formulae: a minority report

I made a series of WAV files with sawtooth waveforms and a smooth envelope
in the following series of intervals, with the lower tone set to 200, 300
and 400 Hz:

(0/0 --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---)

--- 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, 11/1, 12/1, 13/1, ...

3/2, --- 5/2, --- 7/2, --- 9/2, ---- 11/2, ---- 13/2, ...

4/3, 5/3, --- 7/3, 8/3, --- 10/3, 11/3, ---- 13/3, ...

5/4, --- 7/4, --- 9/4, ---- 11/4, ---- 13/4, ...

6/5, 7/5, 8/5, 9/5, ---- 11/5, 12/5, 13/5, ...

7/6, --- --- ---- 11/6, ---- 13/6, ...

8/7, 9/7, 10/7, 11/7, 12/7, 13/7, ...

9/8, ---- 11/8, ---- 13/8, ...
...

Please try this for yourselves! The impression of 'flatness', no particular
increase in dissonance, over the course of each horizontal series was
surprising as were the clear and discrete increases in dissonance from
series to series as d increased.

After a weekend of heavy listening, I'm willing to risk going out on a limb
(something I may very well regret) and support a measure for intervals with
harmonic timbres based on the size of the denominator alone.

Given that the differences in harmonicity over the n/1 series are small to
negligible, which is reasonable for dyads with harmonic timbres, I find
habit to be my only objection to extending the principle to each larger
denominator, and -- aside from reducing to lowest terms -- ignoring factors
altogether. Then consonance could be expressed simply as 1/d, dissonance
as d.

This leads to results that are a bit uncoventional. Ratios of high numbers
have high consonance when d is low. But these are supportable when heard
away from the background noise of existing repertoire.

Of course, it can't be all that easy. The frequency-dependent factors
(roughness, span/remoteness) require a supplemental calculation, as would
tolerance*. But dyads seem to be straightforward, with several plausible
measures floating around; the "holy grail" for tuning theory will be a
workable theory for all n-ads.

This does not mean that I am giving up on complexity measures based on
factoring. After tolerance is taken into account, I suspect that factoring
will play a role in measuring consonance at the triad and higher n-ad
level. The complexity of lattice moves largely define the identity and
coherence of harmonic and melodic progressions. Then there is the role of
factoring in the perception of complex rhythms, but that's a topic for
another list... (is there an 'alternative rhythms list' out there?).

------
* to which Barlow's scale step property should be added (where the
acceptance of one interval identity does not conflict with the
identification of other scale members).

Daniel Wolf <DJWOLF_MATERIAL@compuserve.com> wrote:

>I made a series of WAV files with sawtooth waveforms and a smooth envelope
>in the following series of intervals, with the lower tone set to 200, 300
>and 400 Hz:
>
>(0/0 --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---)
>
> --- 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, 11/1, 12/1, 13/1, ...
>
> 3/2, --- 5/2, --- 7/2, --- 9/2, ---- 11/2, ---- 13/2, ...
>
> 4/3, 5/3, --- 7/3, 8/3, --- 10/3, 11/3, ---- 13/3, ...
>
> 5/4, --- 7/4, --- 9/4, ---- 11/4, ---- 13/4, ...
>
> 6/5, 7/5, 8/5, 9/5, ---- 11/5, 12/5, 13/5, ...
>
> 7/6, --- --- ---- 11/6, ---- 13/6, ...
>
> 8/7, 9/7, 10/7, 11/7, 12/7, 13/7, ...
>
> 9/8, ---- 11/8, ---- 13/8, ...
> ...
>
>
>Please try this for yourselves! The impression of 'flatness', no particular
>increase in dissonance, over the course of each horizontal series was
>surprising as were the clear and discrete increases in dissonance from
>series to series as d increased.
...
>Then consonance could be expressed simply as 1/d, dissonance
>as d.

Try extending those series backwards too, so they become:

1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, 11/1, 12/1, 13/1, ...

1/2, --- 3/2, --- 5/2, --- 7/2, --- 9/2, ---- 11/2, ---- 13/2, ...

1/3, 2/3, --- 4/3, 5/3, --- 7/3, 8/3, --- 10/3, 11/3, ---- 13/3, ...

1/4, --- 3/4 --- 5/4, --- 7/4, --- 9/4, ---- 11/4, ---- 13/4, ...

1/5, 2/5, 3/5, 4/5, --- 6/5, 7/5, 8/5, 9/5, ---- 11/5, 12/5, 13/5, ...

1/6, --- --- --- 5/6, --- 7/6, --- --- ---- 11/6, ---- 13/6, ...

1/7, 2/7, 3/7, 4/7, 5/7, 6/7, --- 8/7, 9/7, 10/7, 11/7, 12/7, 13/7, ...

1/8, --- 3/8, --- 5/8, --- 7/8, --- 9/8, ---- 11/8, ---- 13/8, ...
...

For the ratios less than one, the tone set to 200, 300 or 400 Hz will now
be the higher tone. I expect you still to experience no change in
dissonance across each row and an increase with increasing denominator d.
But then we realise that the interval 2/3 is exactly the same size as the
interval 3/2 etc. They are the same interval. And yet we have decided that
2/3 is more dissonant than 3/2. A contradiction? No. Paul Erlich has
already explained this, although I think the full implications have not
been appreciated.

Those who say dissonance is proportional to d are keeping the lower
frequency constant. If you keep the higher freq constant then it's
proportional to n (when the fraction is in the usual improper form (n>d)).
Those who say n+d (or (n+d)/2) are comparing intervals about roughly the
same center.

This implies that lower frequencies are inherently more dissonant than
higher ones. To use n+d is just to remove the effect of the average
frequency on dissonance, and just leave that which depends on the ratio
itself.

Thanks Dan, for helping to make this clearer.

Regards,
-- Dave Keenan
http://dkeenan.com

Daniel Wolf wrote,

>After a weekend of heavy listening, I'm willing to risk going out on a
limb
>(something I may very well regret) and support a measure for intervals
with
>harmonic timbres based on the size of the denominator alone.

This was exactly the result I posted a long, complicated "derivation" of
some time ago. It was adapted from the appendix of Van Eck's _J. S.
Bach's Critique of Pure Music_; I had to correct an error in that and
proceed from there on my own. My harmonic entropy model is based on
exactly the same framework as this derivation, with the additional
assumptions (also taken from Van Eck) that one's perception of an
interval is "blurred", that this blurring takes the form of a Guassian
probability distrubution centered around the actual interval, and that
the probability of a given just interpretation of the interval is the
area under the Gaussian between the mediants between that just interval
and its neighbors in the Farey series. My only original contribution was
to propose calculating the entropy of the probability distribution
(defined as -sum(p*log(p))) as a measure of dissonance.

Basically the derivation was a proof (original to me) that the
(log-frequency) distance between the mediants between a just interval
and its neighbors in the Farey series, assuming the numbers making up
the just interval are much smaller than the order of the Farey series,
is inversely proportional to the denominator of the just interval. In
particular, the width is approximately equal to 2/Nd, where d is the
denominator and N is the order of the Farey series. (BTW, if d is not
small compared to N, then the actual width may be as small as 1/Nd, but
never exceeds 2/Nd).

Daniel, when I posted that long, complicated derivation, it was part of
a discussion I was having with you. I'm glad you've come to agree with
its result!

Dave Keenan wrote,

>Those who say dissonance is proportional to d are keeping the lower
>frequency constant.

My derivation assumes that the upper frequency is held constant! Did you
mean upper, or do we have something really interesting to discuss?

Paul Erlich wrote:

>Dave Keenan wrote,
>
>>Those who say dissonance is proportional to d are keeping the lower
>>frequency constant.
>
>My derivation assumes that the upper frequency is held constant! Did you
>mean upper, or do we have something really interesting to discuss?

We have something really interesting to discuss. Also, see my response in
Digest 112 to Carl's "huh?" I'm sorry I incorrectly attributed this to you,
Paul.

I understood that Dan made 3 sets. Each set contained all the intervals
that he listed. The first fixed the *lower* tone at 200Hz, the second fixed
the *lower* tone at 300Hz and the 3rd fixed the *lower* tone at 400Hz. And
I understand he perceived that within each set, all ratios with the same
denominator (in lowest terms) had about the same disssonance, and that
dissonance increased with increasing denominator.

Dan, please confirm or deny. (Unlike your government's policy on telling my
State Emergency Service whether its submarines are carrying nuclear weapons
when they visit my city's port)

Regards,
-- Dave Keenan
http://dkeenan.com

Message text written by INTERNET:tuning@onelist.com
>please confirm or deny. (Unlike your government's policy on telling my
State Emergency Service whether its submarines are carrying nuclear weapons
when they visit my city's port)

Regards,
-- Dave Keenan
http://dkeenan.com<

Aye. Yes. Ja.

DJW