Intervals and Scales

In mathematics, we have engineering problems that require careful calculations to measure the stress of weight on a wall, or the maximum safe speed for traffic on a highway, or the proper thickness of a concrete floor in a shopping mall. In each case, the human factor is merged with the math to jointly develop an answer, the human factor provides parameters (input data, requirements), and the math formulas provide a logical path to the solution. The human factor provides the reason to use a formula to derive a solution. The math helps us organize our solution and define it with a communicable language.  The human factor judges the utility of the answer produced by the formula.

Music has many of the same attributes. We have formal and informal rules for the composition and analysis of music. However, the human factor interprets music and forms an opinion of the nature and utility of music. Just as we are unique, our opinion about music is unique and our passion for specific songs is unique. When songwriters build their songs, they draw from the tools of scale, interval, rhythm, beat, and tempo.  Moreover, they add the human factor.

Scale. The most common western chromatic scale is a series of 12 semitones that exist in each octave. The chromatic scale consists of 7 scales of 7 pitches for each scale (also know as modes) that use combinations of half steps and whole-steps to define the mode. From what I can determine, the origin of western major and minor scales starts with the music of the ancient Greeks.  Augustine understood the Greek interpretation of the scales, he wrote on the subject at great length. He acknowledged the chromatic scale and modes came from the Greeks.  To explore what he might have been cognoscente of, we also should discuss the Pythagorean and tempered scale.

Pythagorean Scale. Proper application of the circle of fifths creates the chromatic scale using the interval of the 5ths (the sonic distance between the first note of a scale and the fifth note of a scale.  For example, start with a C note on the piano as the starting frequency. A 5th up from C is G, a 5th up from G is D, a 5th up from D is A, a 5th up from A is E, and a 5th up from E is B. The only note missing from the scale is F, which is a 5th down from the starting note C. This system of jumping five notes at a time, if continued backwards, would also yield more notes on the scale and eventually end up back at C. The Circle of 5th’ defines the 12 notes of the chromatic scale and places them into scales depending upon your starting note.

Tempered Scale. Pythagorean Scale (circle of fifths) is mathematically perfect, but the scale has subtle interval (sonic) issues (a classic collision between real numbers and complex numbers).  Getting the tones to display a uniform relationship for use in any scale required a bit of tweaking to achieve uniformity.  Around 1700, musicians choose to balance these inconsistencies by adding a few cents (minuscule adjustments to the frequency of the note) to some intervals, and to shave off a little from others, until the interval ratio between each of the half-step intervals sounded the same. This alteration of the chromatic scale is called tempering, the result being the tempered scale, or the scale of equal temperament (Bach was so pleased with the change he wrote specific music with a multitude of key changes to celebrate this new scale).  Today, we all use the tempered scale.  However, talking about the scale is only the beginning.  What about terms used to define the sonic distance between the notes in the scale?

Intervals. When we listen to the scale plunked out in sequential order, we develop a sense or perception of sonic distance between the notes, we establish mental notions of the relationships between tones in the scale that are built upon the tonic, a distinct “personality” or identity.  There is more, what about when we move from one note to another and plunk out a few in between, like crumbs between the starting point and the stopping point.

Transition. Tones that cause us to anticipate the next tone in the melody are called tendency or transition tones.  Some people define music in terms of resting and tension.  Maximum resting or release occurs at the tonic.  Maximum tension is created by using the fifth chord of the scale.  Movement from the fifth chord to the tonic (carefully selecting notes and chords that lead us to anticipate the tonic chord) creates anticipation of rest.   Movement is what melody is all about, but to make it work properly, we need to talk about time.    To talk about musical time, we need to discuss rhythm, tempo and the beat.

Rhythm and Tempo. In most melodies or chord arrangements, a given note is ‘alive’ for a short period, perhaps a second or less. The very nature of creating a melody involves an artistic transition from note to note with some lingering (time phrasing) along the way. The relative duration of time consumed by the assemblage of notes becomes rhythm.  For me, the essential definition of rhythm begins with the heartbeat, with our breathing, the progression of thought, expressed in time rather than notes.  I think that breathing, pulse, and tides are all examples of rhythm.  The timing of a melody is the core of a song — the beat.

The Beat. The nature of the beat, experienced when listening to music, could be defined as the recurring ‘foot tap’ that occurs within a given song. From a math perspective, rhythm is movement in time and motion is movement in space. The regularity of movement in time is the number of beats in a measure. Measures are a simple tool to acting as a collector or container of beats. We attempt to contain the same number of beats in measure and then provide some sort of a name to the container of beats such as 4/4 (four quarter notes per measure). The measures are stacked together and tightly coupled into units called songs.

We instinctively engage in rhythm when we walk. We create machines with moving parts that often have defined rhythm; we respond to the rhythm; we breathe in a rhythm. Our heartbeat is a rhythm. The core of our life is an embodied rhythm of events and counter events. Our ears respond to sound and we instinctively classify sound as pleasing or not pleasing. We identify, compare, and replicate sound.  We assign meaning to sound.  We assemble sound into collections and call the collective result a song. We layer sounds into a complex tapestry.

If one person were to say that music is a set of mathematical relationships that can be explained with algebraic equations, and another person were to say that music is a gift from God that mankind will never really totally comprehend (please review my essays titled creating art and defining art), I suggest that both of those individuals would be absolutely correct.

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