source file: mills2.txt Date: Fri, 27 Sep 1996 11:54:30 -0700 Subject: RE: More on spacing: response to PAULE From: PAULE I am going to respond to this as I read it, so this won't be a perfectly formulated reply. I will post the derivation of my tonalness algorithm seperately, as it is mathematically painful and will be difficult to notate in ASCII. A general remark: The central pitch processor is the mechanism by which we perceive a set of harmonic partials as a single note -- the virtual pitch -- with an associated timbre. Just about any instrument in the symphony orchestra serves as a perfect example. Whether this process is inborn or acquired, some claim prenatally, is a matter of debate. Its existence is not. >how is >it that you are able to decide that sine waves are "distinguished and >digestable" when described in harmonic context but suddenly not so when >renotated as subharmonic? A large or infinite harmonic series of sine waves will be interpreted by the central pitch processor as a single complex tone, having a clear fundamental. All combination tones of the sine waves will fall into this same harmonic series, reinforcing the sensation. A large subharmonic series will confuse the central pitch processor, as each pair of tones implies a different fundamental. Often the perfect fifth will win out due to its importance in defining the virtual pitch. The combination tones of the subharmonic series will not conform to the subharmonic series; in the case of an large subharmonic series, they will fill up nearly all of pitch space, confusing the central pitch processor further and creating lots of roughness. >(2) Or do you define as subharmonic any complex without 2^n as the lowest >member? No. A subharmonic complex is one whose relative periods can be expressed with smaller integers than the relative frequencies. Clearly there is no distinction for two-note complexes. >A spacing theory would compare the notations to determine which >characterization is more likely (if neither is, then the neutral >characterization is chosen). Yes, I agree -- if roughness between harmonic partials comes into play. For sine waves, the subharmonic representaion is irrelevant. >Your algorithm - and I do wish to examine it >in detail - strikes me as a root direction theory (and more in the >Hindemith than in the Rameau direction). Yes, I believe it partially is that. I have found that there are already successful algorithms for computing the roughness (Plomp & Levelt) aspect of consonance (K&K, for example). However, I did not come across any algorithms that applied the root-finding function of the central pitch processor to chords in a mathematically consistent way, and tonalness -- or clearness of root perception -- is the other component of consonance. So I put such an algorithm together, and it's a total idealization, but is intended to be a good approximation as long as some partials fall within the optimal frequency range for the central pitch processor. Given more psychoacoustical data, such as how the resolution of the central pitch processor varies for a given set of frequencies and amplitudes, the algorithm can be easily made more flexible. >(3) You cite several psychoacoutistical certainties in support of your >algorithm, I am especially curious to learn of experimental results >supporting (a) "in a purely psychoacoustical sense, a chord played will >evoke certain interpretations in terms of the harmonic series"; this may be >what your algorithm tells me, and is a conjecture that I would like to >support, but I would like to find experimental evidence in this direction >(and I have encountered evidence to the contrary: see Boomsliter and Creel >on the 7th partial); My musical experience certainly differs from this; for example, how do you explain the fact that a 5:6:7 diminished triad evokes the missing root of the dominant seventh chord, if not for the importance of the 7th partial? Terhardt has done much work in this area as well. >(b) "the ears central pitch processor simply doesnt >go that far"; I am completely confused by this: what do you mean by >"central pitch processor", and how far, exactly does it go? The harmonic >progression I presented, and its Pythagorean interpretation are simply too >obiquitous to be dismissed in this way. You yourself stated that the fundamental of a harmonic relationship can't be too low in pitch for the relationship to be perceived; even with that simplistic a model, how can you expect 54:64:81 to be perceived as such, and with what root? >(4) The sequential procedure would be something like the following: given >the complex 500 Hz, >600 Hz 750 Hz, the first ratio interpreted would be the 3/2 fifth of >750/500, the initial interpretation would be 3/2 in a harmonic series over >fundamental 250Hz; however, not finding an intermediate tone between 2 and >3, successive fifths in the series are scanned to find a similar complex, >examining each n(2,3) until an intermediate tone is located that matches >the complex. When, for example 10,12,15 over a fundamental of 50 Hz is >selected, then it is compared with its subharmonic notation /6,/5,/4. If >this characterization is more simple, then it is chosen. Once again, there is no psychacoustic mechanism by which subharmonic relationships of sine waves can be heard! >(Need I add the >fact that the difference tones of this complex are all within the spectrum >of 50 Hz? And this supports which interpretation, pray tell? >That certain tuning >systems, temperaments among them, may map onto harmonic series is probably >true, but it requires a definition of intervallic tolerance that seems to >me to be musically (i.e. culturally, and thus limited to particular times >and places) defined, and probably outside of the domain of psychoacoustics >proper: I disagree. The definitions of intervallic tolerance are nothing more and nothing less than a charcterization of the psychoacoustic phenomena which distinguish just intervals -- namely, consonance. This is subdivided into roughness, whose tolerance has to do with critical bands and beating; and tonalness, whose tolerance has to do with the precision with which frequency information is transmitted to the central pitch processor. You are correct that both aspects are not independent of amplitude and duration, but that just means that an algorithm such as mine would ideally have to take these things into account. In principle, it is still the right way of looking at things. >Your statement _the ear doesnt continue its harmonic series math from >one chord to the next_ is unsupported, and probably unsupportable. I did not mean for it to be taken in such a strong sense. Clearly there are strong elements of commonality between one chord and the next when their relationship is a simple one; the dominant-to-tonic progression is a case in point. And clearly the central pitch processor can extract some information on virtual pitch from non-simultaneous tones -- the virtual pitch mechanism is not dependent on phenomena associated with exact simultaneity, such as difference tones. But while the difference between a 9:8 and a 10:9 is easy to perceive harmonically, when these intervals represent the melodic intervals between the roots of non-adjacent chords in a progression, the difference comes down to how the progressions are handled on paper. >(7) I cannot resist a few more remarks on the subharmonic. First, although >for musical purposes we never need get past fairly small numbers, while >every harmonic complex may be renotated as subharmonic and vice versa, due >to one of those nice paradoxes in the fundamentals of mathematics - and >beyond my ability to explain - the number of members in the harmonic series >is greater than that in the subharmonic series. What does this sentence mean? And I have a good knowledge of mathematics, fundamental and otherwise -- what on earth are you talking about? >Second, the _descent to the >infinitesimal_ represents to me a series of increasingly complex - indeed, >counterintuitive - calculations, hence my speculation about increased >mental activity. The ascription of quantum effects to the infinitesimal >comes from the intuitionist C. Hennix, resident mathematician at the >Institute for Psychoanalysis in Paris; Is this in a context relevant to music, or any other real-world applications? I'll bet my hat it's pseudo-scientific drivel. >Third, musical examples of the >subharmonic are readily available in the repertoire, cf Martin Vogels book >on the _Tristan_ chord, or this little example from the Rondo K.494, m >164-165, where the left hand sustains for two measures the chord c4 d4 f4 >ab4, which moves (does not resolve) to c4 f4 a4, i.e subharmonic seventh >chord from c moving to f major 6-4. Mozart uses this chord, here and >elsewhere, in a voice leading distinctive from his use of diminished and >dominant seventh chords. By my spacing characterization, the fourth c-f >would be too important to overlook, and the traditional (in American >practice) _half diminished_ description seems not very useful. If this >were in just intonation, I would choose a subharmonic characterization /8 >/7 /6 /5 over the harmonic 105 120 140 175. I am curious to learn how >Pauls algorithm will interpret this chord (16 18 21 26? or 24 27 32 38?). I strongly agree that subharmonic characterizations can be useful in musical practice and to the future development of music. But the essence of this characterization lies in the partials, which acheive minimum roughness when the fundamentals are arranged in _either_ a harmonic or a subharmonic series. My algorithm deals not with roughness but with tonalness, and ascribes no harmonic interpretation of c4 d4 f4 ab4 more than a negligible amount of certainty. For a coarse enough frequency resolution, 9:10:12:14 would win out, but still by a negligible amount. Both the 3:4 interpretation of c4 f4 and the 5:8 interpretation of c4 ab4 will be more powerful; therefore the chord will not be heard as a single harmonic complex. Let me reiterate my view of the equal-tempered minor triad with harmonic partials. I would tend to characterize it as a subharmonic complex, since (a) the level of roughness is similar to that of a major triad, and thus a near-local-minimum, due to the individual intervals being close to 3:2, 5:4, and 6:5 and (b) several possible harmonic representations accodring to my algorithm (10:12:15, 16:19:24) lead to essentially the same interpretation: that of a perfect fifth with an added non-harmonic note. Since the exact numbers in (a) are clearly more important than the exact numbers in (b), the former should be included in an overarching representation, thus: 1/6:1/5:1/4. If a chord has favorable roughness characteristics but only tolerable tonalness characteristics, it just might be a subharmonic complex. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 27 Sep 1996 22:39 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02198; Fri, 27 Sep 1996 22:39:33 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA02799 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA00329; Fri, 27 Sep 1996 13:39:32 -0700 Date: Fri, 27 Sep 1996 13:39:32 -0700 Message-Id: <22960927203522/0005695065PK3EM@MCIMAIL.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu