Tuning-Math Digests messages 10878 - 10902

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 11

Previous Next

10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500 10550 10600 10650 10700 10750 10800 10850 10900 10950

10850 - 10875 -



top of page bottom of page down


Message: 10878

Date: Fri, 23 Apr 2004 17:17:46

Subject: Re: 270 equal as the universal temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > Then who came up with the subject line?
> 
> I came up with the subject line; you came up with your own
> interpretation of what you thought it should mean.

So what did you mean, according to your own interpretation, when you 
said "I was not calling 270 a universal temperament"?



________________________________________________________________________
________________________________________________________________________



------------------------------------------------------------------------
Yahoo! Groups Links

<*> To visit your group on the web, go to:
     Yahoo groups: /tuning-math/ *

<*> To unsubscribe from this group, send an email to:
     tuning-math-unsubscribe@xxxxxxxxxxx.xxx

<*> Your use of Yahoo! Groups is subject to:
     Yahoo! Terms of Service *


top of page bottom of page up down


Message: 10880

Date: Sat, 24 Apr 2004 09:45:13

Subject: Re: lattices of Schoenberg's rational implications

From: monz

returning to an old subject ...

during a big discussion i instigated concerning
possible periodicity-blocks which might describe
Schoenberg's 1911 12-tET theory as posited in his
_Harmonielehre_,


--- In tuning-math@xxxxxxxxxxx.xxxx "genewardsmith" 
<genewardsmith@j...> wrote:

Yahoo groups: /tuning-math/message/2848 *

> Message 2848
> From:  "genewardsmith" <genewardsmith@j...> 
> Date:  Sun Jan 20, 2002  7:20 pm
> Subject:  Re: lattices of Schoenberg's rational implications
>
>
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > However, I think the only reality for Schoenberg's 
> > system is a tuning where there is ambiguity, as defined
> > by the kernel <33/32, 64/63, 81/80, 225/224>. BTW,
> > is this Minkowski-reduced?
> 
> Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.



a few messages after that,

Yahoo groups: /tuning-math/message/2850 *

Paul Erlich posted four different possible lattices
based on those unison-vectors.


i've made a rectangular-style lattice based on those
same four unison-vectors, using the Tonalsoft software.  
this is what the software gave me:

~cents..  ratio

0000.000  1/1
0084.467  21/20
0231.174  8/7
0266.871  7/6
0386.314  5/4
0498.045  4/3
0582.512  7/5
0701.955  3/2
0813.686  8/5
0884.359  5/3
0968.826  7/4
1115.533  40/21


here is a screen-shot of the actual Tonalsoft lattice:

Yahoo! - *
tuning-math-2848-minkowski-reduced-schoenberg-12et.jpg


... delete the line-break in that URL, or use this one:

Document Not Found *




-monz







 



________________________________________________________________________
________________________________________________________________________



------------------------------------------------------------------------
Yahoo! Groups Links

<*> To visit your group on the web, go to:
     Yahoo groups: /tuning-math/ *

<*> To unsubscribe from this group, send an email to:
     tuning-math-unsubscribe@xxxxxxxxxxx.xxx

<*> Your use of Yahoo! Groups is subject to:
     Yahoo! Terms of Service *


top of page bottom of page up down


Message: 10881

Date: Sun, 25 Apr 2004 18:15:17

Subject: Re: What's with 14

From: Joseph Pehrson

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

Yahoo groups: /tuning-math/message/10323 *

 wrote:
> 
> Joseph asked on MMM:
> 
> "What's with 14, though... it scores pretty badly on the famed Paul
> Erlich accuracy chart... :)"
> 
> The tuning that the Zeta function likes for 14 has a flat octave, 
and
> corresponds to <14 22 33 39 48|. It has the following TM bases:
> 
> 5-limit: [27/25, 2048/1875]
> 7-limit: [21/20, 27/25, 2048/1875]
> 11-limit: [21/20, 27/25, 33/32, 242/225]
> 
> 27/25 in the 5-limit, 21/20 and 27/25 together in the 7-limit, and
> 21/20,27/25 and 33/32 in the 11-limit give the beep temperament, so
> this 14-et val is closely associated to beep. The top tuning has
> octaves around four cents flat.
> 
> Another val regards 14 as a contorted version of 7 in the 5-limit; 
in
> the 11-limit it is <14 22 32 39 48|. TM bases are
> 
> 5-limit: [25/24, 81/80]
> 7-limit: [25/24, 49/48, 81/80]
> 11-limit: [25/24, 33/32, 45/44, 49/48]
> 
> This involves decimal, meantone and jamesbond, and the TOP tuning of
> the octave is now quite sharp, not flat; 1209.43 cents.
> 
> Other vals are possible; for instance a father version is <14 23 33 
40
> 49|. TM bases for this are
> 
> 5-limit: [16/15, 15625/13122];
> 7-limit: [16/15, 50/49, 175/162];
> 11-limit: [16/15, 22/21, 50/49, 175/162];


***Well, most of this is, admittedly, a bit over my head... but I 
believe I saw that the 3-limit is reflected in 14-tET, at least 
according to the Erlich chart, and I don't see it listed in the 
above... (??)

Thanks!

JP


top of page bottom of page up down


Message: 10883

Date: Sun, 25 Apr 2004 19:56:50

Subject: Re: What's with 14

From: Paul Erlich

Hi Joseph.

As far as 5-limit goes, I have a suggestion.

Remember the big ET chart on Monz's equal temperament page:

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

It's the first chart there . . .

Now mouse over "zoom: 1" above the chart. If you can't see the yellow 
triangular grid, mouse over "zoom: 1" under "negatives".

You'll see 14 occuring three times on that chart . . . once 
overlapping 7.

These are three ways of "using" 14-equal in the 5-limit.

Look at how large the errors are of the basic 5-limit consonances.

Two of the instances of 14, it is true, are fairly close to the "just 
perfect fifths - just perfect fourths" line.

But they both have thirds that are off by around 30-80 cents.

The other instance of 14 (the one at the top) doesn't fare much 
better, and the "perfect fifth" is some 70 cents sharp!!

Am I making any sense?

-Paul


--- In tuning-math@xxxxxxxxxxx.xxxx "Joseph Pehrson" <jpehrson@r...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> 
> Yahoo groups: /tuning-math/message/10323 *
> 
>  wrote:
> > 
> > Joseph asked on MMM:
> > 
> > "What's with 14, though... it scores pretty badly on the famed 
Paul
> > Erlich accuracy chart... :)"
> > 
> > The tuning that the Zeta function likes for 14 has a flat octave, 
> and
> > corresponds to <14 22 33 39 48|. It has the following TM bases:
> > 
> > 5-limit: [27/25, 2048/1875]
> > 7-limit: [21/20, 27/25, 2048/1875]
> > 11-limit: [21/20, 27/25, 33/32, 242/225]
> > 
> > 27/25 in the 5-limit, 21/20 and 27/25 together in the 7-limit, and
> > 21/20,27/25 and 33/32 in the 11-limit give the beep temperament, 
so
> > this 14-et val is closely associated to beep. The top tuning has
> > octaves around four cents flat.
> > 
> > Another val regards 14 as a contorted version of 7 in the 5-
limit; 
> in
> > the 11-limit it is <14 22 32 39 48|. TM bases are
> > 
> > 5-limit: [25/24, 81/80]
> > 7-limit: [25/24, 49/48, 81/80]
> > 11-limit: [25/24, 33/32, 45/44, 49/48]
> > 
> > This involves decimal, meantone and jamesbond, and the TOP tuning 
of
> > the octave is now quite sharp, not flat; 1209.43 cents.
> > 
> > Other vals are possible; for instance a father version is <14 23 
33 
> 40
> > 49|. TM bases for this are
> > 
> > 5-limit: [16/15, 15625/13122];
> > 7-limit: [16/15, 50/49, 175/162];
> > 11-limit: [16/15, 22/21, 50/49, 175/162];
> 
> 
> ***Well, most of this is, admittedly, a bit over my head... but I 
> believe I saw that the 3-limit is reflected in 14-tET, at least 
> according to the Erlich chart, and I don't see it listed in the 
> above... (??)
> 
> Thanks!
> 
> JP


top of page bottom of page up down


Message: 10885

Date: Sun, 25 Apr 2004 22:33:29

Subject: Re: What's with 14

From: Joseph Pehrson

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

Yahoo groups: /tuning-math/message/10326 *

wrote:
> Hi Joseph.
> 
> As far as 5-limit goes, I have a suggestion.
> 
> Remember the big ET chart on Monz's equal temperament page:
> 
> Tonalsoft Encyclopaedia of Tuning - equal-temperament, (c) 2004 Tonalsoft Inc. *
> 
> It's the first chart there . . .
> 
> Now mouse over "zoom: 1" above the chart. If you can't see the 
yellow 
> triangular grid, mouse over "zoom: 1" under "negatives".
> 
> You'll see 14 occuring three times on that chart . . . once 
> overlapping 7.
> 
> These are three ways of "using" 14-equal in the 5-limit.
> 
> Look at how large the errors are of the basic 5-limit consonances.
> 
> Two of the instances of 14, it is true, are fairly close to 
the "just 
> perfect fifths - just perfect fourths" line.
> 
> But they both have thirds that are off by around 30-80 cents.
> 
> The other instance of 14 (the one at the top) doesn't fare much 
> better, and the "perfect fifth" is some 70 cents sharp!!
> 
> Am I making any sense?
> 
> -Paul
> 

***Yes, I can see that a line drawn through the two 14s is 
practically parallel to the just 3:2 line, the thirds, however, being 
way off...

I'd forgotten how nice these charts look in "negative" mode... I 
think certain features come out better that way, too...

JP


top of page bottom of page up down


Message: 10886

Date: Sun, 25 Apr 2004 22:34:42

Subject: Re: What's with 14

From: Joseph Pehrson

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

Yahoo groups: /tuning-math/message/10327 *

wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Now mouse over "zoom: 1" above the chart. If you can't see the 
> yellow 
> > triangular grid, mouse over "zoom: 1" under "negatives".
> 
> I'm getting a "not found" for these. Monz?


***Possibly you did what I first did, Gene, and actually *clicked on* 
the links, rather than just "mousing over" without clicking...  I got 
that error message first, too...

JP


top of page bottom of page up down


Message: 10887

Date: Sun, 25 Apr 2004 23:11:54

Subject: Re: What's with 14

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Now mouse over "zoom: 1" above the chart. If you can't see the 
> yellow 
> > triangular grid, mouse over "zoom: 1" under "negatives".
> 
> I'm getting a "not found" for these. Monz?

Are you clicking on them? You shouldn't. Just mouse over.


top of page bottom of page up down


Message: 10888

Date: Sun, 25 Apr 2004 23:45:58

Subject: another 'hanson' incidence

From: Paul Erlich

http://www.anaphoria.com/hrgm.PDF - Ok *

horagram 9 (p. 11)


top of page bottom of page up down


Message: 10890

Date: Mon, 26 Apr 2004 17:32:56

Subject: Re: More 270

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> > In case anyone is inspired to consider 270 in the light of a 13-
> limit
> > notation system, this might help: the TM basis is 676/675, 
> 1001/1000,
> > 1716/1715, 3025/3024 and 4096/4095. I used this to compute the 
first
> > twenty notes of a Fokker block, and used that to compute all the
> > 13-limit intervals with Tenney height less than a million up to 
> twenty
> > 270-et steps, with the following result:
> > 
> > 1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440, 540/539, 
> 364/363, 729/728, 325/324}
> > ...
> > 8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847}
> 
> This list misses a couple of intervals that Dave and I have found 
to 
> be valuable for notating 11-limit consonances, and we have chosen 
the 
> symbols for these two intervals to notate 1deg and 8deg of 311:
> 
> 1deg  5103:5120 (3^6*7:2^10*5), the 5:7 kleisma, symbol |(, notates 
> 7/5 and 10/7
> 8deg  45056:45927 (2^12*11:3^8*7), the 7:11 comma, symbol (|, 
notates 
> 11/7 and 14/11

Well, these have more than three digits in the numerator and 
denominator, so were explicitly excluded from Gene's list.


top of page bottom of page up down


Message: 10891

Date: Mon, 26 Apr 2004 19:41:19

Subject: Vanishing tratios

From: Paul Erlich

Before I finalize my paper, I'd like to explore the following idea.

What if I take a 3-term ratio ("tratio"?) and have it vanish?

Let's say 125:126:128.

So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 vanishes.

Any two of these three 'commas' of course would be enough to give you 
the result: in 7-limit multibreed/multival/wedgie form,

<<3, 0, -6, -7, -18, -14]],

the temperament formerly known as Tripletone.

Other examples would seem to be:

243:252:256 for 7-limit Blackwood
245:252:256 for Dominant Sevenths
343:350:360 for 7-limit Diminished
441:448:450 for Pajara

Does anyone know a way to find the simplest (lowest numbers) tratio 
for a given codimension-two temperament? How about for the 7-
limit 'linear' temperaments listed here:

Yahoo groups: /tuning-math/message/10266 *
?

And, salivating, I ask, is there a straightforward calculation to go 
from the vanishing tratio to the TOP error and/or complexity -- like 
there is for vanishing ratios in the codimension-1 case?

For a single vanishing ratio n:d, the TOP error is proportional to

log(n/d)/log(n*d),

and complexity [= 'L1 norm' of the wedgie] is proportional to

log(n*d).


top of page bottom of page up down


Message: 10892

Date: Mon, 26 Apr 2004 19:58:10

Subject: Re: Vanishing tratios

From: Paul Erlich

Since I just mentioned Negri on the MakeMicroMusic list, I think it's

672:675:686

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> Before I finalize my paper, I'd like to explore the following idea.
> 
> What if I take a 3-term ratio ("tratio"?) and have it vanish?
> 
> Let's say 125:126:128.
> 
> So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 vanishes.
> 
> Any two of these three 'commas' of course would be enough to give 
you 
> the result: in 7-limit multibreed/multival/wedgie form,
> 
> <<3, 0, -6, -7, -18, -14]],
> 
> the temperament formerly known as Tripletone.
> 
> Other examples would seem to be:
> 
> 243:252:256 for 7-limit Blackwood
> 245:252:256 for Dominant Sevenths
> 343:350:360 for 7-limit Diminished
> 441:448:450 for Pajara
> 
> Does anyone know a way to find the simplest (lowest numbers) tratio 
> for a given codimension-two temperament? How about for the 7-
> limit 'linear' temperaments listed here:
> 
> Yahoo groups: /tuning-math/message/10266 *
> ?
> 
> And, salivating, I ask, is there a straightforward calculation to 
go 
> from the vanishing tratio to the TOP error and/or complexity -- 
like 
> there is for vanishing ratios in the codimension-1 case?
> 
> For a single vanishing ratio n:d, the TOP error is proportional to
> 
> log(n/d)/log(n*d),
> 
> and complexity [= 'L1 norm' of the wedgie] is proportional to
> 
> log(n*d).


top of page bottom of page up down


Message: 10894

Date: Mon, 26 Apr 2004 21:33:26

Subject: Re: More 270

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
> <gdsecor@y...> 
> > wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...> wrote:
> > > > In case anyone is inspired to consider 270 in the light of a 
13-
> limit
> > > > notation system, this might help: the TM basis is 676/675, 
> 1001/1000,
> > > > 1716/1715, 3025/3024 and 4096/4095. I used this to compute 
the 
> first
> > > > twenty notes of a Fokker block, and used that to compute all 
the
> > > > 13-limit intervals with Tenney height less than a million up 
to 
> twenty
> > > > 270-et steps, with the following result:
> > > > 
> > > > 1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440, 
> 540/539, 364/363, 729/728, 325/324}
> > > > ...
> > > > 8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847}
> > > 
> > > This list misses a couple of intervals that Dave and I have 
found 
> to 
> > > be valuable for notating 11-limit consonances, and we have 
chosen 
> the 
> > > symbols for these two intervals to notate 1deg and 8deg of 311:
> > > 
> > > 1deg  5103:5120 (3^6*7:2^10*5), the 5:7 kleisma, symbol |(, 
> notates 7/5 and 10/7
> > > 8deg  45056:45927 (2^12*11:3^8*7), the 7:11 comma, symbol (|, 
> notates 
> > > 11/7 and 14/11
> > 
> > Well, these have more than three digits in the numerator and 
> > denominator, so were explicitly excluded from Gene's list.
> 
> But I assume that you and Gene would both agree that it is 
essential 
> that symbols be provided to notate 7/5, 10/7, 11/7, and 14/11 in 
JI.  
> So if these are the principal ratios that will be mapped to 131, 
139, 
> 176, and 94 degrees of 270, respectively, then the notational 
> semantics should take this into account.
> 
> For example, taking C as 1/1, if a G-flat of 132deg (arrived at by 
a 
> chain of fifths) is lowered by 1deg to arrive at 7/5, then the 
> interval of 1deg alteration will be 5103:5120.
> 
> The only other 15-limit consonances requiring a 1-deg alteration 
> (from tones in a chain of fifths) are 13/11 and 22/13.  Raising the 
A 
> of 204deg by 1deg gives 22/13, with the interval of 1deg alteration 
> as 351:352.  While the interval of alteration has smaller integers, 
> the ratios being notated are less simple.
> 
> I submit that the semantics of a notation should be determined by 
the 
> simplicity of the ratios for the resulting pitches rather than 
those 
> for the intervals of alteration, so I question the imposition of a 
3-
> digit cutoff for the latter.
> 
> --George

I think Gene was just using it for purposes of illustration and 
certainly not considering the question of notation in the same way 
you and Dave have been.


top of page bottom of page up down


Message: 10895

Date: Mon, 26 Apr 2004 22:39:35

Subject: Re: Vanishing tratios

From: Paul Erlich

7-limit Miracle -- 7168:7200:7203?

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> Since I just mentioned Negri on the MakeMicroMusic list, I think 
it's
> 
> 672:675:686
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > Before I finalize my paper, I'd like to explore the following 
idea.
> > 
> > What if I take a 3-term ratio ("tratio"?) and have it vanish?
> > 
> > Let's say 125:126:128.
> > 
> > So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 
vanishes.
> > 
> > Any two of these three 'commas' of course would be enough to give 
> you 
> > the result: in 7-limit multibreed/multival/wedgie form,
> > 
> > <<3, 0, -6, -7, -18, -14]],
> > 
> > the temperament formerly known as Tripletone.
> > 
> > Other examples would seem to be:
> > 
> > 243:252:256 for 7-limit Blackwood
> > 245:252:256 for Dominant Sevenths
> > 343:350:360 for 7-limit Diminished
> > 441:448:450 for Pajara
> > 
> > Does anyone know a way to find the simplest (lowest numbers) 
tratio 
> > for a given codimension-two temperament? How about for the 7-
> > limit 'linear' temperaments listed here:
> > 
> > Yahoo groups: /tuning-math/message/10266 *
> > ?
> > 
> > And, salivating, I ask, is there a straightforward calculation to 
> go 
> > from the vanishing tratio to the TOP error and/or complexity -- 
> like 
> > there is for vanishing ratios in the codimension-1 case?
> > 
> > For a single vanishing ratio n:d, the TOP error is proportional to
> > 
> > log(n/d)/log(n*d),
> > 
> > and complexity [= 'L1 norm' of the wedgie] is proportional to
> > 
> > log(n*d).


top of page bottom of page up down


Message: 10896

Date: Mon, 26 Apr 2004 22:44:36

Subject: Re: Vanishing tratios

From: Paul Erlich

5-limit 12-equal -- 625:640:648?

I'm going tratio-wild, but I have to go! :(



--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 7-limit Miracle -- 7168:7200:7203?
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > Since I just mentioned Negri on the MakeMicroMusic list, I think 
> it's
> > 
> > 672:675:686
> > 
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> > wrote:
> > > Before I finalize my paper, I'd like to explore the following 
> idea.
> > > 
> > > What if I take a 3-term ratio ("tratio"?) and have it vanish?
> > > 
> > > Let's say 125:126:128.
> > > 
> > > So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 
> vanishes.
> > > 
> > > Any two of these three 'commas' of course would be enough to 
give 
> > you 
> > > the result: in 7-limit multibreed/multival/wedgie form,
> > > 
> > > <<3, 0, -6, -7, -18, -14]],
> > > 
> > > the temperament formerly known as Tripletone.
> > > 
> > > Other examples would seem to be:
> > > 
> > > 243:252:256 for 7-limit Blackwood
> > > 245:252:256 for Dominant Sevenths
> > > 343:350:360 for 7-limit Diminished
> > > 441:448:450 for Pajara
> > > 
> > > Does anyone know a way to find the simplest (lowest numbers) 
> tratio 
> > > for a given codimension-two temperament? How about for the 7-
> > > limit 'linear' temperaments listed here:
> > > 
> > > Yahoo groups: /tuning-math/message/10266 *
> > > ?
> > > 
> > > And, salivating, I ask, is there a straightforward calculation 
to 
> > go 
> > > from the vanishing tratio to the TOP error and/or complexity -- 
> > like 
> > > there is for vanishing ratios in the codimension-1 case?
> > > 
> > > For a single vanishing ratio n:d, the TOP error is proportional 
to
> > > 
> > > log(n/d)/log(n*d),
> > > 
> > > and complexity [= 'L1 norm' of the wedgie] is proportional to
> > > 
> > > log(n*d).


top of page bottom of page up down


Message: 10897

Date: Mon, 26 Apr 2004 02:34:00

Subject: Re: another 'hanson' incidence

From: Carl Lumma

> http://www.anaphoria.com/hrgm.PDF - Ok *
> 
> horagram 9 (p. 11)

What about it?

-Carl



________________________________________________________________________
________________________________________________________________________



------------------------------------------------------------------------
Yahoo! Groups Links

<*> To visit your group on the web, go to:
     Yahoo groups: /tuning-math/ *

<*> To unsubscribe from this group, send an email to:
     tuning-math-unsubscribe@xxxxxxxxxxx.xxx

<*> Your use of Yahoo! Groups is subject to:
     Yahoo! Terms of Service *


top of page bottom of page up down


Message: 10898

Date: Tue, 27 Apr 2004 19:34:05

Subject: Re: Vanishing tratios

From: Paul Erlich

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

> 5-limit 12-equal -- 625:640:648?
> 
> I'm going tratio-wild, but I have to go! :(

Let's define the function

weird(a,b,c) = a*b*c/gcd(a,b)/gcd(a,c)/gcd(a,b)

5-limit ETs and lowest-weird tratios (by inspection)

ET........tratio............weird
03-equal: 45:48:50......... 3600
04-equal: 24:25:27......... 5400
05-equal: 75:80:81......... 32400
07-equal: 384:400:405...... 259200
("""""""""240:243:250...... 486000)
09-equal: 125:128:135...... 432000
10-equal: 729:768:800...... 4665600
12-equal: 625:640:648...... 6480000
15-equal: 243:250:256...... 7776000
16-equal: 3072:3125:3240... 259200000 
19-equal: 15360:15552:15625 3888000000  
22-equal: 6075:6144:6250... 1555200000

The monotonic pattern seems to break here. Did I miss any lower-weird 
and/or simpler tratios?


top of page bottom of page up

Previous Next

10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500 10550 10600 10650 10700 10750 10800 10850 10900 10950

10850 - 10875 -

top of page