Consonance and Dissonance, Interval Numbers

We have already been using intervals for some time, but the only measuring rulers we have been using are the Whole and Half steps. Counting large pitch distances–intervals–with this method is very inefficient, prone to errors, and not how musicians talk about them. In addition, building chords (Ch. 6) is not done with W and H steps–you have to understand full intervals.

While this course is more of a technical approach to music, the real goal is to have a better understanding of the aesthetics of music. One concept that people generally judge music on is how “pleasant” sounding it is; that’s a subjective word, for sure, but it evokes a widespread psycho-perceptual concept of a favorable musical experience. It’s not that we always want music to sound pleasant; in fact, we enjoy the contrast of certain “harsher” elements of music being introduced. These subjective words are narrowed down to more precise terms with consonance and dissonance. Now that we are discussing intervals, we need to have a more accurate sense of how notes sound together.

Listen to this song by Pentatonix:

Regardless of whether you like their style (or not), most would agree that their harmony–the blending of notes together—has an overall pleasing effect. Musicians call this consonant harmony.

Consider this piece, by Pat Metheny:

Again, some listeners might find this pleasing as a matter of style, but, generally, most people would find this rough sounding or displeasing, which musicians call dissonant.

There are two aspects to consonance and dissonance: acoustic/psychological and stylistic/historical. Certain intervals, due to the physiological nature of our hearing (literally, the cochlea and nerve cells in our ear) sound “rough” or “smooth,” and are generally accepted across cultures as “harsh” or “pleasing.” Within a certain style or time period, context can affect these judgments. This is reflected in the names used for Western musical intervals; for sure, this is only one perspective on these, but this is the musical system we are operating in. Finally, these descriptions below refer to these intervals primarily as harmonic intervals–when the pitches sound at the same time; how they are used melodically are somewhat different, discussed more below.

Lecture Video, Consonance and Dissonance:

Naming Intervals

The first step in naming the interval is to find the distance between the notes as they are written on the staff. Beginning with the first note, count all lines and spaces until, and including the second note. This gives you the number for the interval.

How to count interval numbers.

1a) 1b)   2a)   2b)

The Number is simply counting up total lines and spaces–1a and 1b. Notice that a clef is not needed. With a clef–2a and 2b–interval number is also the same as counting pitch class letter names.

The interval from 3^rd line to 4^th line (1a), from B to D (2a) is a “third”. The interval from the bottom space to 4^th line (1b), from A to F (2b) is a “sixth”. Note that at this stage the key signature, clef, and accidentals do not matter at all. Interval number is only categorizing the pitch classes, i.e., letter names (lines/spaces).

Interval numbers are also classified by two larger categories: Simple intervals are one octave or smaller.

Compound intervals are larger than an octave. You simply keep counting, as you can see from the examples below. From E up to the F an octave higher, E to E would be 8, so F is 9, etc. To convert a compound interval to simple or vice versa, simply add or subtract seven–not eight. If you add/subtract 8, then you are double-counting the octave pitch in your calculation. Prove this to yourself by counting the examples below, and then subtracting 7 from the compound interval. If you move the higher note down an octave, it will match the simple interval.

Lecture Video: Interval Numbers:

Lecture Video: Compound Intervals:

Classifying Intervals

So far, the actual distance, in half-steps, between the two notes has not mattered. But a third made up of three half-steps sounds different from a third made up of four half-steps.  And a fifth made up of seven half-steps sounds very different from one of only six half-steps. In the second step of identifying an interval, clef, key signature, and accidentals become important: this is known as identifying the quality of the interval.

The pitches A to C natural and A to C sharp are both thirds, but A to C sharp is a larger interval, with a different sound. The difference between the intervals A to E natural and A to E flat is even more noticeable. Be sure to listen to this example. 

While using half steps is adequate to identify the precise differences between intervals of the same number, as mentioned before, using them is inefficient and not practical. Therefore, we have another method, but first, let’s talk about the quality categories.

Interval Quality Categories

Interval qualities can be one of the following, and are put into two separate groups:

Interval Categories by Number and Interval Qualities.

There are two basic categories, grouped by interval number: all prime (1), 4th, 5th, and 8th intervals can only be diminished, perfect, or augmented; all 2nd, 3rd, 6th, and 7th intervals can only be diminished, minor, major, or augmented. Interval number categories have limited “crossover.” You can’t have a Major 5^th, nor will you have a Perfect 7^th. Only Diminished and Augmented apply to both categories.

Lecture Video: Interval Categories:

The double arrows represent ONE halfstep difference between each adjacent quality. As the chart states, to the left is smaller, meaning the pitches are closer together (think about this on the piano to help picture it); to the right is larger, meaning the pitches farther apart.

Look and listen to the examples below; notice that you can make an interval quality smaller or larger by changing the top note or the bottom note. Why? Because you are making either the top note or the bottom note move closer or farther from the other note; regardless, the distance between them is changing. Notice also the importance of the enharmonic spelling of the notes: the number is determined by the pitch name (lines/spaces)–not the enharmonic equivalent on the piano. Interval names are about how it is spelled on the page! [Notice the double sharp on the bottom d3 interval–that’s enharmonic to an A natural on the keyboard.] Enharmonic intervals will be explained in more detail below.

Lecture Video: Changing Interval Qualities:

In order to classify intervals more quickly and accurately, we need a starting point, a reference point, to quickly figure out what they are. This is very important to understand: memorize a starting point for interval qualities, and then make adjustments for ones that are not memorized.

Pitch Class Interval Method

This method uses the white keys on the piano, or the generic pitch-class intervals, i.e., no accidentals to start with.

Press the play button in the Noteflight version and listen through the entire chart. 

Lecture Video: Pitchclass Interval Method:

Enharmonic Intervals

The diminished fifth and augmented fourth sound the same on the keyboard. Why? Both are six half-steps, or three whole tones, so another term for this interval is a tritone. In Western Music, this unique interval, which cannot be spelled as a major, minor, or perfect interval, is considered unusually dissonant and unstable (tending to want to resolve to another interval). It’s also unique because it splits the octave exactly in half (6/12 HS). The name, tritone, comes from when it is spelled an Augmented 4^th (see the Pitch Class Interval Quality chart above: F-B, broken down as three whole steps = F-G, G-A, A-B = 3 (whole) tones. In a much earlier system, the tetrachord (4^th scale), played an important role in music theory, and “tone” was the common term for what we call a whole step.

You have probably noticed by now that the tritone is not the only interval that can be notated in more than one way. In fact, because of enharmonic spellings, the interval for any two pitches can be written in various ways. A major third could be written as a diminished fourth, for example, or a minor second as an augmented first (practically, though, enharmonic intervals are not that common for beginners). 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 (Ch. 4), the chord (Ch. 6), and the harmony (Ch. 8).

Inverting Intervals

Knowing how to invert intervals is important for two reasons: 1) it can be used as a shortcut to identify and build intervals; 2) to understand how inversion works with chords (Chapter 6). 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 (this works in both directions!).

While inverting, the numbers and the qualities maintain certain pattern, as shown here:

Patterns for Interval Inversion. The numbers always add up to 9, and the Quality patterns are easy to remember: P stays P, the “M’s” swap, and then the “extremes” swap–the largest A and the smallest d.

Lecture Video, Interval Inversion: