Dynamic tonality is a paradigm for tuning and timbre which generalizes the special relationship between just intonation, and the harmonic series to apply to a wider set of pseudo-just tunings and related[1] pseudo-harmonic timbres.[2]
The main limitation of dynamic tonality is that it is best used with compatible isomorphic keyboard instruments and compatible synthesizers, or with voices and instruments whose sounds are transformed in real time via compatible digital tools.[3]
The static timbre paradigm
Harmonic timbres
A vibrating string, a column of air, and the human voice all emit a specific pattern of partials corresponding to the harmonic series. The degree of correspondence varies, depending on the physical characteristics of the emitter. "Partials" are also called "harmonics" or "overtones." Each musical instrument's unique sound is called its timbre, so an instrument's timbre can be called a "harmonic timbre" if its partials correspond closely to the harmonic series.
Just tunings
Just intonation is a system of tuning that adjusts a tuning's notes to maximize their alignment with a harmonic timbre's partials. This alignment maximizes the consonance of music's tonalintervals.
Temperament
The harmonic series and just intonation share an infinitelycomplicated – or infinite rank – pattern that is determined by the infinite series of prime numbers. A temperament is an attempt to reduce this complexity by mapping this rank-∞ pattern to a simpler, finite-rank pattern.
Throughout history, the pattern of notes in a tuning could be altered (that is, "tempered") by humans but the pattern of partials sounded by an acousticmusical instrument was largely determined by the physics of their sound production. The resulting misalignment between "pseudo-just" tempered tunings, and untempered timbres, made temperament "a battleground for the great minds of Western civilization".[4][5][6] This misalignment, in any tuning that is not fully Just (and hence infinitely complex), is the defining characteristic of any static timbre paradigm.
Instruments
Many of the pseudo-just temperaments proposed during this "temperament battle" were rank2 (two-dimensional) – such as quarter-comma meantone – that provided more than 12notes per octave. However, the standard piano-like keyboard is only rank1 (one-dimensional), affording at most 12notes per octave. Piano-like keyboards affording more than 12notes per octave were developed by Vicentino,[4]:127 Colonna,[4]:131Mersenne,[4]:181Huygens,[4]:185 and Newton,[4]:196 but were all considered too cumbersome / too difficult to play.[4]:18
The dynamic tonality paradigm
The goal of dynamic tonality is to enable consonance beyond the range of tunings and temperaments in which harmonic timbres have traditionally been played. Dynamic tonality delivers consonance by tempering the intervals between notes (into "pseudo-just tunings") and also tempering the intervals between partials (into "pseudo-harmonic timbres") through digital synthesis and/or processing. Aligning the notes of a pseudo-just tuning's notes and the partials of a pseudo-harmonic timbre (or vice versa) enables consonance.
The defining characteristic of dynamic tonality is that a given rank-2 temperament (as defined by a period α, a generator β, and a comma sequence)[7] is used to generate, in real time during performance, the same set of intervals[2] among:
Generating all three from the same temperament solves two problems and creates (at least) three opportunities.
Dynamic tonality solves the problem[4][5][6] of maximizing the consonance[8] of tempered tunings, and extends that solution across a wider range of tunings than were previously considered to be consonant.[7][2]
Dynamic Tonality solves[9] the "cumbersome" problem cited by Isacoff[4]:18,104,196 by generating a keyboard that is (a)isomorphic with its temperament[7] (in every octave, key, and tuning), and yet is (b)tiny (the size of the keyboards on squeezeboxes such as concertinas, bandoneons, and bayans). The creators of dynamic tonality could find no evidence that any of Isacoff's Great Minds knew about isomorphic keyboards or recognized the connection between the rank of a temperament and the dimensions of a keyboard.[7]
Dynamic tonality gives musicians the opportunity to explore new musical effects (see "New musical effects," below).
Dynamic tonality creates the opportunity for musicians to explore rank-2 temperaments other than the syntonic temperament (such as schismatic, Magic, and miracle) easily and with maximum consonance.
Dynamic tonality creates the opportunity for a significant increase in the efficiency of music education.[10]
A rank-2 temperament defines a rank-2 (two-dimensional) note space, as shown in video1 (note space).
Video 1: generating a rank-2 note space
The syntonic temperament is a rank-2 temperament defined by its period (just perfect octave, 1/2), its generator (just perfect fifth, 3/2) and its comma sequence (which starts with the syntonic comma, 81/80, which names the temperament). The construction of the syntonic temperament's note-space is shown in video2 (Syntonic note-space).
Video2: generating the syntonic temperament's note space
The valid tuning range of the syntonic temperament is show in Figure1.
Figure1: The valid tuning range of the syntonic temperament, noting its valid tuning ranges at different p-limits and some notable tunings within those ranges.
A keyboard that is generated by a temperament is said to be isomorphic with that temperament (from the Greek "iso" meaning "same," and "morph" meaning "shape"). Isomorphic keyboards are also known as generalized keyboards. Isomorphic keyboards have the unique properties of transpositional invariance[11] and tuning invariance[7]:4 when used with rank-2temperaments of just intonation. That is, such keyboards expose a given musical interval with "the same shape" in every octave of every key of every tuning of such a temperament.
Of the various isomorphic keyboards now known (e.g., the Bosanquet, Janko, Fokker, and Wesley), the Wicki-Hayden keyboard is optimal for dynamic tonality across the entire valid 5-limit tuning range of the syntonic temperament.[2]:7-10 The isomorphic keyboard shown in this article's videos is the Wicki-Hayden keyboard, for that reason. It also has symmetries related to Diatonic Set Theory, as shown in Video 3 (Same shape).
Video3: Same shape in every octave, key, and tuning
The Wicki-Hayden keyboard embodies a tonnetz, as shown in video4 (tonnetz). The tonnetz is a lattice diagram representing tonal space first described by Euler (1739),[12] which is a central feature of Neo-Riemannian music theory.
Video4: the keyboard generated by the syntonic temperament embodies a tonnetz.
Non-Western tunings
The endpoints of the valid 5limit tuning range of the syntonic temperament, shown in Figure1, are:
Perfect5 = 686cents (7 TET): The minor second is as wide as the major second, so the diatonic scale is a seven-note whole tone scale. This is the traditional tuning of the traditional Thai ranatek, in which the ranat's inharmonic timbre is maximally consonant.[8]:303 Other non-Western musical cultures are also reported to tune their instruments in 7TET, including the Mandinka balafon.[13]
Perfect5 = 720cents (5 TET): The minor second has zero width, so the diatonic scale is a five-note whole tone scale. This is arguably the slendro scale of Java's gamelan orchestras, with which the gamelan's inharmonic timbres are maximally consonant.[8]:73
Dynamic timbres
The partials of a pseudo-harmonic timbre are digitally mapped, as defined by a temperament, to specific notes of a pseudo-just tuning. When the temperament's generator changes in width, the tuning of the temperament's notes changes, and the partials change along with those notes – yet their relative position remains invariant on the temperament-generated isomorphic keyboard. The frequencies of notes and partials change with the generator's width, but the relationships among the notes, partials, and note-controlling buttons remain the same: as defined by the temperament. The mapping of partials to the notes of the syntonic temperament is animated in video5.
Video5: Animates the mapping of partials to notes in accordance with the syntonic temperament.
Dynamic tuning
On an isomorphic keyboard, any given musical structure—a scale, a chord, a chord progression, or an entire song—has exactly the same fingering in every tuning of a given temperament. This allows a performer to learn to play a song in one tuning of a given temperament and then to play it with exactly the same finger-movements, on exactly the same note-controlling buttons, in every other tuning of that temperament. See video3 (Same shape).
For example, one could learn to play Rodgers and Hammerstein's "Do-Re-Mi" song in its original 12tone equal temperament(12 TET) and then play it with exactly the same finger-movements, on exactly the same note-controlling buttons, while smoothly changing the tuning in real time across the syntonic temperament's tuning continuum.
The process of digitally tempering a pseudo-harmonic timbre's partials to align with a tempered pseudo-just tuning's notes is shown in video6 (Dynamic tuning & timbre).[3]
Video6: Dynamic tuning & timbre.
New musical effects
Dynamic Tonality enables two new kinds of real-time musical effects:
Tuning-based effects, that require a change in tuning, and
Timbre-based effects, that affect the distribution of energy among a pseudo-harmonic timbre's partials.
Polyphonic tuning bends, in which the pitch of the tonic remains fixed while the pitches of all other notes change to reflect changes in the tuning, with notes that are close to the tonic in tonal space changing pitch only slightly and those that are distant changing considerably;
New chord progressions that start in a first tuning, change to second tuning (to progress across a comma which the second tuning tempers out but the first tuning does not), optionally change to subsequent tunings for similar reasons, and then conclude in the first tuning; and
Temperament modulations, which start in a first tuning of a first temperament, change to a second tuning of the first temperament which is also a first tuning of a second temperament (a "pivot tuning"), change note-selection among enharmonics to reflect the second temperament, change to a second tuning of the second temperament, then optionally change to additional tunings and temperaments before returning through the pivot tuning to the first tuning of the first temperament. An example would be:
Tuning from 12-TET (P5=700) up to P5=701.71, and continuing to play with the schismatic temperament's note-choices, before
Returning to the "home" quarter-comma meantone tuning, temperament, and note-choices via the 12-TET pivot tuning.
The fact that the Major Third and Diminished Fourth are enharmonic in 12-TET – enabling the fundamental's 5th partial to be sounded by either interval – is what makes 12-TET suitable as a pivot tuning between the syntonic and schismatic temperaments.
At both endpoints of this temperament modulation (syntonic, P5=696.6; and schismatic, P5=701.71), the 5th partial is exactly the just 5:4 ratio (386.31 cents) above its fundamental, as per the Harmonic Series, via different intervals (syntonic, Major Third; schismatic, Diminished Fourth).
Timbre-based effects
The developers of dynamic tonality have invented novel vocabulary to describe the effects on timbre by raising or lowering the relative amplitude of partials.[15] Their new terms include primeness, conicality, and richness, with primeness being further subdivided into twoness, threeness, fiveness etc.:
Primeness
The overall term primeness refers to the level to which overtones or partials of the fundamental tone whose harmonic order is a multiple of some prime factor; for example:
The order of partials 2, 4, 8, 16, ..., 2n (for n = 1, 2, 3 ...) only contain the primefactor2, so this particular set of partials is described as having twoness, only.
The of partials numbered 3, 9, 27, ..., 3n can only have their order divided evenly by the prime number3, and so can be said to only demonstrate threeness.
Partials of order 5, 25, 125, ..., 5n can only be factored by prime5, and so those are said to have fiveness.
Other partials' orders may be factorised by several primes: Partial12 can be factored by both 2 and 3, and so shows both twoness and threeness; partial15 can be factored by both 3 and 5, and so shows both threeness and fiveness. If yet another appropriately-sized comma is introduced into the syntonic temperament's sequence of commas and semitones it can provide for a 7thorder/ partial (see video5), and thus enable sevenness.
Consideration of primeness of a sound is meant to enable a musician to thoughtfully manipulate a timber by enhancing or reducing its twoness, threeness, fiveness, ..., primeness.
Conicality
Specifically turning down twoness produces timbre whose partials are predominantly odd order – a “hollow or nasal” sound[16] reminiscent of cylindrical closed bore instruments (an ocarina, for example, or a few types of organ pipes). As the twoness increases, the even partials increase, creating a sound more reminiscent of open cylindrical bore instruments (concert flutes, for example, or shakuhachi), or conical bore instruments (bassoons, oboes, saxophones). This perceptual feature is called conicality.
Richness
The term richness is close to common use for describing sound; in this context, it means the extent to which a timbre's spectrum contains partials whose orders include many different prime factors: The more prime factors are present in the orders of a timber's loud partials, the more rich the sound is. When richness is at minimum, only the fundamental sound is present; as it is increased, the twoness is increased, then the threeness, then the fiveness, etc.
Superset of static timbre paradigm
One can use Dynamic Tonality to temper only the tuning of notes, without tempering timbres, thus embracing the Static Timbre Paradigm.
Similarly, using a synthesizer control such as the Tone Diamond,[17] a musician can opt to maximize regularity, harmonicity, or consonance – or trade off among them in real time (with some of the jammer's 10degrees of freedom mapped to the tone diamond's variables), with consistent fingering. This enables musicians to choose tunings that are regular or irregular, equal or non-equal, major-biased or minor-biased – and enables the musician to slide smoothly among these tuning options in real time, exploring the emotional affect of each variation and the changes among them.
Compared to microtonality
Imagine that the valid tuning range of a temperament (as defined in Dynamic Tonality) is a string, and that individual tunings are beads on that string. The microtonal community has typically focused primarily on the beads, whereas Dynamic Tonality is focused primarily on the string. Both communities care about both beads and strings; only their focus and emphasis differ.
Example: C2ShiningC
An early example of dynamic tonality can be heard in the song "C2ShiningC".[18][3]
This sound example contains only one chord, Cmaj, played throughout, yet a sense of harmonic tension is imparted by a tuning progression and a timbre progression, as follows:
Cmaj 19 TET harmonic
⟶
Cmaj 5 TET harmonic
⟶
Cmaj 19 TET consonant
⟶
Cmaj 5 TET consonant
The timbre progresses from a harmonic timbre (with partials following the harmonic series) to a 'pseudo-harmonic' timbre (with partials adjusted to align with the notes of the current tuning) and back again.
Twice as rapidly, the tuning progresses (via polyphonic tuning bends), within the syntonic temperament, from an initial tuning in which the tempered perfect fifth (p5) is 695cents wide (19tone equal temperament, 19 TET) to a second tuning in which the p5 is 720cents wide (5 TET), and back again.
As the tuning changes, the pitches of all notes except the tonic change, and the widths of all intervals except the octave change; however, the relationships among the intervals (as defined by the syntonic temperament's period, generator, and comma sequence) remain invariant (that is, constant; not varying) throughout. This invariance among a temperament's interval relationships is what makes invariant fingering (on an isomorphic keyboard) possible, even while the tuning is changing. In the syntonic temperament, the tempered major third (M3) is as wide as four tempered perfect fifths (p5‑s) minus two octaves – so the M3's width changes across the tuning progression
from 380cents in 19 TET (p5 = 695cents), where the Cmaj triad's M3 is very close in width to its just width of 386.3cents,
to 480cents in 5 TET (p5 = 720cents), where the Cmaj triad's M3 is close in width to a slightly flat perfect fourth of 498cents, making the Cmaj chord sound rather like a Csus 4.
Thus, the tuning progression's widening of the Cmaj's M3 from a nearly just major third in 19 TET to a slightly flat perfect fourth in 5 TET creates the harmonic tension of a Csus 4 within a Cmaj chord, which is relieved by the return to 19 TET. This example proves that dynamic tonality offers new means of creating and then releasing harmonic tension, even within a single chord.
This analysis is presented in Cmaj as originally intended, despite the recording actually being in Dmaj.
History
Dynamic tonality was developed primarily by a collaboration between William Sethares, Andrew Milne, and James ("Jim") Plamondon.
A prototype of the Thummer
The latter formed Thumtronics Pty Ltd. to develop an expressive, tiny, electronic Wicki-Hayden keyboard instrument: Thumtronics' "Thummer."[19][20][21][22][23] The generic name for a Thummer-like instrument is "jammer." With two thumb-sticks and internal motion sensors, a jammer would afford 10degrees of freedom, which would make it the most expressive polyphonic instrument available. Without the expressive potential of a jammer, musicians lack the expressive power needed to exploit dynamic tonality in real time, so dynamic tonality's new tonal frontiers remain largely unexplored.
Related Research Articles
An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.
In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.
In music, there are two common meanings for tuning:
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the fundamental frequency of a periodic signal. The fundamental frequency is also called the 1st harmonic; the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a harmonic series.
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifths which are "pure" or perfect, with ratio . This is chosen because it is the next harmonic of a vibrating string, after the octave, and hence is the next most consonant "pure" interval, and the easiest to tune by ear. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈ 702 cents wide.
Meantone temperaments are musical temperaments; that is, a variety of tuning systems constructed, similarly to Pythagorean tuning, as a sequence of equal fifths, both rising and descending, scaled to remain within the same octave. But rather than using perfect fifths, consisting of frequency ratios of value , these are tempered by a suitable factor that narrows them to ratios that are slightly less than , in order to bring the major or minor thirds closer to the just intonation ratio of or , respectively. A regular temperament is one in which all the fifths are chosen to be of the same size.
In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.
In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including Pythagorean and most meantone temperaments.
A regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most commonly used in the Western world, the generator is a tempered fifth, which is the basis behind the circle of fifths.
In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.
In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness, unpleasantness, or unacceptability, although there is broad acknowledgement that this depends also on familiarity and musical expertise. The terms form a structural dichotomy in which they define each other by mutual exclusion: a consonance is what is not dissonant, and a dissonance is what is not consonant. However, a finer consideration shows that the distinction forms a gradation, from the most consonant to the most dissonant. In casual discourse, as German composer and music theorist Paul Hindemith stressed, "The two concepts have never been completely explained, and for a thousand years the definitions have varied". The term sonance has been proposed to encompass or refer indistinctly to the terms consonance and dissonance.
In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Each step represents a frequency ratio of 21⁄53, or 22.6415 cents, an interval sometimes called the Holdrian comma.
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET or 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Each step represents a frequency ratio of 31√2, or 38.71 cents.
In music, 19 equal temperament, called 19 TET, 19 EDO, 19-ED2 or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps. Each step represents a frequency ratio of 19√2, or 63.16 cents.
In music, 22 equal temperament, called 22-TET, 22-EDO, or 22-ET, is the tempered scale derived by dividing the octave into 22 equal steps. Each step represents a frequency ratio of 22√2, or 54.55 cents.
In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the process of altering the size of an interval by making it narrower or wider than pure. "Any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds is called a temperament." Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation, Pythagorean tuning and meantone temperament meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.
A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.
An isomorphic keyboard is a musical input device consisting of a two-dimensional grid of note-controlling elements on which any given sequence and/or combination of musical intervals has the "same shape" on the keyboard wherever it occurs – within a key, across keys, across octaves, and across tunings.
In music, 17 equal temperament is the tempered scale derived by dividing the octave into 17 equal steps. Each step represents a frequency ratio of 17√2, or 70.6 cents.
William A. Sethares is an American music theorist and professor of electrical engineering at the University of Wisconsin. In music, he has contributed to the theory of Dynamic Tonality and provided a formalization of consonance.
↑ Euler, Leonhard (1739). Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae (in Latin). Saint Petersburg Academy. p.147.
↑ Jessup, L. (1983). The Mandinka Balafon: An introduction with notation for teaching. Xylo Publications.
Spectral Tools home page, with examples of and tools for creating music that takes advantage of dynamic tonality
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