Your response clarified your position in some detail; it will be useful if certain other features in your approach are spelled out (I am writing in parallel to re-reading your text):
(1) Does your central pitch processor depends on harmonic partials? Real instrumental sounds deviate significantly from a ideal series. Does your processor make tolerance decisions at both the individual instrument (pitch) and the instrumental ensemble (chord) levels? It was for precisely this complication that I made my initial assumptions of a sine wave orchestra.
(2) Are amplitude and phase differences among the partials ignored?
(3) Would it be terribly difficult to couple your root-finding function with some harmonic distance function in order to address chord progressions?
(4) The limits on amplitude and durational aspects limit the algorithm to a part of the world�s music; would you define this more precisely? In general, I find this to be the most difficult part of your work (and of other research along these lines) in that you have reified aspects particular to a small musical repertoire as psychoacoustical facts.
Some comments:
(1) Naturally, if a triad is tuned 5:6:7, the fundmental frequency of the series on 1 is reinforced; what happens, when it is tuned /7:/6:/5, or in a temperament where the approximation of 7 is poor? If I recall correctly, Boomsliter and Creel demonstrated that in harmonic contexts both trained and untrained subjects had a preference for the real interval of 225/128 over 7/4, thus showing that a lower position within a harmonic series was not necessarily preferable to some other, more complex relationship.
(2) My musical experience suggests, however, that diminished, augmented, and other intervallically symmetrical pitch complexes function in tonal music as ambiguous - neither major nor minor - structures. For this reason, I introduced the _neutral_ characterization for pitch spacings that are neither harmonic nor subharmonic.
(3) 54:64:81 is a chord that I am able to identify immediately in relationship to a previously known tonic of 2^n. I recognize it in both terms of harmonic distance from 2^n, from the size of the melodic intervals which I have learned to distinguish (thus at least part of my recognition is cultural), and from beating. Moreover, my recognition of this chord cues me in on the entire field of relationships including this chord and the chord proceeding it. It is this field that seems to determine musically dynamic characteristics: chord _progression_; tonicity; _resolution_; surprise, etc..
(4) The melodic difference between 9/8 and 10/9 is too real to be dismissed as something "handled on paper". Even if the listener doesn�t get the melodic difference immediately (something heard all the time in good string playing or unaccompanied vocal music), eventually the comma difference will either manifest itself in a _bad_ fifth, which is immediately _heard_ as such, or a _return_ to a tonic that has missed its mark by a comma (the ear is much faster at picking this up than the eye anyways).
(5) Which interpretation does this support? Your own analysis of the minor triad prefers a subharmonic description. You can not have it both ways. For this reason, I find it sufficent to base my spacing description upon fundamental frequencies and a single calculated fundamental or guiding tone.
(6) The harmonic and subharmonic series are two transfinite series with different memberships; the second may be constructed from the first, but a one to one mapping of the second onto the first is not possible. This is however, coincidental to my point, that we process subharmonic structures less well than harmonic due to conceptual and computational difficulties. Example: it is much easier to compare three objects with one object than it is to compare one object with thirds of an object.
(7) Prof. Hennix has worked specifically with musical issues; most closely with the late John Myhill and with La Monte Young. (Her closest purely mathematical collaborations were with Myhill and with A.S. Yesinin-Volpin.) She may represent a different school of mathematical foundations from your own, but her work is anything but drivel; indeed intuitionist proof demands are the most rigorous.
Daniel Wolf
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 29 Sep 1996 17:28 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04120; Sun, 29 Sep 1996 16:27:03 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA02153 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA05024; Sun, 29 Sep 1996 08:26:03 -0700 Date: Sun, 29 Sep 1996 08:26:03 -0700 Message-Id: <199609291719.RAA18439@teaser.teaser.fr> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu
There was at the end of september, in the Abbaye of Royaumont (near Paris, France)an important 'colloque' on microtonality organized by "Voix= Nouvelles"=20 (Marc Texier, director).=20
Here is the programm of the differents events:
*09/19 =20 Marcel PERES (Directeur de l'ensemble Organum): Le probleme du temperament= =20 dans l'interpretation de la musique gregorienne.
Pedro MEMELSDORFF (Directeur de l'ensemble Mala Punica): Microintervalles=20 dans l'oeuvre de Matteo da Perugia.
Klaus HUBER (compositeur): Le tiers de ton chez Guillaume Costeley et de nos jours.
Pierre CAZES (claveciniste): Echelles et temperaments =E0 l'epoque baroque.
Marc TEXIER (Directeur de Voix Nouvelles): Modalite, timbre :=20 les deux vecteurs d'un depassement du temperament egal au XXe siecle.
Pascale CRITON (musicologue): Les approches ultrachromatiques de la premiere= =20 moitie du XXe siecle (Wyschnegradsky, Carrillo, Haba).
At the end of this first day there was a presentation of the followings=20 instruments : Clavicytherium, Piano-Carrilo in 1/16 of tone, microtonal=20 clavicembalo of Alain Louvier and microinterval midi keyboard by=20 Fran=E7ois Paris.
After Dinner, the public presentation of the "Theorie de la Pansonorite" of= =20 Wyschnegradsky realised by Franck Jedrzejewsky and edited by Contrechamps, Geneve.
*09/20 Makis SALOMOS (musicologue): Xenakis et la recherche hors-temps.=20 Theorie des Cribles. Echelles non-octaviantes.
Pierre-Abert CASTANET (musicologue)/ Alain LOUVIER (compositeur): L'ecole microtonale en France (Bancquart, Louvier, Barbaud, Philipot,= Ballif, Marie, Levinas...).
James TENNEY (compositeur): Les nouveaux temperaments aux Etats-Unis.
Christian POCHE (ethnomusicologue)/Klaus HUBER: Les maqams arabes et leur=20 utilisation par Klaus Huber.
Fran=E7ois PICARD (musicologue)/XU SHUYA (compositeur): Echelles dans la=20 musique chinoise et leur utilisation par Xu Shuya.
Pascal GALLOIS (bassoniste): Les nouvelles techniques du basson illustrees par des extraits d'oeuvres de Berio, Hersant et Taira.
Harry STARREVELD (fl=FBtiste): La flute en 1/4 de ton.
*09/21 Philippe LEROUX (compositeur): De l'action sonore au processus.
Fran=E7ois PARIS (compositeur): "Anamorphose et oblique dans la musique d'aujourd'hui". Vers de nouveaux temperaments. Claviers midi microtonaux.
Marc TEXIER: Temperaments, echelles sonores, microintervalles : une nouvelle frontiere de la musique.
Brian FERNEYHOUGH (compositeur)/Trio Contrechamps (I.Magnanat,vl; J.Sulem,= alto; D.Haefliger, vlc): Analyse du Trio =E0 cordes de Brian Ferneyhough. Oeuvre en 1/8 de ton. Execution commentee de cette oeuvre en avant premiere.
This "colloque" was, at my knowledge, the first big event on microtonality= =20 in France with such a diversity of approaches. I wish it is the beginning of a real interest for other tunings...