# Music Theory I
## Introduction

This is the first Music Theory course. It provides an introduction to basic music theory, pitches, pitch collections (scales), key signatures, rhythmic meters, and modes. You should already know music fundamentals such as clefs, staves, accidentals, dynamics, and rhythmic values.
For more fundamental musical theory, see Introduction to Music or Fundamentals of Music.

## What is music?

The definition of music itself is elusive. We want to be general enough to include a wide range of styles and cultures and approaches, yet not so wide that the term "music" can mean anything, and therefore, nothing. Here are some attempts:
* Music is a series of tones
* Music is sound organized by humans
* Music is an art-and-science
* Music is an abstract communication language
* Music is an emotional expression of the soul
* Music is made up of the elements of melody, harmony, rhythm, timbre, texture, dynamics, and form
* Music is the created order of nature
* Music is culturally-defined psychological expression in sound
* Music is love
All of these definitions have merit and truth. For now, let us define music as an art form expressed through the medium of sound, organized aurally (what we hear) and rhythmically (when we hear it).

## What is music theory?

When we say music theory, we usually aren't talking about a universal system of understanding music. Rather, we are focusing specifically on Western Tonal Harmonic music, or "common practice" music. This type of music was developed from 1650 to 1900 in Western Europe. We consider it "common practice" not because it is the only good sounding music, but because it was a widely accepted system of analyzing music for three hundred years and is the foundation for much classical, pop, rock, and some aspects of blues and jazz music.
Let us define this type of music theory a little further. Music theory is merely a system of describing, notating, communicating, and analyzing music. It is not a "theory" like the theory of relativity, but rather a framework for understanding many types of music. It does not describe how music should work or what makes music work, but is simply a method by which composers, musicians, educators and anyone else can communicate and understand Western Tonal music.
While there are certain advanced elements to music theory (such as tuning and the overtone series) which have a "universal" application due to being based on the physics of vibration, the more common elements of music theory that are taught in universities today are these aspects of Western Tonal Harmony. The music theory we discuss here would not be particularly suited to talking about Indonesian Gamelan music or very modern classical pieces, both of which would require a different method of music theory analysis.

## Pitch

All music is based on sound and when we refer to a specific tone we are referring to a pitch. Every key on a piano plays a different pitch. In traditional western music, we focus on twelve pitches to make music. They are as follows:
C, C♯/D♭, D, D♯/E♭, E, F, F♯/G♭, G, G♯/A♭, A, A♯/B♭, and B
It may be helpful to picture a piano keyboard:
If you've seen a piano before, you'll notice that after we get to B, the whole thing just starts over again. Normally, there are eight C-keys on a piano. Each of those keys plays the same sound (in this case, the tone C) but higher or lower depending on the key. Therefore, we would say that the eight C-keys on a piano are the same pitch class but different pitches.
If you consult our pitches above, it may appear that seventeen notes have been listed, but this list actually only contains twelve notes. The notes with a slash between them are enharmonic, meaning that they represent the same pitch with two different names. For example, C♯ (C sharp) and D♭ (D flat) are actually the same. To see this, these notes both get played by the same black key on a piano.
The sharp and flat symbols should be familiar to you by now, but let us have a quick review. A sharp is one pitch higher than the note it is attached to. For instance F♯ (F sharp) is one pitch higher than F. Inversely, a flat is one pitch lower than the note it is attached to. Thus, E♭ (E flat) is one pitch lower than E. You might be tempted to think that any sharp or flat is the same thing as a black key on the piano, but this is not so. For instance, E♯ is enharmonic with F and C♭ is enharmonic with B.
While they are not quite as common, there is also a double sharp (x) and a double flat (♭♭). These symbols only mean that the tone is two pitches higher or lower than the note they are attached to. For example, A♭♭ would be enharmonic with G and Dx would be enharmonic with E.
The majority of Western music has emerged from the twelve pitch classes listed above. Just about every Western scale and chord (and a great many non-Western scales and chords) are founded from these notes.

## Intervals
### The octave

As was briefly mentioned above, all the C-keys on a piano are in the same pitch class, but are different pitches. If you remember, we also said that after we list the twelve pitches, we start over. For example...
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
...and so on and so on. If you look at any pitch "C", what is twelve pitches above it? Another C. What is twelve pitches above that? Yet another C. This distance (of twelve pitches) is called an octave.
Another way to think of it would be that an octave is the distance between two consecutive pitches in any pitch class. So if our pitch class is "C" we might have the following pitches: C1, C2, C3, C4, C5, C6, C7. The distance between C3 and C4 is one octave. The distance between C2 and C5 is three octaves.

### Basic intervals

The octave is a way of measuring musical distance. The distance between any two notes is called an interval. There are twelve basic intervals (corresponding with the twelve pitches) but there are two that particularly important: the half step (or semitone) and whole step (or whole tone). A half step is the distance between two consecutive pitches. For instance, the distance between G and A♭, or F and F♯, or B and C. You could think about a piano; the distance between any key to the keys immediately next to it is one half step. A whole step is two half steps. So the distance between D and E, or A♭ and B♭, or E♭ and F is one whole step.
These intervals are listed in order, beginning at the Unison and progressing up to the octave:
| Interval Name | Interval Distance | Example |
| Perfect Unision or Unison or P1 | The distance between one pitch and itself, in other words, no difference. The same pitch twice in a row. | C7 to C7, or F4 to F4 |
| Minor Second or Half Step/Semitone or m2 | The distance of one half step. | D to E♭, or G to G♯ |
| Major Second or Whole Step/Whole Tone or M2 | The distance of one whole step; the distance of two half steps. | A to B, or E to F♯ |
| Minor Third or m3 | The distance of one whole step and one half step; the distance of three half steps. | D to F, or G to B♭ |
| Major Third or M3 | The distance of two whole steps; the distance of four half steps. | E♭ to G, or A to C♯ |
| Perfect Fourth or P4 | The distance of two whole steps and one half step; the distance of five half steps. | C to F, or A♭ to D♭ |
| Tritone or Augmented Fourth/Diminished Fifth or +4/°5 | The distance of three whole steps; the distance of six half steps. | D to G♯, or F to C♭ |
| Perfect Fifth or P5 | The distance of three whole steps and one half step; the distance of seven half steps. | E to B, or F♯ to C♯ |
| Minor Sixth or m6 | The distance of four whole steps; the distance of eight half steps. | A♯ to F♯, or E♭ to C♭ |
| Major Sixth or M6 | The distance of four whole steps and one half step; the distance of nine half steps. | D♯ to B♯, or F to D |
| Minor Seventh or m7 | The distance of five whole steps; the distance of ten half steps. | G to F, or C to B♭ |
| Major Seventh or M7 | The distance of five whole steps and one half step; the distance of eleven half steps. | E to D♯, or G♯ to F |
| Perfect Octave or Octave or P8 | The distance of six whole steps; the distance of twelve half steps. | E♭4 to E♭5, or D6 to D7 |
The naming conventions come from the following set of rules, which you don't need to learn, but are useful to know about at least.
* The unison, fourth, fifth, and octave are perfect. The term perfect doesn't mean that the notes are superior, but comes from the very clear, resolute, final, "perfect" sound they have.
* The m3, M3, m6, and M6 are considered consonant intervals because they produce a fairly pleasant, stable sound, although not as stable as the perfect intervals.
* The m2, M2, m7, and M7 are considered dissonant intervals because they produce a fairly tension-filled sound that needs to resolve.
* A minor interval is smaller than its major counterpart by a semitone. Conversely, a major interval is larger than its minor counterpart by a semitone.
* An augmented interval is larger than its perfect/major counterpart by a semitone.
* A diminished interval is smaller than its perfect/minor counterpart by a semitone.
Although they weren't listed above, there are more intervals. For instance, as stated above, a diminished interval is smaller than its perfect/minor counterpart by one half step. Therefore, a diminished fourth would be a perfect fourth (5 half steps) minus one half step, ending up with four half steps. You may notice that this is identical to the distance of a major third. The difference between the two is how we would write the interval on the page, but there would be no aural difference. Take care to note that you cannot have a minor fifth or a major fourth, for such terms are impossible. If you were to increase the distance in your perfect fifth, you would have an augmented fifth, and so on.
We can name these intervals in a universal way, consisting of the following:
[Modifying Symbol][Number]
where the modifying symbol denotes perfect (P), major (M), minor (m), augmented (+), or diminished (°), and the number corresponds to the name of the interval - so 2 for second, 6 for sixth, etc.
Let us talk briefly about enharmonics, for when it comes to intervals they do make a difference. If you started with C and wanted a minor sixth, you would go up eight half steps until you got to G♯/A♭, however you wouldn't use the note G♯. The distance between C and G♯ is an augmented fifth. If you wanted a minor sixth, you would say C to A♭. The difference here is, again, written but not aural. There is no discernible change from an augmented fifth or a minor sixth, but for our analytical purposes, it is a very important distinction.

### Compound intervals

There is a system of naming notes past the octave, used commonly in jazz but also to an extent in modern classical and pop music, called compound intervals. These are essentially the same as the basic intervals, but expanded by an octave (12 semitones remember). Our naming and notational conventions carry over, but we add seven to the note number.
For example, a minor sixth (8 semitones) goes up an octave to become a minor thirteenth (now 20 semitones). Where we notate it before at m6, we now notate with m13.
Again, this system is used mostly in jazz. For now, as we study Western Tonal Harmony, we will use our basic intervals. If you were to see a minor thirteenth in a piece of music, we would simplify and call it a minor sixth.

## Meter and rhythm
### Introduction to meter

As you listen to most music, you will notice a discernible beat, or a primary pulse. In common practice music, we can either divide this beat into two parts (and multiples of two, such as four), which we call simple meters, or we can divide it into three parts (and multiples of three, such as six), which we call compound meter. When we say we can divide the beat into two or three, all we are saying is that it is easiest to divide the beat into this number. We could arbitrarily divide the beat into five or seven, but almost all music naturally divides into two or three.
Notice all that we have been talking about is how the beats divide. But beats can also be grouped into larger sections. We could place beats into groups of two (which we call duple), three (which we call triple), or four (which we call quadruple). We could place the beats into as large a group as we want, but for now let us stay with duple/triple/quadruple as they are the most common. These groupings of beats are called measures or bars and are shown on sheet music with thin lines to divide the measures.
Thus, we could have a piece of music with a simple triple meter. Refer to the terms above. A simple triple meter would be one in which we could put the beats into groups of three and the beats themselves divide easily into two parts. The familiar song "Amazing Grace" is in simple triple meter. Take a look at the chart below:
| Beat Division | Beat Grouping | # of Beats in a Measure | Familiar Music |
| Simple | Duple | 2 | The Stars and Stripes Forever by John Philip Sousa Mary Had a Little Lamb (Traditional) |
| Simple | Triple | 3 | An der schönen blauen Donau (The Blue Danube Waltz) by Johann Strauss II Amazing Grace by John Newton/William Walker Caribbean Blue by Enya |
| Simple | Quadruple | 4 | Don't Stop Believin by Journey Yesterday by The Beatles Bad Romance by Lady Gaga |
| Compound | Duple | 2 | The Washington Post March by John Philip Sousa We Are The Champions by Queen |
| Compound | Triple | 3 | Jesus bleibet meine Freude (Jesus, Joy of Man's Desiring) from Herz und Mund und Tat und Leben (Heart and Mouth and Deed and Life) by Johann Sebastian Bach Walkürenritt (Ride of the Valkyries) from Die Walküre (The Valkyrie) by Richard Wagner Clair de Lune (Moonlight) from Suite Bergamasque by Claude Debussy |
| Compound | Quadruple | 4 | Memory from Cats by Andrew Lloyd Weber Lights by Journey |
No doubt you have heard the terms rhythm and meter by now, perhaps interchangeably, however let us separate the two words: rhythm refers to the length of a pitch in a piece while meter is the framework of strong and weak beats in which rhythm is placed.

### Meter signatures

A meter signature or a time signature is located at the beginning of a piece and sets the meter type (Simple/Compound and Duple/Triple/Quadruple) and the beat unit. The beat unit is the note length which gets one beat. A time signature is set up like this:
[Beats per Measure] [Beat Unit]
The top number will tell us what the meter type is while the bottom number will tell us the beat unit. If the upper number is 2, 3, or 4, it will usually mean that the meter is simple duple, simple triple, or simple quadruple respectively. If the upper number is 6, 9, or 12, the meter will likely be compound duple, compound triple, or compound quadruple. The lower number indicates the following: 2 = half note, 4 = quarter note, 8 = eighth note, 16 = sixteenth note. You can reference this chart for easier understanding:
| Meter Signature | Beats per Measure | Beat Unit | Meter Type |
| 2/4 | 2 | Quarter Note | Simple Duple |
| 3/4 | 3 | Quarter Note | Simple Triple |
| 4/4 | 4 | Quarter Note | Simple Quadruple |
| 2/2 | 2 | Half Note | Simple Duple |
| 3/2 | 3 | Half Note | Simple Triple |
| 3/8 | 3 | Eighth Note | Simple Triple |
| 4/8 | 4 | Eighth Note | Simple Quadruple |
| 4/16 | 4 | Sixteenth Note | Simple Quadruple |
| 6/8 | 2 | Dotted Quarter Note | Compound Duple |
| 9/8 | 3 | Dotted Quarter Note | Compound Triple |
| 12/8 | 4 | Dotted Quarter Note | Compound Quadruple |
| 6/4 | 2 | Dotted Half Note | Compound Duple |
| 9/4 | 3 | Dotted Half Note | Compound Triple |
| 6/16 | 2 | Dotted Eighth Note | Compound Duple |
| 9/16 | 3 | Dotted Eighth Note | Compound Triple |

## Pitch collections

The traditional Western music that we are studying does not usually use all twelve of our pitch classes. Instead, we have discovered that it is aurally pleasing to use only some of the twelve pitch classes at any given time. We call this a pitch-class collection. If we were to select all twelve of the pitch classes we would have a chromatic collection. A chromatic scale is one in which all of the pitch classes are played. However, we normally use a diatonic collection ("diatonic" with the root of "tonic", referring to the tones) of seven pitch classes. These diatonic collections usually take the form of the major and minor scales.

### Major scale

No doubt you have heard of the major scale before. As we have said above, the major scale is a diatonic pitch-class collection. The major scale employs seven of the twelve pitch classes and is composed of the unison (P1), the major second (M2), the major third (M3), the perfect fourth (P4), the perfect fifth (P5), the major sixth (M6), and the major seventh (M7) before repeating again. This particular selection of pitches is what gives the major scale its distinct sound and what differentiates it from other pitch-class collections (such as the minor scale).
As an example, if we choose C to be the root (the starting point of a pitch-class collection), the Major scale would be:
C, D, E, F, G, A, B, C, etc.
Note that we repeat the C at the top of the scale. So we use eight notes but only seven pitches (because the C is repeated).
Another way to describe the Major scale is to describe the steps between each note of the scale. We know that the distance between each neighboring pitch is one half step. If we use "H" to name a half step and "W" to name a whole step, the Major scale can be described as WWHWWWH.
One creates a Major scale by first choosing a certain pitch as the root, or starting point. Because it started on "C", the C Major scale was shown above has a "C" root. Other major scales can be created by starting on other notes and then by following the pattern of WWHWWWH from that root. Any of the twelve pitches may be the basis for the major scale.
Some major scales are redundant in the sense that they sound the same as their enharmonic counterparts (e.g. G♭-major sounds the same as F♯-major). Some redundancies are more common while others are not. Generally, if one has to go beyond the use of single sharps or flats, then one has created a redundant scale which can be more simply expressed (e.g. G♭♭-major: G♭♭, A♭♭, B♭♭, C♭♭, D♭♭, E♭♭, F♭, G♭♭ is incredibly redundant compared to F-major: F, G, A, B♭, C, D, E, F).

### Minor scale

The minor scale is another diatonic pitch-class collection. Like the major scale, the minor scale employs only seven notes from the twelve available pitches. There are many types of minor scales but for our purposes, we are going to focus on three. The Natural Minor scale, the Harmonic Minor scale, and the Melodic Minor scale.

#### The natural minor

This is the most common form of the minor scale, and can be represented by the pattern WHWWHWW. A common way to create a natural minor scale is to take a major scale a lower scale degree 3,6, and 7 by a semi-tone. The scale could also be thought of in terms of intervals and is composed of the unison (P1), the major second (M2), the minor third (m3), the perfect fourth (P4), the perfect fifth (P5), the minor sixth (m6), and the minor seventh (m7). Using "A" as our root, the scale would be as follows:
A, B, C, D, E, F, G, A

#### The harmonic minor

The Harmonic Minor scale is almost exactly the same as the Natural Minor scale with the exception of one pitch. A common way to create a harmonic minor scale is to take a major scale a lower scale degree 3, 6 by a semi-tone, leaving scale degree 7 as written in its parallel major. It can be represented as WHWWH(W+H)H. Note the large Whole+Half Step (3 semitone) in parenthesis. This augmented second interval between the sixth and seventh pitches is what distinguishes the Harmonic Minor scale and was created to add more tension and interest to the music incorporating it.
When A is the root, the scale runs:
A, B, C, D, E, F, G♯, A

#### The melodic minor

The Melodic Minor is very distinct from the previous scales discussed in that it has two forms, ascending and descending, which are used respectively according to rise and fall of the scale. The descending form of the scale is the same as the Natural Minor scale:
A, G, F, E, D, C, B, A
Note that the pitches are in reverse order because, obviously, we are descending downwards. The ascending form is a kind of combination between major and minor scales. It can be represented as WHWWWWH or:
A, B, C, D, E, F♯, G♯, A

### Modes

Modes are another type of pitch-class collection. There are seven common modes and they are all diatonic. Let's build a major scale on "C":
C, D, E, F, G, A, B, C
We know that this is a major scale but it is also one of the seven modes, called Ionian. The Ionian Mode and a major scale are one and the same. Let us imagine the same seven pitches (C, D, E, F, G, A, B) but using instead the root of "D".
D, E, F, G, A, B, C, D
This is no major scale, nor is it any minor scale we have learned. It is, however, a mode, called Dorian. If we continue this pattern, using each of the seven pitches as our root, we will get all of our modes. If we use "E" as the root, the mode would be Phrygian. If we use "F", we would have Lydian Mode. If we use "G", we would have Mixolydian. If we use "A", we would have Aeolian Mode. Let us take a closer look at Aeolian Mode:
A, B, C, D, E, F, G, A
Look familiar? The Aeolian Mode and the Natural Minor scale are the same thing, just like the Ionian Mode and the major scale. Finally, if we used "B" as the root, we would get Locrian Mode.
Remember that even though we used only C, D, E, F, G, A, B in our example, a mode can be built on any pitch.
| Mode | Root relative to major scale | Root relative to C Major scale | Interval Sequence | Example I | Example II |
| Ionian | I | C | WWHWWWH | C, D, E, F, G, A, B, C | C, D, E, F, G, A, B, C |
| Dorian | II | D | WHWWWHW | D, E, F, G, A, B, C, D | C, D, E♭, F, G, A, B♭, C |
| Phrygian | III | E | HWWWHWW | E, F, G, A, B, C, D, E | C, D♭, E♭, F, G, A♭, B♭, C |
| Lydian | IV | F | WWWHWWH | F, G, A, B, C, D, E, F | C, D, E, F♯, G, A, B, C |
| Mixolydian | V | G | WWHWWHW | G, A, B, C, D, E, F, G | C, D, E, F, G, A, B♭, C |
| Aeolian | VI | A | WHWWHWW | A, B, C, D, E, F, G, A | C, D, E♭, F, G, A♭, B♭, C |
| Locrian | VII | B | HWWHWWW | B, C, D, E, F, G, A, B | C, D♭, E♭, F, G♭, A♭, B♭, C |

## Key signatures

A key signature shows which pitch classes are to be consistently played sharp or flat throughout the piece. It is placed at the beginning of each line of a score. To help understand why key signatures are beneficial, look at this example:
Even though both images above are of the same music, the second version is much easier to read, analyze, and to play. The key signature, by placing the flat accidentals in a tidy formation, rather than having them dispersed across the page, makes the reading of them much easier.
We have eight sharp key signatures and eight flat key signatures. They look like the following:
You may have noticed that each time we add a sharp, we use the pitch a P5 higher. In other words, the order in which we add sharps is:
F♯, C♯, G♯, D♯, A♯, E♯, B♯
After this, another P5 about B♯ is F♯ again so, therefore, that is as many sharps as we can add. You also may have noticed that once we have B♯, all the pitches have the sharp accidental attached and it would be impossible to sharp any more pitches without using double sharps.
Each time we add a flat, we use the pitch a P5 lower. The order of flats is:
B♭, E♭, A♭, D♭, G♭, C♭, F♭
Similarly, after F♭, the next pitch would be B♭ again. Many students use a mnemonic to remember the order of sharps and flats. A very common mnemonic would be "Father Charles goes down and ends battle" (the order of sharps), or reversed, "Battle ends and down goes Charles' Father" (the order of flats). You are welcome to use whatever you like but the key signatures are something that should be memorized.
Some of the key signatures are enharmonic with each other: D♭ is enharmonic with C♯, G♭ with F♯, and C♭ with B. To better understand this, take a look at the circle of fifths, with shows each of the key signatures, which major or minor key it may represent, the number of sharps or flats it contains, and which keys are enharmonic.
You may be wondering how we can know what the major or minor key of the piece is from the key signature. Well, the key signature with three flats indicates a major or minor key with B♭, E♭, and A♭. The major key that has such flat notes is E♭ Major, and the minor key is C Minor. However, there are a few tricks that make it easier to determine the key without having to consult sheets of key signatures and major and minor scales.
* To determine the major key of a flat key signature, look at the next-to-last flat. The key signature with six flats is G♭ Major. Of course, this does not work with only one flat, but there is only one key signature with one flat: F Major.
* To determine the major key of a sharp key signature, look at the last sharp, then move up one half-step. For instance, the key signature with four sharps in E Major. Take care to note that if you move up one half-step but they key signature contains a sharp on that note, you must include the sharp. For instance the key signature with seven sharps is C♯ Major not C Major, which has no sharps.
Be careful that you do not assume that the key signature automatically determines the actual key of a piece. The key signature might have two flats, but this does not mean that the piece is in B♭ Major. It could be in G Minor, or C Dorian, or E♭ Lydian, or... etc. When determining the key of the piece, we must consider the key signature in combination with the pitch-class collection and the interactions of the intervals within the music.