Music Theory/Scales and Intervals

A musical scale is a set of notes, usually not arbitrary, of which most notes in a piece of music might be chosen. There exist many scales with highly distinctive sounds, though some are much more common than others. the term "scale" comes from the latin word 'scala' meaning 'ladder'. Thus a scale is a ladder of notes.

The chromatic scale

The chromatic scale is the scale of all of the possible notes in Western music. Some traditional systems are not based around the chromatic scale (such as 24-tone scales), but this book will not discuss them. However, many of the principles, such as the concept of tonality, still apply to them.

The chromatic scale was probably invented by Pythagoras. He noted that two strings of equal tension will sound very similar when one string is exactly half the length of the other, but the shorter string is higher in pitch — what we now call an octave higher. The reason, although Pythagoras may not have known this, is that the shorter string vibrates twice as fast: the pitch of a note is directly proportional to the length of the string. (This is how stringed instruments like the guitar work the way they do. When a finger is pressed against the string, the vibrating portion is made shorter by pinning the string against the fingerboard, cutting off a portion of the string from the vibration.)

Next, Pythagoras began devising ratios of string length, and eventually came to a system of twelve tones, each one a mathematical ratio compared to a given length of string. For instance, the seventh tone has a proportion of 3:2, meaning the string must vibrate one and a half times as fast relative to the first tone. This became the mathematical foundation of the chromatic scale, as each note in the scale had a simple ratio such as 9:8. (However, modern instruments usually use a system called "equal temperament" where the ratios are not so simple. The result is that every note in the scale is very slightly out of tune, but without this adjustment, an instrument would be out of tune in any key other than the key it was designed for.)

Here are two versions of the chromatic scale, one with sharps, the other with flats:

 C C D D E F F G G A A B
 C D D E E F G G A A B B

Notice there are no sharps or flats between E and F, nor are there any between B and C. Notes in the same column are enharmonically equivalent: E shares the pitch of F, and F shares the pitch of E. This does not mean they are the same, even though they sound the same. This will be explained later.

By the way, one step up the chromatic scale is called a semitone. C is a semitone up from C, and E is a semitone down from F. Two steps are called a whole tone. They are also often referred to as the half step and whole step, respectively. We will use these terms frequently.

The chromatic scale is rarely used as a scale in the normal sense. Usually, notes are chosen from the chromatic scale (although, historically, many scales existed before the chromatic scale was actually discovered) to form a scale of their own. In Western music, the most important of these is the major scale.

Intervals

Simple intervals

Before we progress, we must discuss intervals. An interval is usually defined as the distance between two pitches, that is, how many semitones lie between them. When the two pitches are the same, they are said to be in unison, and two notes played in unison can be impossible to distinguish from a single note when they are played by the same instrument and the instrument is properly tuned. When they are twelve semitones apart, they are an octave apart (we will learn why it is called an octave shortly). Simple intervals are defined as those intervals that are one octave or less apart.

Intervals are usually named according to the relationship of the higher note to the lower note in the major scale, though they also have alternate names depending upon the spelling of the particular notes on the page of music.

Semitones	Common Name	Alternate Names
0	perfect unison	diminished second
1	minor second	augmented unison
2	major second	diminished third
3	minor third	augmented second
4	major third	diminished fourth
5	perfect fourth	augmented third
6	tritone	augmented fourth, diminished fifth
7	perfect fifth	diminished sixth
8	minor sixth	augmented fifth
9	major sixth	diminished seventh
10	minor seventh	augmented sixth
11	major seventh	diminished octave
12	perfect octave	augmented seventh

This table gives the most common nomenclature for each interval according to its relation to the major scale. For example, the interval of four semitones occurs as the third note of the major scale, and thus it is called a major third. The interval of seven semitones occurs as the fifth note of the major scale, and so it is called a perfect fifth. Whether an interval is "perfect" or "major" depends on mathematical ratios of frequencies as determined by the Greeks. Other possible names are given under "alternate names," and the most common of these are emboldened. One may draw several inferences from this table:

1. If any perfect interval is raised by one semitone, the interval becomes augmented
2. If any perfect interval is lowered by one semitone, the interval becomes diminished
3. If any major interval is raised by one semitone, the interval becomes augmented
4. If any major interval is lowered by one semitone, the interval become minor
5. If any major interval is lowered by two semitones, the interval becomes diminished

Compound intervals

Compound intervals are defined as those intervals greater than one octave apart. These intervals may be considered by exactly the same rules as their simple counterparts.

Semitones	Name(s)	Simple Counterpart
13	minor ninth	minor second
14	major ninth	major second
15	minor tenth	minor third
16	major tenth	major third
17	perfect eleventh	perfect fourth
18	augmented eleventh	tritone
19	perfect twelfth	perfect fifth
20	minor thirteenth	minor sixth
21	major thirteenth	major sixth

The compound intervals work by following the same five rules as the simple intervals above (so the augmented eleventh might also be called a diminished twelfth!). Why even bother giving them separate names? The answer lies in their normal function within music. Complex jazz chords are built around stacks of thirds, and so the terms "ninth," "eleventh," and "thirteenth" are needed to designate intervals larger than a seventh.

Mnemonic memorization examples

The following chart intends to give some mnemonic support in recognising musical interval. For each interval, ascending or descending, a popular song is given that contains it prominently. Capitalized syllables or a ">" mark the stated interval:

Interval	Ascending example	Descending example
minor second	Jaws theme, "I'm Dreaming Of A" white christmas	Für Elise
major second	Happy Birthday, Frere Jacque	Freddie freeloader (miles davis), The way we were, Corcovado
minor third	Rock A Bye Baby, To Dream The Impossible Dream, Brahms' "Lullaby", Greensleeves	Hey Jude, Ring Around The Rosy
major third	Oh When the Saints, Morning Has Broken	Clock Chimes (first two notes, Good night, Ladies;
perfect fourth	Auld Lang Syne, Here Comes the Bride	I've Been Working on the Railroad, Eine Kleine Nachtmusik
Augmented fourth	The Simpsons theme, Maria (West Side Story)	European police siren
perfect fifth	Twinkle Twinkle Little Star (notes 2 and 3), Star Wars Theme (1st 4 notes)	My Girl (Bass part at beginning), Feelings, The Flintstones
minor sixth	Conquest of Paradise (Vangelis), the Entertainer (notes 3 and 4), Black Orpheus	theme from love story
major sixth	NBC theme tune, My Bonny Lies Over The Ocean	Music of the Night (Phantom of the Opera), Nobody Knows the Trouble I've seen
minor seventh	There's a place for us, star trek	Watermelon Man
major seventh	Bali-Hai (1st and 3rd note)	I Love You (Cole Porter) 2nd & 3rd notes
perfect octave	over the rainbow	Bulls On Parade, Willow Weep for Me

Traditional Scales

The major scale

The major scale is a diatonic scale, which was first invented by the Greeks thousands of years ago (thus this is sometimes known as the Ionian scale/mode). The major scale is most simply described as the eight note progression consisting of the perfect and major semitiones, i.e., perfect unison, major 2nd, major 3rd, perfect 4th, perfect 5th, major 6th, major 7th, and perfect octave in that order. You have already seen the major scale: C D E F G A B; do re mi fa sol la ti; 1 2 3 4 5 6 7. Scales may be constructed according to their intervals. You can see that the C major scale consists of two whole tones, then a semitone (moving from E to F), then three more whole tones, then again a semitone (moving from B back to C). If we add the implied C at the end of the scale, we would have eight notes: C D E F G A B C.

The minor scale

The minor scale, or the Aeolian mode, is also a diatonic scale. The C minor scale is C D E F G A B; 1 2 3 4 5 6 7. You can see that it consists of one whole tone, then a semitone (moving from D to E), then two more whole tones, then again a semitone (moving from G to A), and a final whole tone. If we add the implied C at the end of the scale, we would have eight notes: C D E F G A B C.

The intervals of the natural minor scale follow the following pattern: tone, semitone, tone, tone, semitone, tone, tone. The following chart demonstrates this natural minor scale construction.

The minor scale is the sixth mode of the major scale; that is, the minor scale starts on the 6th note of the relative major scale. In the case of the C minor scale, the relative major scale is E major. We can illustrate this with two octaves of the Eb major scale, highlighting the C minor scale. E F G A B C D E F G A B C D E. You will learn more about modes later.

Pentatonic and Blues Scales

The pentatonic scales

A pentatonic scale has five notes. Each note in the major pentatonic scale is a fifth (seven semitones) relative to another note. For example, the C major pentatonic scale starts with C, then from there we can get G, then D, then A, then E. Rearranging the scale to ascending order from C, we get: C D E G A. This is the C major scale with F and B removed! So, why use it? Sometimes less is more, and pentatonic scales are certainly easier to use when improvising.

The major pentatonic is the same as the major scale with the 4 and 7 notes removed, while the minor pentatonic has the 2 and 6 notes removed, that is, the minor pentatonic is relative to the major pentatonic.

So, to use our earlier example contrasting the E major pentatonic with the C minor pentatonic: E F G B C E F G B C E.

Pentatonic scales are abundant in rock and blues music, though these are certainly not their only uses. Traditional Japanese music has defined and named many more pentatonic scales, some of which do not use the western twelve-note basis.

The blues scale

The most common blues scale has six notes, and may be considered a minor pentatonic scale with the diminished fifth added as a blue note. In a major blues tune, the minor third is also considered a blue note.

Therefore, the C blues scale is: C E F G G B. Sometimes the raised seventh degree (B) is added to this scale but most often used as a passing note, much like the diminished fifth. The blues scale is most commonly used in jazz improvisation to create a "bluesy" flavor.

The Symmetric Scales

Symmetric scales include scales such as the whole-tone scale, octatonic scale (also called the diminished scale), and chromatic scale, and their defining characteristic is that they are composed of repeating subunits within an octave. This property allows these scales to be transposed to another pitch (or "key"), yet retain exactly the same notes as the original scale.

The chromatic scale

The simplest of the symmetric scales, the chromatic scale, is composed of repeating semitones (half-steps). Thus, the chromatic scale built on C contains the notes C,D,D,E,E,F,G,G,A,A,B, and B. The chromatic scale built on D contains the notes D,D,E,E,F,G,G,A,A,B,B, and C. Notice that these are exactly the same notes as the chromatic scale built on C. In fact, a chromatic scale built on any note of the twelve-tone western music scale will share these notes, a property which warrants the inclusion of this scale among the symmetrics. Usually chromatic scales are spelled with sharps when ascending and flats when descending.

As noted above, composers will often choose certain notes from this scale to use more frequently than others, thereby providing the listener with a sense of a "home" note, referred to as the tonic. However, many composers in the twentieth century have demonstrated that using all twelve chromatic notes equally can defeat any sense of tonal center. This technique is called atonality or, less commonly, pantonality, and can have a very unsettling effect upon those unaccustomed to this music. An everyday occurance of atonal music would be in the soundtracks to many horror films, documentaries, or other movies where there is a need for extreme dissonance and tension to match the onscreen action.

The whole-tone scale

The whole-tone scale is made of repeating whole tones (whole-steps). Therefore, a whole-tone scale built upon D would contain D,E,F,G,A, and B. Like the chromatic scale, these pitches are the same pitches that one would find in a whole-tone scale built upon E, or any of the pitches in this particular scale. For instance, a whole-tone scale built upon F would be F,G,A,B,D,E, and a whole-tone scale built on B would be B,C,D,F,G,A. These two are really the same scale, since C=D and D=E. For this reason, there exist only two possible whole-tone scales:

1. the scale including the pitches C,D,E,F,G, and A
2. the scale including the pitches D,E,F,G,A, and B.

Any whole-tone scale within the western musical system will fall enharmonically into one of these two categories.

The whole-tone scale was used widely by impressionists to create a floating, ethereal sound. The scale also finds a place in jazz improvisation, as it is among the most colorful scales to use where a raised-fifth scale degree is indicated. Incidentally, the scale contains all of the notes of two augmented chords placed side-by-side, a whole step apart.

The octatonic (diminished) scale

The octatonic, or diminished, scale is among the simplest scales possible, yet has been used to tremendous effect in nearly every genre. This eight-note scale may be conceived in two manners, but both of the approaches use a repeating subunit of alternating whole-steps and half-steps. The first manner, most often used by classical composers and termed octatonic, encourages beginning with a half-step, while the second, used frequently by jazz players and composers who call it diminished, encourages starting with the whole-step. Beginning from C (using the first method), the octatonic scale would include the notes C,D,E,E,F,G,A, and B. As with the other symmetric scales, this scale may be moved to a different starting note yet retain the same pitches as the original. Thus, E,E,F,G,A,B,C,D is an octatonic scale (first method) that shares all eight pitches with the octatonic scale starting on C. There are, then, three different octatonic scales possible:

1. C,D,E,E,F,G,A, and B
2. C,D,E,F,G,A,B, and B
3. D,E,F,G,A,A,B, and C

Any other octatonic scales within the western system will fall enharmonically into one of these three groups.

The use of the octatonic scale in western music can be seen as early as Bach, who used pieces of the scale within his counterpoint to imply diminished harmony. Modern composers of the classical canon use this scale as a colorful alternative to redundant diatonicism or austere chromaticism. Jazz improvisers often turn to the diminished scale to improvise over a dominant seventh harmony to imply the flat-ninth degree of a chord. The octatonic/diminished scale is extremely versatile: a single octatonic scale (C,D,E,E,F,G,A, and B contains the notes of four dominant-seventh chords (C,E,G,B; E,G,B,D; F,A,C,E; and A,C,E,G), two fully-diminished-seventh chords (C,E,G,B and C,E,G,B), and a plethora of major, minor, and diminished chords.

Other "theoretical" symmetric scales

Other collections of pitches may be considered "symmetric scales," even though they are not often used as such. The fully-diminished-seventh chord is made up of repeating subunits of minor thirds (three semitones), and there are three distinct pitch collections:

1. C,E,G,B
2. C,E,G,B
3. D,F,A,C

Any other fully-diminished seventh chords are enharmonically equivalent to one of these three collections.

The augmented chord is made of repeating subunits of major thirds (four semitones), and there are four distinct collections:

1. C,E,G
2. D,F,A
3. D,F,A
4. E,G,B

Any other augmented chords are enharmonically equivalent to one of these four collections.

Finally, the interval of a tritone (diminished fifth, augmented fourth, or six semitones) may be considered with the symmetric scales because there are only six distinct varieties using the subunit of a tritone. A tritone beginning on C (C,F) has the same pitches as a tritone beginning on F (F,C)