The elements of music :: Intervals

9. Interval number (or quantity)

9.1. Semitones and tones are used to measure the distance between two pitches. Intervals are used to measure the distance between two scale degrees, or between two notes. Intervals are more than pitch-distances; they are musical distances.

9.2. For example, the distance from C to C♯ and the distance from C to D♭ involve the same two pitches. In absolute (non-contextual) terms, both distances are a semitone in size. But because the notes are spelled differently, the two distances are not the same musically. Musically (or contextually), they are intervals of an augmented unison and a minor second, respectively. As a compromize between absolute and musical terms, you could also call those two distances a chromatic semitone and a diatonic semitone, respectively.

9.3. An interval has two properties—its number (or quantity) and its quality. If you only know an interval's number, then you know the general interval. If you also know its quality then you know the specific interval.

9.4. An interval's number is found by taking the natural name of the lower note and making that the tonic of a major scale. The natural name of the higher note will fall on a scale degree (ignoring sharps and flats). The ordinal number of that scale degree is the interval's number. The intervals between consecutive major scale degrees are all seconds.

9.5. In the interval C to D, for example, the lower note is the tonic of the C major scale and the upper note is the second degree of the scale. So the interval's number is a second.

9.6. Because sharps and flats are ignored, the intervals C to D♭ and C to D♯ also come out as seconds using this method. They are different-sized seconds, but an interval's number alone does not determine the absolute size of the interval. To do that, as we will see, the interval's quality is also needed.

9.7. An interval's number can also be thought of as the number of note letters in the range spanned by the interval.

9.8. For example, the interval C to D♭ involves the note letters C-D—a range of two note letters—so the interval's number is a second.

9.9. The interval from C to E involves the note letters C-E—a range of three note letters—so the interval's number is a third.

9.10. The interval C to C♯ involves only one note letter. Rather than a 'first', such an interval's number is usually called a unison.

Read the instructions on the training page and take the General intervals between general notes test.

10. Scale formulas revisited

10.1. A scale formula is formed by selecting a subset of the octave's twelve pitches. This selection process is driven solely by a sequence of intervals. A scale formula is, essentially, a sequence of intervals.

10.2. A scale formula with n degrees can be defined by a sequence of n-1 intervals. The nth interval can be inferred, being the interval needed to complete the octave.

10.3. The way scale degrees are spelled is determined by the intervals of the scale formula. The note letter of each scale degree is determined by the note letter of the tonic as well as by the number (or quantity) of the intervals leading to it.

10.4. A heptatonic scale formula can be defined by a sequence of six intervals. The intervals are all seconds, so the intervals between consecutive heptatonic scale degrees are seconds. Because the intervals are seconds, consecutive scale degrees have alphabetically consecutive note names.

10.5. A pentatonic scale formula can be defined by a sequence of four intervals. The intervals are a mixture of seconds and thirds, so consecutive scale degrees sometimes have alphabetically consecutive note names and sometimes they skip a letter.

11. Interval quality

11.1. Knowing the number of an interval only tells us its width in note letters. To determine the actual width of an interval in semitones, we need to know its quality as well as its number.

11.2. In the major scale formula we know that some seconds are a semitone in size, and some are a tone in size. The larger seconds are known as major seconds ('major' meaning 'greater') and the smaller seconds are known as minor seconds ('minor' meaning 'lesser').

11.3. An interval whose number is second and whose quality is major is a tone in size and is known as a major second, or maj2.

11.4. An interval whose number is second and whose quality is minor is a semitone in size and is known as a minor second, or min2.

11.5. Augmented seconds (for example, C♭ to D) also occur and are three semitones in size.

11.6. In theory, diminished seconds (for example, B♯ to C) exist and are zero semitones in size.

12. The major scale formula revisited

12.1. We can now define the major scale formula as the sequence of intervals maj2, maj2, min2, maj2, maj2, maj2. A min2 completes the octave.

13. Interval arithmetic

13.1. The result of adding an interval to an interval is an interval. The number of the resulting interval is the sum of the numbers of the operands minus one. The quality of the resulting interval depends on its size in semitones.

13.2. The result of subtracting an interval from an interval is an interval. The number of the resulting interval is the difference between the numbers of the operands plus one. The quality of the resulting interval depends on its size in semitones.

13.3. The addition or subtraction of intervals can result in a descending interval, the analogy of which in numeric arithmetic is the negative number. While the positive or negative quality of a number is known as its sign, the ascending or descending quality of an interval is known as its direction. An interval whose direction is not called out explicitly is assumed to be ascending.

13.4. Starting from a note and moving higher by an interval is the same as adding an interval to a note (or subtracting a descending interval from a note), and the result is a note.

13.5. Starting from a note and moving lower by an interval is the same as subtracting an interval from a note (or adding a descending interval to a note), and the result is a note.

13.6. Determining the interval between two notes is the same as subtracting the destination note from the source note, and the result is an interval.

14. Thirds

14.1. In the major scale formula some thirds (for example, from the tonic ascending to the mediant) are two tones (or four semitones) wide, and some (for example, from the tonic descending to the submediant) are a tone plus a semitone (or three semitones) wide.

14.2. The larger thirds are known as major thirds ('major' meaning 'greater') and the smaller thirds are known as minor thirds ('minor' meaning 'lesser').

14.3. An interval whose number is third and whose quality is major is four semitones wide and is known as a major third, or maj3. A maj3 is equal to maj2 + maj2.

14.4. An interval whose number is third and whose quality is minor is three semitones wide and is known as a minor third, or min3. A min3 is equal to maj2 + min2 or min2 + maj2.

14.5. Augmented and diminished thirds also occur.

15. Fourths

15.1. In the major scale formula nearly every fourth (for example, from the tonic ascending to the subdominant or from the tonic descending to the dominant) is five semitones wide. Because it sounds so pleasing due to its simple frequency ratio, this is known as a perfect fourth.

15.2. Only one fourth—the one from the subdominant ascending to the leading-tone—is not perfect. It is three tones (or six semitones) wide and, because it is a slightly inflated version of the perfect fourth, it is known as an augmented fourth.

15.3. An interval whose number is fourth and whose quality is perfect is five semitones wide and is known as a perfect fourth, or per4. A per4 is equal to maj3 + min2 or min2 + maj3 or min3 + maj2 or maj2 + min3.

15.4. An interval whose number is fourth and whose quality is augmented is six semitones wide and is known as an augmented fourth, or aug4. An aug4 is equal to maj3 + maj2 or maj2 + maj3.

16. Fifths

16.1. In the major scale formula nearly every fifth (for example, from the tonic ascending to the dominant or from the tonic descending to the subdominant) is seven semitones wide. Because it sounds so pleasing due to its simple interval frequency, this is known as a perfect fifth.

16.2. Only one fifth—the one from the leading-tone ascending to the subdominant—is not perfect. It is three tones (or six semitones) wide and, because it is a slightly deflated version of the perfect fifth, it is known as a diminished fifth.

16.3. An interval whose number is fifth and whose quality is perfect is seven semitones wide and is known as a perfect fifth, or per5. A per5 is equal to maj3 + min3 or min3 + maj3 or per4 + maj2 or maj2 + per4.

16.4. An interval whose number is fifth and whose quality is diminished is six semitones wide and is known as a diminished fifth, or dim5. A dim5 is equal to min3 + min3 or per4 + min2 or min2 + per4.

17. Sixths, sevenths, octaves and unisons

17.1. All sixths in the major scale formula are either major (nine semitones) or minor (eight semitones). Augmented and diminished sixths also occur.

17.2. All sevenths in the major scale formula are either major (eleven semitones) or minor (ten semitones). Augmented and diminished sevenths also occur.

17.3. The numbered groups that piano keys are collected into are named octaves because that term describes their size. The numbered octaves on the piano begin and end on C natural but, in general, any span of twelve semitones is an octave. As a musical interval, the name perfect octave is given to the distance between any note and the note with the same name in a higher or lower octave (for example, C4 ascending to C5). Augmented (for example, C4 ascending to C♯5) and diminished (for example, C4 ascending to C♭4) octaves also occur.

17.4. A perfect unison is the interval between any note and itself (for example, C4 to C4). The analogy in numeric arithmetic is the number zero. Just as it makes no sense to talk of the sign of zero, it makes no sense to talk of the direction of a perfect unison. Augmented (for example, C4 ascending to C♯4) unisons also occur and are one semitone wide. Diminished (for example, C4 ascending to C♭3) unisons do not exist, not even in theory, because intervals less than zero semitones in size are undefined in music. Two intervals do exist between the notes C4 and C♭3, however. One is the augmented unison C♭3 ascending to C4, and the other is the descending augmented unison C4 descending to C♭3.

Take the Intervals between notes test.

18. All the intervals in the major scale formula

18.1. The following tables show all the intervals to be found in an octave of the major scale formula.

Take the Intervals between degrees in the major scale formula test.