I did an Altavista search for Eitz and Tonwort and found the following as a new book at the Eda Kuhn Loeb library at Harvard. If anyone has the time to peruse it, Nadja Schnetzler () would most likely be very grateful and show her gratitude with chocolate .
47.Eitz, Carl, 1848-1924. Das Tonwort : Bausteine zur musikalischen Volksbildung. Leipzig : Breitkopf & Hartel, 1928. -- [HOLLIS# AIP5489]
On Thu, 1 Oct 1998, bram wrote: > My guess, without knowing exactly what those papers say, is that they are > generally based on an algorithm for cancelling out the beats of any two > notes. I know the beats can be essentially calculated using continued > fractions, so I'm guessing there's a rather straightforward way of > calculating, given a*sin(x) + b*sin(y), a and b being volumes and x and y > being wavelengths, there's a rather straightforward way of calculating a > bunch of other sin functions which when added will cancel out all the > beats.
I don't think Bill is canceling out the beats in an antinoise sense; rather, he is matching tuning and timbre so that the intervals of the scale line up with the partials of the sound. That is, he gets rid of the beats the old-fashioned, JI way: by making the overtones match up.
--pH http://library.wustl.edu/~manynote O /\ "Churchill? Can he run a hundred balls?" -\-\-- o NOTE: dehyphenate node to remove spamblock. <*>
On Thu, 1 Oct 1998, Paul Hahn wrote: > On Thu, 1 Oct 1998, bram wrote: >> I know the beats can be essentially calculated using continued >> fractions, so I'm guessing there's a rather straightforward way of >> calculating, given a*sin(x) + b*sin(y), a and b being volumes and x and y >> being wavelengths, there's a rather straightforward way of calculating a >> bunch of other sin functions which when added will cancel out all the >> beats. > > I don't think Bill is canceling out the beats in an antinoise sense;
Actually, thinking further on it, I don't think that would even be possible, considering that the beats don't actually exist physically, but arise because of nonlinearities in the auditory system.
--pH http://library.wustl.edu/~manynote O /\ "Churchill? Can he run a hundred balls?" -\-\-- o NOTE: dehyphenate node to remove spamblock. <*>
> On Thu, 1 Oct 1998, Paul Hahn wrote: > > On Thu, 1 Oct 1998, bram wrote: > >> I know the beats can be essentially calculated using continued > >> fractions, so I'm guessing there's a rather straightforward way of > >> calculating, given a*sin(x) + b*sin(y), a and b being volumes and x and y > >> being wavelengths, there's a rather straightforward way of calculating a > >> bunch of other sin functions which when added will cancel out all the > >> beats. > > > > I don't think Bill is canceling out the beats in an antinoise sense; > > Actually, thinking further on it, I don't think that would even be > possible, considering that the beats don't actually exist physically, > but arise because of nonlinearities in the auditory system.
Ah, but they do :)
After thinking about it a bit more, I realize that the beats need to be 'augmented' rather than 'cancelled', since they correspond to regions of decreased volume.
The very simplest example is sin(x/n)+sin(x/(n+1)) since that only has one beat.
Consider the case where n=4. It starts out at about double the volume of an individual sin wave, then the volume decreases until the two waves exactly cancel out at x=10, then come to complement each other and x=20, and continue to alternate.
If a sin wave is now added which becomes most angular at the place where the other two waves cancel out, it will tend to make the volume consistent throughout. Specifically, the function
sin(x/4) + sin(x/5) + 5*sin((x+10)/40)
should have less of a beat.
I say less instead of none because I'm fairly certain I did something wrong with the volumes. Can someone verify for me whether sin(x) and a*sin(x/a) are of the same volume? If they are, I know how to make it exact.
Note that phase information is very important here - simply adding a third sin wave of the correct wavelength but wrong phase will be unlikely to do a good job fixing the beat.
Unfortunately, I don't have the tools handy to make neat diagrams or sound files to illustrate the above - could anybody suggest some good free ones which work under windows?
I'm still not sure what to do when the phases of the two waves don't line up exactly, and haven't really looked into how to fix ratios with more beats (like 21/17) but I think I'll be able to work that out later - it's an *interesting* problem.
I really doubt that it could be done as bram describes (and if it can, I suspect it would do odd things to the timbre and/or harmony), but hey, go for it. Let us know how it works out.
--pH http://library.wustl.edu/~manynote O /\ "Churchill? Can he run a hundred balls?" -\-\-- o NOTE: dehyphenate node to remove spamblock. <*>
> As Bill Sethares shows in his recent papers on timbre and > consonance,("Local consonance and the relationship between timbre and > scale"1993.
I suggest you get a copy of his book/CD. It's called ... (ooo, I hope I get these words in the right order!) ... Tuning Timbre Spectrum Scale, and published by Springer.
>I have a somewhat better idea of what I mean by 'beats' now. If take the >volume of a sound wave (I forget the three letter acronym off the top of >my head)
RMS?
>and express it as the sum of a bunch of volumes of individual >sine waves, (no, I don't know a simple way of doing that transformation)
Is it unique?
>the beats are the waves in the resulting summation which weren't added >together to get the original waveform.
Sounds like you're talking about interference, which is a much more general phenomenon than beating.
>I think making a sound whose timbre is a bunch of pure tones is even >easier - just take the derivative of the square root of the sum of their >squares, but that just seems *too* easy.
I don't understand what you mean by that, but any sound can be expressed as a (possibly infinite) bunch of pure tones.
Sorry for the delay with my 2 cents. I haven't had access to the internet for over a week, and when I did yesterday I had to do it on Windows, an absurd program called Explorer, and an absolutely idiotic-unintuitive mail (apparently) program called Outlook (which managed to destroy some of my mail). Coincidentally all 3 apps are made by the same obscure North-West American company.
I don't know much about all this yet, but might what Brad is talking about and Sethares' approach be achieved using additive synthesis ?
It just seems the logical solution.
There's bound to be a lot of info on the mighty mighty infobahn about additive. Greg Sandell's SHARC Timbre database also (I think) has some info that is of use, since there's been mention on the Sharc list about people using the data to build sounds in the Kawai K-5000 additive synths (I wonder if they have any microtuning capability) .
If you don't have access to any real-time additive synthesis system (Kyma, K-5000, MSP,etc.) , there's always building samples in any of the free software synthesis systems (Csound, Syd, etc.) and loading them into a sampler like Vsamp (on a PowerMac).
BTW, Kyma is now just $3300 for the base unit (four 80 MHz dsps, 96 MB RAM , digital i/o) and it's new and improved: "Based on the new Motorola DSP-56309 chip running at 80 MHz, the Capybara 320 base unit is now available for US$3300 and provides a minimum of four DSPs (expandable to 28) with multi-channel I/O, synchronization to external clocks, and 96 MB of sample RAM (expandable to 672 Mb) in a low-noise, rack-mountable package connected to a desktop or laptop Macintosh or Windows PC." This beaut is getting more and more affordable (compared to many of the commercial synths and samplers around,it's actually cheap !) and powerful......
-Drew
---------------------------------------------------------------------------- "Out of the ash I rise with my red hair And I eat men like air."-Sylvia Plath
> I don't know much about all this yet, but might what Brad is talking > about and Sethares' approach be achieved using additive synthesis ?
Yep.
> there's always building samples in any of the free > software > synthesis systems (Csound, Syd, etc.) and loading them > into a sampler like Vsamp > (on a PowerMac).
I should probably look into CSound, the sound capabilities for Java (my development environment of choice) appear to not have been written yet.
I've now gotten a copy of Roederer's book, "The Physics and Psychophysics of music" and have to say the first chapter is one of the best philosophical tracts I have ever read.
I've also got a copy of Sethares's book on special order, but it will be a matter of weeks or months before it arrives (grrr). Anybody have a suggestion for how to order it other than just going through amazon.com?
I have a somewhat better idea of what I mean by 'beats' now. If take the volume of a sound wave (I forget the three letter acronym off the top of my head) and express it as the sum of a bunch of volumes of individual sine waves, (no, I don't know a simple way of doing that transformation) the beats are the waves in the resulting summation which weren't added together to get the original waveform.
I think making a sound whose timbre is a bunch of pure tones is even easier - just take the derivative of the square root of the sum of their squares, but that just seems *too* easy.
> Bram wrote, > > >the beats are the waves in the resulting summation which weren't added > >together to get the original waveform. > > Sounds like you're talking about interference, which is a much more > general phenomenon than beating.
I think you're right, although I've done some poking around and haven't been able to find anything which describes interference much past the usual constructive versus destructive interference. Could anyone point me to a more in-depth explanation of it?
-Bram
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End of TUNING Digest 1569 *************************