Background

Musical Intervals

As musical “interval” is the harmonic distance between the pitches of two notes. To take the octave as an example, given a vibration with frequency f (in cycles per second, also known as Hertz, abbreviated Hz), the note that's one octave higher will vibrate with frequency 2f Hz, with successive octaves at 4f Hz, 8f Hz, 16f Hz, and so on. Pythagoras, a Greek philosopher,[2] described this 2,500 years ago.[3]

This doubling of frequency at each octave indicates a logarithmic relationship, which makes discussion and comparison of intervals complex and non-intuitive. In the late 1880's, Alexander Ellis devised a system in which the octave was divided into 1200 “cents”, with each cent denoting 1/1200^th of an octave. Any given interval – not just the octave – can be described as being some number of cents “wide”, or of containing or comprising this or that number of cents, without needing to state any specific pitches. Thus the concept of the “musical interval” is independent of pitch.

In modern Western twelve-tone equal-temperament tuning (12-tet), all twelve semi-tones in an octave are of equal width: 100 cents each.[4]

The Harmonic Series

When a tight string of uniform density and thickness is plucked, it does not sound a single note. An infinite number of notes are sounded, although most are sounded too softly to be detected by the human ear. The length of the string determines the frequency of the loudest note sounded, f. No matter what the pitch of f may be, the other notes sounded always have the same relationship to f, that being 2f, 3f, 4f, 5f, 6f, 7f, 8f, É, ∞f (where ∞ is “infinity”). Each of these notes, called “partials,” adds its “part” to the overall quality or timbre of the plucked string's sound.

The lowest-pitched notes (f, 2f, 3f, etc.) are loudest. You are very unlikely to hear any partials above 12f from any string.

Table 1 below shows the first six partials of the Harmonic Series in which the fundamental frequency f is Middle C (C₄).

Table 1: The Harmonic Series (on C₄)

As you can see, these first few partials sound the C Major triad: CEG. This is not an accident or coincidence. Plucking a string tuned to any given note will sound that note's major triad in its first five partials in the pattern shown above.

Likewise, the human vocal chords also produce sounds that also follow the Harmonic Series, more or less. Each person's voice deviates uniquely from the ideal Harmonic Series, allowing “voice-print identification.” The human ear contains myriad tiny hair cells which are oriented in a specific pattern of spatial intervals which reflect the musical intervals of the Harmonic Series, more or less, thus allowing the ear/brain/mind to act as a “voice-print identifier.”[5]

The pattern of intervals defined by the Harmonic Series is the physical basis of the Western world's music theory.

Patterns of Intervals   

Scales are sequential patterns of intervals, cycling at the octave.[6] Scales are independent of pitch. For example, in the diatonic scale's Ionian mode – the “major scale” – the pattern of intervals is the same for any starting pitch: w‑w‑s‑w‑w‑w‑s, where “w” stands for “whole tone” (two semi-tones) and “s” stands for “semi-tone” (one semi-tone).[7] Change the pitch of the first note (the tonic), and all of the other pitches must change accordingly – but the intervals between the pitches in the scale remain the same.

Even changing to the relative minor of that tonic (Aeolian mode of the diatonic scale) does not change the cyclic sequence of intervals; only the starting point in the cycle is changed (in effect, starting just before the final “w‑s” at the end of the major scale's interval pattern and then wrapping around to the start of the pattern, yielding w‑s‑w‑w‑s‑w‑w). It is this cyclic pattern of intervals – independent of pitch – that defines the diatonic scale. Any sequence of intervals, such as a melody, that is derived from a scale is also just a pattern of intervals, as independent of pitch as is the scale itself.

Simultaneous combinations of notes – that is, chords – are also patterns of intervals. A major triad is simply a root note followed by a major third with a minor third on top. Change the pitch of the root, and the pitches of the other notes must change accordingly – but the pattern of intervals remains the same. It is this pattern of intervals that defines the “major triad.”

The pattern of intervals that defines a major triad is the result of yet another pattern, also related to the pattern of intervals in the diatonic scale. The diatonic scale's cyclical sequence of intervals has 7 modes, each starting the same cyclical sequence in a different place. Taking the starting note of a diatonic mode as its first degree and stacking successive odd-numbered degrees one atop the other, one gets a diatonic “tertian” chord – that is, a chord in which the inter-note intervals are always thirds (either major or minor). This “stacking of thirds” is the basic rule for chord construction across all tonal scales, not just the diatonic scale. It is an inherent characteristic of tonality.

There is another pattern in music that is deeper still, unified by the concept of “meantone tuning.” A meantone tuning (or temperament) is one in which (a) all of the notes are generated by fifths, and (b) the syntonic comma is tempered to unison. The common Western 12-tone equal-temperament tuning (12-tet) is a meantone tuning, but there are many other possible meantone tunings. All of these meantone tunings support (more or less) the fundamental features of tonality – the tonic centre, chords, chord progressions, tension, release, etc.

Western music converged on 12-tet primarily because it was the best meantone tuning for the piano-style keyboard and guitar fret-board. Some “alternative” meantone tunings have been explored in the history of Western music, but most have not, due to their incompatibility with piano-style keyboards and fretted string instruments. However, non-Western cultures make considerable use of meantone tunings which have rarely been exploited in the mainstream of Western music, for reasons that will be discussed later in this document. Facilitating the exploration of non-12-tet meantone tunings could open new doors to cultural understanding and musical creativity.

In short, music is all about patterns of intervals (in rhythm). Unfortunately, these patterns of intervals are not made obvious in the Common Western Music System, which focuses almost exclusively on pitch. Making these patterns of interval more obvious could have significant benefits to music education, cultural exchange, and musical creativity.

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