Today's music theory simply explained

by Matthew Leitch, 30 July 2009

How simple?

This article offers a simple explanation of the basics of music theory and some tips on how to think about notes. It'll be particularly interesting if you like numbers or are in any way put off by words like ‘subdominant’, ‘tonic’, and ‘Mixolydian’.

One reason why there's a place for an introduction like this is that the language and ideas of music have evolved over many centuries and the words and symbols used today are often baffling and even misleading because music has changed but the words haven't.

Another reason is that the fundamentals of music are simpler and more regular than they often seem when presented with all the baggage of past centuries. For some people there is still no alternative to learning the old ways, but you don't have to let the words confuse you. Instead, put a tidy, modern system behind the old words and they will make more sense to you.

Vibrations, octaves, and tunings

You probably know that sound is what we experience when vibrating air buffets our ears. If the frequency of the vibration is between about 20 and 20,000 vibrations a second it sounds like sound. The faster the vibrations the higher the note sounds i.e. the higher its pitch.

If we hear two sounds and one sound is vibrations exactly twice as fast as the other then they seem to fuse together into one sound and it's hard even to hear that there are two. This difference in sound, when the frequencies are doubled, is special and has a special name in music. It is an octave.

The word ‘octave’ is a living fossil. The ‘oct’ part means eight, as in octopus. It comes from a time when musicians divided the gap between one pitch and the pitch an octave higher into seven jumps. That meant there were seven pitches in each octave plus an eighth that was really the first pitch of the next octave.

Today we divide the octave into 12 gaps, using 12 notes per octave. However, the old name remains.

The 12 intervals are called ‘semitones’ and two together are called a ‘tone’, for historical reasons.

Different ways to divide up an octave are called ‘tunings’ and there are many different ways to do it. However, in modern Western music we have just one and it uses 12 notes per octave and 12 identical gaps, where each note's frequency is the same multiple of the one just below it. The technical name for this system is ‘12 tone equal temperament’ or 12TET for short.

Naming pitches

The ancient Greeks began the habit of naming different pitches with letters of their alphabet. The names you hear today are A, B, C, D, E, F, and G but of course that only makes 7 names per octave. As Western music moved on to 12 notes an octave the extra notes were named by adding symbols alongside the original names.

The symbol ‘♯’ is pronounced ‘sharp’ and means ‘up one semitone’. The symbol ‘♭’ is pronounced ‘flat’ and means ‘down one semitone.’

As a system for naming pitches this leaves a lot to be desired so it is worth thinking a bit more about how to name notes precisely and unambiguously. Before I show you some modern methods, here are problems with the old style letters you need to be aware of.

1. The names don't identify an individual pitch. Because music is played over a range of several octaves, if you say ‘C’ that could mean the C in any of the playable octaves.

2. Many pitches have more than one name. For example, notes called C♯ can also be called D♭, D♯ is the same as E♭, and so on.

3. A lot of music theory starts octaves at C rather than the more obvious A, for historical reasons.

4. Music for certain instruments is deliberately written so that the note names do not match those of other instruments. For example, a C on a piano is not the same as what is called a C on a typical saxophone. Saxophonists think they are playing a C but they are not.

There are two recognized systems for naming notes properly and I'd like to suggest another too. Table 1 sets them out side by side.

• The MIDI note number (based on the General MIDI specification agreed worldwide) is the most widely used naming system of all, built into virtually every electronic musical instrument and music software program, playable on any personal computer, and even on some portable music machines. This is the pitch naming system of the digital age.

• The vibrations per second column tells you what a physicist would say.

• The Scientific Standard Notation (SSN) is a simple adaptation of the old system that adds a number to show which octave is involved.

• The numeric version of the Scientific Standard Notation is my suggestion, which links the MIDI note number to the SSN. The first digit is for the octave number, while the number after the dot is for the pitch within the octave. If you take 12 times the octave number plus one, and add the pitch number within the octave, then you get the MIDI note number.

It doesn't take very long to learn the MIDI note number for each place on old style music staves and to learn them for each pitch your musical instrument can play. This needs to take account of any oddities about the way music is written for specific musical instruments. For example, music written for a bass guitar is always written as if it is an octave higher. The lowest note a typical bass guitar can play is MIDI note number 28, but on paper it is written as note 40.

Naming pitch classes

A ‘pitch class’ is simply a set of pitches differing by one or more whole octaves. For example, using MIDI note numbers an example of a pitch class is 12, 24, 36, 48, 60, ... and so on. In Scientific Standard Notation that would be C0, C1, C2, C3, C4, ...

Old style music theory has tended to ignore octave differences between pitches on the grounds that two pitches separated by an octave sound similar in many ways. However, it does matter which octave is involved, particularly when you listen to two or more pitches played together.

Here is a table showing two methods of naming pitch classes.

Naming intervals

Different gaps between pitches have names. The old fashioned names for intervals go back to a time before 12TET, when tunings were different. The intervals described as ‘perfect fourth’ and ‘perfect fifth’ are no longer perfect because the pitches of the notes have changed a little. It's just another of those historical oddities.

It is glaringly obvious that the simplest way to name intervals is by giving the number of semitones they contain. This is also the difference between the MIDI note numbers of the pitches. With this in mind, here are the old style names.

As with pitch names, the old style has some weaknesses:

1. The names are long.

2. There are alternative names for the same interval, though unfortunately you can't just choose your favourite and always use it. Other factors determine which name is correct.

3. As a result of those rules about which name is correct it is actually possible to have no name for an interval.

4. They don't add. For example, two seconds don't make a fourth.

This complexity has nasty consequences if you want to work out, as quickly and easily as possible, what interval lies between two notes. In the old style method, the alternative names for pitches and intervals combined with the rules for naming make the link from the note names to the interval horribly complex.

In contrast, MIDI note numbers translate to numerical intervals using simple arithmetic you already know, e.g. 78 up to 83 is an interval of 5. As an added refinement you can use negative numbers for falling intervals and positive numbers for rising intervals if you want to make a distinction.

The old style of pitch naming only has an advantage with octave intervals, but this is also shared by the Scientific Standard Notation and my numeric adaptation of it. The ability to find notes an octave higher or lower is perhaps most useful to composers, with players needing more fluency with smaller intervals.

If you play a musical instrument you may find it helps to learn to make jumps of any small number of semitones from any pitch, either up or down.

Describing melodies

Nobody knows how people memorize and describe melodies in their heads. The mystery is that we seem to recognize a melody even when it is played a little higher than usual or a little lower. As long as the pattern of intervals between successive notes stays the same we recognize it. How, and how could we write down a melody in a way that is independent of the precise pitch it starts from?

We can't rely on note names for this because they are tied to particular pitches. Here are three possibilities that I can suggest:

Interval sequence

The idea is to write down a sequence of intervals that describes the tune. For example, ‘1 0 2 −3’ means start on any note, then play the next note one semitone higher, the next at the same pitch, then go up two semitones, then down 3.

This is a logical approach but if you make a mistake it's hard to recover, and the shape of a melody can be hard to see. Furthermore, people seem to be able to simplify tunes quite easily by removing some notes, which is an argument against interval sequences.

Relative pitch numbers

This approach involves writing a sequence of pitch numbers that are relative to whatever starting pitch is used. For example, the interval sequence ‘1 0 2 −3’ becomes the relative pitch numbers ‘0 1 1 3 0’ meaning to start on any pitch, then play the next note a semitone higher, the next also a semitone higher than the starting note, the next three semitones higher than the starting note, and then play the starting note again.

(There is a recognized style for writing melodies that is similar to relative pitch numbers and it is explained briefly below.)

This is slightly more intuitive, but there seems to be more to it than that. For example, in a typical 12 bar Blues song there is a basic melody that is repeated a number of times, then repeated again but lifted exactly 5 semitones, then later is repeated again but lifted 7 semitones. This suggests a third scheme, which is the one I think is closest to the truth.

Relative pitch numbers with reference shifts

This is like the relative pitch numbers but allows the pitch that is taken as zero to be redefined. The redefinitions might use interval sequences or relative pitch numbers. For example, those Blues shifts might look like this:

Shifts using interval sequences: ‘0 1 1 3 0 (+5) 0 1 1 3 0 (−5) 0 1 1 3 0 (+7) 0 1 1 3 0 (−2) 0 1 1 3 0 (−5) 0 1 1 3 0’

Shifts using relative pitch numbers: ‘0 1 1 3 0 (5) 0 1 1 3 0 (0) 0 1 1 3 0 (7) 0 1 1 3 0 (5) 0 1 1 3 0 (0) 0 1 1 3 0’

The numbers in brackets are the reference shifts, represented in semitone counts of course. A lot of classical music contains repetition, or near repetition, that would be easier to think of using reference shifts.

Naming and describing chords

A chord occurs when two or more notes are played together, usually starting and ending at the same time. Chords are categorized by the pattern of intervals between the notes in them, sometimes known as chordal quality. A particular chord is identified by this pattern of intervals and by a particular pitch designated its ‘root’.

Similar to melodies, the notes that define a type of chord can be specified in terms of relative pitch numbers (relative to the root pitch) and interval stacks. With interval stacks the intervals take the notes from low to high. Here is a table that links the traditional names and symbols to their numerical equivalents. Work through the arithmetic of some examples and you'll soon get the hang of it.

Unfortunately, chord names used in music books today do not use a common standard of notation. The table below shows only one possible name for each. The ‘X’ is replaced by the name of the pitch class containing the note at the bottom of the chord, e.g. ‘C’, which in these examples is the root of each chord. For example, Csus4 is an Xsus4 pattern chord with C as the tonic (though which particular C this is the old style notation does not show).

If you focus on the relative pitch numbers or intervals stacks then you can see how some chords are similar to others. I have put them into order in the above table to bring this out.

Any chord can be varied by lifting the lowest note up by one octave. This is called an ‘inversion’. For example, the first inversion of a chord using relative pitch numbers 0 3 7 would be 3 7 12. The second inversion would be 7 12 15. To invert again would give the same chord but all one octave higher than it started, i.e. 12 15 19.

The above table of chords (and their inversions) covers nearly all the recognized types and in fact only a handful of these types account for most that you hear in music. And yet, many possible combinations of pitches are not included. The reason for this is that the others don't sound very good. They create various unwelcome auditory effects. Playing pitches that sound like music is like walking on a pavement and trying not to walk on the cracks.

For the same sort of reasons, it is quite common for one instrument (or hand) to play a chord while another instrument (or hand) plays a tune using the same notes shifted one or two octaves higher. Consequently, the ability to play tunes using pitch classes found in particular chords is useful.

Harmony, consonance, and dissonance

The full scientific explanation of why some pitches sound good together and others do not is still controversial. Most music theory tends to be on the basis that only pitch class matters, not the octaves involved, and it doesn't matter what instrument you are playing. In fact it is clear that both of these are important.

If you listen to two notes sounded together at exactly the same frequency it is virtually impossible to tell that they are two notes. However, if their frequency differs very slightly, by just a fraction of a vibration per second, they sound like one slowly changing note. As the difference in frequency is increased that movement seems to speed up.

What is happening is that the vibrations are interacting so that sometimes they tend to cancel out and sometimes they tend to combine in strength. The resulting variation in loudness is called beating and the number of beats per second is the same as the difference in vibrations per second between the two notes.

Beating at a rate of 2 to 60 per second is quite unpleasant to listen to (worst at around 35 beats per second) and is the main reason that two pitches a semitone apart sound nasty. However, this sound differs depending on how high the two pitches are. For example, between MIDI notes 12 and 13 (C0 and C♯0) the beating is at about one beat per second, but between MIDI notes 118 and 119 (A♯8 and B8) the beating is at 443 per second, which is too fast to sound like beating at all, and can sound like a third pitch roughly equivalent to MIDI note 69 (A4).

Musical notes have a definite pitch, which is based on the number of vibrations of the air that occur per second. However, this is a simplification. Most musical instruments actually produce a bundle of frequencies together, with most energy being in the pitch of the note (called the fundamental) but also energy in the frequency that is twice the basic pitch, some energy in the frequency three times the basic pitch, four times it, and so on. These other frequencies are called harmonic overtones.

The amount of energy in each multiple of the basic frequency is different between different instruments and can vary with the pitch played too. Some sounds have a lot of energy in the multiples while others do not.

Let's go back to those two imaginary notes played at slightly different frequencies and keep increasing the difference. After a while something really amazing happens, which is that the pitches seem to meld together again. A bit more adjustment of the frequency and that effect disappears. Further on and there's another melding, which also goes as the difference in frequencies is increased further.

In all there's a consistent pattern of consonance and dissonance as the frequency gap increases.

The points of ‘melding’ correspond to places where the ratios of the frequencies are simple. The points in descending order of consonance are: 1:1 and 2:1 (an octave),3:2, 5:3, 4:3, 6:5, and 5:4. This is probably because, with these ratios, the multiples are in agreement.

The octave differences have the most agreement, e.g. for 200 Hz and 100 Hz, 200 = 2 × 100, 2 × 200 = 4 × 100, 3 × 200 = 6 × 100, etc. These are strong because there tends to be more energy in the lower multiples. Differences of 3:2 are next, e.g. for 150 Hz and 100 Hz, 2 × 150 = 3 × 100, 4 × 150 = 6 × 100, 6 × 150 = 9 × 100, etc. Differences of 4:3 should be next (though they aren't), e.g. for 100 Hz and 75 Hz, 3 × 100 = 4 × 75, 6 × 100 = 8 × 75, 9 × 100 = 12 × 75, etc.

Certain musical intervals in 12TET are very close to these simple ratios, and in older tuning systems they were often exactly the same as some of these intervals, by design. For example, a 7 semitone interval used to be exactly the ratio 3:2 and could justifiably be called a ‘Perfect’ fifth. In 12TET today the ratio is different by 0.11%, but that is enough to make it a less than perfect interval, and at higher frequencies 0.11% is much more noticeable. The old ‘Perfect’ fourth is now a difference of 5 semitones, which is also 0.11% off perfect. A difference of four semitones is now a 0.79% away from the smoother sound of 5:4.

Unfortunately, tuning using exact ratios creates a musical system that is very hard to work with, so the current situation is the lesser of two evils.

The characteristic sounds of intervals (and chords) depend to some extent on the pitch at which they start. They also depend on the instruments used because some instruments have much more energy in the higher multiples than others and this makes the consonance/dissonance effects much stronger.

(In theory, two notes created using ‘sine waves’, which have no energy in multiples at all, should have no consonance or dissonance when we hear them together other than the beating between their basic frequencies. However, we can still hear the melds at higher differences, though they are not so obvious. The reason for this is unknown but I speculate that there may be distortions created by our ears that add the multiples but which our brains correct for. In other words, we can't hear the imperfections in our ears because our brains remove them, but the brain fails to remove the beating between multiples, which remains as the only evidence of our ears' limitations.)

Naming and describing scales and modes

A musical scale is not usually a smooth progression of pitches; it is really a tune. The most famous is the ‘Major’ scale. We're so accustomed to its familiar ‘Do Re Me...’ escalation of pitches that it is rather difficult to experience this as anything other than a smooth progression of notes but it is not a smooth progression and its distinctive pattern of intervals is how we recognize it.

(Strictly speaking, a musical scale is a tune with a starting pitch, its ‘tonic’. The Major scale, for example, is a type of scale, whereas C Major is a particular scale of the type Major.)

In relative pitch numbers the Major scale going upwards is:

0 2 4 5 7 9 11 12

In the traditional way it ends on the note one octave above the starting note, which could also be the start of another octave of the Major scale if you wanted to carry on.

The Major scale going downwards is just the reverse.

A scale can be started from any pitch. In interval sequence terms the Major scale is:

2 2 1 2 2 2 1

In relative pitch numbers with reference shifts this could be done in a variety of ways and some are more convenient than others with particular musical instruments. One way is:

0 2 4 5 (+7) 0 2 4 5

An alternative that works well with a guitar is:

0 2 4 (+5) 0 2 4 (+5) 1 2

Modes are ways to play the notes of a scale starting from different starting notes. They have traditional names that go back to ancient Greece.

The successive pitches of a scale are called the ‘degrees’ of the scale. The degrees of a Major scale have old names and Roman numbers.

Obviously it is easier to use the simple numbers, either the relative pitch numbers or Roman numerals, rather than the old style names. The names are long and not particularly memorable. It would be easier for most people if they were abandoned.

The Roman numerals (with sharp and flat signs if necessary) are sometimes used to write down melodies. The method is like relative pitch numbers so, for example, the melody ‘0 2 2 4 0’ would be ‘I II II III I’ and sequences of chords can also be written in a similar way e.g. ‘II7 IIIm I5’.

The Roman numerals are also used as a shorthand to identify certain chords of three pitches. These are the chords created by starting on a degree of the Major scale, adding a pitch two degrees up, and adding another pitch two degrees up from that. For example, the triad starting on II includes II, IV, and VI and is a Major chord (because of its pattern of intervals). In more modern numbers this is a chord using degrees 1, 3, and 5, and could also be described as a 0 3 4 chord starting on relative pitch number 2. Not all triads on the Major scale are Major chords. The fact that the same word is used with different meanings is rather confusing.

Here are some other well known scales.

The so called chromatic scale is all 12 pitches of an octave, plus the first of the next octave. In other words, it is just all the pitches. The reasons for the name are lost in the complicated and broken history of music and may be linked to a system for showing the duration of notes using their colour.

Another way of looking at scales is to see them as chords with extra notes added in between and to get up to the note an octave above the starting note. This helps to make sense of the names of the scales. In the following analysis the numbers represent gaps between successive pitches in scales or their abbreviated cousins, arpeggios (which are the notes of chords played one after the other, like on a harp). For example, the number 12 means playing a note followed by the note an octave higher. Sequences indented below a sequence are the same sequence but elaborated by dividing jumps into more than one smaller jump. Hyphens have been added to help you see which numbers are breakdowns of numbers at a higher level. Look at some examples to really understand how it works.

• 12   Octave jump
• 8-4
• 4-4 4
• 2-2 1-3 1-2-1   Hamonic Major scale
• 3-1 3-1 3-1   Augmented scale (jazz)
• 7-5
• 4-3 5   Major arpeggio (note the major chord 4 3)
• 2-2 3 2-3   Major Pentatonic scale
• 2 2 1-2 2 2-1   Major scale
• 2 2 1 2 2 1-1 1   Bebop Dominant scale (jazz)
• 2 2 1 2 1-1 2 1   Bebop Major scale (jazz)
• 3-4 5   Minor arpeggio (note the minor chord 3 4)
• 3 2-2 3-2   Minor Pentatonic scale
• 2-1 2 2 1-2 2   Natural Minor scale
• 3 2 1-1 3 2   Minor Blues scale
• 2-1 2-2 1-3-1   Harmonic Minor scale
• 2-1 2-2 2-2-1   Melodic Minor (up) scale
• 2-1 2-2 1-2-2   Melodic Minor (down) scale
• 2-1 1-3 2-3   Major Blues scale
• 2 1 1 1-2 2 1-2   Bebop Minor scale (jazz)
• 6-6
• 3-3 3-3   Diminished 7th arpeggio
• 1-1-1 1-1-1 1-1-1 1-1-1   Chromatic scale
• 5-7
• 2-3 2-3-2   Suspended Pentatonic scale (jazz)

Here's the same analysis but this time presented as relative note numbers. As you go down the roots of the tree extra notes are added, which are shown in bold to make them easier to pick out.

• 0 12   Octave jump
• 0 8 12
• 0 4 8 12
• 0 3 4 7 8 11 12   Augmented scale (jazz)
• 0 2 4 5 8 9 11 12   Harmonic Major scale
• 0 7 12
• 0 4 7 12   Major arpeggio (note the major chord 0 4 7)
• 0 2 4 7 9 12   Major Pentatonic scale
• 0 2 4 5 7 9 11 12   Major scale
• 0 2 4 5 7 9 10 11 12   Bebop Dominant scale (jazz)
• 0 2 4 5 7 8 9 11 12   Bebop Major scale (jazz)
• 0 3 7 12   Minor arpeggio (note the minor chord 0 3 7)
• 0 3 5 7 10 12   Minor Pentatonic scale
• 0 2 3 5 7 8 10 12   Natural Minor scale
• 0 3 5 6 7 10 12   Minor Blues scale
• 0 2 3 5 7 8 11 12   Harmonic Minor scale
• 0 2 3 5 7 9 11 12   Melodic Minor (up) scale
• 0 2 3 5 7 8 10 12   Melodic Minor (down) scale
• 0 2 3 4 7 9 12   Major Blues scale
• 0 2 3 4 5 7 9 10 12   Bebop Minor scale (jazz)
• 0 6 12
• 0 3 6 9 12   Diminished 7th arpeggio
• 0 1 2 3 4 5 6 7 8 9 10 11 12   Chromatic scale
• 0 5 12
• 0 2 5 7 10 12   Suspended Pentatonic scale (jazz)

Using scales to identify sets of pitch classes

A scale can be used to pick out certain pitch classes. For example, the Major scale starting with any member of the C pitch class selects C, D, E, F, G, A, and B. It leaves out others such as C♯ and B♭.

A tune that only uses pitches within the pitch classes of a scale is said to be written in the scale e.g. a piece ‘in A Major’ uses the pitches picked out by the Major scale starting on an A.

A lot of music conforms to this kind of pattern for historical reasons and because of the inconvenience of writing music that does not conform. Some composers deviate more than others, and the success of Cole Porter and David Bowie, to mention just two, suggests that a bit of deviation can work well.

Notes on staves, accidentals, and key signatures

As you know, there are 12 intervals in an octave, and 12 pitches per octave. Unfortunately, there are only 7 old style names for the notes (A to G) and the others are identified by adding sharp and flat symbols. The same issue arises with music written using the familiar horizontal lines i.e. staves.

The staves and gaps between them represent pitches, but there are only enough of them to show 7 pitches per octave (A to G) with no space for the others.

If you are writing down music in this style there are two techniques for indicating the other pitches. One is to write a ♯ or ♭ symbol just before the note. For example, if you want to show an A♯ then write the appropriate dot on an A line/gap and just before it write the ♯ symbol. The sharp and flat symbols written this way are called ‘accidentals’ though of course there is nothing accidental about them.

The other technique is used when you consistently want to use the ♯ or ♭ version of a pitch. Instead of writing the ♯ or ♭ symbol just before every relevant note you just write it once at the start of the line. This then stays in force throughout unless over-ridden by a symbol just before a note. In addition to the usual ♯ and ♭ symbols there is another called ‘natural’ (♮) that means to just play the pitch indicated by the position of the dot, with no adjustment. These over-ride symbols stay in force to the end of the bar, unless over-ridden by another one.

Learning to read sheet music fast enough to play from it takes hundreds of hours of practice and unfortunately getting used to having sharps or flats at the beginning of a line vastly increases the learning time needed. It is very hard to learn to see the same symbol on the page but interpret it as something different depending on a symbol that is outside the focus of your gaze. Add to that the extra complication of accidentals that are in force to the end of a bar and you can understand why so much practice is needed to master sight reading.

(Yes, even within the community that uses this style of notation there have been many, many attempts to introduce more user friendly notations, but the dominant style has an inertia that is hard to shift. People who have come to music via computers or guitar tend to rely on the much more logical piano roll style of a music sequencer or the wonderfully literal guitar tab style.)

Beginner players generally start with music that has no sharps or flats at the beginning of the line and gradually add more complicated situations, learning to cope with each one.

In theory this trick of putting sharps and flats at the start of a line could be applied to any pitch in either direction, but learning to read them quickly is so hard that it would be unbearable to allow any combination.

What is actually done is to allow only the specific bundles of sharps or flats that pick out the pitches of a Major scale. These are called key signatures.

Here is a table showing the combinations that are allowed and the scales they pick out.

In the above table the scales defined by each signature are identified by the pitch class of the first note, using old style letters and the more logical numbers. If you look carefully at the numbers you will see that there are patterns there that can make it easier to remember the link between the key signatures and scales.

To get to the scales from the key signature, start by counting the sharps or flats. If you have flats then take that number away from 12. If the number you have now is even then this is the number of the starting note of the major scale. For example, with four flats you get 12 − 4 = 8, which is even so the major scale starts on 8 (Ab). If the number is odd then add 6 and if that comes to 12 or more take 12 away to reach the starting note for the major scale. For example, three sharps is odd, so 3 + 6 = 9 (A).

To find the starting note of the associated natural minor scale just add 9 then take away 12 if needed. For example, four sharps means the major scale starts on pitch 4 (E) and the natural minor scale starts on pitch 4 + 9 = 13; that's more than 11 so take away 12 to get 1 (C♯).

To go the other way, working from the starting pitch class of the major scale to the key signature, think like this: if the starting pitch is an even number then this is the number of sharps to use, but if it's more than 6 it is usually converted into flats by taking the number away from 12. For example, starting on 8 (G♯) would give 8 sharps, but this is usually taken as 4 flats. If the starting pitch is an odd number then add 6 and take away 12 if necessary. This again gives the number of sharps, but you might need to convert it to flats by subtracting from 12. For example, starting on 9 (A), add 6 giving 15, so take away 12 giving 3, and that's the number of sharps needed.

The pitches denoted by the notes written on staves depend on what instrument the music is written for. Some instruments, like saxophones, are called ‘transposed’ instruments because they produce a note that is consistently different from what appears on the printed page. The universal standard is ‘concert pitch’, which is with no transposition and is used by instruments such as the piano.

In the following table the notes associated with each line or gap in the treble and bass staves are given for some common instruments.

Written in the key of ...

Music theorists often write about music being written in keys. For example, if a piece is written ‘in the key of C Major’ then that means that there were no sharps or flats at the beginning of the line and most if not all of the pitches used in the piece were written without accidentals. In other words, the composer stuck to the pitch classes identified by the C Major scale.

The theorists also talk about key changes and about the imagined qualities of different keys. I think this needs to be regarded with some skepticism. Consider these points:

• Most people can identify the familiar tune of a Major scale when it is played, but they cannot tell which pitch it started from. Therefore, music played ‘in the key of C Major’ will sound pretty much the same as music played ‘in the key of D Major’, for example.

• The pitch classes identified by a Major scale are the same as those of one of the Natural Minor scales. (The Natural Minor scale is just a Major scale started from a different place.) Therefore there should be no perceptible difference between Major and Minor keys unless there is some convention of always starting on a particular degree of the scale.

• Not all music uses enough of the pitches to uniquely identify the scale it fits into. In the simplest case, a tune that uses just one pitch could be from several scales. Even with more notes there is no guarantee of getting enough information to uniquely identify the scale.

However, there are many pieces of music where a key change seems to be obvious at once even to the untrained ear. These are where the tune is transposed (i.e. lifted or dropped by a fixed number of semitones). Most people recognize immediately when the shift has occurred. The transposition is what people recognize and the fact that it also corresponds to a key change is irrelevant.

Timing of notes

Having spent a long time on pitch it is time to deal with the problems of naming and writing rhythms. The system that has arisen over the centuries and is now regarded as standard for a lot of Western music takes some getting used to because pieces of music using different notation for rhythm can sound the same, some notation is ambiguous, what you see on the page is not always exactly what has to be played, and some of the arithmetic appears confusing.

There are really two main elements to the system. One concerns the pattern of time differences between notes actually played. The other is not directly audible but provides a mental guide to players and composers.

The note duration system

We recognize the rhythm of a piece of music even if it is played faster or more slowly than we have heard it before. This shows that it is the relative lengths of notes that is most important to our understanding of music.

Accordingly, the lengths of musical notes do not have a standardized absolute duration, but they have a duration relative to each other. The standard symbols, names, and durations relative to the longest note in the system are shown in this table.

Relative duration	Number name	Old style name	Individual note symbol	Grouped symbol
1	Whole	Semibreve		None
1/2	Half	Minim		None
1/4	Quarter	Crotchet		None
1/8	Eighth	Quaver
1/16	Sixteenth	Semiquaver
1/3	Third	-
1/6	Sixth	-

The notation has two other tricks that together allow a huge variety of durations to be written: (1) A dot just after a note increases its duration by 50%. (2) Two or more successive notes at the same pitch joined by a curved line (called a tie) are to be played as one note whose duration is the sum of the individual notes.

At the start of a piece of music written in the old fashioned style is a symbol consisting of two numbers written one on top of the other, or a capital C or a capital C with a vertical line through it. This is called the ‘time signature’.

Music is divided by vertical bars that appear at regular intervals. One function of the time signature is to specify the number of relative duration units to put into each bar.

For example, the notation 4/4 means to put 4 quarter units in each bar. The notation 6/8 means to put 6 eighth notes per bar.

I have seen some explanations of this state that the relative durations are relative to the length of a bar. For example, a crotchet is said to be a quarter of a bar long. This is not correct because, for example, 3/4 means a bar with three ‘quarters’ in it. If those were three quarters of a bar it would not make arithmetic sense.

The mental guide system

The note duration system defines the timing of the notes you hear but behind this is a mental guide system that influences composers and players.

Imagine a single drum beat, sounding regularly without any variation in speed or loudness. The sound would be ‘bang bang bang bang bang’ and so on. Some Western music has an underlying rhythm like this but more often we stress notes periodically in some way. For example, it might sound more like ‘bang!! bang bang bang!! bang bang bang!! bang bang’, with the stress on every third beat.

This gets taken further with subsidiary stresses. For example ‘bang!! bang bang! bang bang!! bang bang! bang’ has a primary stress every four beats, with a secondary stress beat in between.

The time signature is used for a second purpose, which is to indicate the stresses. Unfortunately, some time signatures could be interpreted in more than one way, giving more than one pattern of stresses. The table below gives the interpretation that is most common

Stress can only be applied if there are notes beginning at the required time. They can then be stressed by making them louder, or having more notes playing together, or by starting important sections of melody on those notes.

On computer music software it is often possible to specify exactly what stresses you want the computer to apply when it plays back music from your notation.

This is not the whole story. The speed of a piece is set by a message above the first line of a piece that shows a picture of a note next to an ‘=’ sign and a number. This means that there are to be that many notes of that type per minute.

Furthermore, special words appearing at the top of the piece also change how the mental rhythm and stress guide is to be interpreted. For example, the word ‘shuffle’ means to give it a blues feel by systematically lengthening and shortening certain beats in each bar.

How the human brain processes all this and decides when to start and finish notes is unknown. However, it seems that:

• We do not rely on the absolute timing of notes. If a piece is played a bit slower or faster than we have heard it before we still think of it as the same piece.

• We are more interested in the start of notes than their ending. If you hear a rhythm played with the notes all shortened, or some shortened, then it is still recognizable as ‘the same’ but if the starting times of notes are varied by the same amount then the piece is less recognizable.

Treating the start and end of notes as events that are in some ways separate may seem an odd concept, or it may make a lot of sense to you. In the MIDI system for electronic music the start and end of the same note are treated as separate events.

Finally

If you have any comments on this article, questions, or ideas for new topics, please let me know.

About the author: Matthew Leitch has been studying the applied psychology of learning and memory since about 1979 and holds a BSc in psychology from University College London.

Words © 2009 Matthew Leitch