Consonance measures and the physical aspects of prime limits

I'm sure this has been discussed before, so I don't need real discussion unless people are interested. Pointers to resources would be appreciated, though.

I'd like to understand more about why we use prime limits the way we do in creating tunings, and specifically why we don't differentiate between primes in the numerator and primes in the denominator. It seems that this would involve discussion of how the harmonic series and overtones relate to prime limits.

The reason I'm asking is that, at a surface level, it appears that having a 7/6 and an 8/7 would describe significantly different physical situations.

A 7/6 is 7/6 the frequency of a 1/1, which means it's 6/7 the wavelength. Physically, that means that I can get that tone if I stop a string 1/7th along its length.

Similarly, if I have a 8/7, I can get that tone if I stop a string at 1/8 of the way along its length.

If I stop a string at 1/8 of the way along its length, its endpoint is already a node (point of non-vibration) of the 1/1's 8th harmonic; its first harmonic has a node at (7/8)/2 = 7/16 of the way along the string, its second harmonic has a node at (7/8)/3 = 7/24 of the way along the string, and its third harmonic has a node at (7/8)/4 = 7/32. 16, 24, and 32 are all multiples of two, which means -- I may be being naive here, but this seems right to me -- that they relate to the fundamental tone closely: 16ths of the string length = 4 octaves, 24ths = 4 octaves + 5th, 32nds = 5 octaves.

If I stop a string at 1/7 along its length, its endpoint is a node of the 7th harmonic, but its first, second, and third harmonics are at nodes of (6/7)/2, (6/7)/3, and (6/7)/4 = 3/7, 2/7, and 3/14. None of those are as closely related to the fundamental tone in the way the 1/8 situation was: 7ths and 14ths of the string length only yield harmonic 7ths.

I know that not every instrument generates tones in the same way, but by the time a wavelength hits your ear, the compression wave could be modeled in a similar fashion.

So it seems to me that not all prime limits are created equal: The 8/7 should seem more closely related to 1/1 than 7/6 should, or at least more closely than its Tenney Height suggests, and in consonance measures like Tenney Height we should handle prime factors in the numerator differently from prime factors in the denominator.

Incidentally, that could fit with 16/9, being 2-limit in the numerator and 3-limit in the denominator, sounding "more normal" (Michael's words) than 7/4, and 15/8 sounding not much worse than 11/6 despite the fact that it's double the Tenney Height.

This is about as far as I've thought this out. Does anyone know where this has been discussed before, or have any opinions on the matter?

Thanks,
Jake

On Thu, Apr 21, 2011 at 1:30 PM, Jake Freivald <jdfreivald@...> wrote:
>
> If I stop a string at 1/8 of the way along its length, its endpoint is
> already a node (point of non-vibration) of the 1/1's 8th harmonic; its
> first harmonic has a node at (7/8)/2 = 7/16 of the way along the string,

7/16 of the way along the string from the bridge, you mean. From the
nut, it'd be 1/8 + (7/8)/2 = 9/16, unless I'm misunderstanding
something. Also, to eliminate mass chaos and panic, the convention is
to generally refer to the octave as the second harmonic around these
parts, because otherwise everyone gets extraordinarily confused.

> its second harmonic has a node at (7/8)/3 = 7/24 of the way along the
> string, and its third harmonic has a node at (7/8)/4 = 7/32.

Likewise with the above, you end up with 1/8 + (7/8)/3 = 10/24 = 5/12
of the way along the string, and also 1/8+(7/8)/4 = 11/32 of the way
along the string.

> 16, 24, and 32 are all multiples of two, which means -- I may be being naive here,
> but this seems right to me -- that they relate to the fundamental tone
> closely: 16ths of the string length = 4 octaves, 24ths = 4 octaves +
> 5th, 32nds = 5 octaves.

That the 8th harmonic of 1/1 and the second harmonic of 8/7 share a
node on a string isn't, I don't think, something that's going to
affect the consonance and dissonance of intervals. This just stems
from the fact that 1/1 and 8/7 are both undertones of 8/1, so you've
rediscovered utonal intervals.

> I know that not every instrument generates tones in the same way, but by
> the time a wavelength hits your ear, the compression wave could be
> modeled in a similar fashion.

How do you see this as being related to the resulting compression wave?

> So it seems to me that not all prime limits are created equal: The 8/7
> should seem more closely related to 1/1 than 7/6 should, or at least
> more closely than its Tenney Height suggests, and in consonance measures
> like Tenney Height we should handle prime factors in the numerator
> differently from prime factors in the denominator.

It seems like what you want is a measure of consonance such that any
interval and its utonal equivalent are rated as being equal. Some sort
of octave-equivalent Tenney Height where you remove all powers of two
would handle that well enough. So 8/7 would reduce to 1/7, and would
have Tenney Height equivalent to 7/4, which would also reduce to 7/1.
But once you get away from dyads, I don't think things will work quite
as smoothly.

> Incidentally, that could fit with 16/9, being 2-limit in the numerator
> and 3-limit in the denominator, sounding "more normal" (Michael's words)
> than 7/4, and 15/8 sounding not much worse than 11/6 despite the fact
> that it's double the Tenney Height.

I don't see how it fits in, can you explain?

> This is about as far as I've thought this out. Does anyone know where
> this has been discussed before, or have any opinions on the matter?

I'm not sure this has been discussed before.

-Mike

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> I'd like to understand more about why we use prime limits the way we do
> in creating tunings, and specifically why we don't differentiate between
> primes in the numerator and primes in the denominator.

It's helpful to think in terms of groups, and I don't see much upside in not doing so, which has always been an objection to Michael's approach which does not allow for inversion in the octave. What, exactly, would you expect to get out of distinguishing primes in the numerator from primes in the denominator, other than in chord construction, where you might want all your 5s or 7s in one or the other? What's the practical use for it?

Gene>"It's helpful to think in terms of groups, and I don't see much upside in
not doing so, which has always been an objection to Michael's approach
which does not allow for inversion in the octave. "

   Which approach and how not so?   The only problems I see in an inversions are
A) That it may go from ratios that are more tolerable to error IE 10/9 to ones that are less IE 9/5 (the nearby 20/11 sounds MUCH more dissonant while between 10/9 and 9/8 sounds fairly much like either 10/9 or 9/8).
B) That having two notes repeated on different octave IE C5 and C6 at once does not change the mood but, rather, slightly elaborates it, if anything.

   Far as prime limit...I don't like it as a measure.  If anything it seems to improve the likelihood of odd-limit matches IE 9/8 and 27/16 form 3/2 even though 27/16 is "high limit".  But since how much gain of this sort you can get from using prime limit varies widely, I'd argue it only very vaguely tells you how capable a scale is.