Intro

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

People played music long before someone wrote down music theory. The rules of music are loaded with exceptions.

We will ignore many extras and just scratch the surface of music theory. The impractical goal is to cover a semester of music theory in an hour, with precise introductions to these music terms:

Major and Minor scales

and scales intervals

chords

The small size of this document means one paragraph may cover large chunks of theory. The goal is for each sentence to contain instructive information. Rereading may help.

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 and musical note names use the first 7 letters of the alphabet. The names repeat so that the 8th white key is the same name as the first.

The smallest difference in pitch on a keyboard is called a half step (semitone). Adjacent keys are a half step apart. Don't skip the black keys. Refer to the keyboard diagram: 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. Refer to the keyboard diagram: 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.

Review: 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. There are numerous types of musical scales. We will spend more time on Major and Minor scales, less on chromatic and pentatonic scales, and none on dozens of other scales. Chromatic scale The chromatic scale encompasses 13 notes, includes white and black piano keys and can start with any note. When starting at C, the notes in order by pitch are: C C♯ D♭ D D♯ E♭ E F F♯ G♭ G G♯ A♭ A A♯ B♭ B C The "C" on the right is 12 half steps higher in pitch than the "C" on 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. The open C string sounds a C note, the first fret is C♯. The open E string sounds an E note, the first fret is F. Review: Keep a chromatic scale chart handy for counting intervals when constructing scales and chords and to locate specific notes on the fret board. Question:

What note is a half step above F♯?

What note is a whole step above C♯? Diatonic scales Major and Minor scales are the most common scales in the family of seven diatonic scale modes. Each scale has eight notes. The first note of a scale is called the tonic and gives the scale its name, e.g. C Major scale. The eighth note of the scale has the same letter name as the tonic and is an octave higher in pitch. "Octave" comes from Latin for "eighth."

Major scale The C Major scale consists of the notes C D E F G A B C. It's the only Major scale with no accidentals (sharps or flats) and that makes it simpler for examples. Why are there eight notes in a Major (or Minor) scale? And why these eight? 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 of why an octave has twelve half steps, and how the eight notes are selected. Accept the fact that there are eight notes in a major scale and leave the "why?" until later. A song 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 ends with a C note. Interval Pattern of a Major Scale Adjacent notes in the C Major scale are either a half step or a whole step apart. Look at the white piano keys starting at a C. 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. By examining the intervals between each adjacent pair of white keys, we determine that the interval pattern for the C Major scale ( C D E F G A B C ) is: whole, whole, half, whole, whole, whole, half. Meaning: C to D is a whole step (black key between), ..., E to F is a half step (no black key between), ...

Other major scales A major scale can start and be named for any of the 12 notes, so there are 12 major scales . The C Major scale is the only Major scale built with all white piano keys. The other 11 major scales include one 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 later. Technically either name can be used. The notes in the F♯ Major scale are the same as in the G♭ Major scale. All Major scales have the same interval pattern! Exercise: Find the notes in one of the 12 major scales: Pick any note to start, then use the interval pattern for a major scale (whole, whole, half, whole, whole, whole, half). The 2nd note in the scale is a whole step higher in pitch, 3rd note is a whole step higher, 4th note is a half step higher, etc. The second row below represents a template for the major scale interval pattern of whole steps and half steps. The template is positioned to show the F Major scale: F G A B♭ C D E F. Slide the template left or right on the chromatic scale of all notes and determine the notes in any major scale. C C♯ D♭ D D♯ E♭ E F F♯ G♭ G G♯ A♭ A A♯ B♭ B C C♯ D♭ D D♯ E♭ E F F♯ G♭ G G♯ A♭ A A♯ B♭ B C F G A B♭ C D E F When you transpose a song, say from F to D, picture the template above as sliding 3 half steps to the left. In the transposed song, F is changed to D, G is changed to E, A is changed to F♯, etc. Qualifiers travel with the letter name, e.g. F7 is changed to D7, Adim is changed to F♯dim. Review: If you understand the concept of interval pattern in a Major scale, you will be able to pick any note name and write down the eight notes in the Major scale with that name.

Intervals Half steps and whole steps refer to the interval between adjacent notes in a scale. A more general definition of interval, describes the distance between any two notes in a scale. 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 (we'll use a Major scale other than C this time), the notes in the G Major scale are G A B C D E F♯ G, we say that C is a third above A. We can also say that an F♯ is a third above D. In both cases, the interval encompasses three note names. The interval from B to C is a second, encompassing two note names. The interval quality may be perfect, major, minor, diminished or augmented. The quality and interval together correspond to a specific number of half steps in an interval. This table lists the quality and number for various intervals. "Number" refers to the number of note names encompassed within a scale, NOT the number of half steps. For example, in the C scale, the major third from C to E encompasses notes C, D, E. The minor third from D to F encompasses notes D, E, F. quality number half steps perfect unison 0 minor second 1 major second 2 minor third 3 major third 4 diminished fourth 4 perfect fourth 5 augmented fourth 6 diminished fifth 6 perfect fifth 7 augmented fifth 8 minor sixth 8 major sixth 9 diminished seventh 9 minor seventh 10 major seventh 11 perfect octave 12

A major interval is a half step larger than a minor interval. Adjacent notes in a scale are either a minor second (half step), or a major second (whole step). A minor third is 3 half steps. C is a minor third above A. A major third is 4 half steps. F♯ is a major third above D. Refer to the piano keyboard or the chromatic scale to count the steps. Unison (a note compared with itself) and octave (encompassing eight note names) intervals are perfect. Fourth and fifth intervals may be perfect, augmented or diminished. The adjective perfect distinguishes a perfect interval from a diminished (half step smaller) or an augmented (half step larger) interval. As you've seen, intervals also may be described in half steps, e.g. an E is 4 half steps above a C. Review: Intervals have a quality and a number. For example, in the A♭ Major scale: ( A♭ B♭ C D♭ E♭ F G A♭ ),

C is a major third above A♭, encompassing the notes A♭ B♭ C, 4 half steps apart.

D♭ is a minor third above B♭, encompassing the notes B♭ C D♭, 3 half steps apart. Refer to the chromatic scale to determine the number of half steps and interval name. Question: In the A♭ Major scale, what note is a perfect fifth above C?

Chromatic scale, repeated for reference C C♯ D♭ D D♯ E♭ E F F♯ G♭ G G♯ A♭ A A♯ B♭ B C Why do the black keys have two names? Each letter A-G is used once to name a note in a scale. That determines if and when a sharp or flat designation is used for a note, as explained below. For example, the 4th note in the F Major scale F G A B♭ C D E F is B♭. The 4th note is named "B flat" instead of "A sharp" because the letter "A" is 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 used as the name of the 1st (and 8th) note in the scale. One more example of black key names: Construct a Major scale starting with C♯. Following the interval pattern for a major scale (whole, whole, half, ...), the notes are 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. There is not a black key named E♯ It is preferable to avoid confusion by using the alternate names for black keys and naming the scale D♭ Major with these notes: D♭ E♭ F G♭ A♭ B♭ C D♭. Both sequences represent the same piano keys. D♭ Major satisfies the goal of using 7 separate note names. In any one Major (or Minor) scale, any accidentals will be either all sharps or all flats. The takeaway concept in this section sheds light on the mystery of sharps and flats in scales. The starting note and the interval pattern determine when a sharp or flat appears in a scale.

Minor scales The interval pattern for a Natural Minor scale is found by choosing white keys starting at A. A B C D E F G A has the pattern

whole, half, whole, whole, half, whole, whole.

Use this pattern to determine the notes in any Minor scale. Notice that the C Major scale and the A Minor scale start at different notes, but contain the same notes ... white keys on a piano. 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 - same notes, but different starting position on the keyboard or chromatic scale. Notice that thescale and thescale start at different notes, but contain the same notes ... white keys on a piano.is referred to as theof thescale. The two scales just start at different notes. All Major scales have a relative minor - same notes, but different starting position on the keyboard or chromatic scale. The term Natural Minor is used to distinguish it from two variations to the Minor scale: Harmonic Minor and Melodic Minor. Review: Pick any note of 17 and determine the notes in the corresponding minor scale using the interval pattern for a Minor scale. What are the notes in the D Minor scale? Extra: The other diatonic scales 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.

Chords We have discussed Major and Minor scales and intervals. Now we will look at building or dissecting chords of 3 and 4 notes. A musical chord is a set of 3 or more notes played together. A 3-note chord is called a triad. For reference, remember where to find the following three definitions. The common triads are major, minor and diminished. 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). The adjective Major is optional for Major chords, i.e. "F" means "F Major." 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). You'll see below that a lower case "m" is shorthand for "Minor", i.e. "F#m" means "F# Minor". 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, or 2 sequential Minor thirds), e.g. Bdim (B-D-F). Chromatic scale, repeated for reference C C♯ D♭ D D♯ E♭ E F F♯ G♭ G G♯ A♭ A A♯ B♭ B C The notes in a D Minor chord, for example, are always D-F-A. The key of a song doesn't affect it. The key of a song will make some chords more likely to be used. Exercise: Determine the individual notes for any major or minor 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. For example, F♯m is a minor chord. A minor chord has a root, a minor third and a perfect fifth. A minor third above F♯ is A. A perfect fifth above F♯ is C♯. Thus, the F♯m chord consists of the notes F♯, A and C♯. Question: Which notes define a C Minor chord? Other triads you may encounter are augmented (2 sequential major thirds: perfect fifth is replaced by an augmented fifth note), suspended second (third is replaced by a major second) and suspended fourth (perfect fourth replaces a major or minor third). Caug has notes C E G♯. Csus2 has notes C D G. Csus4 has notes C F G.

Harmonizing a scale Harmonizing a scale means, use each note of the scale as the root of a chord and then complete the chord using notes that belong to that scale. We will harmonize the C Major scale. We know which notes are in the C Major scale. Starting with the root note in question, simply pick the 1st, 3rd and 5th notes. So for a chord with a root of C, we pick C E G. Then referring to the definitions for triads, and determining that C E G is a root, a major third and a perfect fifth, we find that C E G fits the definition of a Major chord. Thus the harmonizing chord for the C note is C Major. Starting with D, pick the 1st, 3rd and 5th notes or D F A. Determining that D F A is a root, a minor third, and a perfect fifth, we find that D F A is a D Minor chord. Thus Dm is the harmonizing chord for the D note in the C Major scale. 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, C Major. Reference: Harmonizing chords for any Major scale are always in this sequence:

Major, minor, minor, major, major, minor, diminished, major. Exercise: Pick a major scale. Write down the harmonizing chords for the scale. For each of eight notes in the scale, append the corresponding chord descriptor. For the E Major scale, E F♯ G♯ A B C♯ D♯ E, the harmonizing chords are: E F♯m G♯m A B C♯m D♯dim E Question: What are the harmonizing chords for the G Major scale? Often, the chords in a song belong to the list of harmonizing chords for the scale corresponding to the key for the song, but song writers achieve interesting effects by using chords that don't follow the rules.

Harmonizing a minor scale To find the harmonizing chords for an A Minor scale, use the same technique that we did for a Major scale, that is: for each note of the scale, build a chord with the 1st, 3rd and 5th notes of the scale. Then determine if the resulting chord is a major, minor or diminished chord. The harmonizing chords for the A Minor scale are: Am Bdim C Dm Em F G Am. Review: To find harmonizing chords for a scale, for each note in the scale, pick a triad chord whose notes are members of the scale. Reference: The pattern of harmonizing chords for all natural minor scales is: minor, diminished, major, minor, minor, major, major, minor. Think about this: The C Major and A Minor (A B C D E F G A) scales contain the same 8 notes, as do any major scale and its relative minor. The chords that harmonize the notes in both scales will also contain the same 8 notes. Since the two scales have the same notes, it makes sense that the same chords are used to harmonize both scales. The difference is that the sequence starts and ends at different chords. Harmonizing chords for C Major scale: C Dm Em F G Am Bdim C Harmonizing chords for A Minor scale: Am Bdim C Dm Em F G Am

Chords with four notes Chords with 4 notes are called tetrachords, or tetrads, or sevenths. Triads have a root, a third, and a fifth. The 4th note of a tetrachord has an interval of a seventh when compared to the root. As shown in the chart of interval definitions, a minor seventh is 10 half steps above the root, and a major seventh is an interval of 11 half steps. Chords with 4 notes are called tetrachords, or tetrads, or sevenths. Triads have a root, a third, and a fifth. The 4th note of a tetrachord has an interval of a seventh when compared to the root. As shown in the chart of interval definitions, ais 10 half steps above the root, and ais an interval of 11 half steps. Reference: The common seventh chords are: (no need to memorize) 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 (Maj7) or (M7) or (Δ7), e.g. CMaj7 or 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 (e.g. C°7) or with dim7 (e.g. Cdim7). Chords can be defined with more than 4 notes, but those chords would be missing a note if played on a 4-string ukulele. You have the theory and rules of finding the notes in triads and tetrads. www.pianochord.org shows the notes and repeats the rules for almost any chord you will encounter.

Harmonizing 7th chords Rules for harmonizing 7th chords over a scale are similar to triads. For each root note in the scale, find the seventh chord whose 4 notes are members of the scale. As an example, find the 7th chord that harmonizes G in the C Major scale C D E F G A B C.

The root note is G. Starting from G, the third in sequence is B. The fifth is D. The seventh is F.

The desired 7th chord is G-B-D-F.

Referring to the chromatic scale, B is 4 half steps above G, or a major third.

D is 7 half steps above G, or a perfect fifth.

G-B-D fits the definition of a major triad: a major third and a perfect fifth.

F is 10 half steps above G, or a minor seventh.

A major triad and a minor seventh form a dominant 7th chord.

Thus the harmonizing seventh chord for G in the C Major scale is G7. The good news is, we only have to find the harmonizing seventh chords for a major scale once. After that we just refer to the pattern which is the same for all Major scales. Reference: 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, Bm7♭5 and CM7. Harmonizing seventh chords for the A minor scale are Am7, Bm7♭5, CM7, Dm7, Em7, FM7, G7, Am7. The generalized sequence for Minor scales is: m7, m7♭5, M7, m7, m7, M7, 7, m7. Just as was noted when harmonizing a scale with triads, a major scale and its relative minor contain the same notes. The seventh chords which harmonize the A Minor scale are the same as the ones which harmonize the C Major scale.

Ukulele C C♯ D♭ D D♯ E♭ E F F♯ G♭ G G♯ A♭ A A♯ B♭ B C The 20 or so frets on a ukulele are each a half step apart like the chromatic scale. 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 interval pattern of a Major scale, and by referring to a chart of the chromatic scale. The fret markers or dots are often on the 5th, 7th, 10th and 12th frets. These frets correspond to the intervals perfect 4th, perfect 5th, minor seventh and octave. 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. To play the 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 to sound an E, 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. Pluck these strings to practice the C scale: C0 C2 E0 E1 E3 A0 A2 A3. Practice playing the C Major scale or any scale.

Ukulele chords On a piano, to play a chord you strike multiple keys simultaneously. To play a chord on a ukulele, you strum multiple strings. All 4 strings usually participate in a chord. There is a 4 digit shorthand for describing which frets are pressed to strum a chord. The 1st digit refers to the G string, 2nd to C string, 3rd to E string and 4th 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 (C-E-G) is 0003, meaning to strum a C chord, the G, C and E strings are open, while pressing the 3rd fret of the A string. You can use any fingering pattern that sounds the 3 notes in a triad chord. The extra string sounds a second occurrence of any of the 3 notes. With enough time and 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 fingering positions - 1402 (G♯ E E B), 4442 (B E G♯ B), 4447 (B E G♯ E). Each fingering position strikes each note of the triad at least once. One of the three notes is played on 2 strings. The different fingering patterns generate slightly different sounds. but in this example all are E Major chords. Some may be inversions, a subject for another day.

Roman numeral numbering For purposes of analysis, harmonizing chords are assigned Roman numerals, I thru VIII, starting with "I" for the chord built on the tonic note of a scale. Major chords are upper case, e.g. IV. Minor chords are lower case, e.g. iii. You may have heard someone talk about a I-IV-V-I chord progression. 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). In the C Major scale or key of C, the Roman numeral chords I, IV, V represent C, F and G chords. Transposing to the key of F, the I IV V chords are F B♭ C. Reviewing the terms major and minor Major and minor describe intervals, 4 vs 3 half steps. Major and minor describe triad chords. The 2nd note in a triad is either a major or minor third. Major and minor describe scales. The 3rd note in the scale is either a major or minor third from the tonic.

What makes music sound good? Sound is pressure waves of air impinging on our eardrums. Imagine two 3-year-olds banging drums. We might call it noise. Imagine a drummer sounding a bass drum twice a second, and another drum four times a second. The beats are regular and synchronized. The resulting sound is more pleasing. Imagine two 3-year-olds banging drums. We might call it noise. Imagine a drummer sounding a bass drum twice a second, and another drum four times a second. The beats are regular and synchronized. The resulting sound is more pleasing. Speed up the frequency of beats from a couple of times a second to say, 400 hertz (cycles per second), in the range of a piano note. At this frequency, we don't recognize individual beats. Instead we hear a musical tone. Add in a second tone of 800 cycles per second, twice as fast, just as with the drum example. Alternate beats of the higher frequency line up with a beat of the lower frequency tone. Just like the drum example, our ears and brain find the synchronized combination pleasing. Slow down the higher frequency so that beats don't line up, and the resulting sound is musically "off". The above example is an example of two notes with an interval of an octave. An octave is the most harmonious interval of two notes. An octave interval is an important concept in the topic of harmonic overtones. Pitch Frequency or pitch measures the vibrations of a string or column of air and is measured in hertz (cycles per second). 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. Overtones are multiples of the tonic frequency. For example, if the tonic frequency is 200 hz, the harmonic overtones will be at frequencies of 400, 600, 800, 1000, 1200 ... The frequency of a perfect 5th is 50% higher than the tonic note. If the frequency of the tonic note is 200 cycles per second (cps), a perfect fifth would have a frequency of 300 cps. I I I I I I I I I I I I I I I tonic frequency V V V V V V V V V V V V V V V V V V V V V V perfect fifth frequency Notice that every other pulse of the tonic (I) note lines up with every third beat of the perfect fifth (V). That is pleasing to the brain, harmonious, and why it "sounds good". A tonic note and a perfect fifth played together sound good. In a nutshell, that explains why certain combinations of tones sound good. In reality each note has many overtones, multiples of the tonic note. Tonic overtones in this example would be 400, 600, 800, 1000, 1200, 1400, ... Perfect fifth overtones would be 600, 900, 1200, 1500, 1800, ... The important concept here is overtones, multiples of the tonic frequency. Harmonic overtones When two notes produce some of the same overtones, the sound is pleasing to the ear, or harmonious. This helps explain 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. 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 (hertz) The tonic and its overtones are 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, etc. (multiples of the tonic). 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. The important takeaway from this section is that two notes sound harmonious when some of their overtones are the same. In this case we refer to them as harmonic overtones.

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 loudness of the series of harmonic overtones.

Math and music Skip this section if the math is too much. The higher pitched ending note on the right in a major or minor scale has a frequency of twice the lower pitched starting note on the left. Why are there 12 piano keys in an octave? Why are there 8 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. 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. A half step is not a fixed frequency difference. Rather it is a fixed ratio equal to approximately 1.06, or more precisely, the twelfth root of 2. There are 12 half steps in an octave. If you multiply a tonic frequency times 1.06 twelve times, you will get the value that is double that of the tonic note. 1.06 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 = 2 (approximately) 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 almost in tune. If you tune the piano to be perfect in one key, it would be farther off in another key. Some performers have their piano tuned to be close to perfect in the key of the song(s) they will play. Here is a web page that goes into the math in detail: "Pitches, Intervals, and Scales": http://msp.ucsd.edu/syllabi/170.13f/course-notes/node4.html