Measuring Intervals Using Ordinal Numbers
We have developed a basic understanding that pitches are high or low in relation to one another, but we haven’t yet investigated the specific distances in pitch space between them. In comparing the pitches of two notes, the distance between them is called an INTERVAL.
One way to measure the interval between two pitches is to count the number of scale degrees spanned by those pitches. Thus, for example, the interval from 1 up to 3 is called a 3rd (since it spans three scale degrees — 1, 2, and 3). Using this method, musicians use ordinal numbers (“2nd…3rd…4th…”) to name intervals, with several exceptions: (1) the interval from one scale degree to itself (no higher, no lower) is not called a “1st,” but rather a “unison”; (2) the interval that spans eight scale degrees is not called an “8th,” but rather an “octave” (as we noted above); (3) all intervals larger than an octave may be referred to as a combination of an octave and some smaller interval, called a compound interval (for example, “an 8va and a 5th”); (4) musicians almost always use compound-interval names to refer to intervals that span more than twelve or thirteen scale degrees (for example, you might hear of either a “10th” or “an 8va and a 3rd” but you will probably never hear of a “14th” — it would be called “an 8va and a seventh”)
You can know the number of the interval without paying attention to the clef. Below, even though the staff has no clef, we know the first interval is a 1st (or “Unison) because the notes appear on the identical line. The second interval is a second, because the notes are now removed by one step. Regardless of whether we are in treble, bass, or any kind of clef – and for that matter whatever key signature is introduced – these intervals will have the same number.
1st 2nd 3rd 4th 5th 6th 7th 8th
Similarly, the distance between letters provides a similar generic number: you simply need to count the letters you are traversing. For instance, any A up to any B is a second, since you traverse two letters: A and B. Any A ascending to any D is a fourth, since you are traversing four letters: A, B, C, and D.
Scale degrees work the same way. The distance traveled between scale degree 2 up to scale degree 4 will be a third, since you traverse three scale degrees: 2, 3, and 4. The distance traveled from scale degree 2 down to scale degree 4 will be a sixth, since you traverse 6 scale degrees: 2, 1, 7, 6, 5, and 4.
The number of the interval, therefore, can simply be found by figuring out how many steps you traverse, whether they be in the musical alphabet, on the staff, or through scale degrees.
Harmonic and Melodic Intervals
An interval is called a HARMONIC INTERVAL when the two pitches are sounded at the same time. An interval is called a MELODIC INTERVAL when the two pitches are sounded at different times.
For the rest of this unit, we’ll represent each interval harmonically, as two whole notes. But you should realize that our work with intervals can apply to any two notes, harmonically or melodically, in any rhythm.
When we use only ordinal labels (“second,” “third,” etc.), we are being general rather than specific: there are several different sizes of each ordinal interval. For example, we have already learned (in Unit 3) that not all seconds are the same size. Some seconds (for example, from C up to D) span two half steps, whereas others (for example, from B up to C) span only one half step. This means that there are at least two different kinds—called qualities—of seconds. We express the quality of an interval by preceding its number with an adjective. For example, the interval from C to D is called a major second, whereas the interval from B to C is a minor second. The five adjectives used for this purpose are MAJOR, MINOR, PERFECT, AUGMENTED, and DIMINISHED.
Methods for Calculating Interval Qualities
There are various ways to calculate the qualities of intervals. Let’s investigate four widely-used ones.
Method 1 Comparison to Intervals above the Tonic in the Major Scale
This method treats the lower pitch in any interval as if it were the tonic in a major scale, then compares the upper pitch to the pitch with the same letter name that would appear in that major scale. To use this method, you must learn four rules:
If the upper note occurs in the major scale of the lower note (which means you’ll have to imagine the key signature of the lower note—without ignoring the key of the piece), then a unison, 4th, 5th or octave is called PERFECT — represented by an upper-case P — and a 2nd, 3rd, 6th or 7th is called MAJOR — represented by an upper-case M.
Here are some examples:
(1) is a perfect 5th (P5) because G occurs in the C major scale (but Gb or G# would not).
(2) is a major 3rd (M3) because E also occurs in the C major scale.
(3) is a major 3rd (M3) too, because Gs (not G) occurs in the E major scale.
(4) is a perfect 4th (P4) because Bf is the fourth note in the F major scale.
(5) is a major 6th (M6) because Gs occurs in the B major scale.
(6) is a perfect 5th (P5) because Af is the 5th note in the Df major scale.
If the upper note is one half step lower than the note with the same letter name in the major scale of the lower note, then a unison, 4th, 5th, or octave is called DIMINISHED — represented by a lower-case d — and a 2nd, 3rd, 6th or 7th is called MINOR — represented by a lower-case m. In order to clarify the distinction between upper- and lower-case, this text adopts the common practice of drawing a_ line over the lower case m. Thus, we write “M” for major and “m” for minor.
Here are some examples:
(1) is a diminished 5th (d5) because Gb is one half step below the G in the C major scale.
(2) is a minor 3rd (m 3) since Eb is one half step lower than E that is in the C major scale.
(3) is also a minor 3rd (m 3) because G is one half step below the G# in the E major scale.
(4) is a diminished octave (d8va) since Af is one half step below A in the A major scale.
(5) is a minor 6th (m 6) because G is one half step below the G# in the B major scale.
(6) is a minor 7th (m 7) because Cf is a half step lower than the C in the Db major scale.
If the upper note is one whole step below the note with the same letter name in the major scale of the lower note, then a unison, 4th, 5th or octave is called DOUBLY DIMINISHED — represented by “dd” — and a 2nd, 3rd, 6th, or 7th is called DIMINISHED — represented (as above) by “d.”
Here are some examples:
(1) is a diminished 3rd (d3) because Gb is one whole step below the G# in the E major scale.
(2) is a diminished 3rd (d3) because Eb is one whole step below the E# in the C# major scale.
(3) is a doubly diminished 5th (dd5) because Cb is one whole step below the C# in the F# major scale.
(4) is a diminished 7th (d5) because Gb is one whole step below the G# in the A major scale.
(5) is a diminished 6th (d5) because Db is one whole step below the D# in the Fs major scale.
(6) is a doubly diminished octave (dd8va) because Cb is one whole step below the C# in the C# major scale.
If the upper note is a half step above the note with the same letter name in the major scale of the bottom note, then any interval is called AUGMENTED — represented by an upper-case A.
Here are some examples:
(1) is an augmented 5th (A5) because G# is one half step above the G in the C major scale.
(2) is an augmented 3rd (A3) because E# is a half step above the E in the C major scale.
(3) is also an augmented 3rd (A3) because G# is a half step above the G in the Eb major scale.
(4) is an augmented 4th (A4) because B is a half step above the Bb in the F major scale.
(5) is an augmented 6th (A6) since G# is a half step above the G in the Bb major scale.
(6) is an augmented 5th (A5) since A is a half step above the Ab in the Db major scale.
Here is a chart that summarizes the four rules for Method 1:
Any condition not indicated on the chart is named by adding the “doubly” or “triply”, etc. to the closest condition on the chart. Thus, an originally major interval with an upper note one whole step above the note with the same letter name in the major scale of the bottom note would be called “doubly augmented” (AA), an originally perfect interval with an upper note one whole step below the note of the same letter name in the major scale of the bottom note would be called “doubly diminished” (dd), and so on.
This method may also be summarized in the following rules:
MAJOR SCALE RULE
- In PERFECT and MAJOR intervals, the upper note would appear in the major scale of the lower note. We’ll call these major-scale intervals.
RULES of ALTERATION
- If a major-scale interval has been made smaller by a half step, perfect becomes DIMINISHED and major becomes MINOR.
- If a major-scale interval has been made smaller by a whole step, major becomes DIMINISHED and perfect intervals become DOUBLY DIMINISHED. Any further reduction in an interval’s size results in it becoming doubly diminished, triply diminished, and so on. (This is rare.)
- If a major-scale interval has been made larger by a half step, both perfect and major become AUGMENTED. Any further increase in an interval’s size would make it doubly augmented, triply augmented, and so on. (This is rare.)
A few intervals require you to apply these rules in slightly different ways. These intervals contain a lower note that is not a tonic in the system of major-mode key signatures we’ve studied thus far.
Here’s one such interval:
It might seem that Method 1 offers no means by which you can identify this interval because there is no D# major scale. However, such an interval can be dealt with by imagining an “artificial” key signature necessary for such a tonic and then proceed with the rules above. In the case of D#–G, one could do the following:
(1) Imagine the key signature for D# major:
(2) Calculate that D is one half step lower than the G# in the D# major scale, making this a diminished 4th.
Of course, this would require you to be able to calculate some rather abstruse (and otherwise useless) artificial key signatures.
Another approach to dealing with intervals whose lower pitch is not a traditional major-mode tonic is to determine the size of the interval without the sharp (or flat) on the bottom note, and then observe how much the sharp (or flat) reduces (or increases) the size of the interval (see Method 3, below).
Here is the D#–G interval (from above) printed without the sharp on the lower note:
This interval is a perfect 4th (G would appear in the D major scale). Since a D# would make this interval one half step smaller (by moving the lower note closer to the upper one), the interval from D# up to G is a diminished 4th.
In any case, never respell either of the pitches enharmonically (for example, respell D# as Eb) — you’ll change both the number and the quality of the interval.
Method 2 Absolute Size in Half Steps
We can create tables of intervals and their qualities measured in terms of their sizes in half steps:
Using either of these tables, you can calculate the number and quality of any interval. For example, the interval from D up to A spans 7 half steps and is therefore a perfect fifth; the interval from Bb down to G# spans 2 half steps and is therefore a diminished third.
If you encounter an interval with fewer or more half steps than those listed for a particular number on this table (a rare occurrence), simply add the adverb “doubly” or “triply,” etc., as needed to the adverb “diminished” or “augmented.” For example, a fifth spanning 9 half steps would be a “doubly augmented fifth.”
Although this method is completely accurate, it involves a good bit of memorization (you must be able to recall the entire table from memory), and requires a great deal of computation, particularly for larger intervals. It is also divorced from the tonal context for which the interval labels were designed.
Method 3 White-Key Intervals
Some musicians find it useful to memorize all the intervals formed between pairs of notes that have no sharps or flats (the so-called “white-key” pitches — those produced by the white keys on the piano). If you have these memorized, you can identify any interval by first determining its number and quality without sharps and flats, and then adjusting its quality on the basis of its sharps or flats using the Rules of Alteration from Method 1. Here is an example:
If you have already memorized the interval from E to G — a minor 3rd — then you can quickly calculate that the flat lowers the E and therefore makes the interval larger by one half step. That turns the minor third E–G into the major third Eb–G.
Since we would have already memorized the interval from F up to C as a perfect 5th, it is easy to see that each flat lowers its pitch by the same amount thereby keeping the size (and number and quality) of the interval the same. Therefore, Fb–Cb is a perfect 5th just as F–C is.
The most difficult aspect of this method is that you must memorize all of the intervals formed between all possible pairs of white-key notes. If you adopt this method, take the time necessary to study and drill these white-key intervals so that you have them at your disposal at any time.
Method 4 Diatonic Intervals
When working within a key, it is useful to instantly know the interval formed between any two scale degrees in that key. A great deal of music is diatonic, so if you’ve memorized the intervals between diatonic pitches you’ll be able to identify the distance between any two pitches within a particular key signature. In addition, knowledge about the intervals between scale degrees will help you to understand more about the relationships between those scale degrees.
Here is an example:
Think of these two pitches in the major key given to you by the key signature. In Ab major, those two pitches are 2 and 7. If you have already memorized that the interval from 2 up to 7 in all major keys is a major 6th, then you know immediately that this particular interval is also a major 6th.
Any alterations of diatonic pitches can be dealt with by applying the Rules of Alteration from Method 1. For example:
In this case, since we would have already memorized the interval from 3 up to 5 (here in D major) as a minor 3rd, it is easy to see that the sharp on A increases this to a major 3rd.
The most difficult aspect of this method is that you must memorize all of the intervals formed between all possible pairs of diatonic pitches. If you adopt this method, take the time necessary to study and drill these diatonic intervals so that you have them at your disposal at any time.
Musicians follow several principles when referring to intervals larger than an octave:
- All intervals larger than an octave may be referred to as a combination of an octave and some smaller interval, called a compound interval. For example, an interval spanning 12 letter names may be called an “8va and a 5th” (rather than a “12th”).
- Musicians almost always use compound-interval names to refer to intervals that span more than twelve or thirteen letter names. For example, you might hear of either a “10th” or an “8va and a 3rd” but you will probably never hear of a “14th” — it would be called an “8va and a 7th.”
- Musicians sometimes treat compound intervals larger than a 10th as if the upper note were really in the same octave as the lower note, thereby removing the extra octave(s) from the calculation. For example, an interval spanning 14 letter names might simply be called a “7th.”
These guidelines also hold true when attaching qualities to interval numbers. Thus, you may refer to the following interval as either a “major 9th” or as an “8va and a major 2nd.”
However, you should refer to the following interval as simply “three octaves and a major 2nd” or a “compound major 2nd” (and most definitely not as “a major 23rd”).
But even referring to multiple octaves can be cumbersome, so — when working with such large compound intervals — musicians often strip away multiple octaves and refer intervals as follows:
The Circle of Half Steps
In Unit 6, we learned about enharmonically equivalent pitches. We wrote out a list of all the pitches, with enharmonically equivalent ones occupying a single location in the list:
You may have noticed that the beginning and end of that list are the same pitch, an octave apart. In fact, the list could continue, repeating itself over and over as it moves through each octave. This means that, ignoring octave differences and spelling differences, there are only twelve unique PITCH CLASSES. Because the list repeats in this modular fashion, we can arrange the twelve pitch classes in a circle, like this:
Musicians number each of the unique pitch classes starting arbitrarily with C as 0. This allows you to perform mod 12 arithmetic in calculating the number of
half steps between pitch classes. So, for example, the number of half steps from B (11) up to D (2) is 3.
Enharmonically Equivalent Intervals
Just as there are enharmonically equivalent pitches, there are enharmonically equivalent intervals. For example, the interval from C up to Eb (3 half steps) is a minor 3rd (m 3) whereas the interval from C up to D# (3 half steps) is an augmented 2nd (A2). Any two intervals that span the same number of half steps but are labeled differently are ENHARMONICALLY EQUIVALENT.
Musicians find it useful to keep track of the results when we move the lower pitch in an interval above the higher one, or move the higher one below the lower one. For example, if we start with the interval from G up to B, we can move the lower note (G) to the upper position:
Note that when we invert a the G–B major 3rd it becomes a B–G minor 6th. This is true of all major 3rds — they invert into minor 6ths. This is also true in reverse: When we invert a minor 6th it becomes a major 3rd.
In fact, there are principles that apply to all intervals when we invert them:
QUALITY UNDER INVERSION
- A major interval becomes minor
- A minor interval becomes major
- A perfect interval will remain perfect
- A diminished interval will become augmented
- An augmented interval will become diminished
NUMBER UNDER INVERSION
- Subtract the interval number from 9 to yield the number of the interval under inversion (2 becomes 7; 3 becomes 6; and so on)
Another way to view the effects of inversion is as follows: