"converting" ET to JI

All this thread about how a given ET will be heard as JI reminded me of an
article I wrote for "Interval" way back in 1979, in which I gave tables for
various popular ET's, such as the following for 11-ET:

11-tone equal temperament: just approximations to 1 cent
1 ( 109.1): 16/15 (+2.6) 33/31 (-0.9)
2 ( 218.2): 8/7 (+13.0) 17/15 (-1.5) 76/67 (+0.0)
3 ( 327.3): 6/5 (-11.6) 29/24 (+0.3)
4 ( 436.4): 9/7 (-1.3) 193/150 (+0.0)
5 ( 545.5): 4/3 (-47.4) 11/8 (+5.9) 26/19 (-2.4) 37/27 (+0.0)
6 ( 654.5): 3/2 (+47.4) 16/11 (-5.9) 19/13 (+2.4) 54/37 (-0.0)
7 ( 763.6): 14/9 (+1.3) 143/92 (-0.1)
8 ( 872.7): 5/3 (+11.6) 48/29 (-0.3)
9 ( 981.8): 7/4 (-13.0) 30/17 (+1.5) 67/38 (-0.0)
10 (1090.9): 15/8 (-2.6) 77/41 (+0.2)

where numbers in parens are cents or cents difference. Of course, this does
not show all JI intervals near each ET tone, only successive approximations
from an algorithm I got from John Chalmers.

I have just made the program (executable for DOS and source code in C)
available at:
http://www.mbay.net/~anne/david/et2ji
The program has an option to include intermediate results as well, which for
11-ET gives:

11-tone equal temperament: just approximations to 1 cent (detailed)
1 ( 109.1): 12/11 (+41.5) 13/12 (+29.5) 14/13 (+19.2) 15/14 (+10.4) 16/15
(+2.6) 17/16 (-4.1) 33/31 (-0.9)
2 ( 218.2): 7/6 (+48.7) 8/7 (+13.0) 9/8 (-14.3) 17/15 (-1.5) 25/22 (+3.1)
42/37 (+1.3) 59/52 (+0.5) 76/67 (+0.0)
3 ( 327.3): 6/5 (-11.6) 11/9 (+20.1) 17/14 (+8.9) 23/19 (+3.5) 29/24
(+0.3)
4 ( 436.4): 5/4 (-50.0) 9/7 (-1.3) 13/10 (+17.9) 22/17 (+10.0) 31/24
(+6.7) 40/31 (+4.9) 49/38 (+3.8) 58/45 (+3.0) 67/52 (+2.4) 76/59 (+2.0)
85/66 (+1.6) 94/73 (+1.4) 103/80 (+1.1) 112/87 (+0.9) 121/94 (+0.8) 130/101
(+0.6) 139/108 (+0.5) 148/115 (+0.4) 157/122 (+0.3) 166/129 (+0.2) 175/136
(+0.1) 184/143 (+0.1) 193/150 (+0.0)
5 ( 545.5): 4/3 (-47.4) 7/5 (+37.1) 11/8 (+5.9) 15/11 (-8.5) 26/19 (-2.4)
37/27 (+0.0)
6 ( 654.5): 3/2 (+47.4) 10/7 (-37.1) 13/9 (-17.9) 16/11 (-5.9) 19/13
(+2.4) 35/24 (-1.4) 54/37 (-0.0)
7 ( 763.6): 8/5 (+50.0) 11/7 (+18.9) 14/9 (+1.3) 17/11 (-10.0) 31/20
(-4.9) 45/29 (-3.0) 59/38 (-2.0) 73/47 (-1.4) 87/56 (-0.9) 101/65 (-0.6)
115/74 (-0.4) 129/83 (-0.2) 143/92 (-0.1)
8 ( 872.7): 5/3 (+11.6) 13/8 (-32.2) 18/11 (-20.1) 23/14 (-13.3) 28/17
(-8.9) 33/20 (-5.8) 38/23 (-3.5) 43/26 (-1.7) 48/29 (-0.3)
9 ( 981.8): 7/4 (-13.0) 9/5 (+35.8) 16/9 (+14.3) 23/13 (+5.9) 30/17 (+1.5)
37/21 (-1.3) 67/38 (-0.0)
10 (1090.9): 11/6 (-41.5) 13/7 (-19.2) 15/8 (-2.6) 17/9 (+10.1) 32/17
(+4.1) 47/25 (+2.0) 62/33 (+0.9) 77/41 (+0.2)

The basic algorithm used to convert a decimal fraction to a ratio is below,
in fake C:

frac2rat ( double frac, double tolerance ) {
double big, small = 1., td;
int bn = 0, bd = 1, sn = 1, sd = 0, t;
/* bn,bd,sn,sd are big numer & denom, small numer & denom */

big = frac; /* decimal fraction (freq. ratio) */
do {
while ( big >= small ) {
big -= small; bn += sn; bd += sd;
}
/* here, bd/bn is approx.; also big < small, time to swap */
td = big; big = small; small = td;
t = bn; bn = sn; sn = t;
t = bd; bd = sd; sd = t;
}
while ( error((bd/bn)/frac) > tolerance );
print(bd/bn);
}

By the way, I agree that "punning" is musically useful, and not only for ET
scales, even for JI scales. For example, I might use a triad 5/4 14/9 15/8
as if the 14/9 were really a 25/16, though it's 7 cents off. But I would not
resolve to such a triad.

David Canright (831) 656-2782 (or -2206)
Math. Dept., Code MA/Ca (831) 656-2355 (FAX)
Naval Postgraduate School DCanright@NPS.Navy.mil
Monterey, CA 93943 USA http://www.mbay.net/~anne/david/

David Canright wrote,

>All this thread about how a given ET will be heard as JI reminded me of an
>article I wrote for "Interval" way back in 1979, in which I gave tables for
>various popular ET's, such as the following for 11-ET:

[. . .]

4 ( 436.4): 5/4 (-50.0) 9/7 (-1.3) 13/10 (+17.9) 22/17 (+10.0) 31/24
(+6.7) 40/31 (+4.9) 49/38 (+3.8) 58/45 (+3.0) 67/52 (+2.4) 76/59 (+2.0)
85/66 (+1.6) 94/73 (+1.4)

[. . .]

David, why does you algorithm follow a very good approximation (9/7) with
many that are not as good? The standard continued fraction algorithm
wouldn't do that.

Paul,
The algorithm I use (for which I credit John Chalmers) has an inner and an
outer loop. By default, only the results of the outer loop are shown, which
give successively better approximations (is this equivalent to the continued
fraction method?), such as in my first example:
4 ( 436.4): 9/7 (-1.3) 193/150 (+0.0)

I included an option to show the intermediate results of the inner loop,
which gives poorer approximations on the way to a good one, just to show
some other intervals in the neighborhood. The result you quote is from that
option (showing many steps leading from 9/7 to 193/150). In either case, I
only print out those that are closer to tone #n than to #(n+1) or to #(n-1).

David Canright (831) 656-2782 (or -2206)
Math. Dept., Code MA/Ca (831) 656-2355 (FAX)
Naval Postgraduate School DCanright@NPS.Navy.mil
Monterey, CA 93943 USA http://www.mbay.net/~anne/david/

> -----Original Message-----
> From: tuning@onelist.com [SMTP:tuning@onelist.com]
> Sent: Tuesday, August 24, 1999 9:05 PM
> To: tuning@onelist.com
> Subject: [tuning] Digest Number 291
> Message: 14
> Date: Tue, 24 Aug 1999 15:27:53 -0400
> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
> Subject: RE: "converting" ET to JI
>
> David Canright wrote,
>
> >All this thread about how a given ET will be heard as JI reminded me of
> an
> >article I wrote for "Interval" way back in 1979, in which I gave tables
> for
> >various popular ET's, such as the following for 11-ET:
>
> [. . .]
>
> 4 ( 436.4): 5/4 (-50.0) 9/7 (-1.3) 13/10 (+17.9) 22/17 (+10.0) 31/24
> (+6.7) 40/31 (+4.9) 49/38 (+3.8) 58/45 (+3.0) 67/52 (+2.4) 76/59 (+2.0)
> 85/66 (+1.6) 94/73 (+1.4)
>
> [. . .]
>
> David, why does you algorithm follow a very good approximation (9/7) with
> many that are not as good? The standard continued fraction algorithm
> wouldn't do that.
>
>
> __________________________________________________________________________
> _____
>

David Canright wrote,

>The algorithm I use (for which I credit John Chalmers) has an inner and an
>outer loop. By default, only the results of the outer loop are shown, which
>give successively better approximations (is this equivalent to the
continued
>fraction method?), such as in my first example:
> 4 ( 436.4): 9/7 (-1.3) 193/150 (+0.0)

I believe it is equivalent. The continued fraction expansion for 2^(4/11)
begins

1 + 1/(3 + 1/(2 + 1/(21 + 1/(25 + 1/(12 + 1/(6 + 1/(2 + 1/(1 +
1/(13)))))))))

so the successive approximations are

4/3
9/7
193/150
4834/3757
58201/45234
354040/275161
766281/595556
1120321/870717
15330454/11914877

at which point the approximation is correct to .00000000001 cent.

By the way, I really like your music!