Abstracts of Conference Papers and Other Presentations

PAPER SESSION I: Friday, April 6, 3:00-5:45

Thatcher Music Building Room 109

3:00 19-tET in a Renaissance Chanson by Guillaume Costeley
Presenter: Ralph Lorenz

One of the earliest uses of 19-tET occurs in "Seigneur Dieu ta pitié," a chanson by the Renaissance composer, Guillaume Costeley. Composed about 1558 and first published in 1570, the four-voice "Seigneur Dieu" calls for nineteen tones that are equally tempered, producing especially consonant thirds and sixths. While other contemporaneous theorists such as Zarlino and Vicentino had also proposed unusual divisions of the octave, the division into nineteen tones was original with Costeley, predating its use by Salinas in De Musica (1577).

Previous studies by Levy (1955), Dahlhaus (1963), and Godt (1969) have focused on this work's general background and historical context, issues of solmization, and proper interpretation. However, technological advances and the advent of new analytical methodologies have made it possible to view "Seigneur Dieu" from a fresh perspective. In this paper I will present a recording in 19-tET that was realized on a Kurzweil 2VX synthesizer. A modified version of the set theory that was originally developed by analysts such as Forte, Morris, and Straus is used to present harmonic materials. In classical set theory, pitch-class sets are reduced to equivalence classes based on transposition (Tn) and inversion (TnI); in this system, major and minor triads reduce to the same set-class type. For music of Costeley's era, however, it is more musically sensible to restrict equivalence classes to transposition (Tn). In this way, sonorities that would be considered equivalent in 12-tET can be classified and differentiated in 19-tET.

3:30 Microtonality in Berlin and Vienna in the early 1900s
Presenter: Joe Monzo

Around the turn of the last century, there was a fair amount of experimentation and composition using microtonality, especially "quarter-tones", in the Austrian and German capitals, Vienna and Berlin respectively, which were then (and still are) the largest musical centers in the German-speaking world. Examples from the latter half of the 1800s are Josef Petzval and Shohe Tanaka, and after 1900 Busoni, Schoenberg, and Webern all flirted with the idea of using microtones, and while none of them made real use of non-12-tET tunings, all three had significant influence on other composers. Richard H. Stein published a quarter-tone piece in 1906, and during the next decade Willi Möllendorf patented a quarter-tone keyboard which made a great impact on both Alois Haba and Ivan Wyschnegradsky, the two most significant European microtonalists of the earlier 20th century. There has been little written about this activity even in German, and almost nothing at all so far in English. There was much contact between many of these composers, and this lecture attempts to trace the lines of influence. I also speculate on the significance non-12-tET may have had for Mahler, since he was a strong supporter of Schoenberg and all of his larger works were written for the flexibly-tunable orchestra, and also on the relevance of microtonality to the development of both atonality and sprechstimme in Vienna at this time.

4:15 Grammatical and Syntactic Aspects of Sonido Trece: The Microtonal System of Julian Carrillo
Presenter: Alejandro L. Madrid

Although the Mexican composer Julian Carrillo is somewhat known as one of the pioneers of microtonal music, his actual musical output has remained in obscurity for almost 80 years. Some musicians -- including Mexican composer Joaquin Gutierrez Heras- have accussed Carrillo's music of structural and stylistic archaism in similar ways to what Boulez did when he declared Schoenberg a reactionary in his article "Schoenberg is Dead". In this paper I will account for Carrillo's use of microtones within traditional frameworks as the result of the Hegelian ideology that permeated his musical thought since his early romantic compositions through his microtonal phase. I will study Carrillo's music within Robert Morgan's description of modernist languages as based on a new relationship between musical background and foreground. I will consider Carrillo's use of microtones an enlargement of the grammatical (foreground) possibilities of a modernist language and will study the relationships between them and the syntactic (background) elements found in his formal structures through an analysis of three microtonal works written between 1924 and 1929: Preludio a Colon for chamber ensamble, Meditacion for string quartet and En secreto also for string quartet.

4:45 Visual Music: A Graphic Appreciation of Partch's Tuning
Presenter: Dudley Duncan

5:15 Alexei Ogolevets, Master Russian Microtonalist
Presenter: Brian McLaren

The presentation "Alexei Ogolevets, Master Russian Microtonalist" discusses the theories and work of the Russian microtonal theorist who died in 1967. Centering on his monumental 970-page 1936 book Foundations of Harmonic Language, this presentation offers an overview of Ogolevets' expansion of Western tuning systems to include equal temperaments based on 2-interval extended Pythagorean tunings 5, 7, 12, 17, 29, 41 note Pythagorean, etc. and also Ogolevets' exploration of equal divisions of the octave defined by a*7 + b*5 where a and b are integers 0,1,2,3... N. The presentation will include a brief discussion of Ogolevets' life and tragic experiences under Stalin, as well as a few words about his final posthumous book on microtonality published in 1969.

PAPER SESSION II: Saturday April 7, 9:00-11:15

Thatcher Music Building Room 109

9:00 An Acoustically-Based Theory and Approach to Quarter-Tone Piano Music
Presenter: Peter Adamczyk

Quarter-tone piano music has often been approached either through the writing of extended textural gestures, or through the composition of relatively independent, linear articulations of the altered tunings.

Proposed here is an acoustically-based theoretical approach which works with the precisely measured and careful composition based on intervals and their associations, building on the innovations of Ives and Cowell, and expanding on the use of register and proportion as refined by composers such as Messiaen and Takemitsu.

There are several observations to consider with quarter-tone piano music: 1. The number of intervallic combinations, when compared to traditional 12-tone equal-temperament, is expanded geometrically. 2. The nature of the added quarter-tone intervals directly engages the effects of acoustic dissonance as distinct from traditional tonal dissonance. This acoustic dissonance complexifies a larger proportion of possible intervallic combinations when compared to the combinations in 12-tone equal temperament. 3. Combinations of pitches often result in very present resonances that are essentially inharmonic, and thus suggest careful treatment.

From these observations is formed the view that inasmuch as quarter-tone music moves away from traditional tonality, it also resists standard atonal treatment. Due to the nature of the piano, additional consideration is to be given the attack envelopes and resonances -- both more tonally rich and potentially dissonant when compared to traditional music.

Presented will be a method of composition for this music which centers around interval organization, and, especially, externally-derived rhythmic and formal structures that take into account the acoustic phenomena unique to quarter-tones. At an intermediate architectural level, a middle-ground is suggested, wherein takes place the exploration of the continuum between "note," "chord," and "timbral fusion," -- a continuum which can govern the range of melodic-harmonic successions as well as define larger aural contrasts.

What emerges from this method is a sound-language which emphasizes inflection and nuance, and which allows a multitude of stylistic approaches. This music's performance, requiring two pianists, can be characterized as a special experience of chamber music, where the ear can hear unique and singular sounds resulting from two performers playing mutually independent keyboards.

9:30 The Importance of the Whole and the Interdependence of Its Parts
Presenter: Drew Lesso

Mictotonal Music involves Tuning Systems exclusively. The field is lucratively based upon Ancient World Philosophies. These are generally founded upon quasi-mystical/mathematical principles. It was not by accident that Plato integrated the Lambdoma in Timaeus, Chapter 6, "The World Soul." The Republic and Timaeus stand as surviving documents and representations for Greek tonal scales while examining political, philosophical and scientific thought 2300 years ago (Ernest G. McClain, Pythagorean Plato, Nicolas-Hays 1978).

It would be absurd to abandon current "n-tone tet" descriptions for microtonal music. After all, we are dealing with a music discipline. The opportunity exists to include proportional information as a microtonal harvest from the worlds of science, graphics & color mixing, biology, even stock market speculations and in other words: tuning, sonic and time/tonal applications into the real world. While these pursuits may lead to the use of sound as a conveyor of information (Earcons), they also allow the microtonalist entry to active comparative fantasy; both real and or imagined. An example of this is found in the work of Charles Lucy, a microtonalist who has based his scales on the number PI.

This brings me to a formula for microtonal research. Which incidentally is also the title for this presentation: "The Whole and the Interdependence of Its Parts." A modern microtonalist's world exists for the application of limits. Whether it be Socrates limit of 12,960,000 or the intimate bendings of 24 tet, each have their dependance or inner sense of balance upon which musicians and investigators can compare their discussions, because of a determination of the whole through a musicalization of its parts.

10:15 Continually Variable Tunings and Timbres Realizable by Computer
Presenter: Christopher Dobrian

Computer sound synthesis provides great potential for experimention with novel musical tuning systems, a fact which has been known for decades. Yet only a fraction of this potential has been explored, due to conceptual restrictions which are inherent in most paradigms of tuning based on acoustic instruments. Computers provide the opportunity to explore abstract concepts not readily realizable by physical means, such as tuning systems that change gradually and continually during the course of a piece of music, scale steps that change continually in size, and timbres that can be designed specifically to suit the tuning system in use and can defy physical acoustic principles.

The author presents selected methods of tuning uniquely realizable by computer, focusing on tunings that can change gradually during the course of a musical passage. He also makes a case for the use of computer synthesis to design timbres that contain frequency relationships which correspond to the tuning system in use in the music. The presentation includes sound examples and musical excerpts.

10:45 Common Tone Adaptive Tuning Using Genetic Algorithms
Presenter: Lydia Ayers and Andrew Horner

The search for optimal tuning of musical scales dates back to antiquity, before Pythagoras and the early Greeks. We can tune some simple chord sequences without beats using a fixed just tuning. Considerable previous work has focused on non-adaptive just, Pythagorean, meantone, Werkmeister, Vallotti and equal-tempered tunings. The best fixed tuning for a mostly diatonic piece staying close to a home key is usually close to 1/4-comma meantone tuning, and the best tuning for an adventurously chromatic piece will tend to be close to 12-tone equal temperament.

Some musical situations force us to choose between a wandering pitch center and mistuned intervals. To avoid pitch shifts, we can sacrifice the tuning of the thirds and fifths. On the other hand, the adaptive just solution keeps all the intervals in tune, but by allowing the pitch to drift. Since adaptive tunings have been much less investigated, we will focus exclusively on them in this paper. (Another option is to allow both some wandering of the pitch center, and some beating intervals, an interesting topic for future research.)

If we are willing to sacrifice a fixed pitch center, can we do better than 1/4-comma meantone tuning for diatonic chords? Can we do better than equal temperament when chords have some chromatic pitches? Simple adaptive just tuning can probably tune most musical examples without beats. However, when intervallic cycles exist, just tuning is forced to lump a big comma into one bad harmonic interval. This paper introduces a genetic algorithm (GA) method of adaptive tuning for these situations.

Our method tunes harmonic progressions to make the harmonic intervals as beat-free as possible. Our assumption is that minimizing beats results in better tuning. The algorithm retains the tunings of common tones from previous chords, and adjusts the tunings of the other pitches. The genetic algorithm method avoids problems associated with simpler enumerative, greedy and distributive tuning strategies.

We will give results for several musical examples using several different error measures, with extended results for minimizing the worst tuning error among harmonic thirds and fifths. We present results for examples ranging from simple diatonic progressions to chord sequences with extended triadic harmonies, and compare the tuning found by the genetic algorithm to standard tunings that would appear appropriate, such as just, 1/4-comma meantone and 12-tone equal temperament. We will also show the GA performs better than least squares under a worst deviation error measure, and as well as least squares under an average squared error measure. We have admittedly contrived the examples to show situations where standard tunings have problems, and they include some unusual voice leading. As we will show, sometimes the most consonant tunings are not obvious, and differ significantly from the standard tunings.

PAPER SESSION III: Saturday April 7, 1:15-2:45

Thatcher Music Building Room 109

1:15 Conjuring the Phantom: Practical and Poetic Use of a Missing Fundamental
Presenter: Sean Griffin

Difference tones are psychoacoustic illusions resulting from the ear's nonlinear response to simultaneously sounding tones. Ideally, when 2 tones are played together, tuned to the ratios of the upper partials of a fundamental, the human ear provides the difference of the tones and, with the right acoustic conditions, the missing fundamental.

The poetic lure of a missing fundamental that controls pitch relationships led me to compose pieces derived from the upper partials of a single fundamental frequency never played but inferred as a difference tone. The results were negligible. This led me to construct a series of psychoacoustic tests to see which tunings would produce the best conditions for hearing them.

The experiment was intended to both test the salience of the phenomenon and answer some important questions. Is there a harmonically related strengthening of difference tones by means of the missing fundamental? Do difference tones produced by means other than overtone related ratios, or relating to a missing fundamental outside the audible spectrum, appear weaker?

The tests consisted of producing difference tones by means of two sine tones tuned to different kinds of ratios. The sine tones were tuned to the ratios of the overtones of a fundamental both within and out of the audible rage and ratios not related to the overtone series. Ostensibly, the repetition rate of the overtone related frequencies would strengthen the related periodic components of the missing fundamental while other ratio relationships would not. Results show a slight strengthening of the difference tone with overtone ratios but point to a an almost prohibitive conditionality of production and show a need for proper context and support from other musical variables.

In this paper I discuss my findings and the problems in utilizing this phenomenon with examples from the test and my recent music. I will also present examples of composing with a missing fundamental by composers Tenney, Grissey, and others.

1:45 Transposed Hexanies
Presenter: Greg Schiemer

Just intonation tuning systems that have no tonal centers provide a starting point for this discussion on tuning associated with a work entitled Transposed Hexanies. The composer describes the hexanies used, their transformation through pitch multiplication and connections between rhythms and tunings used in the piece. The lecture will demonstrate the significance of transposition as a way of extending the scope of the hexany.

2:15 Navigating the Infinite Web of Pitch Space
Presenter: Kyle Gann

"Navigating the Infinite Web of Pitch Space" outlines an approach to just-intonation composition which attempts to make complex and distant pitch relationships intelligible by linking them on the multidimensional matrix of consonances. For example, the famous "wolf fifth" with a frequency ratio of 40/27 can be rendered intelligible by a series of ratios made up of factors of 40 and 27: for instance 4/3, 10/9, 5/3, and so on. By thinking of all possible just-intonation ratios as existing in a web, it is possible to link a virtually infinite number of pitches and create a new kind of tonality firmly centered yet extending beyond the usual reaches of pitch perception. Examples from Gann's works, Custer and Sitting Bull and How Miraculous Things Happen, will illustrate his approach to just-intonation composition.

PAPER SESSION IV: Sunday April 8, 9:00-10:45

Thatcher Music Building Room 109

9:00 Non-Temperament: Pitch Relationships of the Harmonic Series as Interrelated Diatonic Sets for Composition
Presenter: Peter Hulen

This paper explores a systematic approach to musical composition applying the collected pitch relationships through octave IV of the harmonic series (i.e. 1/1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8) to the creation of individual diatonic sets. In this system of tone order, eight diatonic sets (each containing pitches in the same relation to one another as the harmonic series through octave IV) have as their fundamentals pitches which also have the same relationships to one another as octave IV of the harmonic series. When tones are ordered in this way, all eight of the sets each have at least one pitch (usually more) in common with each of the other sets. These interrelationships can be demonstrated by calculating the ratios themselves, cents deviation from 12-tone equal temperament against a common standard pitch, or frequency values in Hertz.

There are a number of procedural implications for composition given by such a system of tone order. Musical examples will be provided which illustrate some of them. Sonic events by themselves obviously have a certain material consistency with respect to tone, being restricted tonally to the sets as described, but the interrelatedness of the sets also allows access to the effect of modulation. Tones for e.g. melodic events can be applied from a given diatonic set while its fundamental is continuously or intermittently sounded, or merely assumed. When an event reaches a certain tone common to another diatonic set, the fundamental can be changed to that of the new set, thereby effecting modulation. Events can then continue, with their tonal materials now applied from the new diatonic set. The resulting texture could be considered analogous to early organum (e.g. of Léonin), with the succession of fundamental tones analogous to the long-breathed chant melody in the tenor, and more complex concurrent events analogous to the comparatively florid duplum.         Intervallic relationships within the sets can be assigned relative categories (e.g. "empty consonance," "perfect consonance," "imperfect consonance," "dissonance") based on their relationship to just intervals, 12-tone equally tempered intervals, historically established practices, or the arbitrary decision of the composer. Criteria for categorization are suggested in the paper. Intervals within events can then be applied according to a scheme of rhythmic relationships analogous to counterpoint. The application of tonal materials as described reflects implications for polyphonic procedure, but homophonic textures can also be effected through the superimposition of intervals as well. Simultaneities can also be categorized by their relative overall dissonance, and applied accordingly. The paper also suggests criteria for their categorization.

As the relationships connecting tonal materials in this system are derived from the pitch ratios of the harmonic series, the term ratiotonic has been coined to describe it. Its implications for composition have only begun to be explored. Yet to be considered are the possible primacy of certain tones within given simultaneities, a means of managing dissonance through progressions of simultaneities, the possible construction of acoustic instruments, etc. Conventional fretless string instruments, voices, and of course, electronic instruments capable of producing precise frequencies seem to be the most immediately fruitful avenues for exploring and refining the application of this system within the crucible of practice.

9:30 A Practical Microtonal Pitch Space; Theoretical and Psychoacoustic Issues, Compositional Applications
Presenter: Thanassis Rikakis

This paper describes a microtonal pitch space that:

  • is rooted in contemporary understanding of the sensory procedures involved in pitch perception
  • functions within the confines of the "standard" 12 semitone equal temperament (ET) system ubiquitous in Western music
  • uses intonation to show the pitch hierarchy(ies) active in a passage, clarify the function of each note in the active hierarchy and denote intention and progression course
  • gains for composers the ability to make systematic use of microtonal information to enhance the expressivity of a musical work

A practical microtonal vocabulary should be easily perceivable. This paper traces the sensory, mathematical and cultural forces that influence/determine the pitch perception processes of a Western listener, finds the pitch space that represents as accurately as possible the relationships and structures promoted by these forces and then chooses a microtonal vocabulary that takes advantage of/complies with the structures of that pitch space.

Overtones prominent in pitch perception, consonance, and tonalness are used in this paper to establish a hierarchy of intervallic relationships. Since the starting point and structural unit of this hierarchy is a JI major triad, a P5s/JIM3s pitch space appears as the best possible representation of the hierarchy of the pitch/intervallic relationships of a note with the other 11 semitone categories. Since the JI major triad is abstracted from the overtone structure and since we know that perception of the pitch of pure tones (such as harmonics) is influenced by even such parameters as volume, we can safely assume that we can round-off the intonations of the P5 and JIM3 to their nearest 5 cent subdivision without loosing much information. We thus increase the practicality of the space by limiting the kinds of intonation variations being used without compromising accuracy.

The microtonal intonations of this space are easily perceivable as they originate from prominent pitch perception processes. The microtones are also structural and thus useful at every level of composition. The microtonal intonations of this space denote and enhance the reference note/center of gravity and the overall structure of the 5 level pitch/intervallic relationships hierarchy, establish unique, structure revealing relationships between notes and allow for unique collections of notes at all 5 levels of our collections hierarchy for each different reference note/gravitational center.

This obviously means that our P5s/JIM3s space is an open space. It shows very accurately the hierarchy of pitch/intervallic relationships of one reference note at a time but can not show accurately the pitch/intervallic relationships for all its notes. Depending on which note is the central note the intonations of the other notes have to change accordingly. This results in an open pitch space with many more than 12 notes. However Balzano has shown that if we are to structure a pitch space of P5s and M3s, mathematical consistency would promote a 12 semitone equal temperament representation as the natural third representation of this space and vice versa.

Instead, the paper comes up with a hybrid solution that maintains the advantages of both kinds of spaces. It proposes to allow only the 12 ET semitones to act as gravity centers/reference notes for the building of hierarchies and provides all microtones necessary for the creation of JIM3s axes by allowing the 12 ET notes of a P5s/M3s space to move up or down 15 and/or 30 cents. The space also maintains all hierarchy enhancing, structure revealing microtonal intonations of the P5s/JIM3s space. The space includes all JI tunings, at an approximation of 4 cents or less, for all intervals for all notes included in the 5 level hierarchies of all 12 possible reference notes. The space has a large number of microtonal intervals (for every interval there are 9 possible expressions). They allow for powerful microtonal passages with each move denoting a possible intention. Denoted intentions can be in agreement to stabilize and focus a passage or be contrasting and thus create ambiguity.

10:15Mental Representations of Pitch-Structures
Author: Gennadiy Kogut
Presenter: Brian McLaren

This presentation will offer an overview of work done by Gennadiy Kogut, senior music professor at the State Academy for Theater and the Arts in Kiev, Ukraine, circa 1980. Mclaren will read from his translation of Professor Kogut's paper with additional explanations, along with diagrams and photographs of the non-12-tone keyboard used in Professor Kogut's experiments. Professor Kogut found that music students could form accurate mental representations of the Western diatonic scale but became less accurate when they tried to reproduce the Western chromatic scale. Attempts to reproduce the 17-tone equal scale indicated that students tended to confuse 17-equal with the 12-equal chromatic scale, and students proved largely unable to form accurate mental representations of the 24 tone equal scale. All students improved with exposure to these unfamiliar scales, however, and showed a marked tendency to form mental representations of all tunings using unequally spaced musical intervals (even in the case of the western chromatic 12-tone equal scale).

PANEL: Sunday April 8, 10:45-11:15

Thatcher Music Building Room 109

10:45 Panel: A Discussion of Real-World Composition in Just Intonation, Equal Temperaments, and Non-Just Non-Equal-Tempered Tunings
Presenters: Warren Burt and Brian McLaren

Warren Burt and mclaren will present a freewheeling 50-minute panel discussion in which they will alternately play and discuss representative microtonal compositions in each the 3 basic tuning categories: microtonal equal temperaments, just intonation, and non-just non-equal-tempered. Burt and mclaren will use slightly less than 3 minutes to discuss how each composition is put together, as well as remarking on the general musical uses and characteristics of each tuning used in each composition. 18 microtonal compositions all told will be excerpted (9 compositions each by Burt and mclaren) in order to give the audience an idea of the possibilities and uses of actual (as opposed to merely theoretical) microtonal tunings in real-world hands-on microtonal compositions.

PAPER SESSION V: Sunday April 8, 1:30-4:15

Thatcher Music Building Room 109

1:30 Rational Microtonality: Some Approaches to Microtones in Seven-Limit (and Greater) Extended Just Intonation
Presenter: David Doty

Microtones (defined as intervals that are perceptibly smaller than a 12TET semitone) are an inescapable feature of music in seven-limit (and greater) Just Intonation. In particular, such intervals arise when consonant harmonies (seventh chords, ninth chords and their subsets) progress by simple-ratio intervals. How such intervals are to be incorporated in various compositional styles is one of the more interesting challenges facing composers working in extended Just Intonation. For example, the effect of these microtones may be minimized by restricting them to inner voices in a musical texture, or they may be exploited for their specific musical affects by placing them in more prominent positions in a composition. Both approaches will be represented with compositional examples, taken mainly from the CD Uncommon Practice.

2:00 Microtonalities of the Lambdoma Harmonic Keyboard
Presenters: Barbara Ferrell Hero and Robert Miller Foulkrod

My colleague Robert Miller Foulkrod and I will demonstrate the Microtonalities of the Lambdoma Harmonic Keyboard. We will invite members of the audience to choose their favorite keynote and program the matrix to that frequency, and will suggest theories of the psychoacoustic effects upon individuals, i.e. the drone, the cue, and other factors. Overheads of the origins, research and development of the steps leading to the application of a mathematical matrix to a prototype Lambdoma Keyboard will be shown. We use a laser-scanner device to illustrate the characteristic Lissajous shapes of the musical intervals. We will demonstrate how the matrix of frequencies below the audible range may create an unusual soothing effect to the listeners.

2:45 Making the Recorder Microtonal
Presenter: Donald Bousted

This paper will describe an analysis, using the IRCAM software programme AudioSculpt, of a section of the CD recording of the author's 25 minute recorder duet 'A Journey Among Travellers'. This will demonstrate, and put into context, some of the properties of 'elastic tuning'. Additionally, a template for future activities will be presented, including a technique of using difference tones to access a range of equal tempered and just scales.

3:15 The Euler Genera and an Hyperdimensional Tone-lattice
Presenter: Erv Wilson

We explore the nested sequence of Euler Genera -- (0), (3), (3, 5), (3, 5, 7), (3, 5, 7, 9), (3, 5, 7, 9, 11) and (3, 5, 7, 9, 11, 13) -- and the corresponding pentahedral tone-lattices. The extrapolation of the seven master-sets to their fully projected genera/tone-lattices will be studied. The reverse parallelism (khiasmos) of the Euler Genera is elucidated through the geometric symmetries.

3:45 The Hackleman-Wilson 19-tone Clavichord
Presenter: Scott Hackleman

In 1975 Erv wilson and I decided to collaborate in the construction of a clavichord using one of his generalized keyboard designs carried out to 19 tones per octave. The generalized keyboard takes us out of the tonal compromise of 12-to-an-octave, provides us with expanded tonal resources to explore, and serves as a tool or physical model in depicting the spatial relationships of intervals. Much like a shape can represent a molecule, so too, this keyboard literally is a model of the ratios and proportions that the scales are actually constructed of. The clavichord, an ancient, private instrument, fosters expressiveness, subtlety, and musical finesse. One does not play a solo on a clavichord, so much as a soliloquy. It is from all these perspectives that I've come to call this 19-tone clavichord a "philosopher's instrument" -- an instrument where we can experience the sensation of music through the physical act of playing it and, through the nature of the keyboard, speculate on the construct of tonality and tuning as well.

Go back to preliminary schedule