Yahoo Tuning Groups Ultimate Backup tuning Re: xenharmonic bridges

I've just done a command line utility to search for xenharmonic bridges.

Example of use:

xbridges 7 0 10 1000

= look for close pairs of ratios using prime limit 7, max exponent for the
primes 0 (for no limit), max diff of 10 cents between the two ratios, max of
10000 for denom.*denum

It looks for ratios in range 1/1 to 2/1.

Here are some of the results:
7 limit, diff at most 10 cents, denom*denum at most 1000:

28/27, 25/24 (7.71 cents); 16/15, 15/14; 28/25, 9/8; 32/27, 25/21; 32/25,
9/7; 14/9, 25/16; 16/9, 25/14; 28/15, 15/8;
http://www.robertwalker.f9.co.uk/xbridge_225_224.mid
(midi file adds 1/1 and 2/1 at the ends)

Factorised in Tune Smithy:
2^2*7/3^3 5^2/(2^3*3); 2^4/(3*5) 3*5/(2*7); 2^2*7/5^2 3^2/2^3;
2^5/3^3 5^2/(3*7); 2^5/5^2 3^2/7; 2*7/3^2 5^2/2^4;
2^4/3^2 5^2/(2*7); 2^2*7/(3*5) 3*5/2^3

These are all examples of the 5==7 bridge 225:224 = 2^5*7 : 3^2*5^2 (7.71152
cents)
- see http://www.ixpres.com/interval/dict/bridging.htm

11 limit, diff at most 5 cents, denom*denum at most 10000:
22/21, 21/20 (3.93 cents); 16/11, 35/24 (4.5 cents); 40/21, 21/11 (3.93
cents);
http://www.robertwalker.f9.co.uk/xbridge_11_lim_max_5_cents.mid

Factorised in TS:
2*11/(3*7) 3*7/(2^2*5); 2^4/11 5*7/(2^3*3); 2^3*5/(3*7) 3*7/11;

bridges:
441:440 = 3^2*7^2 : 2^3*5*11 ( 3.93016 cents)
385:384 = 5*7*11 : 2^7*3 (4.50256 cents)

13 limit, diff at most 3 cents, denom*denum at most 10000:
28/27, 27/26 (2.38 cents); 27/26, 26/25 (2.56 cents); 26/25, 25/24 (2.77
cents);
39/25, 25/16;
http://www.robertwalker.f9.co.uk/xbridge_13_lim_max_3_cents.mid

Factorised in TS:
2^2*7/3^3 3^3/(2*13); 3^3/(2*13) 2*13/5^2; 2*13/5^2 5^2/(2^3*3);
3*13/5^2 5^2/2^4;

bridges:
729:728 = 3^6 : 2^3*7*13 (2.37644 cents)
676:675 = 2^2*13^2:3^3*5^2 (2.56289 cents)
625:624 = 5^4:2^4*3*13 (2.77219 cents)

7 limit, diff at most 1 cents, denom*denum at most 10000:

50/49, 49/48 (0.721 cents); 60/49, 49/40; 64/49, 98/75; 75/49, 49/32; 80/49,
49/30; 96/49, 49/25;
http://www.robertwalker.f9.co.uk/xbridge_2401_2400.mid

Factorised in TS:
2*5^2/7^2 7^2/(2^4*3); 2^2*3*5/7^2 7^2/(2^3*5);
2^6/7^2 2*7^2/(3*5^2); 3*5^2/7^2 7^2/2^5;
2^4*5/7^2 7^2/(2*3*5); 2^5*3/7^2 7^2/5^2;

These are all examples of the 2401:2400 xenharmonic bridge: see
http://www.ixpres.com/interval/td/schulter/septimal.htm

7 limit, diff at most 0.5 cents, denom*denum at most 10000:

126/125, 245/243 (0.396 cents); 36/35, 250/243; 27/25, 175/162; 144/125,
280/243; 81/70, 125/108; 243/200, 175/144; 216/175, 100/81; 162/125, 35/27;
324/245, 250/189; 243/175, 25/18; 36/25, 350/243; 189/125, 245/162; 54/35,
125/81; 81/50, 175/108; 288/175, 400/243; 216/125, 140/81; 243/140, 125/72;
324/175, 50/27; 243/125, 35/18;
http://www.robertwalker.f9.co.uk/xbridge_4375_4274.mid

Factorised in TS:

2*3^2*7/5^3 5*7^2/3^5; 2^2*3^2/(5*7) 2*5^3/3^5;
3^3/5^2 5^2*7/(2*3^4); 2^4*3^2/5^3 2^3*5*7/3^5;
3^4/(2*5*7) 5^3/(2^2*3^3); 3^5/(2^3*5^2) 5^2*7/(2^4*3^2);
2^3*3^3/(5^2*7) 2^2*5^2/3^4; 2*3^4/5^3 5*7/3^3;
2^2*3^4/(5*7^2) 2*5^3/(3^3*7); 3^5/(5^2*7) 5^2/(2*3^2);
2^2*3^2/5^2 2*5^2*7/3^5; 3^3*7/5^3 5*7^2/(2*3^4);
2*3^3/(5*7) 5^3/3^4; 3^4/(2*5^2) 5^2*7/(2^2*3^3);
2^5*3^2/(5^2*7) 2^4*5^2/3^5; 2^3*3^3/5^3 2^2*5*7/3^4;
3^5/(2^2*5*7) 5^3/(2^3*3^2); 2^2*3^4/(5^2*7) 2*5^2/3^3;
3^5/5^3 5*7/(2*3^2);

All examples of
4375:4374 = 5^4*7^2 : 2*3^7*7 (0.395756 cents)

7 limit, max exponent 3, max diff 0.33 cents, any denominator and
denumerator less than 100000:
200/189, 1323/1250 (0.325 cents); 100/81, 3087/2500; 250/189, 1323/1000;
5000/3087, 81/50; 50/27, 9261/5000; 2500/1323, 189/100;
http://www.robertwalker.f9.co.uk/xbridge_250047_250000.mid

2^3*5^2/(3^3*7) 3^3*7^2/(2*5^4); 2^2*5^2/3^4 3^2*7^3/(2^2*5^4);
2*5^3/(3^3*7) 3^3*7^2/(2^3*5^3); 2^3*5^4/(3^2*7^3) 3^4/(2*5^2);
2*5^2/3^3 3^3*7^3/(2^3*5^4); 2^2*5^4/(3^3*7^2) 3^3*7/(2^2*5^2);

all: 250047:250000 = 2^4*5^6: 3^6*7^3 (0.325441 cents)

7 limit, max exponent 6, max diff 0.08 cents, any denominator and
denumerator less than 100000
2187/2000, 78125/71442 (0.0703 cents); 17496/15625, 40000/35721;
35721/31250, 2500/2187; 4374/3125, 50000/35721; 35721/25000, 3125/2187;
2187/1250, 62500/35721;
35721/20000, 15625/8748;
http://www.robertwalker.f9.co.uk/xbridge_517058_517037.mid

Factorised in TS:
3^7/(2^4*5^3) 5^7/(2*3^6*7^2); 2^3*3^7/5^6 2^6*5^4/(3^6*7^2);
3^6*7^2/(2*5^6) 2^2*5^4/3^7; 2*3^7/5^5 2^4*5^5/(3^6*7^2);
3^6*7^2/(2^3*5^5) 5^5/3^7; 3^7/(2*5^4) 2^2*5^6/(3^6*7^2);
3^6*7^2/(2^5*5^4) 5^6/(2^2*3^7);

i.e. xenharmonic bridge
517058:517037 = 2^3*5^10 : 3^13*7^2 ( 0.0703145 cents)

Anyway, you can try the program yourself and find more:

http://www.robertwalker.f9.co.uk/xbridges.exe [48 Kb]

You can also use the prog. to search for closest pair, by setting cents
diff. arg to 0:

Run without args for info about how to use it and what the args mean

c-code:
http://www.robertwalker.f9.co.uk/xbridges.c [19 Kb]

My idea for making these searches is that perhaps if one plays close
together ratios with differing harmonic basis, the resulting music might
have a particularly lively feel to it, even if one can't conciously separate
the ratios. A bit like the impressionist idea of placing coloured patches
close together, so that at a distance they merge, but have more vibrancy
than if one were to just mix the colours together on the palette.

I think perhaps these midi clips are particularly "lively". What do the
other members think?

Also I feel the 5==7 bridges are much easier to distinguish than the 7==11
or
11==13 bridges, even with closer spacing for the ratios.

Compare the 7==11 bridges
http://www.robertwalker.f9.co.uk/xbridge_11_lim_max_5_cents.mid
(3.93 to 4.5 cents)
with the 5==7 bridges
http://www.robertwalker.f9.co.uk/xbridge_2401_2400.mid
(0.721 cents)

Are the bridges in the second clip are easier to distinguish than the
ones in the first?

What do you think?

:-)

I've done a fractal clip with the 7-limit 517058:517037 bridges (with 1/1
and 2/1 added)
- it is rather a fractal like scale as well, with wide distribution of step
sizes:
154.744 cents 0.0703145 cents 40.9885 cents 0.0703145 cents 35.6265 cents
0.0703145 cents 350.547 cents 0.0703145 cents 35.6265 cents 0.0703145 cents
350.547 cents 0.0703145 cents 231.5 cents

N.B. the resolution of MIDI pitch bend at default pitch bend range, as used
in the clips, is 4096 steps to a semitone, which works out at 100/4096 =
0.024414 cents.

http://www.robertwalker.f9.co.uk/impressionist_palette.mid

All the instruments are playing the same tune at varying speeds.

The tune begins with the musical seed, which also generates the complete
tune:
http://www.robertwalker.f9.co.uk/impr_palette_seed.mid
(this first instance of seed includes two of the close ratios, though not
adjacent to each other - second and third highest of the notes played, both
around the middle of the seed, to either side of the lowest note played)

Robert
http://www.robertwalker.f9.co.uk/fts_beta/fts_beta_download.htm

(Min req Win 95/98 + soundcard)

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

Hi Paul,

>Only the first clip worked for me, and I heard absolutely no
>difference between the pairs of notes. Actually, it's well-
>established that the _melodic_ just-noticeable difference for human
>hearing is at least 6 cents (depending on register).

I didn't know that. However, maybe it could depend on melodic context too?

I did a test: did the ratios 64/49 98/75 in isolation, then did them
prefixed by 1/1

I did .WAV files for them, and did a random selection, some in order 64/48
98/75, and some in reverse order.

With the two ratios in isolation, got no better than chance, so I don't
think I can resolve them in isolation.

But when prefixed by 1/1, got 7 right out of 9.

I also felt that I could tell which was which, though with some hesitation.

(I know 9 is a small sample, but result is promising)

However I know it is easy to fool oneself about things like this.

I then tried looking at the WAV files in Goldwave, and counted 60 waves in
each of the notes. Both came to 0.083 milliseconds.

Notes should have been 724.0699 Hz 724.3716 Hz
which for 60 waves works out as 0.082865 and 0.082830

Can't really expect to distinguish them by counting waves, and the frequency
analysis in CoolEdit doesn't seem to have fine enough resolution.

What I'll do is try a more scientific test - I'll do a little prog. to make
a batch file to copy the two files to a random sequence of files, then
listen to them and see if I still get better than random results.

I find 7 cents easy in melodic context, e.g. between 9/8 and 28/25, not
"just-noticeable" - 10/10 and no hesitation whatsoever.

I can't distinguish those either in Goldwave / CoolEdit.

It could also vary from person to person? Like night sight - keen amateur
astronomers learn to see really faint galaxies that nobody else can see
through the telescope. I've had that experience, shown galaxies by a really
keen amateur astronomer - if I spent ages looking I eventually could just
see it, but he could see it at once, to the extent that when slewing the
telescope to find it, he could tell instantly when it was in the field of
view.

I think my sense of pitch in the melodic context got more acute as a result
of learning to play the cello a bit some years back - I got really into fine
tuning it, added fine tune adjustments to all the strings at the bridge end,
and spent ages just tuning it and listening to the harmonics etc. (Don't
play it at present though - sold it a few years back as rather large when
continually moving from one place to another, and because I wasn't playing
it so much anymore.)

Anyway, hope the .exe file is of use,...

Maybe I should use larger pitch intervals for the "musical impressionism"
experiments....

Robert

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Carl Lumma <CLUMMA@NNI.COM>

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

Hi Paul,

I'm in the middle of testing the soundcard etc.

I've shown my one is overall sharp by 2 2/3 cents in absolute pitch, and
relative pitch is correct to within about 0.2 cents (not as good as I
expected, but good enough for useful experiments).

Found possible to actually count number of waves in Goldwave. (My soundcard
has two synths, one of which producew regular waves for some voices, which
are easy to count, and one which produces the more lifelike ones that are
much harder to count since the wave shape varies so much as the note
continues. so this is for the basic synth).

Method was to show high resolution (individual samples shown as steps),
count 10 waves, measure exactly, use that to predict where the 100th wave
will cross the 0 position, go to that place and find exact crossover pos for
100th wave, and use that to predict 1000th wave, and check that, by which
time one has a pretty accurate value for the frequency (In practice, helped
to also look for 50th and 500th wave as an double check).

Ex. of this method to find frequency of a note in a .WAV file:

1.5601 secs = 1000 waves
so frequency is 640.984552 Hz
This was supposed to be 16/11 from 440 hz
so should be 640
- sharp by 2.66 cents

N.B. really pleased to discover this - could also use it to find frequency
of bird song for really accurate transcriptions of pitches, I hope.

I also did a prog to make a batch file that copies 2 (or more if you like)
files at random to the files
0001.mid, 0002.mid,...0020.mid (up to whatever number you choose)

It just chooses one of the 2 files at random each time, so as you work
through the copies, you have no idea how many more to expect of either type.

Did prelim results:

For 1/1 22/21 21/20
was supposed to have diff of 3.93 cents
Actually, because of soundcard resolution, diff was 3.74 cents
Out of 20 copies of 1/1 22/21 21/20 and 1/1 21/20 22/21 in random order, got
17 right, 2 not sure, 1 wrong.
(not sure = put the file in both lists, after going through a second time to
check. However in future, will make sure all the files go in one list only)

On binomial model, s.d. is sqrt(n/4), i.e. sqrt(5) = 2.236068
so 17 is 3.130495 s.d.s away from the mean, so significant at the 99.74
percent confidence level.

So I think I can prob. distinguish those.

For
1/1 16/11 35/24 diff should be 4.503 cents, because of soundcard res., was
4.78 cents
got 8 correct, 12 wrong
So doesn't look as if I can distinguish those at all, even though separation
is wider (but I did this experiment 2nd, so could just be tiredness?? Or is
it a real effect of more complex ratios
??)

Also, a rather nice test is to play
1/1 21/20 35/24 40/21 2/1
and
1/1 22/21 16/11 21/11 2/1
- first is 7 limit, 2nd is 11 limit, and the ratios of 2nd are all within 5
cents of ratios of 1st.

I tried those, and couldn't tell which
was which. (9/20 right)

However those are just kind of dry runs, to test the method. I'll do it
properly now. Want to know what's going on - it is rather intriguing.
Especially, whether the interval from the 1/1 helps or not, and if so, in
what context it does.

Anyway, need to do more tests. Also need to redo the tests using the basic
synth, - seems to me that to be thorough one needs to rule out the poss.
that the the timbre may change a bit with frequency. With the more realistic
synth, so much going on when one looks at the wave file. I haven't noticed
any such effect, but seems a poss. to consider, while the basic one has such
a simple wave form that isn't likely to be affected at all in that way.

I've done update of the xbridges prog. It's now a bit faster, and I've added
option to look specifically for xenharmonic bridges

http://www.robertwalker.f9.co.uk/xbridges.exe (win 95/98 .exe file)
and
http://www.robertwalker.f9.co.uk/xbridges.c (c-code)

You add the prime factors for the other side of the bridge like this:
13;5
means look for bridges between 5 limit, and 13 limit or lower (looks for any
that are 5 limit on one side, and not 5 limit but 13 limit on other side)

Ex. of use:

XBRIDGES 13;5 4 6 1000

Prime factors (max exponent 4):
2 3 5 7 11 13
Prime factors for other side of xenharmonic bridge:
2 3 5
Max denominator * denumerator 1000
Will show all pairs with diff at most 6 cents
14/13, 27/25 (4.939 cents); 27/25, 13/12 (5.335 cents);
35/26, 27/20 (4.939 cents); 18/13, 25/18 (5.335 cents);
36/25, 13/9; 27/16, 22/13 (4.925 cents);
39/22, 16/9;
(ratios shown in increasing order, and diff shown whenever it changes)

I also did a search for the bridge mentioned by Manuel Op de Coul in
http://www.ixpres.com/interval/td/schulter/septimal.htm

Prime factors: 2 3
Prime factors for other side of xenharmonic bridge:
2 3 7
searching all ratios with denominator and denumerator less than 1000000
Will show all pairs with diff at most 4 cents
537824/531441, 531441/524288 (2.79 cents);
531441/524288, 64/63 (3.804 cents);
137781/131072, 256/243;
2187/2048, 16384/15309;
567/512, 65536/59049;
19683/16384, 2048/1701;
5103/4096, 8192/6561;
81/64, 524288/413343;
177147/131072, 256/189;
45927/32768, 1024/729;
729/512, 65536/45927;
189/128, 262144/177147;
413343/262144, 128/81;
6561/4096, 8192/5103;
1701/1024, 32768/19683;
59049/32768, 1024/567;
15309/8192, 4096/2187;
243/128, 262144/137781;

(ratios shown in increasing order, and diff shown whenever it changes)
First one is
2^24*7^5 3^24
i.e.
ratio of (3/2)^24 to 7^5

Rest are various versions of
7*3^14 : 2^24

I'll do some more tests, and perhaps post the .wav files and the prog. for
making batch files to copy any pair of files randomly to a list of 20 files,
so anyone else can have a go at them too, if they like - rather fun if you
like that sort of thing,...

Robert

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

Hi Paul,

>Putting aside the soundcard resolution, I thought you were earlier making
>the claim that lower-prime-limit unison vectors were easier to distinguish,
>(or was it harder?), but these have the same prime limit: 11. I think the
>true explanation in this case is that in the first case you were comparing
>small melodic intervals -- small semitones -- while in the second case you
>were comparing larger melodic intervals -- small fifths -- which are a bit
>harder to judge.

The ones in the example were
1/1 22/21 21/20
and
1/1 16/11 35/24

In each case, one ratio is 7-limit, one is 11 limit.

I thought that might make them easier to distinguish. For wider spacing than
this, where one can distinguish them, but recognize them as pretty much the
same note, might give effect of extra vividness to the note like putting
yellow and blue together in an impressionist painting.

Just a nice kind of idea, not really a theory as such.

Could be either chords or melodic intervals, investigating melodic intervals
at present.

However I have a lot of experiments to do before I know where all this is
going, if anywhere - will let you know if anything interesting comes up
later on...

Can't really comment yet on whether it's an effect of wideness of interval
until I do some more tests, though that sounds like a possible explanation
to bear in mind, thanks.

Perhaps I'll include one of the impressionist palette ts files with
next release of TS beta, or the one after that, as they are rather fun and I
think rather vivid, whatever the theory might be, if any. (Maybe would be
just as vivid if the ratios were all the same limit as each other).

Thanks for your comments; it helps me to see things more clearly!

Robert