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.

Intervals

Simple intervals

Before we progress, we must discuss music intervals. An interval in music is usually defined as the distance between two notes, pitches or tones 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 alternative names depending upon the spelling of the particular notes on the page of music.

Semitones	Common Name	Alternative 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 becomes minor
5. If any major interval is lowered by two semitones, the interval becomes diminished

Interval	Diminished	Perfect		Augmented
		Minor	Major
Unison		0
Second		1	2	3
Third	2	3	4	5
Fourth	4	5		6
Fifth	6	7		8
Sixth	7	8	9	10
Seventh	9	10	11
Octave		12

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
22	minor fourteenth	minor seventh
23	major fourteenth	major seventh
24	double octave	octave

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: Trainear is an online ear trainer that's specifically for associating intervals to songs. Here are some examples for each interval:

Interval	Ascending example	Descending example
minor second	Jaws theme, Trouble by Coldplay, "I'm Dreaming Of A" White Christmas, opening of 4th movement of Dvorak's 9th Symphony,	Für Elise, Godfather theme, "It's Not Safe To Swim Today" by Veil Of Maya, Bohemian Rhapsody (“I see a little silhouetto of a man”)
major second	Frere Jacques, Happy Birthday (notes 2 and 3), Blind Melon "No Rain" guitar riff	Freddie freeloader (miles davis), The way we were, Corcovado, "Trouble" by Coldplay, "Never Gonna Give You Up" by Rick Astley, "Pull me Under" by Dream Theater
minor third	Greensleeves, Seven nation army (the white stripes), Rock A Bye Baby, To Dream The Impossible Dream, Brahms' "Lullaby", guitar riff from "Whole Lotta Love" - Led Zeppelin, "Smoke on the Water" by Deep Purple	Hey Jude, Ring Around The Rosy, "The Star-Spangled Banner", 50s bass progression, Peter Gun Theme
major third	Oh When the Saints, Morning Has Broken, "It's a Small World", verse and guitar riff of "Blister in the Sun", "Boston" by Augustana, "The Number of the Beast" by Iron Maiden, Pokémon main theme	Clock Chimes (first two notes), Good night, Ladies; Super Mario Bros. Overworld Theme, Massenet "Meditation" from Thaiis, "Heavens Divide" from Metal Gear Solid: Peace Walker where the lyrics begin
perfect fourth	Auld Lang Syne, Here Comes the Bride, Hi Ho, Amazing Grace, The Love Boat Theme (second), "The licc"	I've Been Working on the Railroad, Eine Kleine Nachtmusik, "Anchor" by Misery Signals, The Legend of Zelda Overworld theme, My Girl (Bass part at beginning)
Augmented fourth	The Simpsons theme, Maria (West Side Story)	European police siren, YYZ by Rush
perfect fifth	Twinkle Twinkle Little Star, Star Wars (Main Theme), Forrest Gump, In the end by Linkin Park, Also Sprach Zarathustra (2001 Space Odyssey), trumpet in "Snake Eater" from Metal Gear Solid 3: Snake Eater, The Love Boat Theme (first), Qual guerriero in campo armato (Broschi)	Feelings, The Flintstones
minor sixth	Chopin's Waltz Op. 64 No. 2, "Sit Down. Stand Up" by Radiohead, Batman Theme(1st and 4th note), Conquest of Paradise (Vangelis), the Entertainer (notes 3 and 4), Black Orpheus, The Incredible Hulk theme	Theme from love story
major sixth	My Bonny Lies Over The Ocean, Chopin's Nocturne op.9 No.2, NBC theme tune	Music of the Night (Phantom of the Opera), Nobody Knows the Trouble I've seen
minor seventh	Somewhere (from West Side Story), star trek	Watermelon Man
major seventh	O terra addio (Verdi's Aida), Bali-Hai (1st and 3rd note), the chorus melody of "Take On Me" by a-ha, the first and third note of the "aaah" in Led Zepplin's the immigrant song	I Love You (Cole Porter) 2nd & 3rd notes
perfect octave	over the rainbow, What Is Hip (bass) – Tower Of Power	Bulls On Parade(RATM), Willow Weep for Me,

Here is a graphic interpretation of intervals. Major count down from the top. Minor count up from the bottom. Naming starts top to bottom.

DO	Major Second		Major Third		Perfect Fifth	Major Sixth			Major Seventh	Perfect Eighth
RE	Major Second		Major Third		Perfect Fifth	Major Sixth		Minor Seventh	Major Seventh	Perfect Eighth
MI			Major Third		Perfect Fifth	Major Sixth	Minor Sixth	Minor Seventh	Major Seventh	Perfect Eighth
FA					Perfect Fifth	Major Sixth	Minor Sixth	Minor Seventh	Major Seventh	Perfect Eighth
SO				Perfect Fourth	Perfect Fifth	Major Sixth	Minor Sixth	Minor Seventh	Major Seventh	Perfect Eighth
LA				Perfect Fourth		Major Sixth	Minor Sixth	Minor Seventh	Major Seventh	Perfect Eighth
TI		Minor Second		Perfect Fourth			Minor Sixth	Minor Seventh	Major Seventh	Perfect Eighth
DO		Minor Second		Perfect Fourth			Minor Sixth	Minor Seventh		Perfect Eighth

Traditional Scales

The major scale

The major scale is a diatonic scale. Originally a Church mode named by Heinrich Glarean in 1547 as the Ionian scale/mode. There is some confusion with beginners as to why the Church modes are named after Greek tribes. A common belief is that the Ionian scale was invented by the Greeks. The Ancient Greeks did not have an Ionian scale but they did lay the foundation for its existence. The Ancient Greeks did have a system of music and we know quite a lot about their music theories from the Ancient Greeks themselves. The real origin of the Ionian mode is the period known as the Renaissance which started in Italy with the 13th century writer Petrach. Renaissance is the French word for "rebirth". The "rebirth" in this case was the rediscovery of lost Ancient Greek texts that had been brought back from the East (including Jerusalem) by Crusaders and other parties. A lot of these books had been unavailable in Western Europe for centuries. These books sparked a desire to rediscover the roots of European civilization. The Renaissance scholars took the music practices of their time which already had been in use for hundreds of years and codified them in reference to the Greeks. You can read about the Ancient Greek system of tetrachords and scales if you're interested. It must be pointed out that we have no idea about the sounds of the instruments of the Ancient Greeks and only a few fragments of music exists. It is now believed that the Church modes are derived from early Christian mass chants which themselves were adapted from Jewish chants. It may be that modern Western Music has for its origins Eastern Jewish liturgical chants and Ancient Greek theories codified from the view of a Renaissance mindset a thousand years later.

The major scale (Ionian mode) 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, 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 Chinese and 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 occurrence 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 diminished, encourages beginning with a whole-step, while the second, used frequently by jazz players and composers who call it octatonic, encourages starting with the half-step. Beginning from C (using the second 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 (second 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 (=A)
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)