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Message: 1675 - Contents - Hide Contents

Date: Sat, 29 Sep 2001 20:01:07

Subject: Re: Commas and the ABC conjecture

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., graham@m... wrote:


>> If b is a 
>> fixed small integer, it follows from a very deep and recent 
>> conjecture in elementary number theory called the ABC conjecture 
that 
>> in the p-limit, there will be only a finite number of ratios 
(a+b)/a. 


> Can you define the p-limit?


What I mean by p-limit is that all of the intervals on the list can 
be factored by primes less than or equal to p. I was looking at 
ratios (a+b)/a in the particularly interesting case where b=1. The 
first list of 7-limit epimoric ratios are all such numbers which 
require a 7 to factor, the ones not requiring a 7 having already 
appeared on a previous list. In the second list I had only those 
ratios with a square numerator, but searched much farther.


>> 7-limit
>> 
>> 36/35, 49/48, 64/63, 225/224, 2401/2400
> 

> These are linearly dependent.  I think 
49/48*63/64=225/224*2401/2400.


Well, that's certainly not true. :) But it's equally true there must 
be some dependency, since we have five numbers and only four primes.

  So 

> the first four can be inverted to give
> 
> [10 12  9  5]
> [16 19 14  8]
> [23 28 21 12]
> [28 34 25 14]
> 
> which I think means we can define either Paultone or meantone from 
these 
> unison vectors.


Paultone? Are those 15, 22, 27?


> [ 46 26  8 -14 15]
> [ 73 41 13 -22 24]
> [107 60 19 -32 35]
> [129 73 23 -39 42]
> [159 90 28 -48 52]
> 
> So all bar the largest define 46-equal.


Anyone want a 46 PB? This would be a good way to get one.

Message: 1676 - Contents - Hide Contents

Date: Sat, 29 Sep 2001 20:30:53

Subject: Re: Commas and the ABC conjecture

From: Paul Erlich

--- In tuning-math@y..., 
genewardsmith@j... wrote:

> --- In tuning-math@y..., graham@m... wrote:
> 

>>> If b is a 
>>> fixed small integer, it follows from a very deep and recent 
>>> conjecture in elementary number theory called the ABC conjecture 
> that 

>>> in the p-limit, there will be only a finite number of ratios 
> (a+b)/a. 


John Chalmers published a list of 
all the superparticular ratios, 
with numbers up to ten billion or 
something, in every limit up to 
the 23-limit, in Xenharmonikon 17 
(the same issue my TTTTTT paper 
is in). You've reproduced the first 
few entries. By the way, Rami 
Vitale responded to you on the 
main list . . . you should check it 
out.

> 

>> Can you define the p-limit?
> 

> What I mean by p-limit is that all of the intervals on the list can 
> be factored by primes less than or equal to p. I was looking at 
> ratios (a+b)/a in the particularly interesting case where b=1. The 
> first list of 7-limit epimoric ratios are all such numbers which 
> require a 7 to factor, the ones not requiring a 7 having already 
> appeared on a previous list. In the second list I had only those 
> ratios with a square numerator, but searched much farther.
> 
>>> 7-limit
>>> 

>>> 36/35, 49/48, 64/63, 225/224, 2401/2400
>> 

>> These are linearly dependent.  I think 
> 49/48*63/64=225/224*2401/2400.
> 

> Well, that's certainly not true. :) But it's equally true there must 
> be some dependency, since we have five numbers and only four primes.
> 
>   So 

>> the first four can be inverted to give
>> 
>> [10 12  9  5]
>> [16 19 14  8]
>> [23 28 21 12]
>> [28 34 25 14]
>> 
>> which I think means we can define either Paultone or meantone from 
> these 
>> unison vectors.
> 

> Paultone? Are those 15, 22, 27?


Hmm . . . Paultone means 
commatic unison vectors of 50:49, 
64:63, and 225:224; and chromatic 
unison vectors of 25:24, 28:27, and 
49:48. So what set of ETs does it 
correspond to? 10, 22, and . . . ?


> 

>> [ 46 26  8 -14 15]
>> [ 73 41 13 -22 24]
>> [107 60 19 -32 35]
>> [129 73 23 -39 42]
>> [159 90 28 -48 52]
>> 
>> So all bar the largest define 46-equal.
> 

> Anyone want a 46 PB? This would be a good way to get one.


Awesome! I'm thinking that 22, 31, 
46, and 72 would be a good choice 
of four cardinalities to base my 
future instruments on. My 
present plan for the four 
fingerboards that come with the 
Rankin system is 22-tET, 31-tET, 
22-out-of-46-tET, and 31-out-of-
72-tET.

Message: 1677 - Contents - Hide Contents

Date: Sat, 29 Sep 2001 20:53:47

Subject: Re: Digest Number 124

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., jon wild <wild@f...> wrote:


>>> 16/15, 25/24, 81/80


>> And these are the usual intervals for defining a 5-limit diatonic
>> scale!


> I know where these intervals crop up in comparing combinations of
> intervals in the usual J.I.  scale to one another, but how, 
precisely, do
> the three intervals "define" the scale? 


These three intervals and 9/8, 10/9, 16/15 are very closely related. 
We have 10/9 = 25/24 16/15 and 9/8 = 81/80 10/9 = 81/80 25/24 16/15.
Looked at in matrix form, we have an upper diagonal transformation 
matrix:

[1 1 1]
[0 1 1]
[0 0 1]

Transforming the basis (81/80)^a (25/24)^b (16/15)^c, which 
represents everything in the 5-limit, to another basis 
(9/8)^a' (10/9)^b' (16/15)^c' which does the same (as we can see by 
the fact that the transformation is unimodular, with determinant 1.)
The first system has jargon wherein 81/80 and 25/24 are "commatic 
unison vectors" and 16/15 is a "chromatic unison vector" in a 
situation where we are seeking a 7-note "periodiity block" scale; the 
7 can be deduced from

    [-4  4 -1]
det [-3 -1  2] = 7a + 11b + 16c,
    [a   b  c]

where we have replaced the largest of the three "unison" intervals 
with indeterminates a, b, c. The result represents the 7-equal 
division of the octave, or "7-tet = 7-et"; it has 7 steps in an 
octave, takes 11 steps to approximate 3 and 16 steps to approximate 
5. The scales we get are approximations to this 7-et, and hence are 
reasonable as scales. We can always pass in this way from a step 
system of representation like 9/8, 10/9, 16/15, to a comma system 
like 16/15, 25/24, 81/80, and back; the comma system is more useful 
in a number of ways and tends to be taken as fundamental here, though 
most people focus on the steps instead.

Are they like a "basis" for

> 5-limit diatonic space, and I could just as well take three other 
linearly
> independent intervals like 9:8 10:9 4:3 to define the scale, in a 
similar
> way to however Gene's three do?


No; to say the scale is "defined" by the fact that 16/15 is 
a "chromatic unison vector" (something of a misnomer, I fear) and 
81/80 and 25/24 are "commatic unison vectors" tells us how to 
construct "periodicity block" scales in a way even more tightly 
controlled than "defining" a system by requiring it to be constructed 
from steps of size 9/8, 10/9 and 16/15; the two ideas are closely 
related but are not the same and do not always lead to the same 
results.


> 
>>> 11-limit
>>> 

>>> 100/99, 121/120, 441/440, 3025/3024, 9801/9800
>> 
>> These give
>> 

>> [ 46 26  8 -14 15]
>> [ 73 41 13 -22 24]
>> [107 60 19 -32 35]
>> [129 73 23 -39 42]
>> [159 90 28 -48 52]
>> 
>> So all bar the largest define 46-equal.


> How do you obtain the matrix from those intervals?


He factored the intervals I gave into primes, wrote them out as row 
vectors (so that 100/99 = 2^2 3^(-2) 5^2 7^0 11^(-1) becomes 
[2 -2  2  0 -1]) and then took the matrix inverse of the resulting 
square matrix. The columns of this matrix can be thought of as ets, 
applied to the rows they are the ets that represent one of the 
intervals by a step (downward in one case, 14) and the rest by a 
unison.

 What, then, can you

> read off of it (i.e. what does it tell you)?


For one thing, we can look at the linear span of subsets of the 
columns, which gives us ets with certain properties, having some 
elements as commas in common--for instance 46+26=72, and adding the 
first two columns together gives us the 72-et; like 46 and 26 it will 
have 441/440, 3025/3024 and 9801/9800 in its kernel (or nullspace.) 
This tells us we can "temper out" these intervals, leading for 
instance to 46 and 26 tone subsets of the 72 et, and tells us how to 
construct such temperings. We can also immediately see how to 
construct a 46 tone periodicity block, other blocks and tempered 
scales, and so forth.

 What other operations can you

> do with the matrix?


Stick around and I'm sure you'll see more! Maybe I'll do that 72-et 
calculation for Joseph Pehrson's benefit.

Message: 1678 - Contents - Hide Contents

Date: Sat, 29 Sep 2001 21:38:23

Subject: Re: Commas and the ABC conjecture

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:


> John Chalmers published a list of 
> all the superparticular ratios, 
> with numbers up to ten billion or 
> something, in every limit up to 
> the 23-limit, in Xenharmonikon 17 
> (the same issue my TTTTTT paper 
> is in). You've reproduced the first 
> few entries. 


Wonderful news! That means I can publish in mathematical forums, and 
cite this as proof that the question is, in fact, musically 
interesting. 

By the way, Rami 

> Vitale responded to you on the 
> main list . . . you should check it 
> out.


I cancelled that about a minute after posting it, but I'm not 
surprised.


>> Paultone? Are those 15, 22, 27?


> Hmm . . . Paultone means 
> commatic unison vectors of 50:49, 
> 64:63, and 225:224; and chromatic 
> unison vectors of 25:24, 28:27, and 
> 49:48. So what set of ETs does it 
> correspond to? 10, 22, and . . . ?


The dual group to the one generated by 50/49, 64/63 and 225/224 is 
generated by h2 and h10, and contains h12 and h22; that's about the 
size of it--we get two elements since we have four primes and a 
linear dependency 50/49 = 225/224 64/63.

Message: 1679 - Contents - Hide Contents

Date: Sat, 29 Sep 2001 23:24 +0

Subject: Re: Digest Number 124

From: graham@xxxxxxxxxx.xx.xx

jon wild wrote:


> I know where these intervals crop up in comparing combinations of
> intervals in the usual J.I.  scale to one another, but how, precisely, 
> do
> the three intervals "define" the scale? Are they like a "basis" for
> 5-limit diatonic space, and I could just as well take three other 
> linearly
> independent intervals like 9:8 10:9 4:3 to define the scale, in a 
> similar
> way to however Gene's three do?


You could use 9:8, 10:9 and 16:15.  I'd prefer not to include 4:3 because 
it means you can't get the diatonic scale by only adding intervals.  
(That's *only* adding, not subtracting or inverting and then adding.)  
(Also means "adding intervals" is multiplying ratios.)  The thing about 
16:15, 25:24 and 81:80 is that you have a "diatonic" step, a "chromatic" 
step and a comma.


>>> 11-limit
>>> 
>>> 100/99, 121/120, 441/440, 3025/3024, 9801/9800
>> 
>> These give
>> 

>> [ 46 26  8 -14 15]
>> [ 73 41 13 -22 24]
>> [107 60 19 -32 35]
>> [129 73 23 -39 42]
>> [159 90 28 -48 52]
>> 
>> So all bar the largest define 46-equal.
> 

> How do you obtain the matrix from those intervals? What, then, can you
> read off of it (i.e. what does it tell you)? What other operations can 
> you
> do with the matrix?


First step is to convert the intervals to matrix form.  100/99 is [2 -2 2 
0 -1], 121:120 is (-3 -1 -1 0 2] and so on.  The full matrix is

[ 2 -2  2  0 -1]
[-3 -1 -1  0  2]
[-3  2 -1  2 -1]
[-4 -3  2 -1  2]
[-3  4 -2 -2  2]

Then you take the matrix inverse.  The first column of the result defines 
46-equal.  So the top entry tells you there are 46 steps to the octave.  
The next one that there are 73 steps to the approximation of 3:1.  Then 
that there are 107 steps to the approximation of 5:1, and so on.  This is 
the temperament you get by tempering out all the unison vectors except the 
first one, [2 -2 2 0 -1] or 100/99.

The second column defines 26-equal.  You get that by tempering out 
everything except 121:120.  Temper out everything except 121:120 and 
100:99 and you have a linear temperament covering 26- and 46-equal, and 
therefore 72-equal (because 26+46=72).  A while back on the big list, I 
probably suggested a notation for this temperament, but I can't remember 
myself.

Oh, and it's second on my list of 11-limit temperaments

18/59

basis:
(0.5, 0.152673565758)

mapping:
([2, 0], ([5, 8, 5, 6], [-6, -11, 2, 3]))

primeApprox:
([72, 46], [(114, 73), (167, 107), (202, 129), (249, 159)])

highest interval width: 15
notes required: 31
highest error: 0.002008  (2.409 cents)


                            Graham

Message: 1680 - Contents - Hide Contents

Date: Sat, 29 Sep 2001 00:31:41

Subject: Commas and the ABC conjecture

From: genewardsmith@xxxx.xxx

I had thought of examining the Farey sequence for useful intervals, 
but it occurred to me that the same thing may be accomplised more 
easily simply by looking at ratios belonging to a p-limit. If b is a 
fixed small integer, it follows from a very deep and recent 
conjecture in elementary number theory called the ABC conjecture that 
in the p-limit, there will be only a finite number of ratios (a+b)/a. 
From this we get the following list:

3-limit

2, 3/2, 4/3, 9/8

5-limit

5/4, 6/5, 10/9, 16/15, 25/24, 81/80

7-limit

7/6, 8/7, 15/14, 21/20, 28/27, 36/35, 49/48, 50/49, 64/63, 126/125, 
225/224, 2401/2400, 4375/4374

11-limit

11/10, 12/11, 22/21, 33/32, 45/44, 55/54, 56/55, 99/98, 100/99, 
121/120, 176/175, 243/242, 385/384, 441/440, 540/539, 3025/3024, 
9801/9800

There is no guarantee that these lists are complete, or even finite, 
but I went out much past the final value in all cases without finding 
another one. If the ABC conjecture is true, these lists are finite, 
and an effective form of it would tell us when they are complete.

If we look only at numerators which are squares, which cover a lot of 
these cases, it is easy to carry this out farther; these lists will 
at least contain all the jumping jacks:

3-limit

4/3, 9/8

5-limit

16/15, 25/24, 81/80

7-limit

36/35, 49/48, 64/63, 225/224, 2401/2400

11-limit

100/99, 121/120, 441/440, 3025/3024, 9801/9800

13-limit

144/143, 169/168, 196/195, 625/624, 676/675, 729/728, 4096/4095, 
4225/4224, 123201/123200

17-limit

256/255, 289/288, 1089/1088, 1156/1155, 1225/1224, 2500/2499, 
2601/2600, 14400/14399, 28561/28560, 194481/194480

19-limit

324/323, 361/360, 400/399, 1521/1520, 3136/3135, 5776/5775, 
5929/5928, 23409/23408, 28900/28899, 43681/43680, 104976/104975, 
5909761/5909760

23-limit

484/483, 529/528, 576/575, 2025/2024, 4761/4760, 8281/8280, 
25921/25920, 43264/43263, 104329/104328, 152881/152880, 
4096576/4096575

I find it fascinating that deep mathematical conjectures of number 
theory such as ABC and the Riemann hypothesis can have some sort of 
connection with music theory.

The most interesting intervals on these lists are the larger ones, 
and these are a fruitful place to look to find notations, scales and 
the like. For instance, the four smallest 7-limit intervals on the 
list are 126/125, 225/224, 2401/2400, and 4375/4374, and this is the 
basis for the notation [h72, h27, -h19, h31]; from which, for 
instance, we could contruct a 72-note PB, or tempered PBs such as the 
miracle system of h72 and h31 and the meantone system of h31 and h19.

I posted this over on tuning by mistake; it case my cancel doesn't 
work it probably should be responded to here.

Message: 1681 - Contents - Hide Contents

Date: Sat, 29 Sep 2001 12:15 +0

Subject: Re: Commas and the ABC conjecture

From: graham@xxxxxxxxxx.xx.xx

Gene wrote:


> I had thought of examining the Farey sequence for useful intervals, 
> but it occurred to me that the same thing may be accomplised more 
> easily simply by looking at ratios belonging to a p-limit. If b is a 
> fixed small integer, it follows from a very deep and recent 
> conjecture in elementary number theory called the ABC conjecture that 
> in the p-limit, there will be only a finite number of ratios (a+b)/a. 


Can you define the p-limit?



> If we look only at numerators which are squares, which cover a lot of 
> these cases, it is easy to carry this out farther; these lists will 
> at least contain all the jumping jacks:
> 
> 3-limit
> 
> 4/3, 9/8


So these define a Pythagorean system


> 5-limit
> 
> 16/15, 25/24, 81/80


And these are the usual intervals for defining a 5-limit diatonic scale!



> 7-limit
> 
> 36/35, 49/48, 64/63, 225/224, 2401/2400


These are linearly dependent.  I think 49/48*63/64=225/224*2401/2400.  So 
the first four can be inverted to give

[10 12  9  5]
[16 19 14  8]
[23 28 21 12]
[28 34 25 14]

which I think means we can define either Paultone or meantone from these 
unison vectors.



> 11-limit
> 
> 100/99, 121/120, 441/440, 3025/3024, 9801/9800

These give


[ 46 26  8 -14 15]
[ 73 41 13 -22 24]
[107 60 19 -32 35]
[129 73 23 -39 42]
[159 90 28 -48 52]

So all bar the largest define 46-equal.


I haven't looked at the higher limits.



                      Graham

Message: 1684 - Contents - Hide Contents

Date: Sun, 30 Sep 2001 12:59:57

Subject: Re: catching up (was: Digest Number 125)

From: monz

Wow... this post lets me know that there's a lot of stuff
happening on this list which I would find exciting, but
I've missed out on it while traveling.

Is there any sort of coherent index to this, so that I can
catch up fairly easily?  Can someone post a list of posts?
Thanks.


-monz


----- Original Message -----
From: jon wild <wild@xxx.xxxxxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Sunday, September 30, 2001 12:39 PM
Subject: Re: [tuning-math] Digest Number 125



>
> Thanks to Gene and Graham for your explanations, and Paul for your offer
> (I emailed you back). I'm still catching up, and I tried to come up with
> an example to start working through for myself. Let's say I wanted a scale
> where four times 13:1 was once 7:1, that is I want the "comma" 13^4 /
> 7*2^12, or 28561/28672, to vanish.  And I further want two 13s to be three
> 7s, that is I want the comma 338/343 to disappear. So I get the following
> matrix:
>
> [4 -1]
> [2 -3]
>
> whose determinant is 10, so I'm looking at a periodicity block of 10
> notes. And indeed 10-tet has good approximations of 13:8 and 7:4--7/10ths
> and 8/10ths of an octave respectively--and the congruences in the matrix
> are true, 4* 7/10 == 8/10  and  2* 7/10 == 3*8/10.
>
> If I understand correctly I can choose from many possible 10-note subsets
> of the Z^2 lattice to construct my just scale, as long as I don't pick any
> pairs of notes separated by my unison vectors (4, -1) or (2, -3). But this
> can still lead to different just scales, so I imagine I can't say that the
> commas 28561/28672 and 338/343 "define" whichever region of the lattice I
> choose as my just scale.
>
> Right so far?  --Jon








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Message: 1687 - Contents - Hide Contents

Date: Sun, 30 Sep 2001 05:06:38

Subject: Some "ABC good" intervals

From: genewardsmith@xxxx.xxx

I found a web page, ABC-Ratios >1.4 * [with cont.]  (Wayb.), with 
a list of all 148 known "good" ABCs according to the definition you 
will find there of "good" (there are others.) Cutting things off at 
the 53-limit and keeping A small, I found the following, with names 
if Manuel gives one on his web page:

(5) 4375/4374 = 5^4*7 / 2*3^7 "ragisma" (ragisma??)

(17) 32805/32768 = 3^8*5 / 2^15 = shisma

(19) 301327048/301327047 = 2^3*11*23*53^3 / 3^16*7

(30) 2401/2400 = 7^4 / 2^5*3*5^2 = "breedsma", presumably after our 
very own Graham Breed. Congratulations, Graham!

(38) 63927525376/63927525375 = 2^13*11^4*13 *41 /  3^3*5^3*7^7*23

(47) 512001/512000 = 3^5*7^2*43 / 2^12 * 5^3 (How's 2^12*5^3 / 
3^5*7^2 for an approximation to 43 in the 7-limit? It's all of 1/296 
cents flat.)

(82) 128/125 = 2^7/5^3 = enharmonic diesis

(131) 282192773751/282192773120 = 3^3*7^10*37 / 2^26*5*29^2

Numero Uno on the list? 6436343/6436341 = 23^5 / 3^10*109; remember 
that when you work in the 109-limit. Of course, many of these are not 
best thought of in terms of the p-limit, but in the set of numbers 
generated by their prime factors, and toss in 2.

Message: 1688 - Contents - Hide Contents

Date: Sun, 30 Sep 2001 07:36:42

Subject: Baker's theorem and commas

From: genewardsmith@xxxx.xxx

If we have A + B = C for a fixed small A, and large B and C, and we 
look at the ratio C/B = (A+B)/B = 1 + A/B, we have to very close 
approximation ln C/B = A/B. If C/B = p1^b1 * ... * pk^bk, 
then B ~ C ~ (p1^|b1| * ... * pk^|bk|)^(1/2) < pk^(k*b/2), where 
b = max_i |b_i|. Hence ln C/B ~ A/B < A pk^(-k*b/2). Baker's theorem 
on the other hand tells us that for some (very large) constant N, we 
have ln C/B > b^(-N). Hence b cannot grow without limit, since 
ln(A pk^(-k*b/2)) = ln A - (k/2) ln(pk) b is linear in b, whereas 
ln(b^(-N)) = -N ln(b) is linear in log b. For large enough b, 
(k/2) ln(pk) b - ln A > N ln(b), violating the bound of Baker's 
theorem; so b is bounded. This means the exponents b_i are bounded, 
and since they take integer values, there can be only a finite number 
of them.

This actually gives an effective bound, but since Baker's constant N 
is absurdly large it isn't much good in practice.

Message: 1689 - Contents - Hide Contents

Date: Sun, 30 Sep 2001 21:19:03

Subject: Re: catching up (was: Digest Number 125)

From: monz

----- Original Message ----- 
From: <genewardsmith@xxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Sunday, September 30, 2001 5:54 PM
Subject: [tuning-math] Re: catching up (was: Digest Number 125)



> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 

>> Is there any sort of coherent index to this, so that I can
>> catch up fairly easily?  Can someone post a list of posts?
>> Thanks.
> 

> So far as my contrutions go, I plan on cleaning them up and putting 
> up a web page.


Excellent.  Thanks.


-monz


 



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Message: 1690 - Contents - Hide Contents

Date: Sun, 30 Sep 2001 07:57:12

Subject: Tetrachordality and Scala (was Re: semi-periodic scales)

From: Carl Lumma

I wrote...


>> n
>> sum  S(i)*S((i+k) mod n)
>> i=1
>> ------------------------
>> n
>> sum  S(i)*S(i)
>> i=1
> 

>This is exactly what we want.  Except, instead of comparing
>the scale and its rotation to a degree near a 3:2, we compare
>it with its transposition by an exact 3:2, simply taking the
>lowest value for all rotations of the transposed version,


Did I say lowest value?  I meant highest value, or 'best'
value.  =1 means perfect correlation.  Actually, isn't it
possible to get >1 ??

-Carl

Message: 1691 - Contents - Hide Contents

Date: Mon, 01 Oct 2001 18:23:36

Subject: Re: Digest Number 124

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:


> The first system has jargon wherein 81/80 and 25/24 are "commatic 
> unison vectors" and 16/15 is a "chromatic unison vector" in a 
> situation where we are seeking a 7-note "periodiity block" scale;


Something's wrong here . . . in a 7-tone PB, specificially the 
diatonic scale, 81:80 is the commatic unison vector, 25:24 is a 
chromatic unison vector, and 16:15 is not a unison vector at all, but 
a "step vector".


> No; to say the scale is "defined" by the fact that 16/15 is 
> a "chromatic unison vector" (something of a misnomer, I fear) and 
> 81/80 and 25/24 are "commatic unison vectors"


Misnomer because it's incorrect!

Message: 1692 - Contents - Hide Contents

Date: Mon, 01 Oct 2001 18:28:04

Subject: Re: Some "ABC good" intervals

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> I found a web page, ABC-Ratios >1.4 * [with cont.]  (Wayb.), 
with 
> a list of all 148 known "good" ABCs according to the definition you 
> will find there of "good" (there are others.) 


I don't see a definition there of "good", and most of these are not 
superparticular ratios at all . . . so what, really, is the ABC 
conjecture?

Message: 1693 - Contents - Hide Contents

Date: Mon, 01 Oct 2001 18:31:26

Subject: Re: Pauls fingerboard kit

From: Paul Erlich

--- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:


> What does 46 have that everyone seems gaga about?
> 
> Is your 22-out-of-46 a sort of "nontempered" version of
> the PB that you see in 22tet? 


Sort of . . . 22-out-of-46 is a "srutar", for Indian-type music. The 
diaschisma 2048:2025 must vanish. It doesn't in 41.


> Is the 31-of-72 going to be Canasta?


Yes, with split frets, so that standard tuning can be used for the 
open strings.

Message: 1694 - Contents - Hide Contents

Date: Mon, 1 Oct 2001 20:15 +01

Subject: 46 (was Re: Pauls fingerboard kit)

From: graham@xxxxxxxxxx.xx.xx

Bob Valentine wrote:


> What does 46 have that everyone seems gaga about? Is 46
> better at 13 (recognizing that it doesn't do 11 or 13
> uniquely, it does keep them off of each other, which 
> 41 does not).


46 is almost 15-limit consistent, but tends to be overshadowed by its 
neighbours 41 and 58.  Well, 58 isn't casting much of a shadow right now, 
but that's all going to change.  So, every now and then we have to remind 
ourselves that 46 is quite consonant as well, although those of us who 
care about that are probably going back to 41 or 58 or 31 or 72.

Interestingly, fudging my temperament finding script so that 46 gets 
included with the 15-limit ETs does bring 29+46 well into the rankings.  I 
wonder what else we could be missing by insisting on consistency.  
Probably no killer temperaments, but maybe some keyboard mappings.  For 
example, a meantone-31 can do the 11-limit with 19 notes, but it doesn't 
make the list because 31 is the only consistent temperament that works 
with it.  Also it isn't unique, but you can't have everything.


                         Graham

Message: 1695 - Contents - Hide Contents

Date: Mon, 01 Oct 2001 00:02:21

Subject: Re: Digest Number 125

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., jon wild <wild@f...> wrote:


> If I understand correctly I can choose from many possible 10-note 
subsets
> of the Z^2 lattice to construct my just scale, as long as I don't 
pick any
> pairs of notes separated by my unison vectors (4, -1) or (2, -3). 
But this
> can still lead to different just scales, so I imagine I can't say 
that the
> commas 28561/28672 and 338/343 "define" whichever region of the 
lattice I
> choose as my just scale.

> Right so far?  --Jon


It's much more restrictive than that--the notes can't be spread all 
over the lattice, but must be bunched together. Let's construct your 
example using my definition of block.

(1) First check that we have a valid basis; 
(28672/28561 * 343/338)^10 = 1.204 < 2, so we are in very valid 
territory.

(2) Form the matrix <2, 282672/28561, 343/338> and invert it:

[1  0  0]^(-1)   [1.0  0.0 0.0]
[12 1 -4]      = [2.8 -0.2 0.4]
[1  3 -2]        [3.7 -0.3 0.1]

(3) Define a norm on q = 2^a 7^b 13^c by setting

||q|| = max(|a + 2.8*b + 3.7*c|, |-.2*b-.3*c|, |.4*b+.1*c|

This makes the numbers 2^a 7^b 13^c into a three-dimensional lattice; 
we can measure the distance between lattice points p and q by

d(p, q) = ||p/q||.

(4) Find sets such that the diameter is less than 1. This means for 
s1, s2 in S, we have d(s1, d2) < 1.

Without loss of generality, we can assume that the points are 
contained in ||q|| <= 1/2. We may also use integer arithmetic if we 
choose, by rescaling by a factor of 10; this would mean ||q||<=5, 
where now v1=10*a+28*b+378c, v2=-2*b-3*c and v3=4*b+c are integers 
less than or equal to 5 in absolute value. If we look at triples 
[v1,v2,v3] less than 5 in absolute and which correspond to integer 
[a,b,c], we find nine candidates:

169/224 [-4 -4 -2]
13/16   [-3 -3  1]
7/8     [-2 -2  4]
13/14   [-1 -1 -3]
1       [ 0  0  0]
14/13   [ 1  1  3]
8/7     [ 2  2 -4]
16/13   [ 3  3 -1]
224/169 [ 4  4  2]

We now have eight possibilities to complete the scale, [+-5 +-5 +-5]; 
however modulo transposition this becomes four, and modulo inversion 
as well, two; hence there really aren't vast numbers of possibilities.
Our eight choices are 91/64,64/91,91/128,128/91,2197/1568,1568/2197,
3136/2197, 2197/3168. If we pick 91/64, [5 -5  5], we have a block 
corresponding to your choice of commas--enjoy!

Message: 1696 - Contents - Hide Contents

Date: Mon, 01 Oct 2001 20:06:21

Subject: Re: Digest Number 125

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., jon wild <wild@f...> wrote:
> 

>> If I understand correctly I can choose from many possible 10-note 
> subsets

>> of the Z^2 lattice to construct my just scale, as long as I don't 
> pick any

>> pairs of notes separated by my unison vectors (4, -1) or (2, -3). 
> But this

>> can still lead to different just scales, so I imagine I can't say 
> that the

>> commas 28561/28672 and 338/343 "define" whichever region of the 
> lattice I

>> choose as my just scale.
> 
>> Right so far?  --Jon
> 

> It's much more restrictive than that--the notes can't be spread all 
> over the lattice, but must be bunched together.


Well, in the diatonic case, you're free to transpose any note by any 
number of 81:80s, since this is the commatic unison vector and is 
ignored in diatonic notation (I object to the idea of there being a 
JI diatonic scale of fixed specification). As for the 25:24, this is 
the chromatic unison vector, so transposing notes by it will lead to 
chromatically altered diatonic scales (still diatonic, as no 
chromatic semitones will be present between notes of the scale). My 
construction for the unaltered diatonic scale is as follows: divide 
the 5-limit lattice into "bands" or "strips" which run parallel to 
the commatic UV (81:80), and which are exactly wide enough to allow 
one chromatic UV (25:24) to reach from one edge of the strip to the 
other. Then, if you stay within one strip, simply choose any 7 notes 
so that no pair is seperated by 81:80 or a power thereof (i.e. a 
multiple of the syntonic comma).

Message: 1697 - Contents - Hide Contents

Date: Mon, 01 Oct 2001 00:54:30

Subject: Re: catching up (was: Digest Number 125)

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:


> Is there any sort of coherent index to this, so that I can
> catch up fairly easily?  Can someone post a list of posts?
> Thanks.


So far as my contrutions go, I plan on cleaning them up and putting 
up a web page.

Message: 1698 - Contents - Hide Contents

Date: Mon, 01 Oct 2001 20:33:16

Subject: 46 (was Re: Pauls fingerboard kit)

From: Carl Lumma

Graham wrote...


>46 is almost 15-limit consistent, but tends to be overshadowed by
>its neighbours 41 and 58.  Well, 58 isn't casting much of a shadow
>right now, but that's all going to change.


What's going to change it?

-C.

Message: 1699 - Contents - Hide Contents

Date: Mon, 1 Oct 2001 22:09 +01

Subject: Re: 46 (was Re: Pauls fingerboard kit)

From: graham@xxxxxxxxxx.xx.xx

Carl wrote:


> Graham wrote...
> 

>> 46 is almost 15-limit consistent, but tends to be overshadowed by
>> its neighbours 41 and 58.  Well, 58 isn't casting much of a shadow
>> right now, but that's all going to change.
> 

> What's going to change it?


Oh, it's bound to change.  You can't keep a temperament like that down for 
long.  Wait until I get my hands on a ZTar ...


                    Graham

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