>In general, the vector will have a length that is some fraction
>M/N of the width of one strip/layer/hyperlayer, measured in the
>direction of this vector (NOT in the direction of the chromatic
>unison vector). M must be an integer, since after N iterations,
>you're guaranteed to be in a point in the same equivalence class as
>where you started, hence you must be an exact integer M
>strips/layers/hyperlayers away. As a special example, the generator
>has length 1/N of the width of one strip/layer/hyperlayer, measured
>in the direction of the generator.

I don't see why M/N yet, but I can see what's happening.

>Anyhow, each occurence of the vector will cross either floor(M/N)
>or ceiling(M/N) boundaries between strips/layers/hyperlayers. Now,
>each time one crosses one of these boundaries in a given direction,
>one shifts by a chromatic unison vector. Hence each specific
>occurence of the generic interval in question will be shifted by
>either floor(M/N) or ceiling(M/N) chromatic unison vectors. Thus
>there will be only two specific sizes of the interval in question,
>and their difference will be exactly 1 of the chromatic unison
>vector. And since the vectors in the chain are equally spaced and
>the boundaries are equally spaced, the pattern of these two sizes
>will be an MOS pattern.

Spot on, for the 1-D case I can vidi!!!

-Carl


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Message: 1217

Date: Sat, 04 Aug 2001 22:18:34

Subject: Re: Another BP linear temperament

From: Dave Keenan

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> <<But this is just a linear temperament. How is it more specifically 
a
> "generalised meantone"?>>
> 
> Again, it's a generalization of meantone because the "tone" and the
> "mean" are generalized! The process of determining a comma and
> fractionalizing and distributing it are also all exactly analogous 
if
> one strips away the particulars.

Ok. Can you tell me what intervals are analogous to the major and 
minor tone in this temperament. I'm concerned that you're generalising 
"tone" to mean "an interval of any size", which I think would be a bad 
move.

> It's the process, not the specific tuning -- besides, if the octave
> divided into 12 and 13 equidistant parts can rightfully be called
> "equal temperament" in either instance, then I see no reason why 
what
> I'm doing here can't be called "meantone"!
>
> That said, I never was happy with "generalized meantone" as a 
blanket
> term for this sort of thing, and I did post several times a while 
back
> trolling for better, less likely to cause confusion, types of terms.
> Unfortunately this never really went anywhere. So, as I haven't
> thought of anything better or less confusing myself, I've just stuck
> with my initial intuitive feeling as that seems obvious and simple
> enough for communication purposes...

So it's just a particular way of arriving at a linear temperament? Why 
not call the process "comma distribution"?

> Does any of this make any better sense now? If not, well, then I'm
> convinced that Graham was actually right, and I am quite incapable 
of
> communicating these things.

I don't think he said (or meant) that exactly. Yes, it makes better 
sense now and we are quite capable of communicating (it takes two).

Regards,
-- Dave Keenan


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Message: 1218

Date: Sat, 04 Aug 2001 01:47:50

Subject: Re: Another BP linear temperament?

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> 
> > You wouldn't write "ratios of {2,3,5,7,9} and their octave 
> > equivalents", would you?
> 
> Why not? I might even use 4 instead of 2.

I thought I answered "Why not?" in the same message you quote from.

Why would you?


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Message: 1221

Date: Sat, 04 Aug 2001 08:05:31

Subject: Re: Another BP linear temperament

From: Dave Keenan

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Dave,
> 
> The way I look at it, what I did was simply remap the unison vectors
> so that the 2D comma is 118098/117649

Ok <goes and figures> that's (2^1 * 3^10 * 7^-6) = 6.7 cents

> and the primary chroma is 343/324.

That's (2^-2 * 3^-4 * 7^3) = 99 cents. Is this for the 9-note per 
tritave MOS?

> This means that the dimensions are 9/7 and 7/6 with a period
> of 3/1 as opposed to the usual 3/2 and 5/4 with a period of 2/1.

Ok. But for this to be a "generalised meantone" in any sense other 
than the one which is much better described by the term "linear 
temperament", then a chain of _four_ 9/7's would have to be 
tritave-equivalent to a 7/6. This is not the case. It needs a chain of 
five according to your comma.

> The 1/6 comma meantone generalization is exactly analogous to 
standard
> 1/5 comma meantone in terms of fractionalizing a comma and altering
> the size of the generator in a 1D chain.

I suspect you mean standard 1/4 comma meantone.

But this is just a linear temperament. How is it more specifically a 
"generalised meantone"? Is it merely because it seeks to approximate 
only two primes (other than the interval-of-equivalence), in this case 
2 and 7? That certainly isn't enough for _me_ to consider it a 
"generalised meantone". What do others think?

So this BP temperament still has an approximate 7:9 generator but it's 
typically a narrow one, where the previous BP temperament has a wide 
one. The mapping from primes to numbers of generators is:

Prime  No. generators
-----  --------------
2      -6
3       0
5      (don't care, but 2 is best)
7      -1

The MA optimum {1,2,7} generator is 434.0 cents giving a maximum error 
of 1.2 cents. This generator is, as you said, a 7:9 narrowed by 1/6 of 
the above comma.

If 5's are included, but not 4's, i.e. {1,2,5.7}, the optimum 
generator is 436.0 cents (1/7 comma wide) with errors of 12.3 cents.

The optimum for the full 7-limit, i.e. {1,2,4,5,7}, is 435.15 cents 
(in between the previous two) but with errors of 14.1 cents. It has 
MOS cardinalities of 4 (5) 9 13 (22) 35 (48), (improper in 
parenthesis).

-- Dave Keenan


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Message: 1224

Date: Mon, 06 Aug 2001 19:36:16

Subject: Re: Another BP linear temperament?

From: Paul Erlich

--- 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:
> > 
> > > You wouldn't write "ratios of {2,3,5,7,9} and their octave 
> > > equivalents", would you?
> > 
> > Why not? I might even use 4 instead of 2.
> 
> I thought I answered "Why not?" in the same message you quote from.
> 
> Why would you?

So that the resulting ratios would tend to be the familar voicings 
within one octave: 3:4, 4:5, 4:7 . . .


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