# Musical Scales

Background Material for Tuning and Temperament

The building up of a musical scale is based on two
assumptions about the human hearing process:
1. The ear is sensitive to ratios of frequencies (pitches) rather than to differences in establishing musical intervals.
2. The intervals which are perceived to be most consonant are composed of small integer ratios of frequency.
 The octave, fifth, and fourth are the intervals which have been considered to be consonant throughout history by essentially all cultures, so they form a logical base for the building up of musical scales. A typical strategy for using these univerally consonant intervals is the circle of fifths.
Index

Temperament and musical scales

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# Musical Intervals

The term musical interval refers to a step up or down in pitch which is specified by the ratio of the frequencies involved. For example, an octave is a music interval defined by the ratio 2:1 regardless of the starting frequency. From 100 Hz to 200 Hz is an octave, as is the interval from 2000 Hz to 4000 Hz. The intervals which are generally the most consonant to the human ear are intervals represented by small integer ratios. Intervals represented by exact integer ratios are said to be Just intervals, and the temperament which keeps all intervals at exact whole number ratios is Just temperament.

 Examples of just musical intervals: 2:1 octave 3:2 fifth 4:3 fourth 5:4 major third 6:5 minor third
Index

Temperament and musical scales

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# Consonance and Dissonance

Two tones are said to be consonant if their combination is pleasing to the ear, and dissonant if displeasing. The simplest approach to quantifying consonance is to say that two tones are consonant if their frequencies are related by a small integer ratio. The ratio determines the musical interval. For example, the octave 2:1, fifth 3:2, and fourth 4:3 are presumed to be universally consonant musical intervals because most persons in any culture or period of history have considered them to be pleasing tone combinations and have built musical compositions around them.

 For example, in the buildup of a pentatonic scale by a circle of fifths, a natural whole tone of ratio 9/8 emerges, satisfying the condition for consonance. A semitone like E-F also emerges, and the ratio 256/243 suggests dissonance.

When you define "consonance" as "pleasing to the ear", then of course you have to ask "whose ear?". You can get into such intense debate about what is "pleasing" that some have come to define music as "sounds organized by human beings" to accede the endless variety. The use of consonance here is limited to giving a suggestion of a simple rule that yields musical intervals that are pleasing to most people, i.e., "consonant".

Index

Temperament and musical scales

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# Circle of Fifths

A full chromatic scale can be created by using just the perfect fourth and fifth musical intervals. This is characteristic of the Pythagorean temperament. This process can be pictured on the circle of fifths. The outer circle visits all twelve notes on the chromatic scale by going up by fifths (or down by fourths) . The inner circle goes down by fifths (or up by fourths). To create all these notes in the same octave, you could drop down an octave when necessary to stay in the original octave.

 Why fifths?
Index

Temperament and musical scales

Reference
Rossing
Science of Sound, 2nd ed.

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