Free copies of my doctoral dissertation on expanded tunings

Dear Friends of The Tuning Forum,

I'd be happy to send a copy of my recently completed doctoral dissertation (see abstract below) to any interested person on this list. It
consists of 113 sheets of paper (printed on both sides = 226 pages) and a cassette tape with examples. I'm giving it away free of charge, but I can
no longer afford to pay for the copying and the US mail.

Expenses are:

copying: 226 pags. x $ 0.05 = $ 11.30 + cassette tape + US snail = about $ 15 (add 2 or 3 $s if you want it wire-bound).

If you're poverty-stricken but tuning-enthusiastic, just send me a check for whatever you can afford and I'll mail you my tractatus:

Enrique Moreno
724 Arastradero Rd. Apt. 206
Palo Alto, California 94306

************* ABSTRACT: *************************

EMBEDDING EQUAL PITCH SPACES AND THE QUESTION OF EXPANDED CHROMAS:
AN EXPERIMENTAL APPROACH


A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF MUSIC
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


By
Enrique I.Moreno
December 1995

ABSTRACT


One of the most intriguing characteristics of traditional tuning systems is that they are based on the assumption that tones separated by
one or more octaves (powers of two) belong to the same pitch class. In turn, octave similarity is the fundamental axiom of traditional Western tonal
and modal harmony and music theory. In previous work on new tuning systems by Bohlen and by Mathews, Pierce, et al., a new tuning was suggested in
which a basic exponential cycle k produces intervals given by k^(1/13), when k = 3. The concept of equal temperaments with intervals given by
k^(1/n) [where k = 3, 5,6,7...N, n and m, are natural numbers not equal to x^y (where x, y = 1,2,3 ... N)] is generalized and elaborated upon.

The resulting tunings, called "expanded tunings" are consequently organized according to the mathematical coherence given by the periodicity
of k. However, the *perceptual* organization of such tunings may seem unclear, since, in the absence of powers of two (octaves), perceptual
similarity would have to be elicited by k-type intervals (termed "morenoctaves" in the context of an expanded tuning), which, at first hand, seems
theoretically possible in light of the the various similitudes between octave and k-interval perception/repetition rate, similar patterns in
Terhardt's proposed Pitch Processor, and harmonic rate coincidence. Since neither Mathews, Pierce et al. nor any other researchers have investigated
the question of perceptual similarity when k > 2, precise definitions of the problems of chroma spaces and octave and perceptual similarity are
analyzed in the light of: 1) the newly proposed families of expanded tunings, 2) their practical meaning in terms of music composition, 3) current
research on related psychoacoustical-cognitive problems, 4) three experiments specifically designed to explore for the first time the plausibility
of cognitive coherence in all expanded tunings (expanded chromas). To accomplish these goals, the class of expanded tunings is explained as a class
of log-based equal tunings derived from the relationships implied in the harmonic series, and the main theoretical properties of these unexplored
musical tuning systems explained.

A review of the literature reveals furthermore that, psychoacoustically, there is no reason to reject the similarity of morenoctaves
provided that certain conditions hold � for example, that a tuning based on the similarity of morenoctaves does not contain close approximation to
octaves. In reference to composition with these systems, solutions to practical problems are proposed. A system to implement expanded tunings using
current technology is explained, and problems concerning the notation of music in expanded tunings are examined. The use of color, for example, is
proposed as a solution to the dissociation between pitch class and pitch name in tunings with more or less than twelve classes.

In order to illustrate the practical meaning of expanded chromas in reference to music composition, and compositional problems, several
originally composed musical examples are examined, and related problems to composition discussed. The abstract dimensionality of pitch is next
discussed in the light of current models, and a new geometrical, basic three-dimensional model of pitch � instead of the traditional two-dimensional
one � is proposed as a better means to represent expanded chromas.

A paradigm of empirical research, based on this model is explained in relation to the problem of obtaining experimental evidence for the
cognitive reality of expanded chroma, and the assumptions necessary for the specific design of a main experiment and two secondary ones are
examined.

The results of three experiments (described herein) show that, among trained musicians, subjects have a spontaneous relative sense of
expanded chroma because statistically, they are capable of recognizing the similarity of harmonic functions of a fragment of music in expanded
tunings under the operation of rigid transposition and � with much more difficulty � under the operation of rigid chord inversion. This means that
perception of the dimension of chroma in music � hitherto unexplored � is a capability of the human mind, and that expanded tunings can be coherent.

Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Thu, 4 Jan 1996 05:21 +0100
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id UAA06590; Wed, 3 Jan 1996 20:21:07 -0800
Date: Wed, 3 Jan 1996 20:21:07 -0800
Message-Id: <9601040419.AA18880@ ccrma.Stanford.EDU >
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu