Rev: Lamothe SD and "Octave Modularity"

>>>>> From a July 11 2001 private communication to Joe Monzo (posted with >>>>> Monz's permission):
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>>>>>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.
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>>>>>What are your thoughts on this?, J Gill
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>>>>> PS - I am very interested in anyone's (included Mr Lamothe's, if he >>>>> is able to find the time) feedback on this.