Some "middle thirds" for Cameron

Hi Cam,

Ok we know that 32/27 is a minor third 5:6, 34/27 is a major 4:5 and as I said 33/27 is closer to minor 5:6 than major 4:5 (there is only one best match for each interval). Next, raising the 27 to 28 gives 32/28 = 8/7 which is (I believe) already a flattened minor. And 32/30 = 16/15 which is closer to a semitone. This leaves only 32/29 to discover. (29 + 32)/(5 + 6) = 61/11, x 5 = 27.7... and x 6 = 33.23...so its not a good match. Trying 8:9, 9:10 also gives bad results. The answer is 10:11, 10 x 61/21 = 29.047...and 11 x 61/21 = 31.952...Therefore this is not a minor third at all but closer to a second.

Now if we go up an octave, we have the range between 54:64 and 54:68. Again we can raise the denominator or lower the numerator. I've done the latter which need only to find 67/54 which ends up being a major third 4:5 (Again there is only one best match per interval). After you try 64/55 etc...you can then take it up the next octave and so on till your hearts content. I'm betting that 6:7 will be in there somewhere. In any case, you see that it does not necessarily say that these middle thirds don't exist or are not musically valuable. It is what it is.

-Rick

Ric, I'll have to read your paper again. I just don't understand what you are doing- the waveform comparison in the paper is completely dependent on Fourier analysis but you say you're not using Fourier analysis, and I just don't get it. :-(

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> Hi Cam,
>
> Ok we know that 32/27 is a minor third 5:6, 34/27 is a major 4:5 and as I said 33/27 is closer to minor 5:6 than major 4:5 (there is only one best match for each interval). Next, raising the 27 to 28 gives 32/28 = 8/7 which is (I believe) already a flattened minor. And 32/30 = 16/15 which is closer to a semitone. This leaves only 32/29 to discover. (29 + 32)/(5 + 6) = 61/11, x 5 = 27.7... and x 6 = 33.23...so its not a good match. Trying 8:9, 9:10 also gives bad results. The answer is 10:11, 10 x 61/21 = 29.047...and 11 x 61/21 = 31.952...Therefore this is not a minor third at all but closer to a second.
>
> Now if we go up an octave, we have the range between 54:64 and 54:68. Again we can raise the denominator or lower the numerator. I've done the latter which need only to find 67/54 which ends up being a major third 4:5 (Again there is only one best match per interval). After you try 64/55 etc...you can then take it up the next octave and so on till your hearts content. I'm betting that 6:7 will be in there somewhere. In any case, you see that it does not necessarily say that these middle thirds don't exist or are not musically valuable. It is what it is.
>
> -Rick
>

Sorry Cam,

I missed this reply for some reason. Ok, yes by all means the original waves are still dependent on Fourier analysis as always. But what I'm talking about doesn't show up on one. I'm not trying to disprove or replace FA in any way, but merely show that there are other types of analysis which is beyond its scope. Given two sine waves of frequency a and b then their Fourier components are just a and b, which we knew already. Of course I don't want to rewrite the paper again here but will just say that the period of the first cycle of every non-epimoric interval b:a is not the difference frequency (a - b) as we would expect but is in fact the value (a + b)/(p + q), where p and q are lower numbered wholes that are uniquely determined for each a and b. The graphs are meant just as a demonstration of how much these match. So for eg, the minor thirds 16:19 and 27:32 both have q:p = 5:6 and (p + q) = 11. What's important is that this q:p is always given by the maths, not guessed at. The solution between the times between the first two largest maxima is always T = (p + q)/(a + b). Since a and b are known then only one value of p and q will satisfy the equation (while 2 + 9, 7 + 4 etc...also = 11, these will also not give a match). And this math'l fact 'models' how we can group or associate each interval with its JI counterpart. Does that make more sense? (Sorry if it doesn't but it's always hard trying to explain maths in English).

-Rick

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Ric, I'll have to read your paper again. I just don't understand what you are doing- the waveform comparison in the paper is completely dependent on Fourier analysis but you say you're not using Fourier analysis, and I just don't get it. :-(
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > Hi Cam,
> >
> > Ok we know that 32/27 is a minor third 5:6, 34/27 is a major 4:5 and as I said 33/27 is closer to minor 5:6 than major 4:5 (there is only one best match for each interval). Next, raising the 27 to 28 gives 32/28 = 8/7 which is (I believe) already a flattened minor. And 32/30 = 16/15 which is closer to a semitone. This leaves only 32/29 to discover. (29 + 32)/(5 + 6) = 61/11, x 5 = 27.7... and x 6 = 33.23...so its not a good match. Trying 8:9, 9:10 also gives bad results. The answer is 10:11, 10 x 61/21 = 29.047...and 11 x 61/21 = 31.952...Therefore this is not a minor third at all but closer to a second.
> >
> > Now if we go up an octave, we have the range between 54:64 and 54:68. Again we can raise the denominator or lower the numerator. I've done the latter which need only to find 67/54 which ends up being a major third 4:5 (Again there is only one best match per interval). After you try 64/55 etc...you can then take it up the next octave and so on till your hearts content. I'm betting that 6:7 will be in there somewhere. In any case, you see that it does not necessarily say that these middle thirds don't exist or are not musically valuable. It is what it is.
> >
> > -Rick
> >
>

Now I understand- but I think that it is not true that we necessarily identify every interval with a proportion found in the harmonic series, rather, that we subconciously do a whole lot of comparing and contrasting within the field of the complete spectra we hear.

>And this math'l fact 'models' how we can group or associate each >interval with its JI counterpart.

"Group" and "associate" are better. But it is not really "JI" that is the "field", it is the spectrum.

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> Sorry Cam,
>
> I missed this reply for some reason. Ok, yes by all means the original waves are still dependent on Fourier analysis as always. But what I'm talking about doesn't show up on one. I'm not trying to disprove or replace FA in any way, but merely show that there are other types of analysis which is beyond its scope. Given two sine waves of frequency a and b then their Fourier components are just a and b, which we knew already. Of course I don't want to rewrite the paper again here but will just say that the period of the first cycle of every non-epimoric interval b:a is not the difference frequency (a - b) as we would expect but is in fact the value (a + b)/(p + q), where p and q are lower numbered wholes that are uniquely determined for each a and b. The graphs are meant just as a demonstration of how much these match. So for eg, the minor thirds 16:19 and 27:32 both have q:p = 5:6 and (p + q) = 11. What's important is that this q:p is always given by the maths, not guessed at. The solution between the times between the first two largest maxima is always T = (p + q)/(a + b). Since a and b are known then only one value of p and q will satisfy the equation (while 2 + 9, 7 + 4 etc...also = 11, these will also not give a match). And this math'l fact 'models' how we can group or associate each interval with its JI counterpart. Does that make more sense? (Sorry if it doesn't but it's always hard trying to explain maths in English).
>
> -Rick
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > Ric, I'll have to read your paper again. I just don't understand what you are doing- the waveform comparison in the paper is completely dependent on Fourier analysis but you say you're not using Fourier analysis, and I just don't get it. :-(
> >
> > --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > >
> > > Hi Cam,
> > >
> > > Ok we know that 32/27 is a minor third 5:6, 34/27 is a major 4:5 and as I said 33/27 is closer to minor 5:6 than major 4:5 (there is only one best match for each interval). Next, raising the 27 to 28 gives 32/28 = 8/7 which is (I believe) already a flattened minor. And 32/30 = 16/15 which is closer to a semitone. This leaves only 32/29 to discover. (29 + 32)/(5 + 6) = 61/11, x 5 = 27.7... and x 6 = 33.23...so its not a good match. Trying 8:9, 9:10 also gives bad results. The answer is 10:11, 10 x 61/21 = 29.047...and 11 x 61/21 = 31.952...Therefore this is not a minor third at all but closer to a second.
> > >
> > > Now if we go up an octave, we have the range between 54:64 and 54:68. Again we can raise the denominator or lower the numerator. I've done the latter which need only to find 67/54 which ends up being a major third 4:5 (Again there is only one best match per interval). After you try 64/55 etc...you can then take it up the next octave and so on till your hearts content. I'm betting that 6:7 will be in there somewhere. In any case, you see that it does not necessarily say that these middle thirds don't exist or are not musically valuable. It is what it is.
> > >
> > > -Rick
> > >
> >
>

Hi Cam,

Yes I'm still trying to find a mathematically more rigorous way of describing the original intervals a and b other than just saying they are "large-numbered" and "non-epimoric". And then there is the problem of what to call the "lower-numbered" ones p and q. "epimoric" is not quite correct because not all of them are (99/70 for eg gave lower ones as 17/12). So I opted for "JI". As I learn more about how to find p and q for any given a and b, something should jump out at me eventually and I'll go back and redefine them.

The musician part of me has always felt that we needed a more definite way of grouping and associating interval types beyond just saying that they were "close". "How close?" is the question. When does minor 1.2 become major 1.25 etc...? And after years of eliminating many other possibilities which had certain desirable features and not others (mainly exact GCD's and difference tones), the mathematician in me senses that this approach has legs. For example, when below you say "complete spectra", the fact that these associations hold when we add any amount of upper harmonics (in the article under that heading), and also holds when we take ratios between these 'GCD's', shows that WLOG they can be applied to the entire spectrum (and we can now even reassess what a spectrum itself is. On most 'real' instruments for eg harmonics only approximate 1,2,3,... so they might already be these 'GCD' intervals). At any rate what you say here and what I'm doing are not mutually exclusive. It extends the playing field of interpretation without necessarily changing it. I'll let you know more when I know myself.

-Rick

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Now I understand- but I think that it is not true that we necessarily identify every interval with a proportion found in the harmonic series, rather, that we subconciously do a whole lot of comparing and contrasting within the field of the complete spectra we hear.
>
> >And this math'l fact 'models' how we can group or associate each >interval with its JI counterpart.
>
> "Group" and "associate" are better. But it is not really "JI" that is the "field", it is the spectrum.
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > Sorry Cam,
> >
> > I missed this reply for some reason. Ok, yes by all means the original waves are still dependent on Fourier analysis as always. But what I'm talking about doesn't show up on one. I'm not trying to disprove or replace FA in any way, but merely show that there are other types of analysis which is beyond its scope. Given two sine waves of frequency a and b then their Fourier components are just a and b, which we knew already. Of course I don't want to rewrite the paper again here but will just say that the period of the first cycle of every non-epimoric interval b:a is not the difference frequency (a - b) as we would expect but is in fact the value (a + b)/(p + q), where p and q are lower numbered wholes that are uniquely determined for each a and b. The graphs are meant just as a demonstration of how much these match. So for eg, the minor thirds 16:19 and 27:32 both have q:p = 5:6 and (p + q) = 11. What's important is that this q:p is always given by the maths, not guessed at. The solution between the times between the first two largest maxima is always T = (p + q)/(a + b). Since a and b are known then only one value of p and q will satisfy the equation (while 2 + 9, 7 + 4 etc...also = 11, these will also not give a match). And this math'l fact 'models' how we can group or associate each interval with its JI counterpart. Does that make more sense? (Sorry if it doesn't but it's always hard trying to explain maths in English).
> >
> > -Rick
> >
> > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > >
> > > Ric, I'll have to read your paper again. I just don't understand what you are doing- the waveform comparison in the paper is completely dependent on Fourier analysis but you say you're not using Fourier analysis, and I just don't get it. :-(
> > >
> > > --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > > >
> > > > Hi Cam,
> > > >
> > > > Ok we know that 32/27 is a minor third 5:6, 34/27 is a major 4:5 and as I said 33/27 is closer to minor 5:6 than major 4:5 (there is only one best match for each interval). Next, raising the 27 to 28 gives 32/28 = 8/7 which is (I believe) already a flattened minor. And 32/30 = 16/15 which is closer to a semitone. This leaves only 32/29 to discover. (29 + 32)/(5 + 6) = 61/11, x 5 = 27.7... and x 6 = 33.23...so its not a good match. Trying 8:9, 9:10 also gives bad results. The answer is 10:11, 10 x 61/21 = 29.047...and 11 x 61/21 = 31.952...Therefore this is not a minor third at all but closer to a second.
> > > >
> > > > Now if we go up an octave, we have the range between 54:64 and 54:68. Again we can raise the denominator or lower the numerator. I've done the latter which need only to find 67/54 which ends up being a major third 4:5 (Again there is only one best match per interval). After you try 64/55 etc...you can then take it up the next octave and so on till your hearts content. I'm betting that 6:7 will be in there somewhere. In any case, you see that it does not necessarily say that these middle thirds don't exist or are not musically valuable. It is what it is.
> > > >
> > > > -Rick
> > > >
> > >
> >
>