Re: [tuning] Digest Number 1106

Thanks for responding. I've had the feeling lately that my postings were totally for nought.

> Dave Keenan wrote,
>
> >Do you have a response to Dan Wolf's smallest-number rule which seems
> >to handily encapsulate both complexity and span in a very simple rule?
>

Paul Erlich answered:

> Although the smallest-number rule agrees quite nicely with the original
> harmonic entropy formulation using a Farey, rather than Tenney, series, I'd
> ask Daniel what he would find when holding the _upper_, rather than the
> _lower_, note constant. I'd conjecture that he will have to invoke a rule
> that is more complex than just a largest-number rule or anything like that .
> . .

Yes, it does get more complex due to an asymmetry in hearing, but perhaps it can be fixed with a rule related to absolute frequency that would, in turn, have the side benefit of more accurately defining the range and fixed pitch criteria left diplomatically undefined in my post.

With the lower tone fixed, the numerator can increase arbitrarily until the two tones are so far apart that no rational relationship is perceived, but with the upper tone fixed the denominator rule seems to hold until (a) the lower tone descending below normal hearing or (b) the interval itself crosses a threshold of discrimination. Unfortunately, in normal musical contexts either of these circumstances could easily occur. A range for the lower limit of hearing is known and easy to incorporated. It's my guess that the discrimination threshold is related to the absolute frequency of an implied fundamental, from which a harmonic series or approximation thereof could be constructed, from which the lowest absolute frequency for a given class of intervals could be derived.

i.e. let's say for argument's sake that that lowest fundamental, among whose partials the intervals are consonant has a frequency of 27.5 Herz. Thus a major third of 137.5Hz/110Hz would be the lowest appearance of 5:4 as a sensory consonance. I won't commit myself now either to the exact frequency or to an exact harmonic series (perhaps the scale of subjective equal intervals (barks* or hectomels) is more accurate), but it sure starts to sound like what I've experienced in real, existing musics, and corresponds, for example, with the rule of thumb of orchestrator John Prince that winds cannot play a Major third below A-c# or Bb-d without sounding muddy.

I'm a practical musician, so I'm after a method for evaluating intervals that can be done on the quick in my head while composing. It strikes me that the combination of a smallest denominator rule scaffolded onto a harmonic series over a fundamental around 25-30 Herz is probably more than adequate.

What about triads and n-ads? Again, I suspect that a scaffolding of a lowest harmonic series is a good place to begin.

Daniel Wolf
Budapest
http://home.snafu.de/djwolf/

* I've recommended Klarence Barlow's _Bus Journey to Parametron_ (available from Frog Peak) too many times on this list, without, as far as I can tell, ever having anyone take up my suggestion. I hope that the recent recommendation of this text by Georg Hajdu will not fall on deaf ears either. It's a highly idiosyncratic document, wholly that of a professional composer and casual theorist, but it's a brilliant, and often very funny, read.

Daniel Wolf wrote,

>I'm a practical musician, so I'm after a method for evaluating intervals
that can be done on the quick in my >head while composing. It strikes me
that the combination of a smallest denominator rule scaffolded onto a
>harmonic series over a fundamental around 25-30 Herz is probably more than
adequate.

Would you have a "limit" above which the rule begins to break down, due to
confusion with simpler ratios? Perhaps a denominator of 13? Harmonic entropy
with a Farey series could model this situation quite well . . . Two high
notes about 30 Hz apart could come from high in the harmonic series above 30
Hz, but unless combinational tones were present, who would know?

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Daniel Wolf wrote,
>
> >I'm a practical musician, so I'm after a method for evaluating
intervals
> that can be done on the quick in my >head while composing. It
strikes me
> that the combination of a smallest denominator rule scaffolded onto
a
> >harmonic series over a fundamental around 25-30 Herz is probably
more than
> adequate.
>
> Would you have a "limit" above which the rule begins to break down,
due to
> confusion with simpler ratios? Perhaps a denominator of 13?
Harmonic entropy
> with a Farey series could model this situation quite well . . . Two
high
> notes about 30 Hz apart could come from high in the harmonic series
above 30
> Hz, but unless combinational tones were present, who would know?

I don't think you read my initial post closely. I specifically
allowed for tolerance but did not define it. I would not
characterize the rule as breaking down, but rather say that intervals
falling within the range of tolerance about a simpler ratio are heard
as out-of-tune versions of that simpler ratio.

This tolerance is in part frequency-dependent. A 9:7, for example,
will be heard at lower frequencies as an out-of-tune 5:4, but
beginning with a lower tone of around 175 Hz, it emerges as a
distinct interval with its own denominator class of sensory
consonance (don't believe me? compare 9:7s with harmonic spectra
where the lower tones are 150 and then 175 Hz - it's clear-cut). This
is predicted by the harmonic series scaffolding.

But the part of tolerance your after has to do with the approximation
of simpler ratios by much more complex ones. Just for kicks (Erv
Wilson's favorite phrase), let's try to define this tolerance band
using only denominator size and ignoring barks, critical band width
et al:

Since this approach is about lining up composite spectra, tolerance
will have a lot to do with the strength of those spectra. With
modern orchestra instruments, spectral strengths tend to fade out
sufficiently around the 16th partial, with early music instruments
extending, perhaps, to the 24th. (Wolfgang von Schweinitz's epic
piano trio "Franz and Morton" is all in natural harmonics through the
cello's 24th, and projects those relationships quite coherently; Paul
Zukofsky uses the 24th violin harmonic as an upper limit). So,
spectral considerations suggest that approximations of simpler ratios
by ratios with denominators greater than 24 will not be heard as
belonging to distinct denominator classes.

Here's another approach. Let's agree that we hear 81:64 as an out-of-
tune 5:4. The tolerance limit must be somewhere between 4 and 64.
Let's try to narrow that further. I don't happen to hear 32:21 as an
out-of-tune perfect fifth, but rather as having already crossed into
the domain of narrow minor sixths, so the limit is now somewhere
between 21 and 64. What about 6:5 and 32:27? Okay, I can buy 32:37 as
an out-of-tune 6:5. Now the limit's between 21 and 27.

24 sounds like a reasonable limit.

>Two high
> notes about 30 Hz apart could come from high in the harmonic series
>above 30
> Hz, but unless combinational tones were present, who would know?

Under the smallest denominator rule, each succesive pair of tones 30
Hz apart over the 30 Hz fundamental is an increase in denominator
thus a decrease in sensory consonance, although, since these
intervals are based on a high enough fundamental, these intervals
will be in voicings that are as consonant as intervals of the same
denominator class can be.

Daniel Wolf

Daniel Wolf wrote,

>I don't happen to hear 32:21 as an
>out-of-tune perfect fifth, but rather as having already crossed into
>the domain of narrow minor sixths, so the limit is now somewhere
>between 21 and 64.

I don't see how the former clause necessarily implies the latter. Wouldn't
you have to hear 32:21 as a distinct sensation, rather than merely "in the
domain of narrow minor sixths"? In fact, doesn't using the term "domain of
narrow minor sixths" already imply a coarseness of categorical perception
that would prevent you from distinguishing, say, 20:19, 19:18, or anything
in-between?

Daniel Wolf wrote,

>What about 6:5 and 32:27? Okay, I can buy 32:[2]7 as
>an out-of-tune 6:5.

But you've skipped over 13:11, which is closer. So perhaps things start
getting a bit fuzzy around 11?

I think they do, in some listeners and in some registers. Witness Dan
Stearns' opinion that 11:9 is more concordant than 9:7 or 11:8. And harmonic
entropy, my little model of "fuzziness", seems to predict this quite
clearly.