Tag: xenharmonic

Understanding the Polychromatic System

Posted on January 11, 2018 by Dolores Catherino - Columns, NewMusicBox

We seem to be at an accelerated phase in our musical evolution, where isolated methods of music practice are rapidly multiplying without a framework of integration and orientation for musicians and listeners to grasp. The polychromatic system is one framework of integration for the various scale configurations of micro-pitch music.

Isolated methods of music practice are rapidly multiplying.

The polychromatic system is oriented toward exploring the outer limits of micro-pitch awareness and its expression in music, from a perceptual rather than conceptual (i.e. mathematical, theoretical, analytical) perspective.

The polychromatic system is based on principles of associative synesthesia: learned associations and conceptual/perceptual integration of audible pitch with visual color. Recent research has shown that associative synesthesia can be developed with practice. Music is an optimal area in which to extend the possibilities and potential of synesthetic awareness, both aesthetically and scientifically.

Intentions:

To simplify and encourage musical curiosity—exploration and discovery of infinite possible pitch scales and their sonic combinations (intervallic or other).
To open new worlds of musical expression, experience, and composition.
To aid in the development of new ways of ‘hearing’ sound/music and the world.

Imagine a chromatic keyboard where each key is split. The front half of the key plays the conventional chromatic semitone pitch while the back half of the key plays the quartertone in between each front-key pitch (24 pitches per octave). We could distinguish these pitches clearly by assigning the quarter tones at the back of the key to a pitch-color (let’s say violet).

Moving up to 36 edo (equal divisions of the octave) brings new complexity. Now we can imagine a pitch-color above (say, blue) and a pitch-color below (say, red) each chromatic pitch. One way to describe them would be to say that, in the key of ‘C’, C-red is flat-ish and C-blue is sharp-ish relative to the chromatic pitch. The problem here is that the terminology of flat and sharp are embedded in the chromatic language both as a pitch definition (Db, C#) and as a pitch modifier (bb, x). By applying the concept of pitch-color, we can avoid both the confusion in terms and extreme notational complexity (countless, incompatible pitch-modifier symbols) ‘bolted-on’ to chromatic black and white notation.

This describes how I proceeded in trying to work with very high pitch-resolutions. The polychromatic system evolved to allow the creation of a simple notation and theoretical language for writing, learning, and memorizing micro-pitch music within a pitch division scheme of 106 and 72 edo.

The pitch-color concept is not absolutely defined, so the values (in Hertz) of C-red are different depending on the scale division method used. In this way, the language says remains generally consistent, while being adaptable to any conceivable pitch scale. Unique pitch definitions are defined at the beginning of each composition. This enables an efficiency of learning and possibility of using of multiple pitch scales (simultaneously or via ‘modulations’), all under a unifying and intuitive pitch-color concept.

The pitch-color concept is intuitive by way of an easy association of micro-pitch to the visual color spectrum—from infrared to ultraviolet. A simple association can be made with a continuum of flatness to sharpness for each reference pitch (chromatic or other) to the visual color sequence we already know from the experience of seeing rainbows and other light diffraction phenomena.

I use equal divisions of the octave as a method of pitch division because it is a rudimentary and self-explanatory element to begin from in my early explorations of auditory limits of pitched-sound differentiation. My approach is to use the highest possible edo scales within each new keyboard design.

While there is one option for a major 3rd in a chromatic system, the polychromatic system offers several pitch-color variations of that interval.

With regard to intervals, the polychromatic system uses relatively defined pitch-color definitions and is based on an idea of intervallic relativity. So, while there is one option for a major 3rd in a chromatic system, the polychromatic system offers several pitch-color variations of that interval. This intervallic flexibility has audibly compelling implications and effects when exploring variations of each component within increasingly complex harmony.

When listeners hear microtonal music as ‘out of tune’, this impression arises from a chromatic frame of reference. A new foundational frame of reference or perspective needs to be established in order to appreciate microtonal music on its own terms and not in comparison to the macro-pitches of the chromatic system. This foundation is what I have been seeking in developing a polychromatic system. In general, microtonal music can seem extremely abstract for unprepared minds and ears, especially without a new framework of understanding (for reference and comparison).

The polychromatic system builds upon the fundamental concepts of the chromatic system (note definitions, harmonic principles, and music theory) as a common point of reference – and departure. From this common framework, increasingly refined levels of micro-pitch discrimination can be explored within a known system of musical understanding. As greater refinements of pitch and harmony recognition are developed, increased awareness can enable the recognition of further pitch-colors, as entities in themselves, without trying to force them into the conventional frame of reference: a coarse pitch-resolution, chromatic system.

In analogy, imagine the chromatic system as an old monochrome (black & white) dot matrix printer, with its chunky, quantized images. If you input a high-resolution image into that framework, it is ‘processed’ into a chunky, quantized image. The foundational framework of the coarse resolution, dot matrix system design must be addressed first.

Paradoxically, ‘tradition’ has vital importance in the creation of new musical systems. Here, we are talking about tradition as our accumulated experience of the past, a shared frame of reference, an implicit basis and context of listening to and composing music—and not in the sense of calcified conventions of the past and present.

Paradoxically, ‘tradition’ has vital importance in the creation of new musical systems.

Polychromaticism is an approach and practice which uses chromatic music theory as an initial basis for conceptual anchors. In this way, extensions of established tonal and harmonic principles assist in the understanding of the new possibilities and implications of the polychromatic system. This creates a conceptual and perceptual bridge between our current and many future systems of music.

The use of chromatic conceptual anchors comes into play when I am trying to memorize a piece for performance. Using the conceptual anchor of a dominant 11th chord, I remember the constituent pitch-colors of the chord as minor modifiers within this larger conceptual structure, instead of having to remember each pitch-color individually.

My composing process is much more intuitive than analytical. As a result, I have neglected doing harmonic analyses, after the fact, to determine possible new theoretical models. My hope is that others with a passion for analysis will create new music theory models to aid in teaching the polychromatic system more efficiently. Philosophically, I see this as part of a larger 21st-century process which is based on collaborative innovation in music, science, technology, and the sound arts.

As a musician, it seemed impossible to gain a fundamental grasp of microtonal music because it consisted of so many discrete pitch-scale methods, all separately existing without a fundamental, underlying system.

With a polychromatic basis in notation and generalized pitch-color concept, any micro-pitch language could be quickly assimilated without having to relearn scale-specific notations. The current expansion of esoteric and fragmented micro-pitch scale methods, of reinventing-the-wheel (in notation) with each, can make wide exploration within microtonal music intimidating and difficult.

The use of non-chromatic color schemes:

While the color schemes here are different, my impression is that these are examples of a modally optimized layout – in the sense that adjacent keys are 3rd, 4th, 5th, and 6th intervals rather than chromatic minor 2nd’s (or smaller) intervals. In the polychromatic system there are multiple pitch-color variations of each interval, rather than the single fixed-interval value in these layouts. Why be limited to just one major 3rd interval, when multiple intervallic pitch-color choices—singularly or simultaneously—could be available?

Ultimately, the polychromatic system exists to make the process of micro-pitch exploration and creation easier. And just as the polychromatic system has evolved from the chromatic system, it too will eventually become a (legacy) reference and conceptual point of departure for many increasingly sophisticated (non-chromatic) music systems of the future. The polychromatic system is one way of making the musical evolution toward triple-digit, pitch-scale resolutions easier to understand and create with.

As a musician and a listener, I experience music as a dynamic and evolving process, a creative interaction that we choose to engage in. Ultimately, the meaning and value of music comes from the quality and depth of this creative interaction.

Art doesn’t come to us, we must come to art.

The possibilities of growth and awareness gained through our engagement with art remind me of the idea that art doesn’t come to us, we must come to art. This idea expresses both a necessary receptivity to new perspectives as well as an active personal involvement which engages the listener’s creativity and imagination, in a similar way to that of the composer. In this process, the listener becomes a receptive-artist interacting with the compositions of the expressive-artist (composer). If we become what we practice, then exposure to, and interaction with challenging art can help us expand our integrated perpetual/conceptual awareness, and expose us to new dimensions of emotion and insight beyond the limits of spoken language.

In the next article, I will describe technical (physical technique and technology) aspects of my approach to polychromatic music, as well as some discoveries and implications that I have become aware of in my early explorations.

Polychromatic Music

Posted on January 4, 2018 by Dolores Catherino - Columns, NewMusicBox

Music seems to be at the forefront of an increasingly pervasive process where technological simulation is cheaply and efficiently substituted for authentic human creation and expression. Further, a technological aesthetics of ‘perfection’ has arisen which values flawless quantization, pitch correction, and production as primary elements over the power of unique, imperfect vitality of human creative expression. Polychromatic music embodies a new paradigm and aesthetic: a humanistic counterbalance to rapidly emerging technological trends which, when they don’t replace human involvement, seem to minimize and/or trivialize it.

Even as a child I was aware that the chromatic/modal tonal languages were nearing the point of exhaustion as far as new areas of exploration and creation, and this stoked a curiosity and an intuitive seeking of the possibility of new dimensions of musical language. As an undergraduate music major, many of the developments of the late 19th (chromaticism) and 20th (stochastic, aleatoric, spectral, microtonal, algorithmic) centuries made sense from this perspective. Yet they seemed difficult to assimilate and understand without a conceptual framework to anchor these new perceptual experiences in a larger foundational context and aesthetic.

With the emergence of AI (artificial intelligence) ‘creativity’ now being used to ‘compose’ music, many new questions and concerns have arisen. Any process that can be formalized in rules or clear, quantifiable instructions, can be efficiently executed by a computer. To me, it seemed that the innovations of stochastic (random operations), aleatoric (chance operations; i.e. dice rolling), serialistic (predefined patterns), and algorithmic (step-by-step procedures) composition were likely candidates for being subsumed within AI generative computation systems.

The human process of creativity lies on a continuum between compositing and composing.

A further distinction is necessary here between creative ‘composing’ and ‘compositing’. Artificial Intelligence generativity (so-called “creativity”) is based on a compositing process; it’s basically all just recombinations of pre-existing data. While it is clear that the human process of creativity lies on a continuum between compositing and composing, a salient aspect of human creativity involves the creation of new ‘data’ rather than the novel recombination of prior ideas.

This leaves us with the contemporary methods of new spectral/timbral and pitch languages as wide open frontiers for exploration and creation. With respect to new timbral languages, I think of spectral music broadly as a paradigm and aesthetic where an emphasis is on the exploration of the timbral aspects and implications of complex sounds. This would encompass harmonics, harmonic (overtone) interactions, and new frontiers in harmony (polyphony). This is an immense world of its own where technology has provided endless possibilities for exploring sound design and works of sonic creation (sound arts).

Another compelling area of exploration lies within the realm of new pitch languages—the xenharmonic philosophy and microtonal/macrotonal pitch definition methods. For the past century, the creation and use of many microtonal methods has been an exciting development in music. This presents new possibilities for differing, extended explorations of ‘tonality’. It seems that the main hindrance to the wider understanding and use of these methods is the result of a lack of any underlying conceptual framework.

At present, we have a growing number of mutually exclusive microtonal pitch definition methods, each with its own notation. As a musician coming from an empirical perspective (practice vs. theory), the impractical situation of learning a new notation system for each microtonal pitch method is a persistent impediment to a larger, unified progress beyond merely creating new microtonal scales. This is where polychromatic music, as a system, comes in.

One way of understanding and distinguishing our contemporary musical terminology of xenharmonic, polychromatic, and microtonal is by a rudimentary differentiation of philosophy, system, and method:

Xenharmonic refers to a philosophy which regards the infinite pitch scale division methods applied to the pitch continuum as equally valuable. Also, it expresses an aesthetic of freedom and openness toward any and all methods of pitch scale division and the exploration of their melodic, harmonic, rhythmic, timbral, etc. implications in new musical compositions.

We have no words for many perceptual aspects of hearing.

The polychromatic system is an intuitive, unifying conceptual framework for exploring any conceivable pitch division method. Our language is grounded in visual concepts: we have no words for many perceptual aspects of hearing: imagery, visualization, dimension, space, etc. As a result, we are faced with communicating auditory concepts in analogy or metaphor. My perspective is to link visual and auditory perceptual concepts into an idea of ‘pitch-color’. The visual basis here is the color spectrum: red, orange, yellow, green, blue, indigo, violet. From this intuitive basis, we can move from a vague flat/sharp conception of pitch to more refined and distinct conceptual ‘pitch-color’ anchors. So, with yellow as a basis of reference, orange and red would be progressively flatter, and green, blue, violet would be progressively sharper. Using a color spectrum with integrated visual/audible associations on a scale from (infra/flat)red to (ultra/sharp)violet. The distinctions of flat and sharp become an increasingly refined spectrum relative to the chromatic (macro)pitch division method, i.e. C, Db, C# etc.

The polychromatic system uses the chromatic language as a common point of departure. In this context, the chromatic language is characterized by the use of letters as pitch names, and by the representation of musical intervals numerically (and modally: C-B as a major 7th rather that a 12th). Also, since the pitch-colors of the system are relatively defined (by the method of pitch division), it creates an intuitive bridge between differing microtonal scale derivation methods.

Microtonality consists of the various, exclusive, and divergent methods of pitch division, notation, and theory. Without a unifying conceptual framework, these methods remain mutually exclusive and excessively difficult to assimilate in a unifying and complementary manner.

A point of clarification: with respect to an integrated philosophy-system-method perspective of music, the chromatic musical language is a system, while the various temperament derivations (meantone, well, just, equal, etc.) are methods (of pitch definition).

The above categories are generalized for preliminary understanding. I see polychromatic music primarily as a system, and secondarily as an aesthetic. For me, this aesthetic involves evolving reflections on humanism in an era of increasing technology. And this is why I devote the effort to physically learn and perform my compositions: to create not only demonstrations of new musical possibilities within the polychromatic framework, but also examples of the human musician utilizing technology in a creatively assistive fashion vs. the human musician creatively assisting (editing, compositing) increasingly sophisticated technological processes.

In the next article, I will focus on describing my approach toward learning and composing within a polychromatic system.

The Science of Sound and Tunings

Posted on August 10, 2017 by Elaine Walker - Columns

As a composer, what drew me to use scales that have more, or less, notes per octave than our standard twelve-tone tuning–or xenharmonic music–was the boredom that crept up on me over the years of using the same twelve notes over and over, plus a curiosity about other possible tunings and what emotional chords they might strike. Many xenharmonic composers are driven by the artistic urge to break down arbitrary barriers of creative expression. And many who are mathematically inclined explore the vast possibilities of xenharmonic tunings because the mathematics is beautiful.

In order to appreciate anyone’s desire to explore the world outside of the common twelve-tone tuning, it helps to understand where this tuning standard came from and how it is somewhat arbitrary and not even mathematically pure. This calls for a short discussion of the science of sound.

Twelve-tone tuning is somewhat arbitrary and not even mathematically pure.

To start with, there is nothing special about any particular note or “frequency” (unless a person has absolute pitch). It’s all relative. That is, the relationships between frequencies are what matters. A musical interval is the difference in frequency–the ratio–between two notes, and the way two frequencies interact has special mathematical and psychoacoustic qualities.

Our twelve-tone tuning was derived from interval ratios between the first sixteen harmonics. Harmonics were not an invention but a discovery about the natural resonant vibrations of musical instruments. Our twelve-tone tuning being related to the spectra of musical instruments results in intervals and chords that sound “in tune.”

The simplest and purest vibration is a single sine wave frequency. Sine waves are common in electronic music. Very low-pitched sine waves are often used as “sub bass” and high sine waves add “sparkle.” However, most sounds contain multiple sine-wave frequencies combined into a complex waveform. Think of tossing differently sized rocks into a pond and observing how the waves combine into an intricate pattern. Some waves reinforce to create larger ones, while others cancel out. Sound combines in this way, whether through air pressure waves interacting, or fluctuating voltage adding and subtracting in a digital mix.

Any sound that we hear, whether “musical” or not, can theoretically be broken down into individual sine waves. But most musical instrument sounds are pitched, and this is because they naturally vibrate at whole number multiples of the main frequency, creating the harmonic series. Any sine waves that fall in-between the neat and tidy harmonics are perceived as noise elements, which is not necessarily a bad thing. The “noise” might be the scrape of a violin bow, or the hammer sound of a piano key, or the pluck of a guitar, or the grit in a synthesizer sound.

As a typical illustration of the harmonic series, think about plucking a guitar string. The entire string vibrates back and forth at a certain speed (the fundamental frequency), and we perceive that vibration as the pitch of the note. At the same time, the string also vibrates in halves, thirds, fourths, fifths, and so on. This series of higher and higher frequencies, at shorter and shorter wavelengths, is the harmonic series.

Figure 1: Harmonic String Vibrations

We don’t have to probe very deep into the harmonic series to see a fundamental relationship to our historical musical preferences. If we approximate the first dozen harmonics on a staff (Figure 2) or piano keyboard (Figure 3), we can quickly see some standard musical relationships. I say “approximate” because the harmonic series doesn’t perfectly align with the intervals in our twelve-tone tuning. If you have a piano handy, try playing these harmonics. In fact, you can do this on any instrument as long as it has a range of a few octaves.

The intervals between the first several harmonics are a very solid basis for our tuning, but the series continues with smaller and smaller intervals ad infinitum. Once we get past the 18th harmonic, the intervals become microtonal, meaning smaller than half steps.

Figure 2. An approximation of the harmonic series on a musical staff, starting with C up to the 12th harmonic.

Figure 3. An approximation of the harmonic series on a piano, starting with C up to the 12th harmonic.

If we begin with a low C (32.703 Hz) as the fundamental frequency, then an approximation of the first seven harmonics would be the notes C₁, C₂, G₂, C₃, E₃, G₃, Bb₃. Just these first five harmonics alone when transposed into the same octave range are enough to build a Major triad–C/E/G–the most used chord in all of Western music. Using the first seven harmonics allows us to build a dominant 7th chord–C/E/G/Bb–the chord most often used for an ending cadence leading into a Major triad.

It’s nontrivial that the first and second harmonics are an octave apart. An octave has the strongest psychoacoustic relationship of any musical interval. It is an extremely interesting musical phenomena that octaves sound like higher and lower versions of the same note. Our twelve-tone tuning is “framed” by this very special interval, as are most (but not all) xenharmonic tunings. This is why our standard tuning includes twelve notes “per octave.” Notice that every time the harmonics double in frequency (harmonics 1, 2, 4, 8, etc.) we have another octave.

The second and third harmonics form an interval of a fifth, which is the next strongest interval to our ears after the octave. Fifths frame our triad chords, and the cadence that I mentioned earlier “resolves down a fifth,” meaning that the root notes move down a fifth interval. The third and fourth harmonics form a fourth, the next strongest interval. The peaceful “Amen” cadence resolves down a fourth, which doesn’t sound as final as resolving down a fifth, but is still a strong cadence. Fourths and fifths are a staple for rhythm guitarists who often strum those intervals as a musical pedal.

The fourth and fifth harmonics form a Major 3rd, and the fifth and sixth harmonics form a minor 3rd. The sixth and seventh harmonics form a slightly smaller minor 3rd. Thirds are another very important interval in Western harmony. Stacking a Major 3rd and a minor 3rd creates a Major triad, and stacking them the other way around creates a minor triad–the second most popular chord in all of Western harmony. So there we have it–at least most of it–as the first seven harmonics provide most of our well-established intervals.

The seventh and eighth harmonics vaguely approximate a Major 2nd (a “whole step”), and the same goes for the next few harmonics, although by slightly smaller intervals each time. The eleventh and twelfth harmonics vaguely approximate a minor 2nd (a “half step”), although it is quite a bit larger than the half steps we use in our twelve-tone tuning. Harmonics 17 and 18 come closest to approximating our standard half step. After that, the harmonics form smaller and smaller microtonal intervals that were simply not chosen to be part of our musical scale.

So, with the first several harmonics transposed into the same octave range, we get these scale degrees: C, __, __, __, E, __, __, G, __, __, Bb, __, C. There are other interval relationships that played a part, such as harmonics 5 and 8 which form a minor 6th (E to C). If transposed down to the root note, we get C to A. Adding the A to our scale then reveals another whole step between G and A, reinforcing the idea of a whole step, and so on. It was found that chopping a whole step into a “half step” was close in pitch ratio to the 17th and 18th harmonics. This interval could somewhat neatly fill in the remaining blanks to form our twelve-tone scale of half steps–C, C#, D, D#, E, F, F#, G, G#, A, Bb, B, C.

Well, almost. We don’t end up with equally sized half steps if we keep the pure harmonic ratios that originally inspired the scale. By around 200 years ago, the scale intervals were adjusted and “evened out” so that every half step had the same frequency ratio of 1:1.05946–not exactly a simple or ideal ratio. This is called “equal temperament,” and it’s how our modern pianos are tuned. It is quite useful in enabling a person to play a song in any key, and to transpose chords during the course of a song without the worry of clashing intervals.

If our ancestors had chosen to base our scale off of the first 36 harmonics, we may have ended up with 24-note-per-octave instruments involving “quarter tone” intervals instead of half steps. Pianos and other instruments would have been much more complicated to build. If we had used 25 or so harmonics, we could have ended up with 19-note-per-octave instruments. In fact, we could have easily ended up with any number of tunings, with plenty of ways to justify them as being the “best” decision.

Twelve notes per octave was probably the best decision for the time, especially considering that it simplified instrument building, yet had plenty of notes for creating a wide variety of expressive musical styles. But just as the ears of average people have adjusted to more and more complexity and variety in musical chords, styles, timbre, and rhythm over the millennia, we can now add new tonalities to the list.

Elaine Walker

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