From a July 11 2001 private communication to Joe Monzo (posted with Monz's permission):

Solving for Pierre's "Sonace_Degree" (when including the prime 2) , and in the general case where the particular unison vectors values would influence the (summation) determinant's terms associated with primes 3 and 5 in various different ways, appears to yield some truly puzzling results. I wondered if you have considered this phenomena, as well. In a recent tuning-math post responding to Pierre's Message #287:

From: "monz" <joemonz@y...>

Date: Fri Jun 22, 2001 1:34 am

Subject: Re: [tuning-math] Re: Sonance degree (DEFINITION)

> ----- Original Message -----

> From: Pierre Lamothe <plamothe@a...>

> To: <tuning-math@yahoogroups.com>

> Sent: Thursday, June 21, 2001 11:10 PM

> Subject: [tuning-math] Re: Sonance degree (DEFINITION)

>

>

>

PL:

> Hi Monz,

>

> I used abstract language to establish a generalized property, but the

thing

> is much more simple than it seems.

> <etc.>

JM:

>But this post is indeed very clear! >Thanks so much. I have

>only glanced quickly at it, but will >study it more fully.

So here's my perplexment. Pierre seems to include the contribution of the prime number 2 in his main tuning post #18625, Feb 12 2001, where he states (page 4, bottom)"thus we have to use only 'reduced modular complexity'" [or,octave-reduced] when he also simultaneously did apply the contribution of the prime number 2 in his calculations of his "complexity" and "sonance" (page 4, top).

Yet in his recent tuning-math post #272, Jun 21 2000, "Sonance degree (DEFINITION)", and post #287, Jun 22 2001, "Re: Sonance degree (DEFINITION)" he clearly utilizes the number 2 (to the absolute value) of the exponent E which exist for the various rational (octave-reduced) possibilities of the "Zarlino" scales rotation in his example. Although his "Sonance_degree" is normalized by a constant which is proportional to the term of the (summation) determinant which is associated with the prime number 2, this value is only a scalar constant.

Here's the rub:

(1) For all distinct rational intervals resulting from such scale rotation as he describes and cites in his example, IF:

(a) The prime 2 of the (octave-reduced, I presume??) interval has a zero or positive value, THEN increasing the power of the prime two by one (equalling transposition of an octave upward for the rational interval) INCREASES the Sonance_degree, thus REDUCING the intervals (relative) "con-sonance" (all positive values, and relative to a value of zero for the 1/1 tonic), ELSE (it must be true that);

(b) The exponent of the prime 2 is of a negative value, AND increasing the power of the prime two by one (equalling transposition of an octave upward for the rational interval) *DECREASES* the Sonance_degree, thus *INCREASES* the intervals (relative) "con-sonance" (all positive values, and relative to a value of zero for the 1/1 tonic).

What are the implications of such characteristics of the function, I ask?

Is Pierre intending to mean by this functional property that certain intervals relative to the tonic become MORE "con-sonant" when raised by an octave, while certain others [depending on their ORIGINAL(octave-reduced)power of the prime number 2], will become LESS "con-sonant"???

The ratio (in the case of my calculation for 17 distinct pitches - which include 1/1, 1/2 and 2/1 - and the 14 intervals centered around the...1/1 tonic in your version of the 22-note Indian sruti)

of these two types of intervals as listed above is one-to-one, 8 of one, 8 of the other! Quite democratic, and non-biased!

Yet, utterly strange. My brain is bent by it, indeed. Could mother nature be coaxed into yielding more than 50% "con" (rather than "dis") - "cordance" by simply, it seems to me, sounding its tones within a (lower) octave (relative to the 1/1), where the exponent of the prime number 2 is assuredly of a *negative* arithmetic value??? Something here does not feel right...Or does it? There are certainly MORE harmonics per unit hertz (CPS) as well as per unit octave in the spectrum,

when considered relative to the bandwidth equal to the tuning of the 1/1 tonic, and when one sounds the tones of any of the intervals of any scale progression at a pitch BELOW the reference 1/1 tonic.

But how can such downward octave transposition magically impart values of the "Sonance_degree" that tend toward "con-sonance" or "con-cordance" simply by trending downward...Am I missing something which is *obvious* to others here???

I suppose (thinking out loud) that if they ALL trended downward in "Sonance_degree" *together*, then all that would matter is each individual value's magnitude *relative* to some sort of averaging,

or summation, of ALL of the various values of "Sonance_degree" (SD) which are present (for the intervals which are being considered).

Ratiometric indicators utilizing the SD of the 1/1 tonic as a reference point(including logarithms) cannot be taken, since the reference value (denominator of the arithmetic argument equal to the SD of the 1/1 tonic) is equal to ZERO...and would be arithmetically undefined.

A linear or root-mean-square average would work nicely, but what would it mean? If (for, at least in the case of the 14 octave-reduced notes, relative to the 1/1 tonic from your 22-note Indian sruti diagram), 50% of the intervals are "con" - "sonant" or "cordant, and 50% are "dis" - "sonant" or "cordant", then an averaging or summation process would seem to record only a slow, and seemingly fairly uniform decrease in the individual various values of SD for each scale interval in the process, as all of those scale intervals (which are relative to some reference tonic) would in such a case be subject to...*equal* (it has been presumed)... decreases in pitch on an octave basis. And, it would seem, thus would follow an uneventful monotonic decreasing trend of each of the individual values of a ("normalized" by average/summation) "Sonance_degree" for each of those individual value considered

(where some averaging/summation process provides a reference value which is responsive to a summation of the variations in the individual values of SD for each scale interval which is undergoing a shift downward (or upward) by 2^(integer) power.

If Pierre meant for SD to be only valid for "octave-invariant" (reduced or increased) ratios between 1/1 and 2/1 (or possibly between 1/2 and 2/1) then WHY does he even...include it (the prime number 2 and it's exponents)...in his example calculations? It seems there must be reason.

If, however, one (or a group of any number which is less than all of the other tones, excepting the 1/1 tonic) of the intervals is reduced in pitch by integer powers of the prime number 2 ONLY, and the others are *NOT* (as in playing the musical 5th, as well as any group of other tones, at a pitch below the tonic by some number of sub-octaves, etc.), then we have in such a (normalized, or relative)"Sonance_degree" (NSD?), as detailed above, an exotic, and *octave-variant*, numerical quantity which would seem to (in some way) represent the "con" or "dis" - "sonance" or "cordance" of multiple simultaneously sounded tones, relative to the 1/1 tonic.

At that point,...why not apply any chord within the scale (with any of the intervals of the chord played in any octave), and...just...watch the "sonance"/"cordance" meter!?!?

I ask you, my friend, is this *too* good to be *true*, or, has mother nature actually been so tamed by Pierre Lamothe's algorithm?

Or has Gill simply stumbled across only what seems intuitively obvious to certain other amazing minds (such as Lamothe's)?

PS- Or, has Gill, in his nieve passion, blithely ignored the total of N^2-N other combinations of intervals present (where N equals the number of degrees of the scale)? Nevertheless, all of the interactions present between the simultaneously sounded tones (and complex tones with energy at all multiples of each fundamental included, at that!) could be calculated. The ultimate mystery is not whether such numerical values could be rigorously calculated by a computer, but, instead, it is the question of whether human auditory perception, being a (rather) highly non-linear (and, in some respects, time-variant coefficient) process actually treats such complex spectral input (as our "wonder-machine" could) as a linear *superposition* of effects, in like manner to our machine, the ear being a logarithmic device in amplitude (log/log slope of apporximarely 2/3), which creates additional sum and difference products at sound levels experienced in performance/listening [this is essentially an "inter-modulation distortion" of harmonic and non-harmonic distortion products (sum and difference frequencies) of the multiplicative co-mixing of each of the individual spectral component with *all* of the other spectral components].

I read that "loudness was found to increase perceptual judgements of dissonance" (Plomp, Levitt, 1969, and Kameoka and Kuriyagawa a few years later), the latter also reporting that the "perceived dissonance of two single partial tones was greater when the low frequency tone was of greater amplitude than the high frequency tone", which consistent with inter-modulation distortion as it typically occurs, due to the sound pressure level (SPL) increasing by an exponent of between 1.5 and 2.0 for each decrease in frequency of one octave (an exponent of 1.5 being a "pink", or equal energy-per-octave, noise spectrum).

I read that in 1939, Sandig found that "dichotic" (stereo) "experiments of consonance perception showed that dissonant tones are perceived as nuetral when split into two sets of harmonics presented to seperate ears, so who's going to ever be able to really state what's going on?

The only thing that seems clear, and this seems problematic for a number of lines of thought which (for simplicitys sake) do not factor in the partials of each individual note simultaneously sounded (as such notes are generated by real-world tone generators, excepting flutes and synthesizers),

is that combinations of two sinusoidal wave forms of various frequency ratios have been reported (by Plomp and Levelt, 1965) to, "not play any significant role when presenting pairs of pure tones", and, in other experiment(s) in the same year, reporting that pairs of sinusoidal tones at "critical band" frequency ratios, "the two sine waves interact to produce dissonant sensations". It would seem to me, based upon the limited set of studies relied upon herein, that, with exception to such "critical band" frequency ratios between two sinusoidal tones, the "sonance" or "cordance" relies entirely on the prescence/absence of energy which is attributible to the partials of those fundamentals existing together with the spectral energy of those two fundamentals (in the two-tone case, at least), and where the octave in which the (complex, real-world) tone is sounded (relative to the 1/1 tonic) cannot ultimately be ignored in analysis.

What are your thoughts on this?, J Gill

PS - I am very interested in anyone's (included Mr Lamothe's, if he is able to find the time) feedback on this.

In the tuning-math message (#499) partially quoted below I wrote:

>Date: Thu, 12 Jul 2001 22:10:55 -0700

>To: tuning-math@yahoogroups.com

>From: J Gill <jgill99@imajis.com>

>Subject: Lamothe SD and Octave Modularity

>

>If Pierre meant for SD to be only valid for "octave-invariant" (reduced or >increased) ratios between 1/1 and 2/1 (or possibly between 1/2 and 2/1) >then WHY does he even...include it (the prime number 2 and it's >exponents)...in his example calculations? It seems there must be reason.

It may be that I have "read too much" into Mr Lamothe's intentions regarding application of his "Sonance_degree".

Possibly (as his main tuning post #18625, Feb 12 2001, page 4, bottom section may suggest), Pierre intends for his "Sonance_degree" (which I have abbreviated as SD in my writings) to be applied exclusively to a group of interval ratios which possess a contribution of the power of the prime number 2 in their prime factorizations which results in that group of interval ratios having a numerical value of between 1/1 and 2/1 (or, perhaps between 1/2 and 2/1) relative to the 1/1 tonic, ONLY. As a result, the algorithm would not address "octave-variance" of scale intervals which exist at higher/lower octaves relative to the 1/1 tonic, and are sounded simultaneously with the 1/1 tonic.

What do you think? I believe that no one, including myself, wishes to erroneously interpret Mr Lamothe's intentions, and I hope that he may see this message and possibly find the time for a brief response. I hope that my writings have clearly framed the questions, and would appreciate it if others have suggestions regarding ways to clearly frame the questions, in addition to speculation as to the possible answers to these questions, that they post them to tuning-math, or email them to me directly.

Best Regards, J Gill