A mathematical idea i guess?

[VicDiesel]

Because chords are in fact ratios,

....you should take a geometric mean rather than arithmetic.

Who know, it could give something interesting.

Victor.

[schnapsglas]

Because chords are in fact ratios,

....you should take a geometric mean rather than arithmetic.

Who know, it could give something interesting.

Victor.

Hm, I know what you mean (pun?) but more complicated. In equal temperament the geometric mean of an octave does turn out to be the tritone and it doesn't generate any more tones beyond that. But if you just take the tritone interval, it doesn't give you the minor third you would expect. The right way is to do it logarithmically or deal with fractions in just intonation.

And that would be a headache because the link Sendy post is practically what I mean.

[ntom]

I was going to make a witty quote response and contribute nothing to this thread, but I was too lazy to.

[Gamma-UT]

Because chords are in fact ratios,

....you should take a geometric mean rather than arithmetic.

Who know, it could give something interesting.

Victor.

The trouble with (Western) music is that an entire mythology of it being based on simple mathematics has built up - it all dates back to the days of musica ficta and musica recta (and stoked by the likes of Kepler and Newton later on). The simple ratios thing only works for the perfect intervals. Once you get past those, you're into exponents and roots and very little that makes thinking about note relationships as arithmetically related worthwhile, unless you're building instruments or software. That's why Guido d'Arezzo 'authorised' the notes of musica ficta - it was impossible to reconcile anything but the blandest of music with the mathematical perfection of musica recta.

[schnapsglas]

I am still thinking about it, whether it deserves more thinking. An octave is basically just a canonical map from cyclic group Z -> Z/12Z = Z_12 and you could have map from Z that maps to frequencies defined by equal temperament, and we could also have exponential map x \mapsto 2^(x/12) as another one if necessary. This way we could just have chord notations like {0, 4, 7} like everyone is familiar with. Now that I've spelt out what everyone knows already,

The problem is we cannot keep generating tones without making it dissonant to something. One day if I have time I should clear this up combinatorially, but that woukd require (my hunch) assuming equal temperament matches just intonation. Which is not a big sacrifice to make, but I wonder if there would be any interesting result at all. Simple glance at Z_12 shows why augmented and diminished 7th are special chords (well, also lydian flat 13th), and why it HAS to be circle of fifths and why it is sufficient. But not sure if we can bring any "musical results" to the table with this method.

[VicDiesel]

The problem is we cannot keep generating tones without making it dissonant to something.

You are trying at the same

1. to come up with a general theory from the ground up,
2. while still holding in the back of your mind your intuitive notion of how music behaves.

Can't have it both ways. You need to define what dissonance is, in terms of your theory, and then you simply generate tones and state "this will be dissonant (in my definition) by such amount".

Victor.

[schnapsglas]

The problem is we cannot keep generating tones without making it dissonant to something.

You are trying at the same

1. to come up with a general theory from the ground up,
2. while still holding in the back of your mind your intuitive notion of how music behaves.

Can't have it both ways. You need to define what dissonance is, in terms of your theory, and then you simply generate tones and state "this will be dissonant (in my definition) by such amount".

Victor.

Hello fellow mathematician!

No, what I mean it is whatever you do, it will end up being dissonant. I don't have any intuitive notion of these "notes." But what I know is that there is some room these needs to have, and if you keep taking some frequency in the middle, you will eventually "hit" something dissonant. Not assuming anything other than the fact that dissonance increases when two frequencies are very close. So for some frequency f, you need (f-c, f+c) to be clear, but c depends on f and given enough f_1, f_2, .... with respective c_1, c_2, ... you cannot have them disjoint.

Well, but I think we don't need to get that far, as in practice the thirds are already quite off just intonation. If you know what I mean, it is impossible to squeeze 6 notes into an octave without having some really weird ratio.

[Gamma-UT]

Well, but I think we don't need to get that far, as in practice the thirds are already quite off just intonation. If you know what I mean, it is impossible to squeeze 6 notes into an octave without having some really weird ratio.

You may want to have a look at Bill Sethares' work. You can squeeze six notes into an octave. You can squeeze in 37. It's all about the timbre.

[schnapsglas]

Well, but I think we don't need to get that far, as in practice the thirds are already quite off just intonation. If you know what I mean, it is impossible to squeeze 6 notes into an octave without having some really weird ratio.

You may want to have a look at Bill Sethares' work. You can squeeze six notes into an octave. You can squeeze in 37. It's all about the timbre.

I would say that's not fair, since we are talking about sine waves (I think we are?). Pitch is a very hard thing to define, after all. And of course, I said "really weird ratio," as in something that doesn't pop up in 12-note just intonation or even something similar. Might need to prove that.

I mentioned six because I did do a proof some time ago about the pentatonic scale and why you cannot fit an extra note without -- actually what was that about? I might need to find it.

(Nevermind, what I found is the proof to: "A scale containing maximal number of notes has no tritone and semitone intervals if and only if it is mode of pentatonic, i.e. pentatonic is unique scale with this property up to modes.")

[Gamma-UT]

I would say that's not fair, since we are talking about sine waves (I think we are?). Pitch is a very hard thing to define, after all.

Sethares deals with that (it might not be on the website but it's in the book). Sines don't tend to sound as grossly dissonant with, say, stacked minor seconds as saw waves.

[schnapsglas]

I would say that's not fair, since we are talking about sine waves (I think we are?). Pitch is a very hard thing to define, after all.

Sethares deals with that (it might not be on the website but it's in the book). Sines don't tend to sound as grossly dissonant with, say, stacked minor seconds as saw waves.

Interesting, thanks for setting me straight. I will have a look at it.

[BERFAB]

Because chords are in fact ratios,

....you should take a geometric mean rather than arithmetic.

Who know, it could give something interesting.

Victor.

The trouble with (Western) music is that an entire mythology of it being based on simple mathematics has built up - it all dates back to the days of musica ficta and musica recta (and stoked by the likes of Kepler and Newton later on). The simple ratios thing only works for the perfect intervals. Once you get past those, you're into exponents and roots and very little that makes thinking about note relationships as arithmetically related worthwhile, unless you're building instruments or software. That's why Guido d'Arezzo 'authorised' the notes of musica ficta - it was impossible to reconcile anything but the blandest of music with the mathematical perfection of musica recta.

Fascinating stuff re: musica ficta and recta. I would've read the entire WIKI, but I was too lazy.

In any event, a couple of observations. First, was it even possible at the dawn of Western music to quantify tonal construction in terms of frequencies? As it was not, it is understandable why ratios would be the prevailing measurement. Sure, NOW we have it down to a science, and anyone with access to electricity is able to exactly replicate precise tone. But this was not always the case.

Second, the more important MATH involved in Western music has less to do with frequency (see above) and more to do with our system of notation and time. BPM, 4/4, 6/8, dotted half, 1/32, all fit conveniently into a very interesting mathematical construct. [Old geezer speaking for a moment:] Many of the seasoned KVR'ers (myself included) look back wistfully (well, maybe not so much) on the days when we would calculate delay times by hand based upon the BPM conversion tables. A fun way to pass the time to say the least.

Cheers
-B

[JumpingJackFlash]

The trouble with (Western) music is that an entire mythology of it being based on simple mathematics has built up ... The simple ratios thing only works for the perfect intervals. Once you get past those, you're into exponents and roots and very little that makes thinking about note relationships as arithmetically related worthwhile

This is all very true, probably the best advice you're going to get on a thread like this. (Though the stuff about musica ficta is largely unrelated).

not sure if we can bring any "musical results" to the table with this method.

No, you won't.
Music isn't an exact science. It is what it is because it evolved that way. Trying to shove a load of mathematics onto it is largely futile and will lead to erroneous conclusions at best.

[Gamma-UT]

Fascinating stuff re: musica ficta and recta. I would've read the entire WIKI, but I was too lazy.

Should you feel the laziness wearing off the Cambridge History of Western Music Theory goes into a lot more detail. A lot more detail. Plus all the stuff on Kepler as well as Schenker. Piling through that lot, you begin to wonder how western music theory didn't implode on itself. Or maybe it did...

[VicDiesel]

the Cambridge History of Western Music Theory

Thank you thank you thank you!

I did not know that book existed. Sounds like just what I need.

Victor.