This is a work in progress...many definitions and elements are missing and will be added in the future, but it is hopefully enough to get one started... enjoy!
We often hear that "music is so mathematical", but we seldom know how to make this fact work in our favor, especially when faced with the years of practice it takes most people to become proficient enough playing an instrument that music can spontaneously flow out of the heart and through the instrument. Dutiful reading of individual notes on a page, while instructive in technique and the historical brilliance of various composers, does not provide the kind of insight needed for such creativity. This tutorial is intended for those like myself for whom sight-reading does not come easily despite practice, but who have a relatively good "ear" for music--that is, the ability to "hear" everything that is going on in a piece and feel the different harmonic lines intuitively (even if in a vague and undeveloped sense).
Why is music mathematical? It seems related to the physiological fact that what tends to sound musical to the human ear is the kind of sound in which frequencies of the comprising tones are related to one another by whole numbers or ratios of whole numbers. For example, notes separated by one octave are related by doubling of frequency (in the sense of acoustical vibration), and the "fifth interval" (say, "C" and "G" played together) has a ratio of 3/2. It turns out that the relationship between frequencies and musical steps is exponential, since going up successive octaves from an initial note gives you 2x, 4x, 8x, 16x, 32x, etc. the frequency of that initial note. And (as will be shown) this is precisely what leads to a profound constancy of structure in music!
Constancy of Structure. To express this concept in another way, the step relationship characteristic of any given chord structure holds true for that structure in any key, and in any octave. For example, a major chord (the 1-3-5 notes of the familiar scale sung "do-ray-me-fa-so-la-tee-do", or the "do"-"me"-"so" from that scale) has the same structure no matter which key or octave it is played in. The "C" chord (c-e-g, obtained 1-3-5 intervals, or 0-4-7 half-steps in the 12-note chromatic scale) is shown below. Note that the "D" chord (d-f#-a) is also obtained by counting 0-4-7 half-steps.
Proof. There is a simple mathematical proof, in which it is only necessary to show that a pair of notes played at the note of frequency, (fo) separated by n half-steps (fn), have the same frequency ratio (fn/fo) as any other pair (f'o and f'n) separated by n half-steps. Using the exponential nature (and 12 half-steps per octave), the relationship is f(s) = fo 2^(n/12) Writing the frequency ratios fn/fo and f'n/f'o, we get fn/fo = 2^(n/12) = f'n/f'o ... which is a function of "n" alone! This means that the number of interval steps (or chromatic half-steps) that characterize a musical interval in any octave or key is the same, and because it has been proven for any pair of notes, it holds true for any combination of (pairs of) notes. In fact, this constancy of structure holds true for different scales--whether they vary in the number of notes per octave or in the sizes of the intervals.
The Scale. That is the starting point for playing music: knowing that the harmonic structure transposes to any key or octave. In actuality, the exponential relationship between frequencies and steps is only an approximation (but an adequate one) because in real life the intervals have been adjusted so as to allow better "circling", or getting back to the same note after ascending by a fixed integral octave after octave until the original note is reached (the "even-tempered" scale). As an aside, it should be noted that what makes a piano sound like a piano (and not a violin) is the unique distribution of overtones (tonal components that are multiples of the frequency of the fundamental tone of the note), and there are a number of excellent articles, such as "The Coupled Motions of Piano Strings" published in the January 1979 issue of Scientific American on how the piano works and why it sounds as it does. But I digress.
As a simple exercise, start with middle "C" (the white key at middle of the piano adjoining a pair of black keys), and play the scale in the key of "C" by ascending the white keys:
Key of "C" (f=256 hertz) .. notes on the major scale
Note Interval Steps Scale Name Frequency Ratio (approx.) C 1 0 do 1.00 f 1/1 D 2 2 ray 1.12 f 9/8 E 3 4 me 1.26 f 5/4 F 4 5 fa 1.33 f 4/3 G 5 7 so 1.50 f 3/2 A 6 9 la 1.59 f 8/5 B 7maj 11 tee 1.89 f 15/8 C 8 12 do 2.00 f 2/1 Notes not on the major scale but "minor" and/or musically interesting Eb 3m 3 n/a 1.19 f 6/5 F# 4+ 6 n/a 1.41 f 7/5 Bb 7 10 n/a 1.78 f 16/9
The minor scale differs from the major scale only in that the 3rd and 7th intervals are one half-step lower. That is, the notes in the key of E are e-f#-g#-a-b-c#-d#-e, and the notes in the key of Em are e-f#-g-a#-b-c#-d-e. Note (an insight helpful in studying musical patterns) that the notes in the key of G are the same: g-a#-b-c#-d-e-f#-g as the notes for the key of Em; only the starting position in the "circle" (of notes on the scale) is different.
Chords. Using the "theorem" (about the constancy of structure) and the understanding of the intervals that go into an 8-note scale, one can construct any chord desired. A basic (unless otherwise specified, "major") chord is constructed by playing the root note ("do") plus the 3rd interval ("me") plus the 5th interval ("so"); in the 12-note chromatic system, that is, the root note plus four "half-steps" up, plus another three half-steps. (See the examples, like D is d-f#-a)). A minor chord has the root note plus three "half-steps" up, plus another four half-steps (Dm is d-f-a). The very first step in learning how to play is teaching yourself these chords and their inversions, becoming familiar not only with playing them, but with transitioning between them. (Good chords to start out with include D, C, F, G, A, B, E, Em, F#m, Am, Bm. These are common in a lot of chord sheets made for guitar, and can be played by any instrument.)
Families of Chords, and Keys. The chords that go with the key of C (chords whose components are part of or compatible with that scale) include "C", "F", "G", "Am", "Dm" (the comprising notes being c-e-g, f-a-c, g-b-d, a-c-e, d-f-a). For notation, capital letters will be names of chords and small letters the names of individual notes. For the names of chords, plain letters ("C", "D", etc.) denote major chords (normal 1-3-5 interval), "m" denotes minor ("Dm", "Em", etc.), and numbers denote extra notes added in to give the chord more fullness. For example, C2 is the regular C-chord with the 2nd-interval note ("d") simultaneously added in. C4 is the regular C-chord with the 4th-interval note ("f") simultaneously added in, Cmaj7 is the regular C-chord with the major-7th-interval note ("b") simultaneously added in, C7 is the regular C-chord with the minor-7th-interval note ("b-flat") simultaneously added in, and "Csus" is the replacement of the 3rd with the 4th interval, ie c-f-g.
The understanding of "musically compatible" groups of chords (e.g. whose components are notes on the scale of the "key" in which the music is written) is crucial for understanding the structure of various musical pieces--the shape of how chords sequentially build upon and transition between one another. For example, in a piece written in the key of G with chords, G, D, Bm, C, Am, G, the chords are based upon the 1st, 5th, 3rd, 4th, 2nd, and 1st intervals in the "G" scale. The 5th intervals of each chord are based upon the 5th, 2nd, 7th, 1st, 6th, and 5th intervals in the scale, and all are compatible with the scale.
Now for an example: the familiar melody from "What Child is This". It will be done in the key of Em (E-minor), which is a related key to G-major (g-a-b-c-d-e-f#-g) because it has the same notes in the scale. But for reasons which will become clear, it is advantageous to think of it in the key of E-minor. The beginning of melody, "What Child is..." is an Em (the first note itself being an f-sharp, 2nd interval on the Em chord), progressing to the major version (G) in "this, who...". The next chord, in "laid..." is D, progressing to a minor version, Bm in "to rest...". Then, the progression continues to a C and Am in "...on Mary's lap is..." at which point the key changes from minor (Em) to major (E) at "...sleeping" and the chord is E. The cycle repeats (with different words), and then again in a major-chord variation (G D C B) that eventually returns to Em.
First, practice this piece (and other pieces) by playing the chords, filling in the melody by voice and/or with the uppermost note being played. Over a period of weeks, as you become more and more comfortable with playing (and transitioning between) chords, you will become able to add flourishes and interpretations all your own. An example of an incredibly intricate and passionate rendition of this piece can be found Liz Story's arrangement of "Greensleeves" (same melody) on the Windham Hill album, "Winter".
For ongoing practice, there are thousands of pieces notated for guitar, meaning that the chord names are written above the lyrics, such as the "What Child Is This" example below:
1. Roots. The bottom-most note is the "foundation"; it sets the tone for the chord (i.e. what's being played by the little finger of the left hand will generally provide the "root" tone of the underlying chord)
2. Melody. The top-most note (i.e. what's being played by the little finger of the right hand) is the "melody"; it is the most easily identifiable lyrical line and can be heard even when there is a lot going on in the chords. Alternatively, one can play a "counter-melody" lyrical line that is harmonically related to the melody.
3. Inversions (chords played with root note not necessarily at the bottom) provide interesting variation, particularly when elements of a piece are repeated. With practice, one can learn to play different inversions with the left and right hands, thus covering more of the keyboard.
4. Modifiers to chords provide musical interest, means of transitions between chords, or means of scale changes (modulations). For example,
5. Arpeggios (playing of the tones of a chord in rapid succession rather than simultaneously) provide beauty and intricacy; with experimentation, one can find various ways of interweaving the the notes between the left and right hands. Passing notes (temporary configurations outside of the strict chord structure but sensible in the larger context, e.g., notes within the key of the piece) can add wonderful expressiveness.
6. Transitions between chords can be made interesting with the use of additional notes played prior to the actual transition which bear some relationship to each chord. For example, one lyrical movement from Em to D can make use of f#, which is 2nd interval for Em and 3rd interval for D; or alternately (depending upon the effect desired) A, which is 4th interval for Em and 5th for D.
7. Augmented Fourths are not in the major or minor scales (they are in between a fourth and a fifth), but they do provide a fascinating musical quality (acoustically based in a 7/5 frequency ratio) and therefore can be added into arpeggios at times to give them a pensively warm touch.
How did this tutorial come about? Well, in 1992, I was asked to start a support group at my church, and without a music leader, I was concerned because I felt that worship (and the refreshing quiet it fosters) was the most important, healing ingredient. One day as I was in prayer, thoughts began coming to mind about chord structures, including the insight about the constancy described above, which led to experimentations on a keyboard. Within a month, I was playing simple worship arrangements for my group, and now play as well on a music team for Sunday services. It is my hope that in sharing this, more people can experience the joy of playing and creating music, particularly music which touches the heart of the Artist Himself.
See also: the Sight-Singers Resource (Marc Rubin's very fine tutorial for sight-reading for piano and singing, which fills in aspects of the subject which I don't address in much detail)
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