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as a diminished fourth, for example, or a minor second as an augmented prime. Always classify the interval as it is written; the composer had a reason for writing it that way. That reason sometimes has to do with subtle differences in the way different written notes will be interpreted by performers, but it is mostly a matter of placing the notes correctly in the context of the key, the chord, and the evolving harmony. (Please see Beginning Harmonic Analysis for more on that subject.)

Figure 4.45. Enharmonic Intervals

Any interval can be written in a variety of ways using enharmonic spelling. Always classify the interval as it is written.

Inverting Intervals

To invert any interval, simply imagine that one of the notes has moved one octave, so that the higher note has become the lower and vice-versa. Because inverting an interval only involves moving one note by an octave (it is still essentially the "same" note in the tonal system), intervals that are inversions of each other have a very close relationship in the tonal system.

Figure 4.46. Inverting Intervals

To find the inversion of an interval

To name the new interval, subtract the name of the old interval from 9.

The inversion of a perfect interval is still perfect.

The inversion of a major interval is minor, and of a minor interval is major.

The inversion of an augmented interval is diminished and of a diminished interval is augmented.

Example 4.9.

Figure 4.47.

Exercise 4.5.6. (Go to Solution)

What are the inversions of the following intervals?

Augmented third

Perfect fifth

Diminished fifth

Major seventh

Minor sixth

Summary

Here is a quick summary of the above information, for reference.

Table 4.1. The examples given name the note reached if one starts on C, and goes up the named interval. Number of half steps Common Spelling Example, from C Alternate Spelling Example, from C Inversion

0 Perfect Unison (P1) C Diminished Second D double flat Octave (P8)

1 Minor Second (m2) D flat Augmented Unison C sharp Major Seventh (M7)

2 Major Second (M2) D Diminished Third E double flat Minor Seventh (m7)

3 Minor Third (m3) E flat Augmented Second D sharp Major Sixth (M6)

4 Major Third (M3) E Diminished Fourth F flat Minor Sixth (m6)

5 Perfect Fourth (P4) F Augmented Third E sharp Perfect Fifth (P5)

6 Tritone (TT) F sharp or G flat Augmented Fourth or Diminished Fifth F sharp or G flat Tritone (TT)

7 Perfect Fifth (P5) G Diminished Sixth A double flat Perfect Fourth (P4)

8 Minor Sixth (m6) A flat Augmented Fifth G sharp Major Third (M3)

9 Major Sixth (M6) A Diminished Seventh B double flat Minor Third (m3)

10 Minor Seventh (m7) B flat Augmented Sixth A sharp Major Second (M2)

11 Major Seventh (M7) B Diminished Octave C' flat Minor Second (m2)

12 Perfect Octave (P8) C' Augmented Seventh B sharp Perfect Unison (P1)

Summary Notes: Perfect Intervals

A perfect prime is often called a unison. It is two notes of the same pitch.

A perfect octave is often simply called an octave. It is the next "note with the same name".

Perfect intervals - unison, fourth, fifth, and octave - are never called major or minor

Summary Notes: Augmented and Diminished Intervals

An augmented interval is one half step larger than the perfect or major interval.

A diminished interval is one half step smaller than the perfect or minor interval.

Summary Notes: Inversions of Intervals

To find the inversion's number name, subtract the interval number name from 9.

Inversions of perfect intervals are perfect.

Inversions of major intervals are minor, and inversions of minor intervals are major.

Inversions of augmented intervals are diminished, and inversions of diminished intervals are augmented.

Note

Thanks to everyone who participated in the survey! It was very useful to me, both as a researcher and as an author, to get a better picture of my readers' goals and needs. I hope to begin updating the survey results module in April. I will also soon begin making some of the suggested