ukelele Music Theory

Intro

The purpose of these pages is to discuss music theory related to chords with a slant toward ukulele. Music notation on a staff and how to read music are not covered.

People were playing music long before someone proposed a theory or tried to write down rules. Definitions of terms change over time. The rules of music are loaded with extras and exceptions that take far more time and space to describe. You could make the argument that one rule of composing is to break the rules ... just enough to make it interesting.

To reduce confusion, and the size of this document, we will mostly ignore the extras, and just scratch the surface of music theory.

The small size of this document means at times one paragraph covers large chunks of theory. Rereading unfamiliar material may help your understanding.

Piano

C♯ D♭ D♯ E♭ F♯ G♭ G♯ A♭ A♯ B♭
... C D E F G A B C ...

We'll start with something that looks familiar, a piano keyboard.

White keys are named using the first 7 letters of the alphabet. The names repeat so that the 8th white key is the same as the first.

The smallest difference in pitch on a keyboard is called a half step. Adjacent keys are a half step apart. Don't skip the black keys. For example, a half step above C is C♯ (or D♭). A half step above E is F.

A whole step is two half steps, to state the obvious. A whole step above C is a D note. Being familiar with the concept of half steps and whole steps will speed your understanding of Scales.

Any white or black key and the next 11 keys form a sequence of 12 note names that repeats over the 88-key keyboard. A black key is named for the adjacent white key plus a ♯(sharp) or ♭(flat) sign to indicate a note raised or lowered by a half step.

Keys to the left are lower in pitch. Keys to the right are higher in pitch.

Black keys are in groups of two and three.

C is the white key to the left of each group of two black keys. Beginning piano students learn this rule to find Middle C near the middle of a keyboard.

The important concepts in this section are half step, whole step and note names.

Scales

A scale is a collection of notes arranged in order by pitch.

The C Major scale (C D E F G A B C) consisting of all white keys is an example.

Chromatic scale

The chromatic scale has twelve notes, plus a 13th note of the same name as the 1st, and includes the notes of both white and black keys on a piano. When starting at C, the notes in order by pitch are:

CC♯ D♭ D D♯ E♭ EF F♯ G♭ GG♯ A♭ AA♯ B♭ B C

The "C" on the right is 12 half steps above the "C" on the left. The 12-half-step interval encompassing 2 notes with the same letter name is called an octave. The higher pitched note, to the right on a piano keyboard, has a frequency of twice the lower pitched note to the left.

The chromatic scale is a handy reference because the frets on a ukulele are also a half step apart. The open strings are commonly tuned to G, C, E and A. Each higher fret is a half step higher in pitch, and one note higher on the chromatic scale.

Diatonic scales

The Major and Minor scales are the most common musical scales. Both are examples of diatonic scales which have seven distinct notes, plus an eighth which duplicates the first, an octave higher in pitch.

Major scale

A common major scale is the C Major scale, consisting of the notes C D E F G A B C. The first note is called the tonic, and gives the scale its name.

Why are there 8 notes in a C Major scale? And why these 8? The short answer is because various combinations of these notes sound good together. The section on harmonic overtones gives a longer answer. A complete answer would discuss the mathematics behind why an octave has twelve half steps, and the related mathematics of a diatonic scale with eight notes.

A song that is written in the key of C Major (or simply C) generally uses notes from the C Major scale. The music often starts and ends with a C Major chord. The melody often starts and ends with a C note.

Intervals

Some notes in the C Major scale are a half step apart and some are a whole step.

A black key between 2 white keys means a whole step from one white key to the next, e.g. D is a whole step above C. If there is not a black key between 2 white keys, the keys are separated by a half step, e.g. F is a half step above E.

These intervals are called seconds. A half step is a minor second. A whole step is a major second.

Looking at the white piano keys starting at a C, the interval pattern for the C Major scale is: whole, whole, half, whole, whole, whole, half.

You can refer to this interval pattern to determine the notes in any Major scale. Pick any of the 12 possible notes to start. The second note in the scale is a whole step higher in pitch, etc.

The term interval, describes the distance between two notes. It includes a quality and a number for example, major third.

The interval number is the number of note names in a scale that an interval encompasses. For example, if the notes in the scale are C D E F G A B C, we say that E is a third above C. We can also say that an F is a third above D. In both cases, the interval encompasses 3 note names.

The quality of an interval may be perfect, major, minor, diminished or augmented.

A major third is 4 half steps. E is a major third above C. A minor third is 3 half steps. F is a minor third above D. Refer to the piano keyboard or the chromatic scale to count the steps.

A major interval is a half step larger than a minor interval.

Unison (a note compared with itself) and octave (encompassing 8 note names) intervals are perfect. Fourth and fifth intervals may be perfect. A fourth that is a half step larger than perfect is called an augmented fourth. A fifth that is a half step smaller than perfect is called a diminished fifth.

The adjective perfect distinguishes a perfect interval from a diminished or an augmented interval.

As you've seen, intervals also may be described in half steps, e.g. an E is 4 half steps above a C.

This table lists the quality and number for various intervals.

qualitynumberhalf steps
perfectunison0
minorsecond 1
majorsecond 2
minorthird 3
majorthird 4
perfectfourth 5
augmentedfourth 6
diminishedfifth 6
perfectfifth 7
minorsixth 8
majorsixth 9
minorseventh 10
majorseventh 11
perfectoctave 12

Other major scales

Major distinguishes the scale from a Minor scale.

A major scale can start and be named for any of the 12 note names, so there are 12 major scales. The C Major scale is the only scale built with white keys only. The other 11 major scales have 1 or more black keys (sharps or flats). The black keys have more than one name. When constructing and naming scales, one of the two names is preferred as explained below.

To find the notes in a major scale, pick any note to start, then use the interval pattern for a major scale (whole, whole, half, whole, whole, whole, half).

CC♯ D♭ D D♯ E♭ EF F♯ G♭ GG♯ A♭ AA♯ B♭ B CC♯ D♭ D D♯ E♭ EF F♯ G♭ GG♯ A♭ AA♯ B♭ B C
F G AB♭ CD EF

The second row represents a template for the major scale interval pattern of whole steps and half steps. Slide the template left or right on the chromatic scale of all notes and determine the notes in any major scale.

For example, the notes in the F Major scale are F G A B♭ C D E F.

When you transpose a song, say from F to C, you can picture the template above as sliding 5 half steps to the left. In the transposed song, F is changed to C, G is changed to D, A is changed to E, etc.

Why do the black keys have two names? Each letter A-G is used once to name a note in a diatonic scale. That determines if and when a sharp or flat designation is used for a note.

For example, the 4th note in the F Major scale above is B♭ . The 4th note is named "B flat" instead of "A sharp" because the letter "A" has already been used as the name of the 3rd note in the scale.

The notes in the G Major scale are G A B C D E F♯ G. The 7th note is referred to as "F sharp" instead of "G flat" because the letter "G" is the name of the 1st (and 8th) note in the scale.

One more example of black key names: Contruct a Major scale starting with C♯. Following the interval pattern for a major scale (whole, whole, half, ...), we start the scale with C♯, D♯, F, F♯, G♯, A♯, B♯, C♯. The letter F appears twice, and E has not been used. It would be technically correct to use E♯ as an alternate name for F. It is preferable to avoid the confusion by naming the scale D♭ Major with these notes: D♭ E♭ F G♭ A♭ B♭ C D♭. Both sequences represent the same piano keys. Db Major satisfies the goal of using 7 separate note names.

The takeaway concept in this section sheds light on the mystery of sharps and flats in major scales. The starting note and the interval pattern determine when a sharp or flat appears in a scale.

Other diatonic scales

Major and Minor are two of seven diatonic scale modes, each with their own interval pattern.

The interval pattern for each mode can be found by starting at a particular key and playing 8 white keys in a row. For a Major scale start at C and play 8 white keys.

The interval pattern for a Minor scale is found by playing white keys starting at A. A B C D E F G A has an interval pattern of whole, half, whole, whole, half, whole, whole.

Notice that the C Major scale and the A Minor scale contain the same notes ... all white keys on a piano. Sometimes A Minor is referred to as the relative minor of the C Major scale. The two scales just start at different notes. All Major scales have a relative minor.

The other five diatonic scale modes are rarely used, but for completeness, the seven modes are Ionian (Major scale), Dorian, Phrygian, Lydian, Mixolydian, Aeolian (Natural Minor scale), and Locrian. To determine the interval pattern for the Dorian mode, use the white keys starting at D, for Phrygian starting at E, etc.

The term Natural Minor is used to distinguish it from two variations to the Minor scale: Harmonic Minor and Melodic Minor.

Chord

A musical chord is a set of 3 or more notes played together. A 3-note chord is called a triad. The common triads are major, minor, diminished and augmented.

A Major triad consists of a root note, a major third (4 half steps above the root), and a perfect fifth (7 half steps above the root), e.g. C Major (C-E-G).

A Minor triad consists of a root note, a minor third (3 half steps above the root), and a perfect fifth (7 half steps above the root), e.g. D Minor (D-F-A).

A Diminished chord consists of a root note, a minor third (3 half steps above the root), and a diminished fifth (6 half steps above the root).

Use this information to determine the individual notes for any triad. Pick any root note and find the major or minor third interval and the perfect fifth to determine the notes in a major or minor chord.

Harmonizing the C Major scale

Harmonizing the C Major scale means, for each note in the scale, identifying a chord that "sounds good" over the scale.

Use each note of the scale as the root of a chord and then complete the chord using notes that belong to that scale.

Starting with C: the notes in a C Major chord are C-E-G. The notes in a C Minor chord are C-E♭-G. Because the notes in a C Major chord are all in the C Major scale, the C Major chord is used to harmonize the C note.

Moving to D: the notes in a D Major chord are D-F♯-A. The notes in a D Minor chord are D-F-A. All of the notes in a D Minor chord are in the C Major scale. We choose the D Minor chord to harmonize a D note in the C Major scale.

When harmonizing, we want the resultant chords to contain notes only from the scale of interest. We list the notes in a couple of chords and pick the chord whose notes are all members of the scale. That's one way.

Another way: We know which notes are in the scale. Starting with the root note in question, simply pick the 1st, 3rd and 5th notes. So for a chord based on E, we pick E G B. Then referring to the definitions for triads, and determining that E G B is a root, a minor third and a perfect fifth, we find that E G B fits the definition of a Minor chord. Thus the desired chord is E Minor.

Continuing to move up the scale a note at a time, and picking 3 notes to form a triad, we determine that the C Major scale is harmonized with these chords: C Major, D Minor, E Minor, F Major, G Major, A Minor, B Diminished.

Harmonized chords for any Major scale are always in this sequence: Major, minor, minor, major, major, minor, diminished.

Are you confused yet?

The terms major and minor are overloaded.

  1. Major and minor describe intervals, 4 vs 3 half steps.
  2. Major and minor describe triad chords. The 2nd note in a triad is either a major or minor interval.
  3. Major and minor describe scales. The 3rd note in the scale is either a major or minor interval from the tonic.
There is a thread of commonality in the various uses of the terms major and minor, but still can be confusing.

While we're on the subject of overloaded definitions, let's mention key, which describes a physical note on a piano, for example. Key is also an attribute of a musical piece, e.g. this song is in the key of C. Key is related to scale but music purists insist that the two terms should not be used interchangably.

Harmonizing a minor scale

Using the same technique that we did for a Major scale, we find the harmonized chords for an A Minor scale: Am Bdim C Dm Em F G. This pattern applies to all natural minor scales: minor, diminished, major, minor, minor, major, major.

Since the two scales contain the same notes, it makes sense that the same chords are used to harmonize the scale, and that is the case. The difference is that the sequence of chords starts and ends at a different chord.

Chords with four notes

Chords with 4 notes are called tetrachords, or tetrads, or sevenths. Triads have a root, a third, and a fifth. Tetrachords have a 4th note which has an interval of a seventh when compared to the root. You won't recall (or do you?) that a major and minor seventh have intervals of 11 or 10 half steps above the root. The common seventh chords are:

A major triad and a minor seventh combine to form a dominant seventh chord, abbreviated with a 7, e.g. D7.

A major triad and a major seventh combine to form a major seventh chord (M7) or (Δ7), e.g. CM7 or CΔ7.

A minor triad and a minor seventh combine to form a minor seventh chord (m7), e.g. Dm7.

A diminished triad and a minor seventh combine to form a half-diminished seventh chord. Half-diminished seventh chords are abbreviated with a slashed circle and a 7. Another notation is m7♭5, e.g. Fm7♭5.

Finally, a diminished triad and a diminished seventh combine to form a diminished seventh chord (or fully-diminished seventh chord). Diminished seventh chords are abbreviated with an open circle and a 7. Another notation is dim7, e.g. Cdim7.

Harmonizing 7th chords

Rules for harmonizing 7th chords over a scale are similar to triads. For each root note, find the chord whose 4 notes are members of the scale.

For major scales the sequence of harmonizing seventh chords is: M7 m7 m7 M7 7 m7 m7♭5, M7. In the C Major scale that is CM7, Dm7, Em7, FM7, G7, Am7 and Bm7♭5, CM7.

Why use seventh chords? I don't know. There are many different answers on the internet, and all who answer are sure of themselves, but there is little consistency between answers. Best answer is "when it sounds good."

Ukulele

CC♯ D♭ D D♯ E♭ EF F♯ G♭ GG♯ A♭ AA♯ B♭ B C

The 20 or so frets on a ukulele are a half step apart. There are no black or white frets to help find our place. There is no convenient way to identify the notes of the C Major scale. Instead we memorize a few fret positions, or figure them out by knowing the step pattern of a Major scale, and by referring to a chart of the chromatic scale.

The chromatic scale diagram above shows the note values for the C string. The first C is the open C string. Pressing on the first fret results in a C♯ (D♭), and so on up the chromatic scale.

There are 12 half-steps in an octave [from C to C]. The note played when pressing the 12th fret is an octave higher and twice the pitch of the open string. Try this: Pick the C string with the 12th fret pressed. Then pick the A string with the 3rd fret pressed, like you do for a C chord. If your uke is tuned they should sound like the same note.

Picking

To pick a C Major scale on the C string, use the pattern for a major scale, which is whole step, whole step, half, whole, whole, whole, half which translates to open (C), 2nd fret (D), 4th fret (E), 5th fret (F), 7th fret (G), 9th fret (A), 11th fret (B), and 12th fret (C).

In practice, instead of pressing the 4th fret of the C string, usually just switch to the open E string, and continue from there. Likewise, instead of playing the 5th fret of an E string, usually switch to an open A string.

This information can be used to check the tuning. The C string 4th fret should sound like an open E string. The E string 5th fret should sound like an open A string.

Plucking an open G string sounds a G note. Moving up the fretboard a fret at a time will sound G♯, A, A♯, B, C, C♯, D, D♯, E, F, F♯ and G, one octave higher in pitch.

The E string and A string function similarly.

Ukulele chords

To play a chord on a piano, you strike multiple keys simultaneously. To play a chord on a ukulele, you strum multiple strings.

All 4 strings participate in a chord (unless you strum only 3 strings). Sometimes they are open, as in the common C Major chord position with 3 open strings, G, C, and E, and the A string pressed at the 3rd fret to play another C note. The C Major chord has 3 notes C, E, G. There is a 4 digit shorthand for describing which frets are pressed to strum a chord. The first digit refers to the G string. The fourth digit refers to the A string. A digit of 0 means the string is open, no fret is pressed. The shorthand for the traditional C chord is 0003, meaning the G, C and E strings are open, while the 3rd fret of the A string is pressed when strumming a C chord.

You have to use a fingering pattern that sounds the 3 notes in a triad chord. The extra string sounds a second occurrence of any of the 3 notes. The second occurrence does not have to be in the same octave. The harmonic overtones of the same note in different octaves still result in a pleasing chord. Test this by plucking 2 strings, the same note in different octaves. Pitch class is the technical term for a note name, and notes with the same name, are said to be the same pitch class.

With enough time and scratch paper, you could devise your own chord fingering positions. Of course, all the easy ones have already been done.

An E Major chord has 3 notes, E, G# and B. Three ways to play an E chord are specified by these patterns - 1402, 4442, 4447.

Pitch

Frequency or pitch measures the vibrations of a string or column of air and is measured in hertz (cycles per second).

Harmonic overtones

When an instrument, say a cello, plays a note, a sound wave is produced by the vibrating string. The sound wave is a combination of a vibration at the fundamental frequency and of vibrations at many harmonic overtones.

When two different notes produce some of the same overtones, the sound is pleasing to the ear, or harmonious.

This principle explains which notes sound good together and to an extent why specific notes are in a scale. In particular, the intervals of fourth, fifth and octave are most harmonious when sounded with the tonic or fundamental note.

The following shows an example of the same pitch frequency appearing in the overtone series of different notes. Overtones are multiples of the tonic frequency. The loudness of each overtone varies with the kind of musical instrument.

Using round numbers to simplify the math, assume a tonic note of 100 hz.

The tonic and its overtones are 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, etc.

An octave note has a frequency 2 times the tonic. The octave and its overtones are 200, 400, 600, 800, 1000, 1200, etc.

A perfect fifth has a frequency 3/2 times the tonic. The perfect fifth and its overtones are 150, 300, 450, 600, 750, 900, 1050, 1200, etc.

A perfect fourth has a frequency 4/3 times the tonic. The perfect fourth and its overtones are 133, 267, 400, 533, 667, 800, 933, 1067, 1200, etc.

These notes in the C Major scale are C (tonic), F (perfect fourth), G (perfect fifth) and C (octave).

The takeaway from this section is that two notes sound harmonious when some of their overtones are the same, the more the better.

Timbre

Why do different musical instruments sound different even though they play the same note? Timbre.

Each musical instrument has its own pattern of relative loudness of individual overtones. A piano, trombone, cello and flute sound different even when playing the same note because of their unique patterns of the series of harmonic overtones.

Roman numeral numbering

For purposes of analysis, harmonizing chords are assigned Roman numerals, I thru VIII, starting with "I" or "i" for the chord built on the tonic note of a scale. Major chords are upper case. Minor chords are lower case. You may have heard someone talk about a I-IV-V-I chord progression.

An augmented chord is upper case with a plus sign. A diminished chord is lower case with a small circle or a minus sign.

Roman numeral numbering is convenient for transposing a song to a different key. The Roman numeral sequence stays the same, but the chord names change depending on the scale (referred to as the "key" of a song).

When working with a C Major scale, the common Roman numeral chords (I, IV, V) represent C, F and G chords.

Pentatonic scale

Another musical scale that is common in various countries and in American folk music is the pentatonic scale with 5 notes plus the repeated tonic. There are both major and minor scales.

The F♯ Major Pentatonic scale corresponds to the black keys on a piano - F♯ G♯ A♯ C♯ D♯ F♯

The nice thing about the pentatonic scale is that any sequence of notes sounds good. Try this out by playing any sequence of black keys to compose a song.

Rhythm

So far we have ignored notation on a musical staff, rhythm, time signature and key signature because this is not a course on how to read music.

A couple of terms are useful.

Time signature - most common are 3/4 (waltz) time and 4/4 (standard) time. The top number indicates the number of beats in a measure. The bottom number, 4 in these examples, indicates that a quarter note gets 1 beat. Other common time signatures are 2/4 (cut) time and 6/8 time.

A verse in Western music - hymns, rock, country - usually has four lines and 8 bars (measures) per line, but not always.

Some melodies begin one or more beats before the first full measure. Such notes are called a pickup or upbeat or anacrusis.

Tempo of a song is measured in beats per minute, commonly in the range of 100-200 bpm.

Math and music

Why are there 12 piano keys in an octave? Why are there 7 notes in an octave?

Why are there no black keys between E and F and between B and C?

Each note on a piano is tuned to a particular frequency. The adopted standard today is that the A above middle C has a frequency of 440 hz. The frequency of an A an octave lower has a frequency half as much or 220 hz. The frequency of an A which is an octave higher has a frequency twice as high or 880 hz. From this example, you can see that the lower octave (8 keys) goes from 220 to 440 hz (a difference of 220), while the upper octave goes from 440 to 880 hz (a difference of 440). So a half step in the upper octave must have a bigger frequency step than a half step in the lower octave.

The answer is a half step is not a fixed frequency difference. Rather it is a fixed ratio equal to about 1.06, or more precisely, the twelfth root of 2. There are 12 half steps in an octave. If you multiply the tonic frequency times 1.06 twelve times, you will get the value that is double that of the tonic note.

Ideally the perfect fifth should have a frequency of 3/2 (or 1.500) times the tonic. A perfect fifth is 7 half steps above the tonic. On a tuned piano, the frequency turns out to be 1.4983, pretty close to 1.500. The perfect fourth, at 5 half steps above the tonic should be 4/3 (or 1.333) times the tonic. On a piano the actual frequency of a perfect fourth is 1.3348 times the tonic frequency, pretty close to 1.3333.

The system of tuning a piano with equal half steps is known as equal temperament. The advantage is that you can switch to a different key and still have the piano be close to in tune.

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