Dissonance Curves as Gen a Rating Devices for Dealing With Harmony

8/3/2019 Dissonance Curves as Gen a Rating Devices for Dealing With Harmony

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DISSONANCE CURVES AS GENERATING DEVICES FOR

DEALING WITH HARMONY

Juan Sebastin Lach Lau

Composer

Morelia, Mxicohttp://web.me.com/jslach

[email protected]

ABSTRACT

Dissonance curves are the starting point for an

investigation into a psychoacoustically informed

harmony. The research comes from the development of

tools for algorithmic composition. These tools aid the

composer by extracting pitch materials from sound

signals, analyzing them according to their timbral andharmonic properties and putting them into motion

through different rhythmic and textural procedures. The

tools are useful either for generating instrumental scores,

electroacoustic soundscapes or interactive live-

electronic systems.

1. INTRODUCTIONDuring the 20th century harmonic theory almost came to

a halt due to the saturation, around 1910, of tonal

harmony, which gave way to a multitude of approachesfor composing music based on aspects of sound other

than pitch relationships. This lead to a broadening of

compositional materials and aesthetic experiences in

which new pitch relations took a secondary role to that

of timbral, textural and rhythmic explorations. Since the

1970s there has been a silent but gradual reawakening of

interest in harmony in a sense that is not limited to the

materials and procedures of the diatonic/triadic tonal

system but that incorporates the widened range of

musical materials and aesthetic concerns of today.

The present study approaches aspects of microtonal

harmony through algorithmic composition tools that

spring out of psychoacoustic research and, through a practice-based compositional approach, develops

insights into the features of the pitch materials produced

by the tools. It delves into theoretical aspects of

harmony, surveying concepts such as harmonic space,

harmonic fields, harmonic islands and rhythmic

harmony, which provide avenues of exploration for the

discovery of new harmonic possibilities. The theoretical

journey is done at three harmonic levels: the micro level

usually understood as timbre, the meso level of texture

and rhythm and the macro level of form.

The tools are available as a library for SuperCollider, a

sound synthesis and algorithmic composition object-

oriented language. It is called DissonanceLib and is

available as an extension package (a quark in

SuperCollider parlance, see [8]).

2. TIMBRAL VS. PROPORTIONAL HARMONYTwo perceptual aspects are normally subsumed under

the term harmony: one which has to do with proportion,

ratio, number, which we'll call its proportional side, and

another that involves sensation and acoustic constitution,which we'll call its timbral facet. These two sides of

harmony are interconnected, without one of them being

able to persist on its own without the other. It is rather

the perspective brought by their setting into context

which can make one aspect stand out from the other and

to this extent, they are both active to different degrees in

different musics and the thresholds and contexts that

produce their mixtures or separation are composable.

This dual aspect has a long history, one of conflict

between proportion and spectrum, stemming from

different orientations towards intervallic qualities. They

are homologous to the divide in mathematics betweenthe study of the continuous and the discrete, its history

going back to the Greeks harmonists, having on one side

a discrete approach in arithmetic with the Pythagoreans

and on the side of the continuous, Aristoxenos and

geometry. These divisions live up till today, emerging as

different approaches to the problem of explaining

consonance and dissonance or the perception of pitch, its

mechanisms being divided into those of pitch-height and

pitch-chroma, for example. These different approaches

represent a polarity or inherent tension that is specific to

pitch relations.

Broadly speaking, it can be said that throughout different

genres of contemporary and electroacoustic music, the

principal pitch techniques deal mostly with aspects of

pitch having to do with higher or lower, that is, with

timbral rather than proportional harmonic relations.

Spectral, atonal and most electroacoustic approaches to

pitch tend towards this timbral aspect. Even most

properly harmonic music, does not go much further from

the harmonic procedures of the early modernist period.

3. DISSONANCE CURVESDissonance curves go back to the psychoacoustics ofHermann von Helmholtz in his book Die Lehre von den

Tonempfindungen of 1862, whose translation and

extension into English as On the Sensations of Tone in

1885 by Alexander Ellis, another important


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psychoacoustician, is one of the most influential books

in the history of acoustic physiology and one of the few

scientific books from the nineteenth century which is

still being published and read in the twentyfirst. (See

[4]. For a comprehensive survey of dissonance curves,

see [7] as well as [2])

Dissonance curves are based on the phenomenon of

roughness or sensory dissonance. Roughness is related

to beatings between sounds, that is, fluctuations in

dynamics produced as a result of interferences between

the amplitudes of two periodic sounds. Furthermore, it

also refers to those interferences that happen between

the partials of a single sound (which is the case of

intrinsic roughness).

When these beatings are slow, they are heard as

amplitude modulations; a common example occurs

when tuning a guitar. At these speeds the beatings are

known as tremolos and their speed, for pure waves, is

the difference between the frequencies of their

fundamentals in cycles per second.

When two sine waves coincide in frequency there are no

beatings. As the frequency of one moves upward, little

by little beatings are produced at progressively higher

speeds. When the two waves are more than 16Hz away

from each other the tremolo is fast enough to become a

continuous vibration, giving rise to an emerging

sensation of a (low) tone, without loosing the rough and

raspy character for which this timbral quality has been

named sensory dissonance.

Helmholtz showed that the interference does not happen

exclusively in the sounding waves themselves, but that

the phenomenon is also produced in perception: it is aproduct of translation, as a consequence of mechanical

processes in the physiology of the ear.

He also showed how roughness reaches a maximum at

around 33Hz for tones of around 100Hz. At higher

speeds, roughness diminishes till it disappears

completely; as the tones move further away from each

other the beatings seem to cease their mutual influence

as they begin to be heard independently.

If instead of interpreting this in cycles per second (Hz)

we see them in the logarithmic scale of cents (invented

by Ellis), we begin to see a pattern: for almost all the

auditory register the interferences happen within the

interval of a minor third. This interval is the limit

between melodic (steps) and harmonic (jumps)

intervals. Even more, if we study this with the bark

scale, which is calibrated to the resolution of the ears

physiology, we see that the interval in barks is the same

for any roughness and in any register.

Helmholtzs theory of hearing models the ear as a bank

of resonators. This model is one of the two types of

psychoacoustic pitch perception theories: those based on

spatial processing (like this one, physiological and

dependent on spectrum), and those based on temporal

processing (which are psychological and depend on

waveform and periodicity, being more relevant to the

proportional aspect of harmony). Some of the current

spatial theories are refinements on Helmholtzs, derived

from discoveries made in the twentieth century, related

to the basilar membrane in the cochlea and known as the

critical bandwidth model. (See [6])

Helmholtzs theory of consonance and dissonance istimbral, based on the spectral content of sound and on

the specific registers in which the partials occur. This is

why timbral harmony depends on register and spectrum;

however, proportional harmony is independent of both

ambitus and timbre.

In a theory based on resonators it is possible to measure

the roughness happening between all partials of a sound

against those same partials transposed by a certain

interval in order to obtain the total roughness

contributed by them all. In terms ofbarks, the maximum

roughness between partials happens between a quarter

and a third of a bark. Sweeping the intervals (in the

manner of a glissando), where for each new intervallic

step the total roughness is calculated, we get a

dissonance curve, which is the roughness profile for the

sound as transposed against itself within a certain range.

The ultimate aim of dissonance curves, at least for our

compositional purposes, lies not so much the

measurement of roughness as in the further analysis of

the curves, which yield, out of their local minima,

intervallic pitch sets with interesting properties (See

Figure 1). These intervals are not only limited to the

partials of the sources spectrum, being more and of

various types and also dependent on the sweeping

interval over which the curve is made. All of them havethe property of being points where the timbre is

minimally rough with itself, which makes them more or

less compatible or concordant with the timbral character

of the source spectrum. It also happens that they

coincide with important harmonic points which is the

reason why a rationalization is performed on the

frequency ratios in order to match them to musically

useful ones. Therefore, they can be used either in

spectral as well as harmonic ways, and this is why they

are further classified and separated by means of an

analysis over the pitch-distance continuum as well as

inside harmonic space.

The pitch sets produce irregular microtonal scales

having variable distances between their intervals with a

different structure for every octave register. The sets are

interpreted in various ways: each interval is represented

as a distance in cents, as a ratio, as a harmonic vector

with a harmonic metric (to choose from harmonicity

(Barlow [1]), harmonic distance (Tenney [9]), gradus

suavitatis (Euler) and geometric norm). On top of this,

each interval stores its roughness measure, which can be

useful for dynamic balances. As pitch sets, they are

partitioned into timbral and harmonic subsets and a

probability and ranking matrix is created, from which

their stochastic harmonic field can be constructed. The


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separation into harmonic and timbral sets is done

according to the periodicity block to which they

belong. (More on harmonic space and fields below. For

more on periodicity blocks, see Fokker, [3])

Different roles can be assigned to the separated interval

sets (and there are many parameters to fine tune the

results). Timbral intervals, holding a close spectral

relationship with the source sound are susceptible, for

example, of being used on top of recorded concrete

sounds, to either reinforce or color them as well as for

instrumental synthesis. Intervals falling within a

periodicity block are prone to be used in a harmonic

setting and can be correlated to certain (12, 19, 22, 31,

41 and 53-tone) equal divisions of the octave which

approximate them and allow for a permutational use of

the intervals by treating them as degrees of those

temperaments. Harmonic intervals are also compatible

with the original source but in a more abstract way and

their settings involve longer time and rhythmic frames

than timbral ones, which focus on the transitory present;

they are less immediate and can function as

fundamentals, tonics, pedals and drones, depending on

their duration1.

Figure 1. A dissonance curve taken from a

mathematical (as opposed to an empirical) spectrum

(that of a sawtooth wave) over an octave. The yielded

ratios are shown beneath each minima, corresponding

to just-intoned intervals.

4. HARMONIC SPACE[C]urrent acoustical definitions of pitch [conceive

it] as a one-dimensional continuum running from low to

high. But our perception of relations between pitches is

more complicated than this. The phenomenon of

octave-equivalence, for example, cannot berepresented on such a one-dimensional continuum, and

octave-equivalence is just one of several specifically

harmonic relations between pitches i.e. relations

other than merely higher or lower. This suggests

that the single acoustical variable, frequency, must give

rise to more than one dimension in sound-space that

the space of pitch perception is itself

multidimensional. This multidimensional space of pitch-

1 Giving a review of possible experimental harmonic strategies isbeyond the scope of this article, though it is a very interesting field of

exploration, still in a compositional more than a theoretical stage at themoment.

perception will be called harmonic space. (James

Tenney, [9])

Harmonic space can be seen to go as far back as

Leonhardt Euler in the XVIII century, when he

proposed visualizing harmonic relations in two

dimensions, arranged by fifths and thirds. Alexander

Ellis, in his appendixes to Helmholtz, devises a more

complete harmonic duodenarium. Later composers such

as Harry Partch ([6]) and Ben Johnston, or scientists

such as Adriaan Fokker and H. C. Longuet Higgins will

also deal with pitch relations in terms of some kind of

multidimensional discrete lattice of points.

Tenney, however, is the first to make a call for arms in

order to rehabilitate harmony from a compositionally

insightful perspective. He also develops the topic from

the standpoint of experimental music, not involving

nostalgia for past musics, though attempting to include

them as well. He proposes this new approach to

harmony basing it on the unlikely figure of John Cage,

who is well known to have disliked harmony because ofits connotations with the German symphonic tradition

and because it forced an a priori (that is, logical and

thus arbitrary) thinking upon sounds themselves.

However, it is by using Cages ideas regarding

composition with any possible sound that the ground is

set for rediscovering a harmony able to link sound- and

note-based composition, capable of incorporating and

pertaining to any type of sound, beyond the worn-out

opposition between noisy and musical sounds.

The axes of the lattice correspond to fundamental

harmonic intervals, defined by prime numbers. Just as

they are of chief importance in arithmetic, primenumbers play a determining role in harmonic qualities,

defining their primary colors or chromas from which

chords and tonal zones are constituted. Number 2

defines octaveness, number 3 the quality of fifths,

number 5 thirds, and 7 sevenths, a dimension which has

been little explored yet because it is not well

approximated by 12 tone temperaments. It is debatable

to which extent chromas higher than 7 (such as 11 and

13, a fourth plus a quartertone and a high minor sixth)

can be perceived without them being confused with

close relationships which are expressible in terms of

lower primes. Each new chroma is weaker than previous

ones and an intervals coordinates define its mixture ofchromas and its relative distance from others. These

relationships also depend on contextual elements to

become audible.

It is well known that an interval does not have to be

exactly tuned in order for it to be identified as belonging

to a certain pitch class. This little understood

mechanism of fault tolerance is at work in chroma

perception and keeps the dimensions in harmonic space

from proliferating indefinitely. It also makes

approximations by temperaments possible. In this

mechanism lies the key for the creation of adequate

contexts for composition with higher and ever stranger

chromas.

2.10.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

170

0

25

50

75

100

125

150

Interval

Roughness

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7/66/5

5/4

4/3

7/5

3/2

8/5

5/37/4

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5. HARMONIC FIELDSIn stochastic harmonic fields the probability of choosing

a note in a pitch set is determined by its harmonicity.

The fields strength is variable and continuous, with

three main zones: at zero all notes have the same

probability (atonal), at one only the most consonant

notes are chosen (tonal) and at minus one only the most

dissonant are chosen (antitonal). This implies the

creation of harmonicity matrixes where the harmonicity

between all intervals is measured. From this matrix all

probabilities are calculated: there is a tonic mode

where only a column of the matrix corresponding to a

certain interval (the tonic) in the pitch set is used; it

works as a sort of mode of the set. An atonic mode is

also possible, using the whole of matrix. It can be

though of as making each new chosen interval the new

tonic and then calculating the probabilities of the next

interval based on this new mode. It has a distinctly

different sonority from the tonic modes.

Other compositional aspects intervene in a harmonicfield, such as the spectral constitution of the generated

notes (which can traverse the consonance and

dissonance polarities of the timbral dimension) as well

as the size and density of the notes (which can be

though of as grains or particles more than traditional

notes, although they do intersect). When the density is

high and size is small, the aggregates tend towards

fusion and coalesce into timbres (harmonic timbres),

while with lower densities/larger sizes the aggregates

tend towards fission and the individuals are heard as

components of chords (timbral harmonies). There are a

few applications built on top of the basic tools that are

used to explore these fields. They have yielded quite afew pieces and algorithmic improvisations already,

although there is still much more to discover about this

approach.

Figure 2. Schematic diagram of harmonic fields. The

horizontal axis pertains to harmonicity, the vertical to

roughness (timbre) and the (supposed) 3rd dimension to

the aggregate dimension of fission/ fusion.

6. CONCLUSIONSThe qualitative and the quantitative, at least in the realm

of perception and the audible, are closely related. In the

case of sonic harmony, consisting mainly in periodic or

quasi periodic sonorities, colors and harmonic qualities

are determined by numeric quantities. These quantities

and qualities correspond formally, not causally.

Although causal factors intervene, the numbers are not

reducible to biophysical processes. The appearance of

numbers does not happen during perception, only after

reflection and theorization (as well as during

composition).

These and other aspects of this research, such as the

relation between rhythm and harmony, have not been

dealt with in the present paper. They have to do with

philosophical (and even metaphysical, in an almost

literal sense of the study of that which is common to all

things (Leibniz), beyond the physical realm, but

springing out from specifically musical questions) and

mathematical issues which are beyond the scope of this

article but which are being pursued as an important

aspect of the authors PhD dissertation, which is in the

writing. Further work is needed in order to develop the

tools towards rhythmic/textural directions in order to

help achieve a generative composition of timbre-pitch-

texture-form. It is also important to let these experiments

permeate and influence aesthetically all aspects of

musical organization in a coherent and integrated way.

7. REFERENCES[1] Barlow, C., Musiquantics, Royal Conservatory, The

Hague, 2006.

[2] Carlos, W., Tuning at the crossroads, ComputerMusic Journal 11/1, 1987.

[3] Fokker, A., Unison vectors and periodicity blocksin the three-dimensional (3-5-7) harmonic lattice of

notes, Dutch Royal Academy of Sciences,

Amsterdam, Proceedings, Series B 72, No. 3, 1969.

[4] Helmholtz H., On the Sensations of Tone as aPsychological basis for the Theory of Music. Dover,

New York, 1960 (1862).

[5] Partch, H., Genesis of a Music, Da Capo Press, NewYork, 1979 (1949).

[6] Plomp, R, Levelt, W., Tonal Consonance and theCritical Bandwidth Journal of the Acoustical

Society of America #38, 548-568, 1966.

[7] Sethares, W., Tuning, Timbre, Spectrum, Scale.Springer, Berlin, 1999.

[8] SuperCollider: http://supercollider.sourceforge.net[9] Tenney, J., John Cage and the Theory of Harmony

Soundings 23, 1984.

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Dissonance Curves as Gen a Rating Devices for Dealing With Harmony