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Basic music theory for schools, music teaching and amateur musicians.
Music theory in a new way
Our aim
The goal of this site is to present music theory in a new way, in order to improve its general understanding at a basic level.
With a few logical tweaks and some new illustrative ideas, we seek to contribute to a pedagogical debate about music education.
Note that our presentation of music theory is not bound to understanding the tangents of the piano.
All materials are developed in dialogue with our pupils (age six upwards) from Danish comprehensive schools and music schools, with amateur musicians and with professional colleagues.

The premise of music education in Denmark combines recognition of social and cultural aspects with a musical learning spiral: Practice/Performance - Appreciation/Comprehension - Creativity. This spiralling and inspiring definition suggests an openess toward definition and redefinition on the basis of experience and new thinking.
None-the-less, there are aspects of music theory as it traditionally is presented that rely more on rote learning than on premises that can withstand logical argumentation. At times we experience angry reactions to the thoughts presented here. Such reactions imply emotion rather than pedagogical reasoning.
Music educators are not doomed to repeat historical praxis. Indeed a more logical presentation of music theory could invite cross-subject educational work. Our interdisciplinary work with music and mathematics has been dubbed musematics by our pupils.

The theory on this site is summarized in our Music Table which we have designed as a practical rebus. The face of the Scale Clock turns, so any key can be decifered. Click on the icons to see the pictures in large format. Examples and exercises are adapted for use with Smart Board/interactive classroom boards.
The witch, Old Toza, has been with us since the beginning of the project.

Enjoy - and feel welcome to contact us with thoughts, questions and constructive critique.
Sonya Christie and Jens Kragh
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Table of contents

Music is the organisation of sound
Pulse – bars – time signatures (time fractions)
The values of notes and pauses in time – the Time Wheel
Pitch – the Herz Column – Stemnotes in the stave/staff system
Naming notes in the musical Clockface
The Scale Clock
Intervals – the distance in pitch between notes
Keys, chords and triad families – the House of Fifths
Musical form
Transposition
A method for reading and playing musical notation

© All rights to the materials on this homepage belong to:Sonya Christie and Jens Kragh - cleek.dk.

Music is the organisation of sound

The intuitive musician
Any child is a musician long before it can walk or talk. Watch them react to the rhythm of music. Listen to their musical ramblings describing their movements in fragments of known song. Hear their experiments with sound production. The young child - before ever receiving formal musical education - subconsciously whirls around in the learning spiral of Practice/Performance - Appreciation/ Comprehension - Creativity. The role of parents and teachers is to nurture this musical spiral, acknowledging and building upon this holistic feel for music. That whole includes tools for analysing, understanding and recombining - and a language with which to communicate with other players. In other words: music theory.

Organization of sound - the musical coordinate system
Music can be defined as the organization of sound. Music as an organizing principal predates formal, written notation. The parameters of music are pitch, time and timbre, no matter the genre. Whether written as orchestral scores, guitar tabs on the internet or through drawings, it is custom to devote the vertical axis to pitch and the horizontal axis to time, read from left to right. Timbre is understood through tradition, instrumentation, dynamics and other musical messages. Being able to read, understand and use musical notation supports musicianship. Reading music is not so different from reading graphs in other situations.
As a musician it is important to recognize which parameter to prioritize. Alone or with an instrument teacher, focus can be on playing the correct notes with proper intonation and nuance. But socially, the horizontal time factor is vital - learn to keep up!

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Pulse – bars – time signatures (time fractions)

Pulse is the heartbeat of music. As with the human heart, musical pulse can be fast or slow. It can be stable, it can quickly accelerate, it can gradually fall to a resting point.

The speed or tempo of a musical piece can be indicated lingually or mathematically. Words in any language can refer to moods, movement, dances. In classical music, standardized Italian phrases are often used. While words are suggestive, tempo can also be expressed with mathematical precision, shown by a note and a number. which signals the type of beat to be counted and the number of beats per minute. Hear different tempo values on the metronome.

As part of the musical organization of sound, certain beats are emphasied more than others. This can be done subtely or pounded out depending on style and genre. Time signatures are a way of describing this organization into regular bars or measures. In a written score, these bars are marked by barlines, vertical lines through the staves. Special barlines indicate the end of a piece of music or that a section should be repeated. Bars are numbered to help orientation.

Time signatures are shown as fractions, where the numerator is the number of beats in the bar and the denominator shows the note value to be counted. There are many different time signatures. Simple time signatures: 2/4, 3/4, 2/2. The first beat in the bar - the downbeat - is the strongest. The underlying subdivision in 3/4 is: one-and two-and three-and. With a compound time signature, the underlying subdivision is a triplet. 6/8 is performed as two triplets in the bar: one-and-a two-and-a. The calypso has a more complex rhythm, its eight quavers played in two long beats and one short: one-and-a two-and-a three-and. There are many more. Create some for yourself.

  
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The values of notes and pauses in time – the Time Wheel

Note values are expressed as fractions of a whole and are therefore often explained in terms of a circle - cutting a cake into halves and quarters.
We have put that circle into a wheel, since an important dimension of music is time. The advantage of the allusion to the wheel is that speed or tempo can be expressed - note values can be shown to have relative lengths. The Time Wheel describes the length of notes as fractions of a whole and as part of a rotation.
In our classroom teaching we have a wheel on a pole. To the wheel face we can attatch note and rest values with velcro. Pupils can clap or play in time by following the wheel. A whole note/semibreve played in a fast tempo will sound for a shorter time than a whole note/semibreve played in a slow tempo. Our animation skills are not advanced enough to steer the speed of the wheel online.

At the bottom of the diagram the whole note/semibreve, its equivalent pause and its fraction are shown marked inside the wheel. The wheel rotates from left to right along the dashed line. One whole rotation is equal to one whole note or its equivalent pause, which in practise depends on the speed/tempo of the wheel.
The next wheel face is evolved by dividing the wheel in half, showing a half note/minim and its rest. Continue to divide the wheel, and create a quarter note/crotchet, an eighth note/quaver and a sixteenth note/semiquaver etc. These notes, their rests and their fractions are shown in the various wheel faces. The length the wheel rotates along the dashed line shows the relative value of the note.

A dot • after a note or pause extends the functional timevalue by half its original length.
A tie between two notes binds them together, so that they sound as one note.
A rhythmic passage is created by combining the various note lengths within a given time signature.

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Pitch – the Herz Column – Stemnotes in the stave/staff system

Pitch and the Hertz Column
Sound is our experience of a band of wavelengths, vibrations from a source: an instrument, a voice, a machine, the wind etc. Sound waves can be long and create deep tones, or short and create high tones. Wavelengths can be too long or too short to be registered by the human ear. We cannot hear the high pitch of a dog-whistle. In old-age we cannot hear the same range or ambitus of waves as in childhood.

Sound waves are measured in Hertz, oscillations per second. Any specific pitch refers to a specific Hertz value. What we hear as a higher octave is an exact doubling of the oscillations per second. The Hertz value of a lower octave is the exact half of the original. Musical intervals can be expressed as mathematical relations between wavelengths. The vertical axis in musical notation is a graphical representation of these relationships.

In referring to deep/low and high notes on a musical instrument, there is an assumption of explanation.
Yet I lift my right elbow up to reach the deep G-string of my violin. My left arm moves towards my face to achieve a higher pitch on any string. On the piano deeper notes lie on the left, higher notes on the right. There is no immediate (unmediated) logik connecting pitch on an instrument with height on a graph. As a pedogogical tool we have created the colour-coded Hertz Column, with reference to seven physical entities from the centre of the planet upwards: lava, underground, earth, plants, sky, sun, universe. See the colour coded instrument tables on this site. Coloured A stickers mark our classroom keyboards.

In the Hertz Column, concert pitch - a standard to which we tune our instruments - is shown as the blue A. It swings at 440Hz. The Hertz-column shows notated A in 6 octaves from 55 Hz in the "Underground" octave (deep, black notes) to 1760 Hz in the "Universe" octave (high, pink notes). We have chosen to group octaves ABCDEFG, rather than the more conventional - but less logical - CDEFGAB. The vertical symmetry of C's is unaffected.

Stemnotes in the stave/staff system
Seven stemnotes subdivide each octave, A B C D E F G (la ti do re mi fa sol).
The stemnotes of the musical alphabet are placed alternately on and between horizontal lines called a stave or staff. When all seven been placed, the lettering series begins again. Both letters and lines are a continuum. To give the system an absolute pitch, we lock groups of five lines with a clef.
In the bass or F clef, the note F is locked to the fourth line, numbered from below. The treble or G clef locks the note G to the second line. Other clefs exist, for example C clefs. There are octavated variations of the bass and treble clefs. The stave system can be extended with ledger lines above and below the system.

We choose to present a stave system with bass and treble clefs joined by a ledger line marking middle C (in the Hertz-column shown in green), rather than starting with the variation specific to piano notation.

Learn the names of the notes in the stave system

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Naming notes in the musical Clockface

Music is defined both by physical laws and by cultural convention. We perceive the physical fact – the exact doubling of frequency. Calling this "same note, but higher" experience an octave is a convention that still gives meaning in western music. This octave is subdivided into not eight but twelve semitones or half steps. We place these twelve semitones in a clock-face to show the continuity of note naming. By association with an ordinary clock it should be clear that pitch does not jump from semitone to semitone, but is a continuum. These in-between notes may sound out of tune or refreshingly "blue", depending on the context. Our Scale Clock presents a logical framework for discussing intonation ‐ the tangents of the piano do not.

The Scale Clock is NOT a circle of fifths. It is a new tool in understanding note naming, scales, intervals and transposition ‐ independent of tangents.
While western convention refers to a subdivision of twelve degrees, equivalent to the tempered piano, the Scale Clock offers a framework translatable to the sixty minutes of an hour or the 360 degrees of geometry. Untempered intonation and the quarter-intervals of world music can be expressed.
Our House of Fifths shows harmonic relationships instead as a table, not to deny the circular relationship, but we find the possibilities of the Scale Clock more important. We can introduce the clockface to young children years before a circle of fifths is relevant.

Basic level: Draw the Scale Clockface (I have hundreds of drawings of the Scale Clockface, full of fansasy.)
Draw a circle. Mark the 4 points: N S E W.
Draw a line towards North. Draw a head.
Cover South with a finger, draw a leg to either side.
Cover one degree to either side of the head with a finger, draw arms.
Create a figure out of the five‐point matchstick man ‐ Toza, the witch ‐ Batman, a princess, a self-portrait.
Avoiding the figure, start at one o'clock and insert the stemnote letters A B C D E F G.
The figure positions the five variations ‐ the piano's black notes.

We name seven stemnotes: A, B, C, D, E, F, G. We find neither logical nor pedagogical reasons for maintaining a historic, germanic tradition of H's.
The remaining five steps occur using the variations sharps and flats. Together the stemnotes and variations form the chromatic twelve-note scale.
A sharp    raises a note by a semitone/ half step, a flat    lowers by a semitone. Both signs can be annulled by a natural    and the altered note returns to its basic form. (Raising or lowering the remaining stemnotes gives enharmonic equivalents.) Sharps or flats used throughout a piece are usually grouped and written on the left of each stave system, after the clef. This is the key signature. They can also appear as accidentals in the course of written piece of music, to embellish the tune or to achieve a particular harmony. In this case they remain in force until the next barline, unless neutralized   . Double sharps and flats can occur.

When a stemnote has been altered, name the stemnote first, thereafter the type of accidental:
F = F sharp / E = E flat / B  = B natural

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The Scale Clock

We put the Clockface with stemnotes and variations into the Scale Clock. There is a semitone/ a half-step between each of the twelve "hours" in the clockface. Two semitones give a whole tone/ step. Scales are particular, regular combinations of whole tones and semitones. Knowing the correct pattern of steps, we can display any scale-type, chord or harmonic function in any key.
Major and minor scales require all seven stemnotes, some altered by sharps or flats. By following the marked rings, scales can be built up from first principals.

Using the Scale Clock
Click on a keynote in major (green) or minor (red) in the table beside the clock. (Turn the clockface disc if you have the physical Scale Clock.)
For example, Place C at the yellow one o'clock. The seven stemnotes form a major scale from C to C. The 7 scale degrees are numbered in the green ring. The notes 1, 3 and 5 of the C major triad are underlined. Chord functions are written beside the numbered steps. Intervals up to an octave are shown.

The relative minor of C major is A minor, the sixth degree of the major scale. It is marked by a red solfa "la".
Click on A minor. The seven stemnotes from A to A form a pure minor scale, the degrees 1 ‐ 7 are marked in the red ring. The leading-note at the 7th step is marked, raised in the harmonic minor. Use major steps 6 and 7 in the upward melodic minor.
The green ring shows the scabelon for major scales. In A major, C is raised, F is raised and G is raised.

The keynotes of all seven modal scales are found in the degrees of the major scale:
1. Ionian (major) – 2. Dorian – 3. Phrygian – 4. Lydian – 5. Mixolydian – 6. Aeolian (minor) 7. Locrian.
Five-note (pentatone) scales appear: in the major ring at steps 1, 2, 3, 5 and 6. – in the minor ring at steps 1, 3, 4, 5 and 7.
Major and minor triads can be read from steps 1, 3 and 5 in the appropriate rings.

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Intervals – the distance in pitch between notes

An interval is the distance in pitch between two notes. These notes can follow each other in a melody or occur simulteneously in a harmony. When we recognize intervals aurally, we are listening to and identifying a physical relationship between frequencies.
This is a musical skill worldwide.

Intervals and chords are a substantial part of the language of Western music theory. Here we move from physics to conventions.
Conventions change over time and are dependant on culture, genre and geography.
We hear the exact doubling of frequency as ‘the same note’ an octave higher. The word ‘octave’ refers to a particular tradition of musical scales, where this interval is divided into eight steps, including the first and last.

Western music is to a great extent built on patterns of intervals.
Understanding how intervals are interwoven into melody and harmony is important in both music theory and music history.

The basic intervals are: 1. prime, 2. second, 3. third, 4. fourth, 5, fifth, 6. sixth, 7. seventh, 8. octave.
As a basic rule primes, fourths, fifths, and octaves are perfect and can be altered to augmented or diminished.
Seconds, thirds, sixths and sevenths are most often found as: major or minor.
Three or more notes with a variety of intervals sounding at one time form a chord.

See basic intervals and scales with a "Cleek" on these icons.
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Keys, chords and triad families – the House of Fifths

Key
The key of a piece of music is determined by the keynote or tonic, around which the melody revolves, and the type of scale on which it is built.
A tune will commonly end on its keynote. Count how many sharps or flats appear at the left of the stave.
Keys in major and minor can be read in the House of Fives based on the keynote and the number of sharps or flats.
N.B.: Accidentals can appear throughout a piece, in a minor key often indicating the leading note a semitone below the tonic. Accidentals can also indicate a modal key or a temporary modulation to a new key. Notice any regular pattern of accidentals.

Triads and chord families
Music across the globe begins in melody and rhythm. Dance and ceremony influence pulse and rhythm. The peculiarities of different languages and instruments influence melodic patterns. This is still clearly seen in folk music. Another parameter that has a huge influence on music inspired in western culture is harmony. Dominating compositional music between the 17th and 19th centuries, these harmonies still form the backbone of popular music today.

Build thirds on a major or minor scale. Using the given scale, add the third and fifth notes above each degree or root.
Triads are often signified by roman numerals, reflecting their degree on the scale. Together these triads form a complementary family of chords.
Any melody note based on the scale will appear in at least one triad. Triads on a scale
Each triad has three positions:



Root position
The root note is at the bottom.
first inversion
The third is at the bottom.
second inversion
The fifth is at the bottom.

Primary triads
The primary triads in a chord family have particular importance:
( I = Tonic "the home triad") ( IV = Subdominant "the away triad") and ( V = Dominant "the homecoming triad"). See the tonal cadence

Parallel chords
In Scandinavia, triads are named according to harmonic function: three primary triads and three parallel triads of opposite gender (major/ minor).
The root of a parallel triad is three semitones deeper in major and three semitones higher in minor.
The English names submediant, supertonic and mediant slur this symmetry.
Major: (VI =Tonika parallel/ submediant) (II = Subdominant parallel/ supertonic) (III = Dominant parallel/ mediant).
Minor : ( III = Tonika parallel/ mediant) ( VI = Subdominant parallel/ submediant) ( VII = Dominant parallel/ leading note) .

In the House of Fifths you can easily read the basic triads of any key. Transfer the skema below to the House of Fifths. Place the tonic over the major / minor column of a key and read the basic chord family above, under and to the side.

The triad family of C major from the House of Fifths
V (D) G major III (Dp) E minor
I (T) C major VI (Tp) A minor
IV (S) F major II (Sp) D minor
The triad family of A minor from the House of Fifths
VII (Dp) G major V (D) E major*
III (Tp) C major I (T) A minor
VI (Sp) F major IV (S) D minor






*In the minor chord family, the dominant triad must be a major chord for reasons of harmony - the raised note is the leading note pointing to the tonic.
The minor chord can be used without the harmonic function of the leading note.

The tonal cadence in C major and C minor.


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Musical form

As music is the organisation of sound, it has a form. Musical form is a description of the sequence of musical ideas combining to create a whole. Being able to describe the form helps orientation within the piece, be it by the composer or others using the music: musicians, dansers, dramatists, DJ's.

In musical notation, form can be signaled by naming sections with letters. For example, music with the form A B C D is a combination of four different musical ideas played in sequence. A common form in songs is verse 1, chorus, verse 2, chorus or A B A B - also common in folk tunes. The musical form rondo, which alternately combines a known musical section with a series of new ideas can be written A B A C A D A E...
A musical idea can be repeated with a slightly different ending or some other slight variation. This can be expressed as A A'.
A pop song with two verses, a chorus, a verse, a chorus, something new, then a double chorus can be expressed as A A B A B C B B - probably starting with an introduction, building up to the choruses with a bridge and ending with a coda.


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Transposition

Transposition means "move to a new position". Musically it refers to changing the pitch of a tune, while maintaining the other parameters. The keynote is higher or lower than the original, but otherwise it is "the same tune". In order to maintain the harmonic structure, the keysignature of the new keynote must be observed.

Use the House of Fifths and the Scale clock as aids.

Certain instruments are built in such a way that when playing notated concert A, a quite different pitch is heard. These so-called transposing instruments include many wind instruments. The naming of the transposing function of these instrumnts refers to the actual pitch when an instrument plays notated C. The pitch heard when a B   trumpet or clarinet plays a given note sounds a whole tone - two semitones - lower than notated. Notated C will sound as B   . An A-clarinet sounds three semitones lower than the notated pitch. Notated C will sound as A.

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A method for reading and playing musical notation

Reading and understanding musical notation requires some fundamental knowledge. Get started by studying the four subjects below:

  1. The time lengths of notes and pauses
  2. Time signatures and the musical pulse
  3. Stem notes and their placement in the stave system
  4. Keys and keynotes
When you are at home with these four subjects, practise reading any songbook written in musical notation. Read through tunes for about quarter of an hour daily. Begin reading each tune by doing the following:
  • Find out in which time signature the tune is written. Notice tempo directions or other rhythmic instructions.
  • Read the tune as a rhythmic sequence. Learn to see bars as complete pictures, notice repetative sequences and variations. In other words: learn to see rhythmic patterns as whole pictures.
  • Teach yourself a "rhythm language" by vocalising the rhythm of the tune using a variety of sounds e.g. (whole- and half notes: ta - aa) - (quarter notes: ta) - (eighth notes: ti - ki) - (sixteenth notes: ta - ke - ti - ki). At the same time, tap the pulse with a finger or toe. Notice the relationship between pulse and the tune's rhythm. Notice also the tune's relationship to the strong beats of the bar.
  • Find the key of the tune using the tune's keynote and its key signature. See the House of Fifths.
  • Read through the tune again, this time naming the notes by lettername or by "solfa" – do re mi fa so la ti do' – from the given keynote.
    As with rhythms, practise seeing sequences and variations.
  • After you have learned these elements, the next step is to transfer the musical code to you instrument:
    Play the rhythm on the tune's keynote. Next, follow the harmonic structure. Try this in real time.
    Play the melody, slowly and carefully at first. Practise so the tune is in real time.
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