Re: converting equal divisions into ratio sequences

[Paul Erlich:]
>But isn't the "outer loop" Canright is referring to equivalent to CF?
I would hope so, since the CF convergents are the only ones for which
one is guaranteed that there is no simpler fraction that yields a
better approximation.

This is (perhaps) another way to see equal divisions of the octave as
sequences of ratios.

If "a" is taken as an equal division of the octave and "b" is taken as
the fraction of "a," D:N=(y-b):(y+b) where y=(x*a) and x=2.9.

Here's an example using 12 (and its divisors):

1e = 1/1 @:
3:3 3:6

2e = 1/1, 7/5 @:
6:6 6:12
7:5 7:10

3e = 1/1, 5/4 @:
9:9 9:18
8:10 10:16

4e = 1/1, 13/11, 7/5 @:
12:12 12:24
11:13 13:22
10:14 14:20

6e = 1/1, 9/8, 19/15, 10/7 @:
17:17 17:34
16:18 18:32
15:19 19:30
14:20 20:28

12e = 1/1, 18/17, 37/33, 19/16, 39/31, 4/3, 41/29 @:
35:35 35:70
34:36 36:68
33:37 37:66
32:38 38:64
31:39 39:62
30:40 40:60
29:41 41:58

Dan

Dan Stearns:
Your method is nice and simple and seems to work pretty well for the
examples you show (esp. 12). Do you have any explanation why it works? or
where the magic number 2.9 comes from? For what n-ET does it no longer work?

And responding to Paul Erlich:
Yes the "outer loop" I mentioned is indeed equivalent to the Continued
Fraction algorithm. (I have proven it to my own satisfaction but would not
be able to convey that proof in e-mail.) I believe the algorithm I posted is
"Viggo Brun's subtractive version of the Euclidean Algorithm" from John
Chalmers, and the "inner loop" gives the "semi-convergents as well".

To approximate an irrational by a rational, there is no one right answer;
there are infinitely many rationals within any distance of any irrational.
When interpreting an ET scale in terms of JI, while the CF algorithm does
give the best approximants for a given size of numerator & denominator,
musicians might well be interested in what _other_ JI intervals are close by
(at least closer than to the neighboring ET interval) as well. Hence the
option of seeing the semi-convergents. For example, in 12-ET, the whole tone
may sometimes function as 9/8 (a CF approximant) but in other contexts as
10/9 (which is a semi-convergent, not a CF approximant).

David Canright http://www.mbay.net/~anne/david/

> -----Original Message-----
> From: tuning@onelist.com [SMTP:tuning@onelist.com]
> Sent: Tuesday, August 31, 1999 9:59 PM
> To: tuning@onelist.com
> Subject: [tuning] Digest Number 299
>
> Message: 13
> Date: Tue, 31 Aug 1999 16:13:47 -0700
> From: "D.Stearns" <stearns@capecod.net>
> Subject: Re: converting equal divisions into ratio sequences
>
> This is (perhaps) another way to see equal divisions of the octave as
> sequences of ratios.
>
> If "a" is taken as an equal division of the octave and "b" is taken as
> the fraction of "a," D:N=(y-b):(y+b) where y=(x*a) and x=2.9.
>
>
> Message: 19
> Date: Tue, 31 Aug 1999 17:31:36 -0400
> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
> Subject: RE: Canright's post on algorithm
>
> In
> such a context, would anything other that CF's make sense? And again,
> isn't
> the outer loop Canright was referring to equivalent to CF's?
>

[Canright, David:]
>Do you have any explanation why it works?

No, not really. I have an ill-attended 9th grade education as a math
background, and as such, I tend to approach these things with an often
clumsy combination of (intuitive) experimentation and (enthusiastic)
naivete.

>or where the magic number 2.9 comes from?

Though I really don't know why it works, it is a variation of (7*5)/12
(i.e., the product of a fifth and a fourth, divided by an octave in
the simplest whole number terms), which in this case is the product of
7 and the arithmetic mean fourth (4&34/35ths) of 5 and 7e (34&4/5ths),
divided by 12.

>For what n-ET does it no longer work?

Though I suppose that's somewhat contingent on what one means by
"work," I've never really had/known a real convenient (non-brute
force) way to check - or even what, exactly, to check against... But
while I'm pretty sure that the errors by which the rounded D:N's
decrease (as the "n-ET" gets larger) would satisfy most conditions,
you do end up with approximations like 1/1, 25/24, 51/47, 26/23,
53/45, 27/22, 55/43, 4/3, 57/41, @:

49 : 49 49 : 98
50 : 48 50 : 96
51 : 47 51 : 94
52 : 46 52 : 92
53 : 45 53 : 90
54 : 44 54 : 88
55 : 43 55 : 86
56 : 42 56 : 84
57 : 41 57 : 82

for something as relatively small as 17e.

Dan

> @:
>
> 49 : 49 49 : 98
> 50 : 48 50 : 96
> 51 : 47 51 : 94
> 52 : 46 52 : 92
> 53 : 45 53 : 90
> 54 : 44 54 : 88
> 55 : 43 55 : 86
> 56 : 42 56 : 84
> 57 : 41 57 : 82

Should have read:

49:49 49:98
48:50 50:96
47:51 51:94
46:52 52:92
45:53 53:90
44:54 54:88
43:55 55:86
42:56 56:84
41:57 57:82

Sorry,
Dan

David Canright wrote,

>When interpreting an ET scale in terms of JI, while the CF algorithm does
>give the best approximants for a given size of numerator & denominator,
>musicians might well be interested in what _other_ JI intervals are close
by
>(at least closer than to the neighboring ET interval) as well. Hence the
>option of seeing the semi-convergents. For example, in 12-ET, the whole
tone
>may sometimes function as 9/8 (a CF approximant) but in other contexts as
>10/9 (which is a semi-convergent, not a CF approximant).

As you may know, I believe it may sometimes function as 9/8 and as 10/9
simultaneously. Anyway, I wouldn't use any of these methods for interpreting
an ET scale in terms of JI. Instead, I'd find the approximations to all the
consonances within a given odd limit, making sure they are consistent with
one another, and then proceed to combine them, often producing many JI
ratios for the same ET degree. For example, 81/64 is 10 steps in 31-tET,
even though it's closer to 11 steps, since 81/64 is always the result of
combining simpler intervals (such as 4 fifths) and never heard on its own. I
doubt 81/64 would show up as a semi-convergent to the 10-step interval in
31-tET. Meanwhile 13/9 does not make sense in 31-tET, since its best
approximation is 16 steps while the best approximation of 13/12 (4 steps)
plus the best approximation of 4/3 (13 steps) is 17 steps.

Alternately, you say that in a 5-limit harmonic context, 9/8 and 10/9 are
dissonances so their melodic function as diatonic seconds is more important
that their JI representations. Thinking this way (as I usually do) it's
really only the ET's approximations to the consonances in a given limit that
are worth looking at, since dissonances function by _not_ being easily
perceived as ratios.