Quantising pitch

Pitch and duration are continuous phenomena. On the pitch dimension, the average human ear can detect sound wave frequencies from about 20Hz to 15,000Hz (15kHz).^[1] On the duration (time) dimension, human cognition tends to interpret events that are separated by less than 100ms as continuous rather than discrete, while events that are separated by much more then 1.5s tend be heard as disconnected, in the sense of not belonging to a single larger phenomenon.^[2]. Within these limits, the values in the dimensions of pitch and duration or timing of musical events each sit on a continuum of value.

Music (with a few exceptions) typically does not operate on these continua. Instead, pitch and duration are quantised.  In this chapter, we will look at how this is commonly done for pitch in Western music. The following chapter will investigate duration.  This will introduce us, also, to the fundamentals of music notation—common ways of writing music in terms of these two dimensions.

In Western music a note with a fundamental frequency of 440Hz is designated “A.” This is only one of several frequencies, however, that are called A.  In standard Western tuning,^[3] every pitch with a fundamental frequency that expresses a simple relationship with 440Hz through repeated doubling or repeated halving is generically called “A” in the Western system.

Why are these notes all called “A”? The answer lies in the very simple mathematical relationships of their fundamental frequencies, which lie on a geometric sequence (1, 2, 4, 8, 16…).  These relationships can be shown graphically by comparing the wavelengths of the fundamental frequencies and by listening to them.  In the graphic below, simple waveforms for the fundamental frequencies of these As are illustrated, and piano notes for these frequencies can be heard by clicking on the numbers on the left. You will hear when you click on these frequencies that there is a strong relationship between them; we perceive that they are alike.

The clear sonic relationship these notes have with one another is called octave equivalence, and this is probably the most fundamental relationship between different musical notes in the Western system (and many others).  Another way of conceptualising this is that these notes belong to a group or class of notes.  In the examples above, this is the pitch-class “A.”

We now have two different ways of conceptualising “A” in music:

1. as any particular given note A (e.g. A440, or A220), or
2. as any one of a class of notes A.

Naming pitch and pitch class: letters A to G

Between any two adjacent notes A,  Western music makes further discrete divisions into intervening notes.  Traditionally there are six additional named pitch classes, following the letters of the alphabet.  This give us seven note names in all: A, B, C, D, E, F and G.  Once we get to G, the pattern returns to A and starts again.  There are two ways to visualise this:

• a line extending infinitely in either direction—higher to the right, lower to the left.
• a circle around which the letters a are arranged—higher going clockwise, lower going counter-clockwise.

In the second representation above, it is possible to imaging rotating the circle in either direction, placing successive letter names at the top.  Completing this action seven times in either direction would represent a complete rotation through all pitch names.

Remember, finally, any of the seven letters from A to G can stand for a pitch (e.g. any given not D) or pitch class (e.g. all notes called D related by octave equivalence)

The piano keyboard—”White” notes

An important and practical way to visualise and understand notes and their recurring patterns of letter names is through the piano keyboard, with its familiar pattern of “white” and “black” keys. Each key represents a note.

In the segment of the keyboard shown above, there are two complete rotations through all of the pitch names (and further incomplete rotations to the left and right).  As you can see, reading from left to right, adjacent white notes read up from A to G and then start again (just like the line or circle of letter names in the previous section.)

At this stage, we are only naming the “white” keys—the “black” keys also have names, but these names are dependent on the white keys adjacent to them (we will get to this later).  However, the as-yet unnamed black keys are important, because they reveal a pattern that identifies where the rotation through the letters A to G recurs.  The black keys group into bunches of two and three, in a regularly recurring pattern (shown above the keyboard).  Going from left to right (which in pitch terms is from lower to higher), you will notice that A is always between the second and third in the group of three black keys.  All the other notes (B to G) also have their own unique position relative to the groups of black keys.

If we count (or play) each white key from one letter to the next instance of that same letter (or, complete one rotation through the letter names), we must use eight keys (including the ones we start and finish on).   Various rotations are shown below: A to A, D to D, C to C and again to the next C. The two over-lapping rotations from C to C and to the next C, shows that we have to restart the count at “1” again for a subsequent rotation.

This means that the “distance” measured in terms of the number of keys (or notes) needed to get from one instance of the same letter to the next is eight and this distance, or interval (to use the correct music theory term), is therefore called an octave (from the Latin octavus, meaning “eighth”).

Notes that are an octave apart always share the same letter name, and always have a fundamental frequency ratio of 1:2, or 2:1 (depending whether the first note is lower or higher).^[4]

If we now go to a complete piano keyboard (88 keys in all), we can see the more-or-less standard range of notes that is used in Western music.

If we wish to distinguish specific notes as opposed to talking about pitch class, a system of numbering can be used, called Scientific Pitch Notation, which gives each note both a letter name and a (usually subscript) number, starting at “0” for the lowest pitch.  In the illustration above, there are several things to note.

The most important of these is that the numbering of pitch names references rotations starting on C, not on A.  In other words, our reference point is C.  This may seem counterintuitive, but it is a standard convention.

C4 is called “middle C”—it sits roughly in the middle of the total range of piano keys.  The first A above it (A4) is the tuning standard A440 (introduced earlier).

The various groups of notes with the same number are understood to belong to a specific range (sometimes also called an octave, slightly misleadingly). These have conventional designations, some of which also relate to the length of organ pipes. These are shown underneath the horizontal brackets.

Scientific Pitch Notation is not the only way of identifying specific pitch. There are a number of other systems, probably the most well-known of which is the Helmholtz system (shown above the keyboard).  It also takes C as a reference.

Finally, for reference, the illustrations also shows the conventional ranges for the standard human voice types (bass, tenor, alto and soprano).

Staff notation

So far we have mapped out a kind of space for pitch, identified a naming system for notes and looked at how the piano keyboard provides a useful and practical way of representing, and playing, notes. Now we are ready to see how notes can be communicated efficiently for the purposes of reading and writing music.  In the Western tradition, the system we use for this purpose is called staff notation, and this introduces the most fundamental element of this system—the staff.

The staff

The staff consists of five horizontal lines, and four spaces.  Both the lines and the spaces are used to indicate the pitch of a note.  The higher the line or space, the higher the note.

To indicate a certain pitch, we place a note head on a line or a space.  We can also place note heads immediate above or below the staff.  If we still need to go further, we can add small individual, leger lines that show the position of each note head precisely above or below the standard five-line staff.  You can think of these as temporary extensions of the stave.

***H5P here: Place note heads on staff in ascending pitch order

The staff is useful for telling us the relative position of notes (higher or lower).  In order to let it tell us precisely what notes are shown, we require an additional piece of information.  This comes in the form of a clef.^[5]

Clefs

A clef is a symbol which shows which lines and spaces of the staff represent specific notes.  There are three different clefs in common use today, the G clef, the C clef and the F clefs.

Each clef identifies a line of the stave wth a particular pitch, thereby also showing all the other associated pitches by relation.

• The lower part of the G clef curls around a line, identifying it as G above middle C (G4)
• The C clef centres around a line, identifying it as middle C (C4)
• The F clef, with its two distinct dots, identifies a line as F below middle C (F3)

It is possible to place any of these clefs on any line of a staff in order to determine the relative pitch of all the others lines and spaces. However, in contemporary practice, most music uses one position for the G and F clef, and two positions of the C clef.

• The G clef is mostly placed on the second line of the staff, and in this position is also called the “treble” clef
• The C clef is mostly placed on the third of fourth line of the staff, and in this position is also called the “alto” or “tenor” clef, respectively
• The F clef is mostly placed on the fourth line of the staff, and in this position is also called the “bass” clef.

The Treble and bass staves

When we also use these terms to refer to staves.  A staff with the treble clef is usually called the “treble staff,” and one with the bass clef, the “bass staff.” Between them, the treble and bass staves are able to represent a large range of notes, from F2 to G5, and even further if we employ leger lines.

When first reading from a staff, it is easy to forget which lines and spaces are attached to which notes. A popular way to remember is to use mnemonics for the lines and spaces for each staff.  Below are some common ones for the treble and bass staves.  There are others ones, and ones for other staves, and you can also make up your own, if that helps.

The great stave

If we look closely at the treble and bass clefs, there is a kind of symmetry around middle C, in the sense that middle C sits on the first leger line below the treble clef and the first leger line above the bass clef.  Now imagine an overlay of the treble, alto and bass clefs, so that the upper two lines (E and G) of the alto clef are also the bottom two lines (E and G) of the treble clef, and the bottom two lines of the alto clef (F and A) are the upper two lines of the bass clef (F and A), the result would be a kind of eleven line “super” staff, as show on the left below.

If we then removed the alto clef’s middle line (middle C) we could also imagine middle C as a note head on a leger line that is a remnant of this alto clef.  Now, if we separate the treble and bass clefs vertically, as shown in the middle part of the illustration below, then middle C must be represented now separately on each clef, with its own leger line.

Finally,  we can show a connection between these two now separated staves by joining then with a line and a brace (as shown on the right).  This is known as the great staff.

The great staff is used extensively in Western music.  Its common uses include:

• Notations of most keyboard music^[6]
• Reductions to two staves of four-part vocal music (so-called SATB)
• Reductions of other instrumental ensembles to a condensed format for playing at the piano or study
• Presenting examples of music for analytical or theoretical discussions

With the use of up to three leger lines, we can show the range of (white) notes from G1 to F6.  Note also, that there is often some overlap and flexibility between the two staves (treble and bass) that comprise the great staff. It is not uncommon to write notes at least two, and often more, leger lines below middle C on the treble staff and above middle C on the bass staff.

The piano keyboard—”black” notes

Returning to the piano keyboard, we can consider the notes of the Western notation system that we have not covered—those identified with the piano’s “black” keys. In the picture below, we can see the inside of an upright piano, showing the action that links the hammers in a row along the upper part of the picture) to the keys. Each hammer strikes a set of similarly tuned strings (usually three).

Note that each hammer links to an adjacent key, black or white.  So only the white keys that are not separated by an interleaved black key strike a set of strings that are directly adjacent.  White keys that are separated by a black key strike a set of strings that are separated by one set. Each adjacent set of strings is tuned to notes that are heard to be the same interval apart.  Therefore, for the various adjacent pairs of white keys, there are two different possible intervals, one smaller than the other:

• smaller (no intervening black key): B to C and E to F
• larger (one intervening black key): C to D, D to E, F to G, G to A, A to B

The smaller interval is called a semitone and the larger interval is called a tone (or sometimes a whole tone in order to distinguish it from a semitone).^[7]

Tones and semitones, sharps and flats

A semitone is the smallest standard interval in the Western music system.  It is the interval that separates two directly adjacent notes on the piano keyboard, black or white.  To the right, we can see at the bottom the pattern of tones and semitones that separate the white-key notes A to G.  We also see, along the top, how every key, black or white, sits a semitone higher or lower than its adjacent neighbour. Adjacent black keys within a group sit a tone apart, and at the end of a group of black keys, skipping the immediately adjacent white key to the next, also is a tone.

Additionally, we are now able to give names to the notes defined by the black keys.  These are defined as “sharp” or “flat” versions of the white-key notes.

• To “sharpen” in music means to raise the pitch of a note
• To “flatten” means to lower the pitch of a note.

When we add the terms sharp or flat to a note name, such as A, it means that we are raising or lowering the pitch by a semitone.  On a piano:

• to play “A sharp” means to play the black key immediately to the right of A;
• to play “A flat” means to play the black key immediately to the left of A

This means that the black-key notes all have two names. For instance, “D sharp,” the black key immediately to the right of D is the same black key immediately to the left of E, meaning it can also be called “E flat.” This is known as enharmonic equivalence.

Also, for some of the white-key notes, there also enharmonic names—for instance, B can also be named C flat, F can be called E sharp, and so on.

Music has five symbols which denote the various alterations possible to notes. These signs are known as accidentals. The name is somewhat misleading, as there is nothing really “accidental” about them.  They are as follows:

The last two will not concern us quite as much immediately, but it is advisable to get to know them.

Writing notes with accidentals

When we write attach these symbols in writing, we add them after the letter name. For instance, A flat is written A, C sharp is written C, and so on.  However, in music notation, the accidental sign is always placed directly in front of the note head it modifies.  Additionally, there are some other rules to follow:

• Always place the accidental close to the note it modifies, but leave a small gap
• Always place the accidental on the same line or space of the note it modifies.
• Make sure the accidental is neither too large, nor too small for reading:  sharp and natural signs are about three spaces high;  flat and double flat signs are about two spaces in size; the double sharp sign is a space in size.

Below are examples of good and poor notation practices for placing accidentals