Reflections on Pitch Relativity
Absolute, fixed, static pitch is an exception in nature. Yet the absolute/fixed pitch concept has been the fundamental assumption in the creation of traditional fixed pitch instruments (keyboard, fretted), enabling efficient consistency in mechanical production and use.
The absolute pitch concept is not a restriction for continuous pitch (fretless, voice) and electronic instruments. The ear, and more generally, perception is based in nonlinear functional processes. Therefore, a musical pitch system based on absolute, static categories (pitch, intervals) and simple linear relationships (integer multiple frequencies) is incomplete and open to the exploration of deeper complexities.
Four compounded perfect fifths theoretically sum to a major 10th (oct + maj 3), yet appreciate the difference in pitch between the pitch formed by ‘pure’ 5ths and the ‘pure’ major 3rd. This is an example that the same theoretical fixed pitch can have more than one unique, relatively defined value. In my early explorations with high pitch-resolution scales (106, 72 edo), I have become aware that intervallic pitch-space increases (expands) as notes proceed sharpward and decreases (contracts) as they proceed flatward.
Pitch relativity is a concept describing the musical adaptation to nonlinear pitch perception characteristics of the human ear.
With it, we can begin to move beyond a reductive and simplifying perspective of *dichotomy with respect to pitch:
in-tune vs. out-of-tune; true/false; 0/1
* dichotomy – a subjective collapse of infinite conceptual possibilities into a few (or 2), based on an binary, either/or, perspective.
Pitch can be reconceived as a contextually interdependent, dynamic, evolving process (verb) with notable tendencies.
Traditional musical pitch perspectives:
A,B,C … G Absolute values (context independent)
Relative values (intervallic context dependent)
2:1, 3:2, 4:3 … Ratios (relative to tonic pitch)
do, re, mi, fa … Solfege syllables (relative to tonic pitch)
maj3, aug4, min7 … Intervals (relative to any given pitch)
Pitch discrimination observations:
- as pitch intervals (audible space between two pitches) progress higher in frequency, intervallic distance (pitch-space) gradually increases and the pitches become relatively sharper.
- as pitches progress lower in frequency, intervallic distance (pitch-space) decreases and the pitches become relatively flatter.
Visualize the pitch continuum as a continuous dynamic phenomena recurring in a spirally ascending, widening (i.e. funnel shaped) fashion with the ascension of pitch.
Note that when playing consecutive pure fifths creates an ascending sharpward pitch spiral in the human audible range (expansion of pitch-space). Conversely, descending consecutive pure fifths create a descending flatward spiral (contraction of pitch-space).
Implications to consider:
- I. Auditory sciences describe physical/perceptual system nonlinearities as ‘distortion products’. A perspective with implicit conceptual assumptions: linear system, optimally output = sum of inputs, passive process (analogy: flat response microphone), structurally/functionally segmented. Thus, deviation from linearity and technological immeasurability (unquantifiable aspects) = perceptual distortion; a defect in the perception vs an indication of higher complexity than the theoretical system can describe.
Consider the perspective of visual color perception: the color magenta is easily perceived and yet exists without any measurable wavelength of light within the color spectrum. In he realm of visual perception, magenta is considered an anomaly and not a visual ‘distortion product’.
Consider the implications of a nonlinear conceptual assumption: output ≠ sum of inputs, interactive and integrated process, gestalt aspects of auditory perception. This infers an incomplete perspective of a richer complexity in which linear and nonlinear (as well as integrated quantitative/qualitative) characteristics are complementary and integral to the gestalt (unified structure beyond sum of parts).
- Development of musical languages which complement the perceptual nonlinearities of the auditory system.
III. Octave displacement relativity – 2nd/9th, 4th/11th, 6th/13th are not exactly the same pitch an octave apart. This characteristic becomes progressively more significant with increasing octave pitch space differentials.
Recognition of the diversity of formerly equivalent interval characteristics (i.e. aug 4th ≠ dim 5th).
Component intervals of traditional chord structures not equivalent even thought they have the same meta-pitch name.
i.e. In the key of C: b7 (Bb) of dominant chord is lower in pitch than b7 (Bb) of minor
chord, maj7 (B) of major chord is lower in pitch than maj7 (B) of min/maj chord.
VI. Hierarchical intonation – in a complex harmonic (polyphonic and overtones) configuration, different intervallic combinations are embedded within the gestalt.
Fundamentally, when compromises need to be made, purity of 5ths seem to take precedence over 4ths, then 3rds, etc.
A closer look at a priori Absolute Pitch assumptions. Current Absolute Pitch distinctions (relative to what?)
Standardized fixed pitch values are determined by the following conventions:
a. pitch A4 set to 440 Hertz.
b. octave pitches (same letter name) = double or one half the frequency of each other adjacent octave. (fixed linear mathematical relationship; octave equivalence)
c. division of octave into 12 equally spaced pitches.
(mathematically not perceptually spaced)
octave = 1200 cents, semitone = 100 cents
A pursuit of defining absolute, fixed pitch values relative to intervallic intonation of component pitches; is analogous fixed categorical concepts like an integer (i.e. 1). This pursuit is followed without regard of the impossibility of defining an absolute pitch value
to irrational numbers (√3), and interdependent pitch/intervallic relativity aspects, similar to the perceptual relativity of color – which is dependent on context of surrounding colors and light.
The current underlying conception of pitch is as fixed whole numbers derived from a logarithmic cents scale (contextually independent relationships).
Pitch relativity expands the above ‘special case’ paradigm to incorporate dynamic, nonlinear, continuous and ‘irrational’ characteristics of intervallic pitch relationships (contextually dependent relationships).
Simile: conceptually absolute ideal harmonic structure (integer-multiple harmonics only) vs perceptually dynamic inharmonicity and subharmonic recognition (inclusion of noninteger-multiple inharmonics, interrelationships and gestalt perception).
Harmonics are defined as integer (whole number) multiples of a fundamental pitch frequency.
The harmonic series is conceptualized as essentially asymmetrical and unipolar in structure.
Harmonics that depart from whole multiples of the fundamental pitch frequency are defined as inharmonics; in recognition of an increasing degree of complexity beyond the ideal harmonic series.
In general, inharmonicity refers to the degree to which an actual harmonic differs from the closest ideal harmonic.
Subharmonics are harmonic frequencies below the fundamental pitch frequency. They are conceived as perceptual sum and difference (combination) tones rather than a pattern of positive integer ratios (1/n).
Combination (sum or difference) tones are believed to be psychoacoustically created additional pitches whose frequency is the sum or difference of harmonic frequencies embedded within the interval (pitch-space) of fundamental pitches.
When viewed from this perspective, absolute pitch conceptual constructs can be appreciated as:
any linearly discrete 2 dimensional quantized framework imposed on a 3 dimensional (or greater) auditory continuum by cultural convention.
Evolving perceptual/conceptual awareness and analysis of pitch continuity (including composite harmonic/inharmonic spectra), and the unique combinatory interrelationships, can pluralistically expand our conventional musical assumptions and allow for expansion toward a new paradigm which incorporates additional dimensions of musically relevant sound phenomena.
A path toward implementation of the pitch relativity paradigm:
Understanding and development of the implications of pitch relativity will require a paradigm shift encompassing new musical perceptual/conceptual categories.
Current music pitch categories are based on a mathematical aesthetic derived from Pythagorean simple ratios with extension into various absolute formulaic pitch divisions and intervallic derivations (usually relative to the octave).
Musical pitch then progressed from mathematical analogy to become a literalization (reification) of number: absolute, contextually static (i.e. ‘A’ = ‘1’ or 440 etc.).
Musical concepts can evolve and gain nuance via increasingly sophisticated, conceptual premises/metaphors.
The metaphor of pitch-color implies a dynamic, context dependent categorization system.
In the same way, the category ‘blue’, to an visual artist, can be a highly nuanced category (dependent on the perceptual, conceptual and technical development of the artist).
Within this type of paradigm, aural perception (phenomenology) would be the basis of musical pitch systems, and intervallic and harmonic conceptualization, as opposed to non-auditory mathematical derivations.
This new paradigm would open currently unimaginable realms of musical exploration, discovery and related implications leading toward new areas of exploration.
New musical constructs may then become understood as essential qualitative interrelationships and dynamic gestalt formations within a multidimensional continuum of auditory perception.
… an expanding awareness beyond the limitations imposed by our current conceptions and comprehension in a mathematical-musical paradigm.