There are so many conflicting accounts of tonality that it will be useful to restate its premises, axiomatically rather than historically for brevity's sake. Tonality is a hierarchical arrangement of the triads based on the natural harmonics of overtone series of a note.Harmony has been one of the very basic principles of Western music in the last few centuries. The roots of harmony go far beyond music, and its most beautiful manifestation is found in the rules governing the movement of stars and planets in space. Before Schoenberg formulated his theory of harmony, tonality was the underlying principle in the theory of form in Western music. Tonality can be thought of as organizing the harmony of structures to a single point of gravity. This idea is itself embedded in harmony, and that is the same way that all harmonics of a tone are integer multiples of the fundamental. Following those principles, many forms and techniques for composition in tonal music have evolved.
Schoenberg says[41, page 19]:
The material of music is the tone; what it affects first, the ear.He further says about tonality[page 27]:
Tonality is a formal possibility that emerges from the nature of the tonal material, a possibility of attaining a certain completeness or closure (Geschlossenheit) by means of a certain uniformity. To realize this possibility it is necessary to use in the course of a piece only those sounds (Kläng) and successions of sounds, and these only in a suitable arrangement, whose relations to the fundamental tone of the key, to the tonic of the piece, can be grasped without difficulty.The ``certain completeness of closure by means of certain uniformity'' which Schoenberg talks about is in fact the uniformity of material and organization, or in other words, sound and music. He further says[page 29]:
It [tonality] is one of the techniques that contribute most to the assurance of order in musical works -- that order, consistent with the material, which so greatly facilitates the untroubled enjoyment of essential beauties in the music.Again here he is talking about ``that order, consistent with the material'' which is the order of tonal music consistent with its material, or the harmonic sound.
Many schools of classical music, for educational purposes, treat harmony and counterpoint as two different and separate elements. It is perhaps a wrong approach to try to find out if harmony evolved according to the mixture of two or more melodies, or if the scales, upon which the melodies are based, were decided according to harmony. The well-tempered scale is obviously a compromise between harmonicity and position independence from the frame of reference. Rosen says[38, page 25]:
Equal temperament absolves us from considering at length whether or not tonality is a `natural' or a `conventional' language. It is quite evidently based on the physical properties of a tone, and it equally evidently deforms and even `denatures' these properties in the interests of creating a regular language of more complex and richer expressive capacities.If in fact all the scale values would have been chosen according to the physics of a tone, then the values of the elements of the scale would have to be changed any time we change our frame of reference (the tonal center). In other words, the scale gives us a constraint on the continuum of frequency, which is created according to the vertical requirement of cohesion in physics of the tone (sound). However, if we fully abide to this constraint, the horizontal plane (the plane for melodies) becomes so constrained that melodies with the same intervals will sound different in different positions in reference to the tonal center, and further, movement of the tonal center will require repositioning of the elements of the scale.
When a singer2.6 wants to learn a melody, there are two orthogonal requirements which have to be learned, namely time and frequency. For now we refrain from involving the sound parameters and ignore such important factors as timbre and dynamics. The simplest case is when the basic structure of the melody is exactly in the well-tempered (or some other) scale and notes all have equal durations. Then, the act of learning the melody is to remember the sequence of the scale values which have to be sung. However, that only makes up for very expressionless melodies, since the structure of the melody does not carry itself into the duration of every note. A singer can create the feeling of this carry over by changing the intonation or duration of notes. In the simplest case, the amount of deviation has a linear relationship with the structure of the melody. However, in reality, this relationship is not linear. An intonation which matches the structure of the melody is a requirement in the vertical plane, and the order of the progression of the notes in time is a requirement in the horizontal plane. When the singer hears the melody for the first time, she gets an impression of the central idea. In formal terms, this central point is the basic structure of the melody, and perhaps on the plane of poetics we can call that the ``emotional meaning'' of the melody. Once the first impression is learned, the singer builds a relationship with the melody, and with every repeated listening or performance of the melody the vertical and horizontal requirements change to better accommodate the basic structure. A simple change in an element of one of the planes may require changes in the values of the elements in the same plane as well as the orthogonal plane. A very subtle change in the intonation of one note may require changes to many other notes as well as changes to the duration of the notes in time. In this scenario we also have to account for the relationship of the mood of the singer and the ``emotional meaning'' of the melody. Obviously that relationship is not by any means linear either. Through this evolution, the melody ``comes alive'', and it finds its own character which specifies the vertical (scale) and the horizontal (durations and operations in time such as vibrato) requirements. Thus, even though the melody was primarily defined by the scale and durations of the notes, once it is subjected to our thought and emotion, it sets its own terms for scale and durations.
Schoenberg says[41, page 23]:
Intuition and inference (Kombination) assisted in translating the most important characteristic of the tone, the overtone series, from the vertical (as we imagine the position of all simultaneous sounds) into the horizontal, into separate, successive tones.
This process is true not only for a melody and a singer or a composer and a piece, but also for a society and a musical culture. This evolutionary point of view is perhaps a much better way of looking at the development of such principles as scale, voice leading, and chord progression which shape the principles of tonality2.7.
The harmony and voice leading rules imply an interrelated network of constraint for prolonging the structures of a tone and at the same time it is the structures of a tone which leads us to realize this plexus of constraints. This technique also creates a paradigm for the interplay of content and form. A melody has to agree with its harmonic context. A piece may be composed of two different themes whose harmonic (and basic structures) are far apart from each other. The first theme sets up its own harmonic context, and through the grammar of harmonic modulations we can accommodate the second theme. However, if these two themes are too far apart from each other, the integrity between the content (the themes) and the form (the harmonic context) of the piece is broken and the relations will not be comprehensible as a unit. Again if the two themes are too close to each other, they cannot stand for themselves and they become variations of each other. Such circular thoughts are part of the process of evolution. The paths for combinations are endless, and here we need a musical intuition to prune the paths. On the evolution of principles of harmony Schoenberg says[41, page 26]:
It is much more correct to say that the development of harmony was not only essentially influenced by melodic principles, that the development of possibility of voice leading was not only essentially influenced by harmonic principles, but that in many ways each was actually determined by the other.
As more and more we try to apply the operations implied by our intuition and inference to the tone, in our mind we derive a different entity (music) from it. The more these operations are applied to the tone, the more distant the new entity is going to be from the tone. However, the closer these operations are to the nature of the tone, the more they will emphasize the structures in the tone itself, and therefore, the closer the entity becomes to what characterizes a tone in our mind.
Chords are instantaneous entities, and melodies are progressive. A chord progression in a sense is a form of melody in itself. On a higher level, key changes, which use pivot chords for their connections, create another sense of melody. All these progressive elements in different layers are heard by the sensitive ear, and in order to have a closure for the piece as a whole, all these melodies have to be related to each other (i.e., be in harmony). The harmony in the structure of a chord is a harmony in sound, and the harmony of the melodies in different layers of time is a harmony in music. The relationship of such logical entities as music, movements, and melodies has to abide by the same rules that govern the relationship of their parts, namely sound, tones, and chords. Schoenberg's theory of harmony is largely based upon this idea on the level of tones and chords, and the continuum which exists between consonances and dissonances.