TONALITY / HARMONY / COUNTERPOINT

. . . music is sound, what happens to sound.
-Thelonious Monk

Tonality is consensus.  Harmony is accumulation. 
 
This is basic music theory. 

Soundwaves travel at 343 meters per second with slight variations depending on altitude and temperature.

Tonality is the field concerning the definitions of and relationships between pitches.

Pitches are defined by their frequencies (periodic oscillations, cycles per second, or Hz.).  Faster (shorter) cycles are higher.  The lowest note on the piano vibrates at 27.5 Hz, the highest at 4186 Hz.  Sounds beyond these ranges can effect how we hear, but exist as pressure rather than pitch.

The ratios of the rates of oscillations between different pitches are what defines their relationships.  If we double the rate (2/1) we will get a pitch an octave higher, if we halve the rate we will get a pitch and octave lower.  If the relationship is 3/2 we will get a "fifth" (these terms are stupid), if it is 4/3 we get a "fourth", 5/4 a "major third", 6/5 a "minor third", 5/3 a "major sixth", etc.  The simpler the fraction is, the consonant the interval.

There are numerous studies of this subject.
Among them:

Campbell & Greated  The Musician’s Guide to Acoustics (this is the clearest and most comprehensive single book on the physics of sound I have come across)
Helmholtz  On the Sensations of Tone (the classic)
Backus  The Acoustical Foundations of Music
Sir James Jeans  Science and Music

The musical use of pitches involves tunings, sets of pitches arranged as scales (modes).

There are numerous studies of this subject.
Among them:

http://www.rdrop.com/users/tblackb/music/temperament/stoess.htm
http://www.kylegann.com/histune.html
http://home.swipnet.se/~w-37192/eng/handbook/Tuning/history.html
Alain Danielou  Music and the Power of Sound
Harry Partch  Genesis of a Music

Just Intonation vs Equal Temperament

The two principle schools of thought about tonality (intonation) reflect the same opposition as the divided whole / additive perceptions of rhythm.  Equal temperament divides the octave into twelve equal steps.  Just intonation builds from a single tone maintaining "pure" fractional integer relationships in each successive step.  The divided whole (equal temperament) is more flexible and inclusive, but the precise –integer based- relationships of the tones are compromised, i.e. all the intervals are slightly skewed so that any one of them can serve any function.  The additive (just intonation) "sounds" better (i.e., sweeter) and appeals to those who like the simplicity of integers, but the tones are not functionally interchangeable, i.e. the relationship between tone #1(1/1) and tone #4(6/5) is not the same as that between tone #4(6/5) and tone #7(45/32), which would be true in equal temperament (i.e., C : Eflat :: Eflat : Gflat).  

Equal temperament is the standard system used in western music.

A lot of music is neither equally tempered nor justly intoned.

Every tuning system has its strengths and weaknesses.

Music that is not constructed in equal temperament is generally called microtonal.  In my experience, all music not synthesized in equal temperament or played on an equal tempered fixed pitch instrument (such as an expressly tuned piano or xylophone) is microtonal simply because the player controls the pitch instinctively and sweetens the sourness of equal temperament or regularizes the sweetness of just intonation.  Each individual has his or her own sense of pitch, often two ears on the same head hear two different pitches when only one is played.  In keeping the twelve tone equal tempered scale as a basis I do not mean that I think it represents what music ought to sound like, only that it is a simple, efficient, way of looking at pitches and their relationships and that it uses the most generally accepted nomenclature.  In my opinion described music (with named notes) is only an approximation of a desired sound anyway.

The standard instruments of the european tradition were designed to accommodate equal temperament, which is what requires the complex mechanisms of pianos and woodwinds and the drill intensive techniques required to "master" all of them.

"World Music" instruments (generally less technologically complex) tend instead to favor one specific set of tonal relationships.

Simply put, music played on european instruments in the equal temperament system can transpose a set of tonal relationships to have its root on any tone, while music played on "world music" instruments cannot.  This flexibility allows an enormous range in ways of thinking about macro-structural tonal relationships that are unavailable otherwise.  A simple and very common example of this is the emotionally manipulative device used in pop-country songs of playing/singing the last verse of a song in a (usually one-step) higher key ("My Girl", "Delta Dawn" –which does it on every verse, et al), to impart a sense of resolution, triumph, or whatever (literally stepping up).  Another common example is the key-change at the bridge (creating tension and/or contrast) found in most tin-pan-alley songs.  Neither of these devices is available to musicians using unequal temperaments.  On the other hand, untempered instruments offer a wealth of micro-structural tonal relationships that are unavailable on tempered instruments.  Simple examples would be bending sitar or oud notes or sliding shakuhachi tones.
   
Elements of Tonality

Vertical / Horizontal.  These terms refer to the perception of time.  Vertical is simultaneous, horizontal is sequential.

Pedal, Row, Chord, and Mode combine to generate tonality.

Each represents a different aspect and/or perception of pitch relationship.
 
Pedal is the note or group of notes in the low register that define or effect the pitch field(s), i.e., resonance, in which the other elements occur.  Overtones, and thus harmony, tend to develop from low to high as oscillation rates are more likely to halve than double.

(The black hole in the Perseus Cluster (M87) sounds B-flat 57 octaves below middle C.  This may be the pedal tone of the universe.)

Row is the relationship of pitches as a sequence.  This corresponds to the traditional, horizontal, concept of counterpoint. 

Chord is the relationship of a group of pitches sounded simultaneously.  This corresponds to the traditional, vertical, concept of harmony.

Mode is the cumulative relationship of a group of pitches over time regardless of sequence.

None of these elements is truly separate from the others.  Each is a single limited perspective from which tonality as a general whole can be observed.

Non-exclusive Tonality

The reason I couldn't be interested was that harmony didn't have anything to say about noise. 
-John Cage

Non-exclusive tonality is the consideration of all sounds as tonal.

Traditional ideas of tonality have excluded most sonic phenomena from music, erecting a hierarchic set of relationships based on various (often dubious) definitions of consonance.  The more one investigates tunings and scales the more one recognizes the subjectivity of their value and the ephemeral quality of their aptness.  This prejudice has been largely broken down over the last fifty years by innovative (though often marginal) musicians in all disciplines ranging from Penderecki and Ligeti in the "classical" field to Celtic Frost in "rock" (with a lot of "free jazz"  and electro-acoustic musicians in between).

The politics of traditional nomenclature tends to devalue tones that aren’t in a diatonic key by requiring them to be designated as exceptions (accidentals) implying that there is a legitimate rule for excluding them (some tones are more equal than others).  Beyond this it treats all non-chromatic tonal relationships as if they do not exist (reminiscent of the Maoist dictionary that left out the words for concepts they didn’t want people to think about).  This approach has its roots in monotheism –my nomination for the single worst idea of human civilization- (for centuries the catholic church banned certain harmonic and rhythmic practices –now commonly used- as heretical) and pythagorean totalitarianism.  The bias of this approach continues to lead to the idea that certain sounds are "non-tonal" and can be dismissed as "noise".

From a non-exclusive point of view all sounds are accepted as tonal with varying degrees of specificity.  This does not exclude the structures of traditional tonalities, but sees each of them as one among many groups of options.  Instead of "notes vs. noise" the concept is "notes out of noise" or "noise out of notes".  In other words one could derive a harmonic structure out of the clanging of a piece of steel, or one could generate a clanging sound from the harmonic implications of a group of less complex (i.e., more regularly periodic) tones. 

Tonality by consensus

Rather than generating tonalities out from a single source, consensus tonalities tend to emerge from a diversity of sources, coalescing around a particularly strong tone or group of tones.  The open ear is capable of hearing (generating within itself) coherence out of almost any group of sounds.

The history of music theory can be seen as a continually broadening acceptance of dissonance or a continually expanding concept of what consonance is.

One did not, like the serialists, have to undermine the concept of tonality in order to escape from the strictures of Diatonic/Chromatic music.  The great contribution of the serial movement was to open our ears out from the restrictive sense of tonality that had prevailed prior to that time, opening the door to the freer tonal thinking of the "new"  european musicians of the sixties.

Following the innovations of Ligeti, Penderecki, and Xenackis and their contemporaries in free jazz and electro-acoustic music old ideas about what comprised a scale were no longer meaningful in an exclusive sense.  If one wanted to use traditional scale ideas (well-tempered or just) they were available, but if one wanted to do something else that was no longer considered anti-musical (except by conservatives).

One of our goals as anarchists should be to wean our listeners from the expectation of accepted ideas of consonance and "resolution".  Our harmony should aim as much to pose questions as to answer them.

Our definition of consonance should be non-hierarchic and non-exclusive.

Every combination of sounds forms a consonance of some kind.  

"Every noise has a note"
-motto of AMM.

Harmony & Counterpoint

Harmony and counterpoint are the traditional european descriptions of the ways tones work together.

The distinction between harmony and counterpoint blurs the more closely one inspects it as both practices consider the combinations and/or accumulations of tones.  The germinal concept of harmony is to analyze music vertically, to perceive the music as a series of single sounds relating to a predominant voice.  The germinal concept of counterpoint is to analyze music horizontally, to perceive the music as the combination of separate lines of tones.  Obviously music happens both ways, the coincidences of separate contrapuntal lines create vertical events and the sequences of vertical events generate separate horizontal lines within them, so the distinction, in the most mechanical sense, can become largely academic. Regardless of how they are arrived at, simultaneous tonal events occur and the descriptions of them are the study of harmony.  One could arrive at the same musical structure by either means.  The elements of tonality (descriptions of the relationships between pitches) –pedal, mode, chord, and row- are the same in either application.

In the simplest practical terms a song sung over a set a chords represents a harmonic (monophonic) approach while a round like "row, row, row your boat" represents a contrapuntal (polyphonic) one.  (The song would exist without the chords, but the round would not exist -as a round- without the contrasting part.)  Jazz improvisors (after the dixieland era –which was essentially contrapuntal) inverted the monophonic model by deriving melodic material from chord progressions (this was also the methodology of "theme and variations" music in the classical style).  Viewed as symbolic representations of society the song, led by the voice and accompanied by the chords, is a hierarchic structure while the round, forming a whole out of equal individual parts, is an anarchic one. 

Anarchist music will naturally tend to be more polyphonic, seeking order in the accumulation of autonomous voices, generating coherence from inclusion rather than imposing it to validate a single musical idea by supporting it with non-autonomous voices.

Historically, western music (for groups of instruments) was essentially polyphonic (Josquin-Bach) until the rise of the "classical" style (Hadyn-Mozart-Beethoven etc.) which (not coincidentally) coincided with the beginnings of the industrial revolution.

Harmony is timbre.  
–Olivier Messiaen

The exact same group of pitches played in exactly the same order for exactly the same durations will sound different when played on a guitar, a flute, a trumpet, a clarinet, etc.  This is because different instruments produce different waveforms.  The simplest waveform, the sine wave (produced by a flute), generates no waves besides its own.  Other instruments, in addition to the primary wave, the fundamental, produce second, third, etc. (fractional) waves that are less audible than the fundamental, but effect the waveform as a whole and thus the sound of the instrument.  In his music, Messiaen used pitches (harmony) to effect the sound (timbre) of other instruments rather than in the traditional sense of structuring chords. 

Representation

Representation is explanation.

The purpose of nomenclature should be to explain and clarify the elements of music to the uninitiated.  The purpose of notation should be to present music silently in the simplest and least confusing manner. 

Current practice achieves neither of these ends and for the most part does exactly the opposite.  The arcane codes of common practice nomenclature and notation empower technocrats at the expense (and very likely for the purpose) of confusing/excluding everyone else.  The only value I find in knowing traditional notation and nomenclature is that it enables me to read examples in music books.  The specialized nature of this skill is similar to knowing latin as a catholic when it was the language of the celebration of the mass.

Common western practice uses the system of equal temperament.  For practical purposes it is the lingua franca for the discussion of tonal ideas.  This is unfortunate for many reasons, most of all because its nomenclature is confusing, incomplete, and inflexible.  Nevertheless, any experienced musician knows what a B-flat is, or a minor seventh chord, while only a specialized few understand a 5-limit minor third or a syntonic comma.

If it were up to me, I would use the twelve tones of the equal tempered scale, but rename them.  Instead of A B-flat B C etc. I would use integers for the names of the notes –0 1 2 3 etc.  I would pronounce them as the French do, un deux trois etc, so that each note would have a one-syllable name, making it sensible to sing the names of the notes.  Instead of Do Re Mi Fa Sol La Ti Do, one would sing oh deux quatre cinq sept neuf onze oh in the key of C (now called the key of oh), and deux quatre six sept neuf onze un deux in the key of D (now called the key of deux).  This system would make the explanation of the names of intervals far less confusing (how many times does one have to explain to a beginner that a "fifth" is seven "half steps" or that some "thirds" are "minor" and others are "major", but a "fifth" can’t be "minor"  but is "diminished", or why two "thirds" make a "fifth" , or why F doesn’t –normally- have a flat?).  The old system wouldn’t lose much by this and the change would enable microtonalists to be as specific as they wanted about a given pitch, i.e, 384.360 instead of Pythagorean "schismatic" third.  Transposing is really easy this way (simple addition or subtraction).  For the beginner, music would make sense instead of requiring years to learn a lot of silly mumbo-jumbo that has been obsolete in practice since the nineteenth century.  Six lines and spaces could provide a traditional looking staff without the use of accidentals, or forgoing the staff entirely, the integers themselves would be used and rhythmic notation would take another form such as metric units / pulse without barlines.

The need to represent music silently has diminished in importance with the advent of recording (we can hear it now) and with each successive generation of musicians fewer bother learning it at all.  I suspect that the old system will be replaced fairly soon, as more academic musicians work with computers, with a more logical, computer friendly version.  This will be inadequate too, but it will have the benefit of attempting to represent the ways in which people are thinking about sound in our own time. 

The downside of any system of visual representation is that it tends to encourage musicians to design sound instead of hearing and playing it.  To my ears this has sucked the life out of pop music over the last decade as producers have worked more at constructing elegant pictures of waveforms, routinely removing any anomalous and random elements, working toward sound rather than from it.