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Music theory book


7.1 Introduction

The term interval signifies the distance between two pitches. The size of an interval may be expressed in different ways.
For example the interval between the notes C and D may be described as:

  • An interval of a tone;
  • An interval of a major second;
  • A ratio of 9/8.

The above expressions describe the same interval from three different points of view. In the same way we might describe the distance between two cities as 100km or 70 miles or 3/4 hour on the train.
The first two expressions - interval of a tone; interval of a major second - are used by musicians. The third definition is used by mathematicians and physicists, who are interested in the numerical relationships between the frequencies of sounds.
Let us take a closer look at these expressions now, starting from the mathematical perspective.

7.2 The auditory sensitivity of the human ear

The human ear measures the relationships between sounds geometrically (1). That is to say it distinguishes the frequency ratio between two pitches rather than their frequency difference. It seems, in effect, not to do subtraction but division. We judge an interval between two pitches to be big or small according to whether the ratio of their frequencies is big or small.
Here is an example:
esempio 1

Braille example

Consider the frequencies of the notes C-F (131Hz-174Hz)
The difference between these two frequencies is 43Hz (174-131=43) and their frequency ratio is 4/3 (174:131 = 4/3).
Compare this with the interval F-G in a higher octave (349Hz; 392Hz). The difference is, once again, 43 Hz but their frequency ratio is 9/8.
If our ears were measuring the difference in frequencies we would hear the two intervals as similar, both having a frequency difference of 43Hz. In reality, however, we hear the first interval, C-F, as being significantly larger than the second one, F-G, as reflected in the relative sizes of the two frequency ratios

7.3 Different ways of describing intervals

Here are two further ways of indicating the distance between two notes:

  1. C - D an interval of a tone;
  2. C - D an interval of a major 2nd.

As already stated, our musical system is based on the subdivision of the octave into 12 equidistant sounds (2).
The distance between one of these sounds and the next is called a semitone. Two semitones make a tone.
It is possible to describe an interval unambiguously by indicating the number of semitones between the two notes concerned. In practice, however, we normally describe intervals not in semitones but with reference to the distance between them based on their alphabetic names.
Consider the interval between the note E and the F immediately above it. This interval is called a 2nd because when we count up the alphabet from the E we go two steps up (the lower note is always counted as 1, not 0). From the E to the G above would, therefore, be a 3rd and so on.
Thus far the naming of an interval is simply a matter of counting up the alphabet but there is a catch. Consider the interval from F to G. This, when we count up the alphabet, is also a 2nd, but it is actually a bigger interval (two semitones) than the interval E to F (one semitone). This means that we need to qualify the numeric description of the interval in some way.
We do this using the following adjectives (3)

  • major
  • minor
  • perfect
  • augmented
  • diminished

The following table shows the classification of all the intervals:

  1. a) The intervals of 1st (unison) (4) 4th, 5th and 8th (octave) may be diminished, perfect or augmented:
  2. b) The intervals of 2nd, 3rd, 6th and 7th may be diminished, minor, major or augmented.

I ese 1 ese 2 ese 3
II ese 4 ese 5 ese 6 ese 7
III ese 8 ese 9 ese 10 ese 11
IV ese 12 ese 13 ESE 14
V ese 15 ESE 16 ESE 17
VI ESE 18 ESE 19 ESE 20 ESE 21
VII ESE 22 ESE 23 ESE 24 ESE 25

Table of intervals in Braille

(1) A geometric progression is a series of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Examples of geometric progressions:
2-4-8-16-32-64 a progression in which the common ration is 2
5-10-20-40 .... a progression in which the common ration is 2
2-8-32-128 ... a progression in which the common ration is 4
(2) Modern acoustics subdivides the octave into 1200 parts, of which the semitone is a 1/100 part: semitone = 100 cents
The cent is an extremely small interval: 2 cents in particular situations can be detected by the human ear but only under laboratory conditions. In the musical context only differences of about ten cents or more are detectable.
(3) For more information about the adjectives used with intervals refer to chapters 10 and 16.
(4) Two notes of the same pitch do not, technically, form an interval so the unison is not always described as an interval. In so far as it is one it must be counted as perfect, like the octave to which it is closely related.

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