Re: Search for consonant diads

This is an idea I'm mulling over for FTS 1.10

Idea is, you enter a timbre in terms of its harmonics (would
default to harmonic series, but one can try inharmonic partials
as well)

Then FTS searches current scale for pairs of notes with
slow beating partials. Counts the consonance in terms
of number of beats (low is best).

If number of beats is larger than some max. value which one can set
then that pair of partials is ignored - it is treated as a difference tone.

Also have option to ignore the partials above a certain point in harmonic series
if one wishes.

Would also weight the number of beats according to the volume
of the partial in the timbre.

Then one can set a limit of the amount of consonance
one wishes to count as a diad.

In many cases, there will be a few slow beating partials,
and the rest all will count as difference tones.

I already have option in FTS to show the number of beats for a diad
(tick New Scale -> beats, and play a diad to see it).

It just uses the normal harmonic series, but will be easy to extend
it to use inharmonic partials as well.

It also just uses a sharp cut off, default, 30th harmonic.

Here is a typical result.

7/4

Frequencies 261.63 Hz, 457.84 Hz.
Difference tone 196.22 Hz, ratio from lowest note 7/4
Beats for all harmonics up to 30 and up to 20 beats per sec.
0 7th (4th) 1831.379 Hz 1831.379 Hz
0 14th (8th) 3662.758 Hz 3662.758 Hz
0 21st (12th) 5494.137 Hz 5494.137 Hz
0 28th (16th) 7325.516 Hz 7325.516 Hz
Key:
beats harmonic (harmonic of upper note) freq of harmonic freq of upper note harmonic.

(
n.b. if you try this in FTS you'll get ratio from lowest note 3/4 (= 7/4 - 1)
- I've found a few bugs in it and I'm debugging it
)

So 7/4 would count as highly consonant, with four entirely consonant
pairs of partials in the range, and no beating partials at all.

7/4 + 1 cent = (7/4)*2^(1/1200)

Frequencies 261.63 Hz, 458.11 Hz.
1.0582 7th (4th) 1831.379 Hz 1832.437 Hz
2.1163 14th (8th) 3662.758 Hz 3664.874 Hz
3.1745 21st (12th) 5494.137 Hz 5497.311 Hz
4.2326 28th (16th) 7325.516 Hz 7329.748 Hz

So it finds it as 7/4 again, though less consonant.

Here is the 19 tone major third:

Frequencies 274.8 Hz, 342.05 Hz.
5.5542 5th (4th) 1308.128 Hz 1302.574 Hz
5.5542 5th (4th) 1308.128 Hz 1302.574 Hz
11.108 10th (8th) 2616.256 Hz 2605.147 Hz
11.108 10th (8th) 2616.256 Hz 2605.147 Hz
16.663 15th (12th) 3924.383 Hz 3907.721 Hz
16.663 15th (12th) 3924.383 Hz 3907.721 Hz

Again, recognises it as a 5/4, and reasonably consonant
(especially if the high partials of the timbre aren't
particularly strong).

19 tone 6/5

Frequencies 261.63 Hz, 313.98 Hz.
0.13437 6th (5th) 1569.753 Hz 1569.888 Hz
0.26874 12th (10th) 3139.507 Hz 3139.776 Hz
0.40311 18th (15th) 4709.26 Hz 4709.663 Hz
0.53748 24th (20th) 6279.014 Hz 6279.551 Hz
0.67185 30th (25th) 7848.767 Hz 7849.439 Hz

- much more consonant than the 5/4.

So far, it has only found one ratio.

17 tone minor:
Frequencies 261.63 Hz, 307.97 Hz.
16.451 7th (6th) 1831.379 Hz 1847.83 Hz
13.444 13th (11th) 3401.132 Hz 3387.688 Hz
3.0071 20th (17th) 5232.511 Hz 5235.518 Hz
19.458 27th (23rd) 7063.89 Hz 7083.348 Hz

- recognises it as a 7/6.

Also found 13/11, 20/17 and 27/23.

It's most consonant as a 20/17.

Try 7/6:
0 7th (6th) 1831.379 Hz 1831.379 Hz
0 14th (12th) 3662.758 Hz 3662.758 Hz
0 21st (18th) 5494.137 Hz 5494.137 Hz
0 28th (24th) 7325.516 Hz 7325.516 Hz

So it recognises that one unambiguously.

Now let's try 20/17

Frequencies 261.63 Hz, 307.79 Hz.
15.39 7th (6th) 1831.379 Hz 1846.769 Hz
15.39 13th (11th) 3401.132 Hz 3385.743 Hz
0 20th (17th) 5232.511 Hz 5232.511 Hz
15.39 27th (23rd) 7063.89 Hz 7079.28 Hz

This time, you see the 13/11 again, and the 27/23

Also you get the same beats with 13/11 and with 7/6,
so in a certain sense, 20/17 is mid way between the two.

Try 27/23:
Frequencies 261.63 Hz, 307.13 Hz.
11.375 7th (6th) 1831.379 Hz 1842.754 Hz
11.375 20th (17th) 5232.511 Hz 5221.136 Hz
0 27th (23rd) 7063.89 Hz 7063.89 Hz

and you get 20/17 and 7/6 as well, and it is
midway between the two.

What about 13/11:

Frequencies 261.63 Hz, 309.19 Hz.
Difference tone 47.568 Hz, ratio from lowest note 2/11
Beats for all harmonics up to 30 and up to 30 beats per sec.
23.784 6th (5th) 1569.753 Hz 1545.969 Hz
23.784 7th (6th) 1831.379 Hz 1855.163 Hz
0 13th (11th) 3401.132 Hz 3401.132 Hz
23.784 19th (16th) 4970.886 Hz 4947.102 Hz
23.784 20th (17th) 5232.511 Hz 5256.295 Hz
0 26th (22nd) 6802.265 Hz 6802.265 Hz

Here, it has found both 6/5 and 7/6, and
both have same number of beats -
13/11 is mid way between 6/5 and 7/6.

+ found 19/16 and 20/17 as well.

Try 19/16
Frequencies 261.63 Hz, 310.68 Hz.
16.352 6th (5th) 1569.753 Hz 1553.402 Hz
16.352 13th (11th) 3401.132 Hz 3417.484 Hz
0 19th (16th) 4970.886 Hz 4970.886 Hz
16.352 25th (21st) 6540.639 Hz 6524.288 Hz
16.352 25th (21st) 6540.639 Hz 6524.288 Hz

Finds 6/5, 13/11 and 25/21, and the 19/16 is
midway between 6/5 and 13/11

25/21:
Frequencies 261.63 Hz, 311.46 Hz.
12.458 6th (5th) 1569.753 Hz 1557.295 Hz
12.458 19th (16th) 4970.886 Hz 4983.344 Hz
0 25th (21st) 6540.639 Hz 6540.639 Hz

Midway between 6/5 and 19/16.

And now we've explored all the ratios that
turned up.

20/17 = midpoint of 7/6 and 13/11
27/23 = midpoint of 7/6 and 20/17
13/11 = midpoint of 7/6 and 6/5
19/16 = midpoint of 13/11and 6/5
25/21 = midpoint of 19/16 and 6/5

So a nice pattern here:

7/6 6/5
| 13/11 |
| 20/17 | 19/16 |
| 27/23 | | | 25/21 |

It's finding the mediants of adjacent notes - you add the top and
bottom : mediant of 19/16 and 6/5 is (19+6)/(16+5) = 25/21.

Let's go back to the 13/11, and set the lowest note to a instead
of c, and you get

Frequencies 440 Hz, 520 Hz.
Difference tone 80 Hz, ratio from lowest note 13/11
Beats for all harmonics up to 30 and up to 30 beats per sec.
0 13th (11th) 5720 Hz 5720 Hz
0 26th (22nd) 11440 Hz 11440 Hz

Now it finds only 13/11, as perfectly consonant, because the
number of beats has increased to over the limit of 30 Hz.

So the amount of consonance you find depends on the pitch
of the chord, but this is in accordance with the way chords
are heard, so that's okay.

The start pitch will be one of the parameters one can choose
in the search for consonant diads.

Then having found the diads in that way, one can
find triads, tetrads etc by looking for chords in which
all the diads are consonant, which is easy to do once
one has found all the consonant diads of the scale.

Nice thing about it is that you can then redo the search
with another timbre, and see what chords turn up.

What would be really nice would be if you could
do the fourier analysis in FTS as well, and just get it to
play a note on your system, and record it, analyse it,
and use its timbre to search for chords. But I'm not sure
how easy it is to write a program to do that, may look into it at
some point,...

Robert

--- In tuning@egroups.com, "Robert Walker" <robert_walker@r...> wrote:
> This is an idea I'm mulling over for FTS 1.10
>
> Idea is, you enter a timbre in terms of its harmonics (would
> default to harmonic series, but one can try inharmonic partials
> as well)
>

Robert,

YOU RULE!!!

This is the coolest thing I've heard of for midi!

Do this! It is a worthy cause.

Thanks,

Jacky Ligon

Hi Jacky,

> > Idea is, you enter a timbre in terms of its harmonics (would
> > default to harmonic series, but one can try inharmonic partials
> > as well)
> >

> Robert,

> YOU RULE!!!

> This is the coolest thing I've heard of for midi!

> Do this! It is a worthy cause.

Thought you'd like it somehow!

Yes I'm very interested to give it a try and see what it comes up with...

Especially, to see what kinds of chords it finds for instruments
with lots of inharmonic partials!

Robert

Hi Robert,

Rather than measuring dissonance by counting beats, which is akin to
Helmholtz's approach in the 19th century, you might want to update your
ideas to take into account the developments of the 20th century by Plomp and
others. Sethares has done a pretty good job in his book -- take a close look
at http://eceserv0.ece.wisc.edu/~sethares/ttss.html
<http://eceserv0.ece.wisc.edu/~sethares/ttss.html> . The essential concept
you need to familiarize yourself with is the _critical band_ -- let me know
if you can "get it" from this website. Now, Sethares gives some formulae for
calculating the total dissonance of any set of partials. I tried
implementing these formulae in conjunction with some sound files I was
making and got some mighty strange results. So I contacted Sethares. First
of all, it turns out that "amplitude" in his formulae is not meant to me
amplitude at all but instead _loudness_ measured in dB. Secondly, there were
some antinomies and ambiguities that come up in trying to use a theoretical
set of dB levels for a given timbre in this formula . . . and Sethares never
got back to me with a satisfactory solution.

So I would say: Read and absorb Sethares's work, familiarize yourself with
his formulae and the critical band concept, and then run a few tests to make
sure you have something that makes sense -- don't take accept the numbers
blindly! Finally, keep in mind that, let alone combination tones, there's a
whole other component of dissonance which Sethares doesn't take into account
-- see Parncutt's _Harmony -- A Psychoacoustical Approach_, or my harmonic
entropy work, for an explanation.

-Paul

Hi Paul,

> Well the thing is we perceive timbre more by the timing of the attacks than
> anything else -- so it would be hard to acheive this in a strictly
> human-controlled performance unless long sustained tones were used. A
> computer-contolled performance should be capable of exploiting this effect
> and completely confusing the listener!

I'll give it a go.

Try:

http://homepage.ntlworld.com/robertwalker/agogo/agogo.mid
pure agogo

http://homepage.ntlworld.com/robertwalker/agogo/agogo_cv.mid
- plays harmonics up to 24/1 on the agogo

Does one hear some extra instruments playing - on my card
I hear a couple more - a high somewhat violin like sound, and a lower one.

http://homepage.ntlworld.com/robertwalker/agogo/agogo_subh_cv.mid
- plays sub-harmonics down to 1/24 on the agogo
I'd say, that it loses the sense of pitch of the notes somewhat.

http://homepage.ntlworld.com/robertwalker/agogo/agogo_subh28_a6.mid
sub-harmonics down to 1/28 - single note

http://homepage.ntlworld.com/robertwalker/agogo/agogo_h28_a5.mid
harmonics up to 28 - single note
I'd say, decidedly more harmonious.

If anyone wants the .ts files, or to see the hex dumps, change ext.
to ts, or hex resp. (all except last two, which I didn't do them for).

Robert

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_17940.html#17979

> Hi Robert,
>
> Rather than measuring dissonance by counting beats, which is akin to
> Helmholtz's approach in the 19th century, you might want to update
your ideas to take into account the developments of the 20th century
by Plomp andothers. Sethares has done a pretty good job in his book --
take a close lookat http://eceserv0.ece.wisc.edu/~sethares/ttss.html

I'm happy to see that Robert Walker is starting to get involved in
some of the Sethares stuff... That should lead to some interesting
results... particularly if the timbres can be altered "fractally" (??)

I was wondering... Is there any meaningful way to map out some of the
Sethares concepts in a kind of visual lattice... or wouldn't that
possibly apply with inharmonic cases (??)

________ _____ _____ _
Joseph Pehrson

Hi Paul,

Thought it might be helpful to spell out the argument:

The utonal chord for the agogo
> http://homepage.ntlworld.com/robertwalker/agogo/agogo_subh28_a6.mid
> sub-harmonics down to 1/28 - single note

and the otonal chord
> http://homepage.ntlworld.com/robertwalker/agogo/agogo_h28_a5.mid
> harmonics up to 28 - single note
> I'd say, decidedly more harmonious.
(for the same somewhat inharmonic timbre)

share all the same diads.

Example:
1/1 2/1 3/1 4/1 5/1
has diads
2 3 4 5 3/2 5/2 4/3 5/3
1/2 1/3 1/4 1/5 2/3 2/5 3/4 3/5
and
1/1 1/2 1/3 1/4 1/5
has exactly the same family of diads.

Only difference is that they are transposed in pitch.
Increasing the pitch makes a chord more harmonious, but
the utonal chord here is already transposed up an octave
- and one could transpose it up more, and still get the same
effect.

Since they have the same diads, then if one tries to measure
the consonance of chords in terms of its componant diads,
both will have to have the same value for consonance.

So therefore there has to be some other component involved
in our perception of consonance. Beating, or critical bands,
or whatever can't be the _only_ component, though it may
well be the most important for chords with small numbers of
notes.

Have I got this right?

If so, I find it pretty convincing!

However, I have a thought:

What if one were to choose the intervals for an "otonal chord"
from an inharmonic series of pitches, and used the virtual
harmonic series to make a recognisable inharmonic timbre,
such as the sound of bells, say.

Might that then too be heard as more consonant, if the
virtual timbre was reasonably recognisable as bell-like?

Or indeed, suppose one combined them to make
the overtone series of the inharmonic timbre itself?

It would be interesting to see if that was as consonant
as the undertone sereis of the inharmonic timbre,
especially if one could find a nice sample that had
no pure harmonic components to it.

Also would be interesting to see if the amount of consonance
depended on how familiar one was with the virtual
inharmonic timbre constructed. For instance, would
a regular gamelan player find a virtual overtone
series made up using one of the typical gamelan timbres
more consonant than others who are somewhat
less familiar with that timbre?

Or indeed, might a regular performer of any musical instrument
with inharmonic timbres be somewhat disposed towards
hearing a bit more consonance in "otonal series" that
use pitches from that timbre?

It is just a thought - one would need some good samples
of inharmonic timbres, and accurately measured partials
to test it out to see if it works.

I could use the agogo patch from my soundcard,
if I had a way of accurately measuring its partials - actually
generating the virtual timbre would be pretty
easy in FTS - just a matter of typing in the ratios
or values in cents for the series, and also the volumes
for each.

Or maybe someone who has everything set up for this
kind of thing already might like to give it a go?
(and use some even more inharmonic timbres)

If anyone wants to use FTS to make an "otonal chord" and an
"utonal chord" for an inharmonic timbre, you can open
http://homepage.ntlworld.com/robertwalker/agogo/harmonic_series.ts
and change the Intervals, and the volumes for the seed.

- see
http://homepage.ntlworld.com/robertwalker/agogo/harmonic_series.txt
for details - also explains how to make a custom voice for the otonal
and utonal chord, which you can use as a timbre in a fractal tune, or play from
MIDI keyboard (except, at present, the custom voice chords require the notes
to be all the same volume - will be changing this later).

N.B. I've just saved
http://homepage.ntlworld.com/robertwalker/agogo/agogo_cv.mid
and
http://homepage.ntlworld.com/robertwalker/agogo/agogo_subh_cv.mid

Before, for some reason, did by recording as the tune was played rather
than by saving to midi.

Prev. version had some delta times 0.01 to 0.03 secs,
because FTS sends the notes one at a time using the multimedia
timer, rather than adding its own timestamps.

I plan to add the MIDI buffer method with timestamps to FTS at some point
in the future, as it will give crisper results for fractal tunes that use chords with lots
of notes
in them. Though actually the very slightly ragged effect one gets sometimes
for the timings is rather nice I think - a bit like a real performance.

Robert

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:

/tuning/topicId_17940.html#18013

Hi Robert!

These are incredible sounds... at least as interesting as anything I
have heard out of CSOUND... How were they done again??

_______ _____ _____ _
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> I was wondering... Is there any meaningful way to map out some of the
> Sethares concepts in a kind of visual lattice...

Hmm . . . not sure what kind of lattice you have in mind . . . if you were thinking of curves
analogous to the harmonic entropy curves and surfaces analogous to the "radioactive
pizza", I was attempting to do these things for some specific timbres, and got some
wacky results so I contacted Bill Sethares. The rest of the story is recounted in my
previous message -- I'm still hoping Sethares gets back to me -- but so far there's no
way to do this without some "fudging". If you like, I can produce the curves and surfaces
corresponding to one of the ficticious "timbres" Sethares uses for the calculations in his
book.

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:

> So therefore there has to be some other component involved
> in our perception of consonance. Beating, or critical bands,
> or whatever can't be the _only_ component, though it may
> well be the most important for chords with small numbers of
> notes.

It seems to fail pretty miserable with just 4 notes, each of which has an ordinary
harmonic timbre.
>
> Have I got this right?

Other than the above caveat, yes.
>
> If so, I find it pretty convincing!
>
> However, I have a thought:
>
> What if one were to choose the intervals for an "otonal chord"
> from an inharmonic series of pitches, and used the virtual
> harmonic series to make a recognisable inharmonic timbre,
> such as the sound of bells, say.

Then the effect would fall apart, since the brain is only looking for "the" harmonic series
(perhaps stretched a little). There's nothing to prevent you from constructing two bells,
with timbres which are the exact mirror-images of one another, such that one bell's
"utonal" is the other bell's "otonal" and vice-versa with respect to their timbres.
>
> Or indeed, suppose one combined them to make
> the overtone series of the inharmonic timbre itself?

That _does_ lead to some consonance for the Sethares reason -- you will have lots of
coinciding partials.

> Also would be interesting to see if the amount of consonance
> depended on how familiar one was with the virtual
> inharmonic timbre constructed. For instance, would
> a regular gamelan player find a virtual overtone
> series made up using one of the typical gamelan timbres
> more consonant than others who are somewhat
> less familiar with that timbre?

Well some theorists have proposed that the virtual pitch phenomenon normally
associated with the harmonic series could be a result of training (including prenatal
training involving the mother's voice) rather than genetics; if they are right, then there
may be something to what you're saying -- though there is far less consistency in what
constitutes a "typical gamelan timbre" than in what constitutes a "harmonic overtone
series". In any case, the effect would require a very extensive training period with the
timbre in question in order to be measurable.

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_17940.html#18035

I'm still hoping Sethares gets back to me -- but so far there's no
>way to do this without some "fudging". If you like, I can produce the
curves and surfaces corresponding to one of the ficticious "timbres"
Sethares uses for the calculations in his book.

Oh yes, Paul! Please do. May not be "pizza," but I would be
fascinated with what you get!

___________ ______ ____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
>
> Oh yes, Paul! Please do. May not be "pizza," but I would be
> fascinated with what you get!

Well of course there will be a pizza, since the triangular shape is merely the boundary of
triads within one octave, and is independent of the method used to calculate the
topography of the pizza. In this case, the topography should be rather similar to that of
the first pizza I produced (well, we'll see) . . . now, we'll have to assume something about
the absolute pitches in order to apply Sethares's formulae . . . should be assume that the
average pitch in each dyad and triad is A-440?

--- In tuning@y..., PERLICH@A... wrote:

/tuning/message

> the absolute pitches in order to apply Sethares's formulae . . .
should be assume that the
> average pitch in each dyad and triad is A-440?

I guess that would seem like a reasonable place to start... Anybody
else??

________ _____ ___ _
Joseph Pehrson

Paul Erlich wrote:

> though there is far less consistency in what
> constitutes a "typical gamelan timbre" than in what constitutes a
"harmonic overtone
> series". In any case, the effect would require a very extensive training
period with the
> timbre in question in order to be measurable.
>

I can't agree with this point more. A very small change in the shape of a
gong or a metallophone can alter the proportions of the partials
dramtically. While one can detect some broadly-drawn spectral patterns
which distinguish the various instrumental types (and distinctiveness is
essential to the orchestration style: witness Widyanto's complaint that Lou
Harrison's aluminum instruments all sounded the same), envelope is a
co-decisive elements here. The more I play and sing Javanese music, the
more I am persuaded that it is the very presence of such a variety of
spectra within single instrumental classes that has made it possible for
listeners to accept such wide temperings (and gamelan tunings are indeed
temperaments, intended to represent three or more tonal transpositions of a
model scale with a limited set of fixed pitches).

To this persuasion, one caveat must be added, and that is that ensembles do
exist in Java and Sunda which are mostly or exclusively composed of
instruments with harmonic spectra. The players in these ensembles have no
difficulty in tuning away from simple just intervals (e.g. van Zanten's
study of Cianjuran showed a preference for major thirds around 100 cents).
But, again, there is some evidence that players are trying to "cover up"
these distances from harmonic series intervals: flute and rebab players as
well as singers use vibrato, and in Javanese citeran, courses are often
tuned to only loosely approximate unisons.

To be complete, I should note that there is an informal, non-systematic, use
of "mistuned unisons" between instruments in Java. This increases volume and
creates a vibrato on the vox humana principle; whether this vibrato is a
desired timbre is difficult to assess, in any case it does provide an
additional degree of the above mentioned intonation cover. It is in Bali
that one finds a systematic use of this principle. I'm less qualified to
write about this, but (c.f. Michael Tenzer's new and excellent study on
Kebyar) the idea is to have a fixed rate of beating throughout the ensemble.
This is obviously a tricky thing to achieve with limited technical means,
but perhaps the diversity of spectral proportions allows considerable
lattitude. However, with cscound or acid wave or mathematica, it's an easy
task to simulate this environment. With two instruments tuned within the
critical band, the apparent pitch is the average. Project a given scale
throughout all the registers, and then assign two instruments to each tone,
differing from one another by a constant number of hertz, whose average
fundamental is equal to the desired pitch. (There -- I gave away my current
project!)

Daniel Wolf