---
id:
aliases: []
title: Music Theory
tags: []
---
# Music Theory

## Note

### Pitch

Refers to the human perception of frequency.
It is usually only appropriate to prefer "pitch" over "note"
when one is specifically referring to frequency.

Notes are **enharmonic** (or **enharmonically equivalent**)
if they represent the same pitch.

#### Pitch Class

| mod 12 | name  |
| ------:| ----- |
|      0 | C     |
|      1 | C♯/D♭ |
|      2 | D     |
|      3 | D♯/E♭ |
|      4 | E     |
|      5 | F     |
|      6 | F♯/G♭ |
|      7 | G     |
|      8 | G♯/A♭ |
|      9 | A     |
|     10 | A♯/B♭ |
|     11 | B     |

[Modular arithmetic](https://en.wikipedia.org/wiki/Modular_arithmetic)

> [!important]
> Adjacent sharps and flats (e.g. C♯ and D♭)
> only represent the _same_ pitch in [equal temperament tuning](https://en.wikipedia.org/wiki/Equal_temperament),
> but they are still considered enharmonic in others.

### Note: Semitones and Steps

It is common to speak of whole or half steps between pitches,
but I find this confusing in a way music educators ought to better anticipate.
In equal temperament tuning, all pitch classes are---
according to human perception---equally spaced.
There is no missing half between E and F or B and C,
because there are no halves at all.
C# is as legitimate a pitch class as C,
and in some disciplines, accidentals are marked on every note,
even "redundantly", in acknowledgement of this fact.

What is meant by the poor terminology is this:

* "half step" = 1 semitone
* "whole step" = 2 semitones

There is no more nuance, E--F is also a "half step".

```lily
\score{
    a~ ais
    b~ c
    \layout {}
}
```

This is an unhelpful convention.
Inexcusable, though, is conflating semitone increments with intervals.
Wikipedia music theory articles frequently link "half step" and "whole step"
to [minor second](https://en.wikipedia.org/wiki/Minor_second)
and [major second](https://en.wikipedia.org/wiki/Major_second) respectively,
which is a [[ambiguity#Category Mistake|category mistake]].

#### Accidentals

| ascii | unicode | lilypond | Name         |
| ----- | ------- | -------- | ------------ |
| bb    | 𝄫       | eses     | double flat  |
| b     | ♭       | es       | flat         |
|       | ♮       |          | natural      |
| #     | ♯       | is       | sharp        |
| x     | 𝄪       | isis     | double sharp |

#### Octave

Octave is a property of staff position, not pitch.
B♯<sup>3</sup> is enharmonic with C<sup>4</sup>

### Duration

* beats
* speed

### Intensity

* pianissimo
* piano
* forte
* fortissimo

## Interval

Refers to the difference between two pitches.
Intervals are notated with an en dash (C♯--E).

### Interval Number

Equal to the difference in staff position plus one.

> [!important]
> The number of an interval is _not_ related to the difference in pitch.
> F♯ and G♭ are enharmonic, but F♯--G♭ is a second.
> F and F♯ are a semitone apart, but F--F♯ is a first.

### Interval Quality

An interval of unspecified quality is called "generic".

difference in semitones

#### Perfect, Major, and Minor Intervals

The common intervals

| dP[^1] | dS[^2] | Short | Name           |
| ------ | ------ | ----- | -------------- |
| 0      | 0      | P1    | Perfect unison |
| 1      | 1      | m2    | Minor second   |
| 1      | 2      | M2    | Major second   |
| 2      | 3      | m3    | Minor third    |
| 2      | 4      | M3    | Major third    |
| 3      | 5      | P4    | Perfect fourth |
| 4      | 7      | P5    | Perfect fifth  |
| 5      | 8      | m6    | Minor sixth    |
| 5      | 9      | M6    | Major sixth    |
| 6      | 10     | m7    | Minor seventh  |
| 6      | 11     | M7    | Major seventh  |
| 7      | 12     | P8    | Perfect octave |

[^1]: difference in staff position (interval number)
[^2]: difference in semitones

#### Augmented and Diminished Intervals

| dP[^1] | dS[^2] | Short | Name               |
| ------ | ------ | ----- | ------------------ |
| 0      | 1      | A1    | Augmented unison   |
| 1      | 3      | A2    | Augmented second   |
| 2      | 5      | A3    | Augmented third    |
| 3      | 6      | A4    | Augmented fourth   |
| 4      | 8      | A5    | Augmented fifth    |
| 5      | 10     | A6    | Augmented sixth    |
| 6      | 12     | A7    | Augmented seventh  |
| 1      | 0      | d2    | Diminished second  |
| 2      | 2      | d3    | Diminished third   |
| 3      | 4      | d4    | Diminished fourth  |
| 4      | 6      | d5    | Diminished fifth   |
| 5      | 7      | d6    | Diminished sixth   |
| 6      | 9      | d7    | Diminished seventh |
| 7      | 11     | d8    | Diminished octave  |

### Compound Interval

A **compound interval** is one greater than number 8.

## Key

A **key** is a [[set-theory|set]] of pitch classes
created by modifying a starting pitch class (the **tonic**)
according to some sequence of intervals.

| Key           | Interval Sequence |
| ------------- |:-----------------:|
| Major         |   W--W--H--W--W--W--H   |
| Natural Minor |   W--H--W--W--H--W--W   |

### Modes

| Mode       | Interval Sequence |
| ---------- |:-----------------:|
| Ionian     |   W--W--H--W--W--W--H   |
| Dorian     |   W--H--W--W--W--H--W   |
| Phrygian   |   H--W--W--W--H--W--W   |
| Lydian     |   W--W--W--H--W--W--H   |
| Mixolydian |   W--W--H--W--W--H--W   |
| Aeolian    |   W--H--W--W--H--W--W   |
| Locrian    |   H--W--W--H--W--W--W   |

### Scale

A **scale** is a progression of the elements of a key
in ascending or descending order.

### Scale Degrees

* degree 1: tonic
* degree 2: supertonic
* degree 3: mediant
* degree 4: subdominant
* degree 5: dominant
* degree 6: submediant
* degree 7: called the **subtonic** if it is 2 semitones/"a whole step"/a major second below the tonic
    or the **leading tone** if it is 1 semitone/"a half step"/a minor second below the tonic
* degree 8: tonic
