Music Theory Interval Classes and the Stability of Harmonic Intervals

Interval Class

An interval class is the result of classifying combinations of two pitch classes. For example pitch classes C and G when combined harmonically could be considered a perfect fifth C up to G or a perfect fourth G up to C. The classification of an interval class is based on the half step distance between the two pitch classes in their smallest combination. In the case of C and G the P4 is the smallest interval in the combination and consists of 5 half steps. A perfect fourth and a perfect fifth are considered as Interval Class (IC) 5. Each interval class consists of an interval smaller than a tritone (augmented fourth or diminished fifth) and its octave pitch class inversion. Perfect Unisons and perfect octaves are considered to be duplicate pitch classes, so their IC number is zero. A table of interval classes would be as follows.

Table 1

Figure 2 - The Natural Harmonic Series Above C2

The numbers (partials or harmonics) represent the position in the series relative to the fundamental pitch (1st partial). The numbers also indicate the frequency ratio between two pitches. For instance, the frequency ratio between partial 4 (C4) and partial 6 (G4) is 6:4. This is the same ratio as 3:2 (G3 to C3) and represents the ratio of a perfect fifth in its pure form. The actual ratio on a tuned piano using equal temperament does not match the 3:2 ratio exactly, but is close enough for most human ears to assume it is "in tune".

The Stability of Intervals: Consonance and Dissonance

Although the classification of harmonic intervals as consonant (stable) or dissonant (unstable) is dependent on cultural preferences and historical context, Paul Hidemith wrote that the stability of intervals could be rationalized from the location of the interval in the natural harmonic series. Intervals which occur lower in the natural harmonic series are more stable than those that occur farther up the series. Intervals with relatively simple frequency rations (3:2 [P5], 4:3[P4], 5:4 [M3], 8:5 [m6], ) tend to be considered consonant (stable) while those with more complex frequency ratios (9:8 [M2], 11:10[m2], 23:16 [A4, d5]) tend to be considered dissonant (unstable). The chart below illustrates interval classes, their sounding roots and their relative stability.

Interval Stability Chart
Interval Class:	5	4	3	2	1	6
Intervals:	P5 - P4	M3 - m6	m3 - M6	M2 - m7	m2 - M7	A4 - d5

Consonant <--------------------------------||---------------------------> Dissonant Stable <--------------------------------||-------------------------> Unstable

The filled in note heads above indicate the sounding root (the most predominant tone) of the interval. Consonant intervals with roots as their bottom pitch (P5, M3, m3) tend to be the most stable intervals and are used frequently to produce resting, pausing or stopping intervals. Notice that the tritone (A4 - d5) has no sounding root and is the most restless, unstable dissonance.

Determining the Stability of a Chord

The relative stability of any chord, regardless of complexity, and the root of any chord can be determined using the interval content of the chord and applying the interval stability chart. The procedure is as follows:

1. Determine the interval content of the chord.
   • Calculate the intervals between every combination of two pitch classes in the chord.
   • Listing the number of each IC present in the chord provides a frequency table of ICs.
   • By listing them in descending numeric order, the table indicates intervals in consonant to dissonant order.
2. Determine the relative level of consonance and dissonance based on the balance of consonant to dissonant intervals.
   • Are there more consonant than dissonant intervals? The chord is more stable than a chord with more dissonant than consonant intervals.
3. The sounding root of the chord is determined by the root of the most consonant interval.

Compare the interval content of the major triad and the diminished seventh chord. As can be seem in the table below, the major triad has only consonant intervals consisting of IC 5, 4 and 3. The illustrated chord has a root of C because the root of the most consonant interval, P5 (IC 5), is C.

The diminished seventh chord on the other hand is very unstable because of the presence of two tritones (D - A^b; F - C^b ). The presence of four IC 3 (m3, M6) interval classes does not offset the dissonant character of the chord. The sounding root of this chord is indeterminate, because there are four IC 3 intervals each with its own root. One could surmise that there are four competing roots in this chord, also contributing to its instability.

Chord 3 is complex chord using a variety of interval classes. The IC frequency content indicates five very strong dissonant intervals (Classes 1 and 6) as well as very stable IC 5 intervals. The strong dissonance again offsets the consonance creating a very unstable dissonant sound. The root of the chord is G# because it is the root of the lowest, most stable interval (P5: G#4 - D#5). Even though the P4 (E4 - A4) is lower than the G#4 - D#5 it is less stable that the P5 according to the Interval Stability Chart.

Chord 1 - Major Triad	Chord 2 - Diminished Seventh Chord	Chord 3 - Complex Chord
IC Freq. 5 1 4 1 3 1 2 0 1 0 6 0 Root C	IC Freq. 5 0 4 0 3 4 2 0 1 0 6 2 Root 4 - roots	IC Freq. 5 3 4 1 3 0 2 1 1 3 6 2 Root G#