Outline of basic music theoryFrom www.oscarvandillen.com

Introduction to music theory: an outline of basic music theory.

This outline offers a concise and complete overview of basic music theory.

Oscar van Dillen 2011

Contents1 Introduction2 Sound and hearing

2.1 Sound2.2 Hearing2.3 Physics of sound

3 Musical notation3.1 Notation of time and rhythm

3.1.1 Symbols for notes and rests3.1.2 Meter3.1.3 Time signature and bar3.1.4 Binary time signatures3.1.5 Dots3.1.6 Ternary time signatures3.1.7 Ties3.1.8 Tuplets3.1.9 Basic tempo indication3.1.10 Correct rhythmical notation3.1.11 Beaming3.1.12 Odd time signatures3.1.13 Complex time signatures3.1.14 Notation of swing rhythm

3.2 Notation of pitch3.2.1 Staff and clef

3.2.1.1 Staff lines and ledger lines3.2.1.2 Commonly used clefs3.2.1.3 Further study

3.2.2 Directions of notestems3.2.3 Basic tones

3.2.3.1 Definition of basic tones3.2.3.2 Systems for basic tones

3.2.3.2.1 Alfabetic nomenclature3.2.3.2.2 German nomenclature3.2.3.2.3 Latin nomenclature3.2.3.2.4 Relative Latin nomenclature3.2.3.2.5 Indian nomenclature3.2.3.2.6 Indian nomenclature transcribed

3.2.3.3 Further study

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3.2.4 Alteration3.2.4.1 Definition of alteration3.2.4.2 Symbols for alteration3.2.4.3 Validity of alteration symbols3.2.4.4 Microtone alteration3.2.4.5 Further study

3.3 Notation of intervals, chords and harmony3.4 Notation in scores3.5 Compact polyphonic notation3.6 Other notation symbols

3.6.1 Tempo indications3.6.1.1 Classical tempo notation3.6.1.2 Metronome numbers3.6.1.3 Counting value changes3.6.1.4 Jazz tempo notation

3.6.2 Dynamic indications3.6.3 Articulation

4 Basic building blocks of melody and harmony4.1 Scales

4.1.1 Definition of scale4.1.2 Different scales4.1.3 Basic major and minor scales

4.1.3.1 Major scale4.1.3.2 Natural minor scale4.1.3.3 Harmonic minor scale4.1.3.4 Melodic minor scale

4.1.4 Definition of mode4.1.5 Church modes

4.1.5.1 History and use4.1.5.2 Basic notation4.1.5.3 Notation on c

4.1.6 Different modes4.1.7 Further study

4.2 Intervals4.2.1 Definition of interval4.2.2 One sound4.2.3 Basic intervals4.2.4 Wide intervals4.2.5 Perception of the interval4.2.6 Basic intervals from c4.2.7 Full names of the intervals4.2.8 Intervals in order of chromatic size4.2.9 Enharmonic equivalence4.2.10 Inversions of the intervals4.2.11 Further study

4.3 Triads4.3.1 Definition of triad4.3.2 Basic notation4.3.3 Types of triads4.3.4 Symbols for triads4.3.5 Inversions of triads4.3.6 Other positions of triads

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4.3.7 Use of triads4.3.8 Further study

4.4 Seventh chords4.4.1 Definition of seventh chord4.4.2 Basic notation4.4.3 Types of seventh chords4.4.4 Symbols for seventh chords4.4.5 Inversions of seventh chords4.4.6 Use of seventh chords4.4.7 Further study

4.5 Consonance and dissonance4.5.1 Definitions of consonance and dissonance4.5.2 Traditional classification of intervals in consonant and dissonant4.5.3 Acoustic order of consonance and dissonance in intervals4.5.4 Consonance and dissonance in scales4.5.5 Traditional classification of chords in consonant and dissonant4.5.6 Use of consonance and dissonance

4.6 Circle of fifths4.6.1 Major scales in order of accidentals4.6.2 Minor scales in order of accidentals4.6.3 Key signatures in order of accidentals4.6.4 Geometry of the full circle of fifths

4.7 Transposition4.7.1 Definition of transposition4.7.2 Use of transposition4.7.3 Example of transposition

4.7.3.1 In C4.7.3.2 Soprano saxophone in B4.7.3.3 Alto saxophone in E4.7.3.4 Tenor Saxophone in B4.7.3.5 Barytone saxophone in E

5 Rhythm5.1 Definition of rhythm5.2 Time, beat, subdivision and feel5.3 Polyrhythm

5.3.1 3 against 25.3.2 2 against 35.3.3 4 against 35.3.4 3 against 4

6 Melody6.1 Definition of melody6.2 Exercise in melodic building blocks

6.2.1 Printable version of melodic building blocks7 Harmony

7.1 Definition of harmony7.2 Functional harmony

7.2.1 Definition of degree7.2.2 Degree as chord7.2.3 Degree as scale7.2.4 Basic degrees as triads

7.2.4.1 Basic degrees in major7.2.4.2 Basic degrees in minor

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7.2.5 Basic degrees as seventh chords7.2.5.1 Seventh chord degrees in major7.2.5.2 Seventh chord degrees in minor

7.3 Further study8 Form9 Footnotes10 Questions11 See also12 Book: "Outline of Music Theory" in preparation13 External links

IntroductionMusicians have always studied music in two ways: both practically and theoretically, and both fields of study stronglydeveloped in the course of history, and are in fact still developing. Though the two are fundamentally different, bothshare a common fate, in that each can only be fully learned from an accomplished master. Neither can be learned frombooks alone, as they both involve practical skills.

The purpose of this text is to provide a concise outline and introduction to basic music theory. I sometimes also speakof common theory as it comprises basic knowledge, tools, methods and models, shared in common by manymusical styles and traditions, which are commonly used in musicology as well. The study of musicology is verydifferent from that of music theory, as this more scientific approach to music leads to knowledge rather than to skills.The masters one needs to study with musicology could as well be found in books, and such is never sufficient formusic and music theory.

The more one's musical skills grow, the more rewarding it is to develop one's musical theoretical skills as well.Ideally, music theory connects four fields of study, as represented in the diagram below:

Music theory interconnects these four fields of study: hearing, singing, reading and writing, in the student's awarenessand brain, both as knowledge and as skill. It is not enough to know how something is or can be done (can be heard,read, sung or written), the practical skills to actually do so oneself should be considered to be included in musictheoretical education and learning.

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With the emergence of ever more complicated models (which are generally presented in either visual or numericalform) for analyzing or even constructing or composing music, it suddenly becomes important to remind oneselfalways that music is made for the ears. When in doubt, the ears will convey the truth or falseness of any proposedsystem of evaluation. Though this has so far been a more or less subjective criteria, basically shared by all knowingears, it probably won't be long before neurophysiology, with contemporary measurement instruments such as theMRI scanner, will scientifically uncover the inner workings of the musical brain. With tools such as these, musictheory can develop into a phenomenological interdisciplinary field of study between art and science.

Sound and hearingSoundSound is the perceived complex phenomenon of vibrations traveling as pressure waves through a medium. For allliving beings on earth this medium is either air or water. Such vibrations are called sound when they are perceived by(our) hearing.

There are thus three components in sound: vibration, medium, hearing. Generally, silence is considered to be theopposite of sound.

A musical sound can be recognized and defined by its pitch (or frequency), duration (or relative rhythmic value),timbre (color or instrument), dynamics (loudness) and position in space (relative to the listener).[1] All music consistsof related sounds and silences, which can be symbolized by notation.

HearingSound is perceived by our hearing, which is a result of our ears picking up vibrations in the air and translating these toour brain, where the perception of hearing comes to awareness.

The ear translates vibrations in the ear into neural signals. Air-vibrations are received by the ear drum (tympanicmembrane) and transmitted by means of the three extremely small bones called ossicles (as the fossil record shows,

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these originally evolved from jaw bones), whose intricate lever-function is to reduce the amplitude of the vibrationwhile increasing the energy transmitted, to the inner ear (labyrinth). The middle ear thus translates pressure waves(back) into mechanical vibrations. The inner ear serves two functions, those of hearing and of balance. The cochlea isthe coiled part of the inner ear which translates the mechanical vibrations into pressure waves in its internal fluid,which are picked up by hair cells that convert their motion into electrical signals. These are in turn interpreted bynerve cells that transform them into electrical impulses that travel through the auditory nerve to structures in thebrainstem for further processing. Hearing is a complicated mechanism!

Apart from having a vested interest in keeping one's ears in perfect condition, what most concerns musicians here, andhence music theory, is basic knowledge about vibrations, especially those that we recognize as tones, as pitches.

Physics of soundA tone is a complex combination of a series of regular vibrations, usually with one prominent frequency which isexperienced as the pitch. A regular vibration is scientifically named a frequency, expressed in Hz (Herz), cycles persecond.

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The amplitude of such a wave corresponds directly to the volume of a sound, the wavelength to its pitch, or frequency.Standard tuning in music today is usually to A=440Hz which means there are 440 vibrations per second to producethis tone. Although human hearing ranges between 25 to 20,000 Hz, the pitches actually used in music are roughlybetween 25 and 5,000 Hz.

The basic regular vibration is a sine-wave, but all musical instruments actually produce a combination of various sinewaves for each tone. Differences in these combinations are perceived as various colors of sound. Some examples ofoscillograms are given below.

Most animals have a very different hearing (and correspondingly: sound producing) range from us humans. Forexample, elephants and whales hear and communicate with subsonic frequencies (infrasound) over vast distances,whereas bats and dolphins use a sonar system employing extremely high frequencies (ultrasound) to locate their preyat short range, especially in conditions where sight will not avail them.

Various tones overlapping or sounding together will produce generations of audible combination tones from theinterference of their compound frequencies, producing complex harmonies and tensions between different tones,described by the concepts consonance and dissonance.

Musical notationMusic is a time-art; music consists of sound and silence, performed by musicians. In musical notation therefore,symbols for both sound and silence are employed, set to a reading basis representing the flow of time. Although anexperienced musician is able to almost read music notation as one reads a book, the actual sound effect of a musicalscore can only be fully appreciated by hearing. Notation is basically an instruction for performance, and less so anactual representation of the sound produced.

Notation of time and rhythmSymbols for notes and restsTo represent different durations of tones, different note symbols were developed. The normal ratio of durations is 1:2,each smaller, faster note being twice as fast, or: half as long, which amounts to the same result, as the larger one. Thissystem begins with the whole note. The smaller values then are the half note, quarter note, eighth note, etc. Olderterms for some note symbols are still in use in some countries, such as quaver, semi-quaver and crotchet, but Iconsider these to be outdated and prefer the numerical values and names.

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A similar system of symbols with relative durations is used for musical silences, called pauses or rests.

The whole rest can be used for two purposes:

A true whole rest, with the exact value of a whole note;1.A full bar rest, not literally corresponding to and therefore independent of time signature.2.

MeterMeter is a usually simple repeating cycle of rhythm originating from poetry. Rhythms are often linked to meters whichare also used in poetry. In Europe, these can be traced back to the ancient Greek meters, some of which have a binarystrucure:

iamb -trochee - spondee - -pyrrhic

others are ternary:

anapest -dactyl - amphibrach

In Ottoman as well as in classical Arabic poetry[2], a related but more complex system of 16 different meters wasused, which explains why the rhythms in the music linked to these traditions are also more complex than those that arecommon in the West.

Example of text and meter:[3]

There is a strong link to language and poetry in most musical traditions, and such naturally leads to the use of meter inmusic. Meter in music is commonly represented by time signatures in notation. With the use of a time signature, it

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becomes mandatory to visually divide the notes into groups separated by vertical lines. Such groups are called bars.Each bar within one time signature has the exact same duration, with no notes nor rests more nor less than its fullduration.

Time signature and barA time signature consists of two numbers, one above the other, without a horizontal line between them (it is not afraction). The upper number indicates the number of beats, the lower number represents the note value used for onebeat. Although time signatures using the quarter or eighth notes as basis are more common, theoretically many moreare possible, and indeed sometimes used. The most common time signatures are two-four, three-four, four-four andsix-eight. Eighth and smaller notes can now be grouped and visually linked together per beat. In grouping thesesmaller note-values thick horizontal lines, called beams, are employed. These are double beams in the case ofsixteenth notes, triple beams with thirty-second notes, etc. It is also possible to thus beam together notes of differentvalues, as long as the beamed groups are a clear representation of the beat-structure. Further details on beaming willbe explained later in this outline.

With the use of time signatures, the notes become grouped in units called bars or measures, each group separated fromthe next by a vertical line, called the barline.

Binary time signaturesBinary time signatures are time signatures that have two beats per beat, and can be recognized by the top numberbeing 2, 3, 4 etc. Usually the lower number is 4, pointing to the 1/4 note (sometimes the 1/2 note occurs in oldermusic). The following examples show how such time signatures can be used in their most simple form:

DotsNote that the last bar of a piece always ends with a double barline. Note also that a dot is used here to make athree-part note out of a two-part note: any dot adds 50% to its duration. A half note is the same as two quarter noteswhich is two beats. The dotted half note however equals three quarter notes, which in these time signatures equalsthree beats. Dots can be equally applied to notes and rests.

The following examples that also include rests show how these dots can be used for more complex rhythms:

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Ternary time signaturesTernary time signatures are time signatures that have three beats per beat, and can be recognized by the top numberbeing 6, 9, 12 etc. Usually the lower number is 8, pointing to the 1/8 note; examples of these can be seen below:

Six-eight may be ternary per beat, it is in itself a binary time signature, as it has two beats per bar.

Note that with ternary beats, it is the dotted notes that represent a one-beat note!

TiesSo far we have seen note-values which fit neatly into a time signature, and durations which are also corresponding tothe basic structure. But not everything can be notated this way. Often it will be necessary to use more irregulardivisions and durations. The use of ties allows for more flexibility in connecting notes. Two notes tied togetherbecome one value, and are performed as one tone.

Note how the second dot adds another 25%. The durations thus represented make sense only when notating sustainedtones, such as of singers or wind instruments. For percussive music, such ties are normally not used, and this verysame rhythm can be notated as below, with rests instead of ties:

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The tie is notated from one note head to the next note head, it cannot tie several notes together with one symbol only.A tie cannot ever connect two different pitches, or tones, only two that are identical. The tie is therefore a rhythmicalsymbol. Still, graphically the tie looks somewhat like a slur, but this is a sign to represent not the rhythmical, but theuninterrupted connection of various different tones being sung, played or bowed in one go. The difference is thus thatties are always used between just two notes representing exactly the same pitches or tones. The slur will be shownlater, in the section on articulation.

As the relative silences of rests are identical, rests are never tied together!

TupletsFinally, to complete the rhythmic possibilities of musical notation, what are called tuplets allow for even more subtlesubdivisions, and even more variations per beat or beat-group become possible.

A tuplet is a temporary deviation from the theoretically normal division or subdivision within a time signature. Tupletscan be notated in various ways, with one number or a ratio (used especially for more unusual ratios). Depending onwhether the group is connected, square brackets can be used to indicate exactly which are the notes and durationsconcerned. Tuplets always change from the time signature (sub)divisions to another regular (sub)division, which couldalso be represented by time signature changes. Tuplets are used for temporary changes in rhythm or when timesignature changes would become too complicated for practical reading. The most common tuplet is the triplet,allowing for three against two notes.

Other examples are the quadruplet (four against three), or quintuplet (five against four or three). It is also possible tocreated nested tuplets: tuplets within tuplets. When writing such things, musicians often overlook simpler ways ofwriting the exact same rhythm. As notation is intended for practical use by performers, overcomplicated notation tendsto create confusion and should be avoided. Besides, very complex ratios are hardly if ever performed mathematicallyexact, except by percussionists. The following slightly complex example contains the same rhythm thrice, notated inthree different ways, ranging from simple to complex notation:

Of course, in these different examples, a slightly different counting and even tempo and feel are used. The indicationof tempo in notation is generally important, especially when the sheet music is to be studied in situations withoutdirect contact to the writer thereof.

Basic tempo indicationBasic tempo indications are done with the one-beat note-value, an = sign and the number of beats per minute (alsocalled bpm). The example above could then be notated in the following way, to produce three times the exact same

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rhythm, because of the tempo and counting adjustments notated:

Note that when changing the one-beat note, the equals sign = actually means becomes.More on tempo indications can be found in the section on other notation symbols.

Correct rhythmical notationBut back to basics now. As stated above, it is mandatory that the structures of the time signatures remain explicitlyvisible in notation. This means that connections through ties and vertical lines with shorter values must follow thebeats: it is a rule that rhythmic notation must always show the beats, the natural accents pertaining to the structure ofthe time signature.

The first bar fails to show the third beat, which is the middle accent of the time signature. Such and similar notation israther unpractical and will easily lead to mistakes in performance. The second bar shows all the beats and istechnically correct. The third bar groups the beats evenly in two groups of two and is used very often in practice,especially in faster tempos, as more of the 'feel' of the bars as a whole is represented in this notation. It is thereforevery important to write rhythm correctly, not only will incorrect notation lead to performance mistakes, but it also cancost valuable rehearsal time.

Ritten stough spelt inca rectly maykes reeding tuff...

So similar to reading text, the easier it is to grasp the music notated at first sight, the less misunderstandings will occur.The structures of the time signatures are therefore important to understand, so notation may follow them.

In all time signatures, the first beat is considered to be an accent. Every beat is accented more than its subdivision, soin 4/4 the first beat is stronger than the third, and both are stronger than beats two and four. In a longer series ofoff-beat notes, usually called syncopations, the beat-structure behind these faster notes should always be visible innotation, like the following example demonstrates:

A time signature adds a cycle, and hence a basic metric feel to a series of beats. Thus this series of twelve eighth-notescan become musically very different, depending on the time signature:

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BeamingNote values smaller than 1/4 can be beamed together with the help of horizontal lines connecting the notestems; thesereplace the value-flags and by connections allow for more visible structuring of the rhythms intended.

When writing or typesetting music notation, great care has to be taken to make the beams that touch each stem pointin the proper direction, which can be either left or right. The preferred direction follows from the rhythmical grouping,where each group or subgroup is made into a visible unit by means of the value flags from the first note onwards allpointing to the right (reading direction), whereas the last one to close the group points left, backwards, to indicated itsbelonging to the prior notes. Any note following a beam pointing left should be on a beat or subbeat.

A further refinement in beaming is possible to be able to visually group the rhythm notated by breaking beams under acommon 1/8 beam, thus clearly showing the subdivision of each beat.

Odd time signaturesA time signature without regular division and subdivision is usually an odd time signature, consisting of a primenumber of beats, e.g. 5, 7 or 11. These are usually also complex time-signatures, and the intended irregularsubdivisions are also shown wherever possible in notation.

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Complex time signaturesComplex time signatures consist of a compound of irregular divisions, such as a 9/8 divided in 2+2+2+3, which is acommon Middle Eastern complex rhythm. In music where such are common, such are usually indicated with a simple9/8 indication, as the irregular subdivisions are an integral part of the music style concerned.

When complex time signatures are not an integral part of the syle musicians perform (such as having the exampleabove performed by classically trained musicians), the subdivisions can better be notated more clearly with theintended subdivision indicated in the time signature itself.

Notation of swing rhythm"Swing" is a term used to denote Jazz interpretation of rhythmic notation, this means: in unequal eighth-notes. Thesixteenth notes are not affected by this feel.

Today swing is usually notated in another font type, and indicated by adding (SWING) at the top left of the sheetmusic.

The 1/8 notes are approximately interpreted in triplets, with every upbeat accented.

Notation of pitchStaff and clefA staff in notation normally consists of 5 parallel horizontal lines. The position of a notehead on a staff determines itspitch: a notehead can be notated either against or through lines.

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Special notation, such as for percussion, can also employ staves consisting of a different number of lines, even only of1 line (e.g. for bass drum or tamtam).

At the beginning of most staves a clef is employed to determine its exact usage and to determine the pitches notated.

A clef (originally from French, meaning: key) determines which pitches are notated on a staff.

Staff lines and ledger lines

The staff lines are numbered upwards, starting with the lowest line:

This numbering is used when describing notation orally, in spoken language, but can also occur in descriptive writing.

Notes outside the staff are notated with the help of (one or multiple) ledger lines. Ledger lines are small horizontallines that act as a local extension of the staff at the position of the note, as can be seen in the next example, whichshows all the steps in basic tones in a four octave scale:

Commonly used clefs

The following four clefs for notating pitch are still in use today, but many more historical clefs can be encountered inolder sheet music[4].

The G clef (sometimes also called violin- or treble clef) is used for most woodwind instruments, violin and themiddle-high register in general. Middle C is notated on one ledger line below the staff.The F clef (also called bass clef) is used for low instruments, such as cello, double bass, bassoon and trombone,and the low register in general. Middle C is notated on one ledger line above the staff.The C clef can occur on the third or fourth line:

On the third line it is called alto clef and used for viola and alto-trombone exclusively.On the fourth line it is called tenor clef and used for the middle-high register of low instruments, such asthe cello, double bass and trombone.In both cases, middle C is notated on the (middle) line where the clef is put.

Further study

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Special:Browse/Staff - browse to find more whiteboards treating staffs.Special:Browse/Clef - browse to find more whiteboards treating clefs.

Directions of notestemsAs seen in the basic forms of notes, those symbols with a notestem have two versions: one with the notestem upwards,one with the notestem downwards. The proper direction of the notestem depends on severals circumtances:

The position of the single notehead (with 1/2 and 1/4 notes): only on the the third line of the staff one canchoose freely, but should still sometimes take some care at producing a logical sequence of notes, depending onthe grouping; noteheads below the third line have stems directing upwards, noteheads above the third line havestems directing downwards (bars 1 and 2)

1.

When the average of the beamed group is neither above nor below the third line, one can freely choose in whichdirection the stems should point. (bar 3)

2.

In more complex groups of notes connected with beams, the amount of "upwardness" of "downwardness" of allthe noteheads is calculated per group; when on average below the third line, stems go up, and when on averageabove the third line, stems go down; (bars 4 and 5)

3.

Basic tones

Definition of basic tones

There are 7 basic tones in most music, from which all other tones are derived by alteration. Together, these 7 basictones make up a diatonic scale, for example a church mode. These basic tones occur in different nomenclatures.

Note that a tone represents a pitch, and is technically not identical to a note.

Systems for basic tones

There are various and different systems and nomenclatures for basically the same 7 basic tones.

The examples in the following sections make use of 3 different clefs.

Alfabetic nomenclature

In the international Western system, the following basic tones are used:

c, d, e, f, g, a, b

This alfabetic nomenclature normally denotes the following tones in notation:

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German nomenclature

In the German nomenclature, the meaning of the letters is slightly deviant, as the b denotes an already altered tone innotation:

c, d, e, f, g, a, h

Latin nomenclature

Latin nomenclature is another widely used and originally Western system, based upon Latin solmisation:

do, re, mi, fa, sol, la, si

Relative Latin nomenclature

The latin nomenclature can also be used relatively, and do can then mean any chromatic tone, for example:

Note that in these last transpositions, some basic tones are altered in the notation.

Alteration is treated in the next paragraph.

Indian nomenclature

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In Indian nomenclature the following basic tones or swaras are used:

SA, RE, GA, MA, PA, DHA, NI

These can be represented in notation as well, in Indian notation:

S R G M P D N

Indian nomenclature transcribed

Transcribed to Western notation, SA is usually rendered as c, even though on different instruments SA can vary, sosuch transcriptions may need retranscription to a certain key if intended for pratical performance.

There are two systems representing the lower and higher octaves, one using a dot above or below the tone, such as .The other system is more easy to use, typing on a computer keyboard, with apostrophes to the right when higher, suchas S' and to the left when in a lower octave, such as 'S. I will use the latter:

Further study

Special:Browse/Basic tone - browse to find whiteboards treating basic tones.

Alteration

Definition of alteration

Alteration in music is the changing of a basic tone, by raising or lowering it's pitch with one or two semitones orother microtones; opening up possibilties of alteration is the first step in the direction of a chromatic system.

Symbols for alteration

In notation the following signs are used for this:

Sharp, to raise the pitch by one semitone;Flat, to lower the pitch by one semitone;Double sharp, to raise the pitch by two semitones;Double flat, to lower the pitch by two semitones;Natural, to correct any of the previous alterations.

In the example below, all these alterations are notated on a b, with the enharmonic equivalents added after each note,but small:

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Validity of alteration symbols

Sharps and flats notated at the clef (notated before the time signature!) are indications of a key signature; see theexplanation on the circle of fifths. Sharps and flats as part of a key signature are valid for the whole staff and for alloctaves, until revoked or changed into a new key signature.

Incidental sharps and flats can occur anywhere within a bar, and are valid for that bar only and only in the octavewhere they are placed. Often, so called cautionary accidentals are added to the following bar, to prevent mistakes.Cautionary accidentals can be

a sharp, to remind one that a former flat or natural of a previous bar is no longer valid;1.a flat, to remind one that a former sharp or natural of a previous bar is no longer valid;2.a natural, to remind one that a former sharp or flat of a previous bar is no longer valid.3.

All this is summarized in the slightly complex example below:

This melody is composed in C minor. When notating music, choosing the right key is indeed very important, as"wrong notation" will confuse the reader, which is all too often also the future performer. The key is stated by the keysignature, and although the key here is somewhat ambiguous due to the abundant use of alterations, this key is stillconfirmed by careful listening; and this impression could easily be reinforced by adding the proper harmonies to it.The key however is therefore not as apparent as it may seem from merely glancing at the key signature: for examplethere are five c's and just two e's, furthermore quite some extra alterations occur, plus it seems to first end in major,before reverting back to minor at last.[5] When in doubt, the hearing takes priority over reading in determining "whatkey is it". Now this example was composed to demonstrate how to read accidentals in practice, and so it contains anumber of details which need special attention:

The bars in detail:

Immediately the 6th and 7th notes of the scale of c minor (natural minor being always taken as a key signature)are altered, to create a temporary minor melodic scale, the c# is notated correctly as such, and not to beconfused with the d, which appears later: here it is a leading tone from c towards d;

1.

The second tone, c, has a cautionary accidental added to it: a natural, reminding the reader that the validity ofthe sharp just before is cancelled by the new bar. Whithout the natural, this tone would still be c natural, but thisfact could easily be overlooked during performance. The cautionary accidental placed before the a is put inparentheses (this is a matter of taste for the engraver[6]), eliminating the likelihood of another a beingerroneously performed here. Parenthesizing cautionary accidentals is merely another possible style, andnormally the use of the two different styles of cautionary accidentals is avoided in one and the same bar (or even

2.

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in one and the same piece of sheet music altogether); however, in this example this is done for obviousinstructional reasons.Two more incidental accidentals: d and b natural; although not a single b appears in the melody, it stillbelongs at the clef, as it is part of the correct key signature.

3.

A most uncommon sight: the second, higher, d has no sign to make clear whether here also a d was intended; apossible cautionary accidental has been omitted. Confusing passages like this will often lead to varyingperformances of the same sheet music. According to the strict rule however, the second d is natural, and shouldbe performed as such. Hearing should confirm its logic (or not, as much modern music can be ambiguous insuch matters).

4.

This bar contains no cautionary accidentals, and the student should try to read it properly him- or herself,applying the rules which have been set out and explained before.

5.

In some modern notation, flats and sharps are never valid beyond the note before which they are placed. In suchnotation often also bars and a time signature are missing (even though such music is not necessarily without rhythm),and sheet music like this should normally mention this out of the ordinary use of accidentals in a footnote or preface.

Microtone alteration

There is is no general standardization as to how to notate the finer-than-semitone intonations sometimes prescribed bycomposers or played by performers. In different books and scores, various alternatives have been used. In any case,notation does allow for precise quarter tone alteration and finer microtones such as shrutis and commas, as well ascompletely different tuning systems such as the Huygens-Fokker 31 tone system[7].

Further study

Special:Browse/Alteration - browse to find whiteboards treating alteration.

Notation of intervals, chords and harmonyWhen notating simultaneous pitches, these are placed in exact vertical correspondance (one above the other), exceptwith neighbouring notes or smaller, as these will partially obliterate each other: these are placed diagonally or next toeach other when identical. The placement of the notes is independent of the presence or absence of notestems, in theexamples below the notestems are present for easier reference.

Notation in scoresMultiple staves can be combined into a staff-system, normal use of a staff-system will assign one staff to eachinstrument. Square brackets are used to group a staff-system (or a part of it) together, curly brackets assign more thanone staff to one instrumental part, as is e.g. customary in piano notation.

In a staff-system multiple staves are vertically synchronized like layers of simultaneous notation, the staves areconnected at least at the beginning, but usually also at the end. Normally the bar lines are drawn through all the stavesof a staff-system, but in case of a large score barlines between groups are not vertically connected.

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As a rule, the staves of higher pitched instruments are put above thos of the lower instruments. If this is done pergroup for all instruments of the classical orchestra, we call this the score-order of the staves.

The following orchestrated example is based upon the same music as the above, yet by its very orchestral soundimplies a slower tempo. It also has a larger staff-system in which the instrument groups are visually linked by the useof square brackets on the left. The groups are also kept visually apart by the interrupted barlines between the groups:

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In the above example there is still a lot missing: there are still no signs added to indicate tempo, dynamics, articulationor other special performance indications such as trills, mutes for the brass, stick type for timpani and so on.

In a score like this it is possible to notate several musical instruments on one staff, to save space. Note however, that inthese cases the notestems of the parts point in different directions (one-staff polyphonic notation).

This score, set in normal score-order for the classical orchestra (top to bottom) is:

Woodwind instruments (pic, 2fl, 2ob, 2cl, bcl, 2bsn, cbn)Brass Instruments (4 hn, 3tp, 3tbn, tb)Percussion (timp)Strings (vi1, vi2, vla, vc, cb)

Any additional solo-instrument would have to be added just above the first violin staff of the string section.

Compact polyphonic notationNotation of chords is a simple way of notating more than one simultaneous tone on a staff. Using this notation ofconnecting several noteheads to one notestem, has one limitation however: all tones must have the same timing andrhythm. In various cases it may be desirable to notate more than one melody on one and the same staff, with differenttiming and rhythm for each changing part. The example below was composed to demonstrate what a simple two-voicenotation could look like: all tones belonging to a melody share the same direction of notestems.

More complex notation can be encountered using this compact polyphonic notation, especially in certain editions ofthe keyboard fugues of Johann Sebastian Bach, where often also rests are included in this type of notation[8].

Other notation symbolsTempo indicationsAn essential part of music, and therefore of notation, is the tempo. The very same music, performed in different tempi,can have vastly different effects and atmospheres. A proper choice of tempo is one of the main things any musicianshould be keenly aware of, and this choice is often dependent on the acoustic properties of the venue where a certainperformance takes place as well.

There are many ways of notating tempo, all basic ways are indicated in the example below:

Classical tempo notation

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Classical tempo notation is done in Italian terms, a short list of which is given here:

Largo - Literally: broad, wide: very slow, one of the slowest tempo indications, only surpassed by thesuperlative term Larghissimo; another variation is the diminutive Larghetto, slightly less slowLento - Literally: slow: generally slow, not further specifiedGrave - Literally: heavy: for serious music, with a procession-like tempo but much slower than AndanteAdagio - Literally: gentle, quietly, carefully: medium slowAndante - Literally: going: slightly slow, like strolling rather than walkin; the diminutive Andantino is slightlyless slow, and generally lighterModerato - Literally: moderate: moderate tempo, neither slow nor fastAllegro - Literally: merry, cheerful: generally fast, but not too fast; the diminutive Allegretto is lighter and lessfastVivace - Literally: lively: fast and livelyPresto - Literally: prompt, immediate: very fast, the fastest tempo indication, only surpassed by the superlativeterm Prestissimo

Other Italian tempo-related terms which are relative to prior parts can also be encountered:

Meno mosso - Literally: less motion: less fast than the passage beforeIstesso tempo - Literally: same tempo: same tempo as before, even though this may seem to not be the caseTempo primo - Literally: first tempo: revert to the tempo of the first section of the piece

Metronome numbers

Beethoven was one of the first composers to use a metronome. Metronomes untilrecently had a limited numer of standard positions, a selection still oftenencountered in scores, ranging from 40-208 in steps of 3, 4, 6 or 8:

40 - 44 - 48 - 52 - 56 - 60 (indicated "Largo")60 - 63 - 66 (indicated "Larghetto")66 - 69 - 72 - 76 (indicated "Adagio")76 - 80 - 84 - 88 - 92 - 96 - 100 - 104 - 108 (indicated "Andante")108 - 112 - 116 - 120 (indicated "Moderato")120 - 126 - 132 - 138 - 144 - 152 - 160 - 168 (indicated "Allegro")168 - 176 - 184 - 192 - 200 (indicated "Presto")200 - 208 (indicated "Prestissimo")

Today digital metronomes are in widespread use and allow for a stepwise choice oftempi in beats per minute, ranging from 40 - 208 in steps of 1.[9]

An interesting fact is that the tempo numbers correspond by and large by the number of beats per minute a livinghuman heart can support: at lower and at higher rates a person dies.

Counting value changes

Counting value changes can be exactly notated by comparing note values, one should take care in reading and writingthis, as the equals sign = should be read as becomes: meaning in the example above, that the counting value of thequarter note is henceforward replaced by the dotted quarter note as counting value, while not changing countingtempo.

Jazz tempo notation

In jazz, tempi are generally loosely notated, and may greatly vary per performance; even one and the same piece can

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be recorded or performed at a wide variety of tempi. For reading notated Jazz, the following short list of commonlyused terms is given here:

(BALLAD) - "Jazz ballad": slow, to very slow, generally with swing notation(MED BALLAD) - "Medium ballad": medium slow, with swing notation(MED) - "Medium tempo": with swing notation(MED WALTZ) - "Medium waltz": medium tempo waltz, in 3/4, usually with swing notation(BOSSA) - "Bossa Nova": medium tempo with non-swing notation(MED UP LATIN) - "Medium up tempo, Latin": with non swing notation(FAST 3) - "Fast in three": fast 3/4 tempo(UP) - "Up tempo": fast or very fast, with swing notation

Dynamic indicationsDynamic indications are indications of the loudness, or sound volume, of the music to be performed.

The basic dynamic indications are usually printed in bold italics, and are given in the example below:

These Italian terms are explained here:

pp, pianissimo: very softp, piano: softmp, mezzo piano: medium softmf, mezzo forte: medium loudf, forte: loudff, fortissimo: very loud

In contemporary composed music, the fourfould indications pppp and ffff are quite common, and even up to a sixfoldpppppp can be found with a composer like Morton Feldman. Nevertheless all these indications are quite relative andone should not be mistaken about the apparent precision they seem to imply. In practice, many more shades anddifferences than e.g. MIDI allows for (128 steps) are possible, but it makes no sense trying to notate such fine nuances.When for example a whole passage is indicated mf it will be only a computer to play all these notes at the samevolume; all performers subtly vary the loudness of individual notes, this practice is a normal organic element ofhuman music in all styles and from all ages.

Other terms are:

fp, fortepiano: a loud beginning, then an immediately soft sustained tone; variations such as fpp or ffmp and thelike are also possbiblesfz, sforzato or rf, rinforzato: reinforced tone, especially at the beginning, which is generally less an accent butmore a quick though gradual temporal loudness growing into and leaving the beginning of a tone

Gradual and sudden transitions of dynamics are also possible, both can be seen in the example below:

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crescendo, or cresc.: getting louder gradually, it is also possible to indicate this by widening linesdiminuendo, or dim. is the same as decrescendo or decresc.: getting gradually softer, it is also possible toindicate this by narrowing linessubito, or sub.: "suddenly" in Italian, indicating a sudden transistion to another, often unexpected, soundvolume; typically, in the example above ff comes unexpectedly after a diminuendo and an unexpected pp after acrescendo; the word subito not only clarifies this beyond doubt but more importantly a performer is more likelyto not oversee this intended sudden dynamic transistion

As a final remark to these indications being relative, let it be noted that for orchestral or bigband use a mf for atrumpet more or less equals a f of a section of violins, (tenor or alto) saxophone or a french horn, and even a ff of aflute or solo violin, if measured in dB[10]. But all these dynamics depend not only on the timbre[11] but also on theregister of the note played; for example a single piccolo playing f in its highest register can always be heard, even with10 trumpets playing ff at the same time.

ArticulationThere is a wide variety of possible articulations, many of which can be notated. But as articulation depends not onlyon style but also on the musical instrument used, these articulations can greatly vary in practice. Even such basicarticulations as staccato are very different in performance between for example a violin and a flute player, bothclassically trained. One should always check how certain players or types of players respond to certain articulationsigns before prescribing these in a musical score.

Some examples of articulations are provided in the example below:

In different styles, there are different attitudes as well towards notated articulation. Jazz performers are used tochoosing most if not all articulations themselves, often on the spot, as improvised, even when they are written outexactly in the sheet music; leading to a lively performance for sure, though perhaps not as a composer originallyintended. In Jazz big bands one can expect more discipline as to the exact performance of notated articulation,especially with regard to slurs and short or accented notes. When working with classically trained musicians on theother hand, one needs to always provide at least a minimum of notated articulation, or none at all will be performed: inthis tradition, players don't usually add their own articulations and are not used to improvising even these, unless theyare soloists and performing as such. Especially string players that perform in sections need to coordinate their bowing,and will always change bowing direction when notes are not slurred. For this reason, at least the slurring of groups ofnotes into phrases can be recommended for string orchestras, because in this case (and only in this particular case:with string groups) it is true that adding slurs is almost mandatory, as writing them will always lead to a better andmore coordinated performance, almost even independent of how the slurs are placed.

When providing articulation however, one should take care not to overburden the sheet music with too many signs andsymbols, as this may lead performers' attention away from the more essential structural elements of the music, such asphrasing and breath, and so as not to provoke unnecessary mistakes.

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Basic building blocks of melody and harmonyScalesDefinition of scaleA scale is a collection of tones arranged in a stepwise ascending or descending order, usually spanning one octave,sometimes two.

The above example in notation is of a theoretical nature, because notation without a clef on the staff does not definethe pitches.

Different scalesDifferent scales can start and end on different tones or tonics, and alteration of tones can be used to achieve an evengreater variety of scales. There are numerous scales used throughout the world, sometimes with different ascendingand descending scales.

A larger set of scales can be derived from basic major, minor and diminished scales.

Basic major and minor scales

Major scale

The major scale consists of two identical tetrachords (4 note groups), each 1 - 1 - separated by a whole tone,together: 1 1 1 1 1

From the major scale the church modes are derived. As a church mode, this scale is called ionian.

Natural minor scale

From the major scale the natural minor scale can be derived.

The natural minor scale is derived from the major scale, which runs from its 6th step: 1 1 1 1 1, and it is calledaeolian as part of the church modes.

Harmonic minor scale

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The harmonic minor scale is derived from natural minor by not lowering the seventh step: 1 1 1 1 .

This scale is not a diatonic scale, as there is an augmented second appearing between the 6th and 7th steps.

From the harmonic minor, the harmonic minor modes can be derived.

Melodic minor scale

The melodic minor scale is derived from both major and minor by combining the lower and upper tetrachords of each,1 1 and 1 1 thus running: 1 1 1 1 1 .

From the melodic minor, the melodic minor modes can be derived.

Definition of modeA mode, like a scale, is a collection of tones arranged in a stepwise ascending or descending order, generally spanningone octave.

More than a scale, in musical practice, a mode can have specific melodic requirements for performance, such as aspecific order of playing the pitches involved in a makam or raga. Music based on the use of modes is generally calledmodal music, and though not excluding the use of harmony, this term is generally used in opposition to music"powered" by functional harmony, which is called tonal music.

Church modesThe church modes are modes which are derived from degrees of the major scale.

"Church modes" is the conventional name for the seven basic theoretical medieval diatonic scales and their relatedderivatives, which were at the time used in European church music. This modal system was used in medieval music,yet their names are older and derived from ancient Greek music theory.

In their basic form the church modes can be notated with only basic tones.

History and use

The church mode system is the predecessor to the Western major-minor diatonic system. The name "church modes"still refers to their usage in christian Gregorian chant and early polyphony, but since they are a result of astandardization process, they cannot be but a theoretically simplified (and thus highly censored) representation of whatwas actually sung in early Christendom, which must have had music and scales taken from a much older music, andlarger collection of scales, used outside of, not exclusively before the existence of, the christian church. The 7surviving modes incorporated into this collection are purely diatonical[12], and derived from 1 basic scale, they eachjust start and end on a different tone.

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Since the development of Jazztheory, the church modes, or shorter:"the modes", have become anintegral part of teachingimprovisation to jazz musicians. Anextraordinary example of a veryelaborate system for modalimprovisation in jazz is GeorgeRussel's Lydian chromatic conceptof tonal organization[13]. Althoughthe practical value of church modesfor modal jazz (mostly withalterations) is undisputed, in fastmusic ("up tempo") with fastchanges of harmony ("changes"),their actual use remains highlyquestionable from a theoretical pointof view, and is of little practical use.

For pedagogical use these modes arestill a wonderful beginners-tool toenhance auditive awareness anddevelop a sense of melody, intervaland harmony, and for developinginstrumental improvisation,independent of style; however, theyhave little practical value forimprovising over fast changes.

Different modal systems exist inother traditions, such as the Arabicmoqqaam-, the Turkish makam-, theNorth Indian (often somewhatethnocentrically called "Hindustani")thaat-, and South Indian (also called"Carnatic") melakarta- systems.

Basic notation

The 7 basic modes are:

Ionian - from c to c - c d e f ga b cDorian - from d to d - d e f g a b c dPhrygian - from e to e - e f g a b c d eLydian - from f to f - f g a b c d e fMixolydian - from g to g - g a b c d e f gAeolian - from a to a - a b c d e f g aLocrian - from b to b - b c d e f g a b

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The basic notation of these modes consists of basic tones and can be done without any alterations (click these musicexamples for playback):

Notation on c

When notating all these modes on c, there appears a different logical order, by the number of alterations needed.

Thus a partial circle of fifths is found with the help of the original tonics: F - C - G - D - A - E - B.

Different modesDifferent modes can start and end on different fundamental tones or tonics, and alteration of tones can be used toachieve an even greater variety of modes. There are numerous modes used throughout the world, sometimes withdifferent ascending and descending scales.

Further studySpecial:Browse/Scale - browse to find more whiteboards treating scales.Special:Browse/Mode - browse to find more whiteboards treating modes.

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IntervalsDefinition of intervalAn interval in music is the sound of exactly two tones sounding together, or one after the other.

All intervals can be measured by the number of diatonic steps and by their semitone distances. More technically, aninterval can also be described as a frequency ratio relationship between two stable vibrations. All intervals can alsooccur on (above or below) all tones.

Many different tunings and hence intervals have been developed and used in various traditions[14], this Outline ofbasic music theory treats the internationally customary tuning in Equal Temperament[15].

One soundThe sounding together of two tones creates a blending of the two into a new, specific sound, is called the interval.Within the sound of each interval, partial tones can be perceived, emerging from the interaction of frequencies as sum-and difference tones. Each interval thus consists of far more than just the two tones played or performed, it is acomplex sound which is at first perceived as a whole.

Basic intervalsDue to notation, intervals are named by the distance between the two basic tones they consist of.

When the lowest tone is counted as 1, the name of the interval follows from counting in diatonic steps upwards(although reversing this direction gives the same result):

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Wide intervalsWider than the octave, music theory describes larger intervals. Though similar, the sound of these is also quitedifferent from that of their related narrower intervals.

The similarity with the narrower basic intervals lies in the fact, that both contain the same harmonic content: the samepitches of tones. Yet the actual sound of these wide intervals, e.g. with regard to consonance and dissonance, is at thesame time so different, that they can be considered as independent musical entities as well: these are intervals in theirown right, and not mere octave transpositions.

Perception of the intervalThe unique sound of each interval can be immediately recognized by hearing, making it in fact a truly fundamentalelement of all music.

When listening to two simultaneous tones, their pitches and frequencies mix and react, creating even more tonessounding together: interference tones. This complex of tones becomes one comprehensive sound, and the individualpitches become acoustically more or less absorbed into the human perception of the single interval: the interval itselfis in fact perceived as louder than the individual tones it consists of. It is human perception which "summarizes" thetones into the single interval, but it has good acoustic reasons to do so. Thus the interval is equally the basicprerequisite for all melody as it is for harmony.

Basic intervals from cThe basic intervals using only basic tones from c are shown below (click to play the examples).

The second series consists in fact of inversions of the first, in reverse order.

Full names of the intervalsA full name of an interval consists of the quality plus the distance, such as minor second (m2), perfect fifth (P5), oraugmented sixth (A6) etc.Each interval thus has a "first" and a "last name".

For the purpose of shorter and easier writing the names of the intervals are abbreviated in the following way: theintervals are represented by numbers, their qualities by preceding letters.

Summary of the abbreviations:

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d m P M A

unisons d1 P1 A1

seconds d2 m2 M2 A2

thirds d3 m3 M3 A3

fourths d4 P4 A4

fifths d5 P5 A5

sixths d6 m6 M6 A6

sevenths d7 m7 M7 A7

octaves d8 P8 A8

Qualities:M = Majorm = minorP = PerfectA = Augmentedd = diminished

Distances:1 = unison2 = second3 = third4 = fourth5 = fifth6 = sixth7 = seventh8 = octave

The whiteboard image shows the semitone relationships between the qualities of intervals:

perfect are the unison, fourth, fifth and octave plus wider versions of these;[16]major is larger than minor;augmented is larger than major or perfect;diminished is smaller than minor or perfect.

Finally, double diminished and double augmented intervals are theoretically possible, but extremely rare.

The table below thus lists all possible basic intervals. Below the table, these are also presented in notation, with c asreference tone (remember, all intervals can occur on -above or below- all tones!).

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semitones -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13

interval d1 P1,d2A1,m2

M2,d3

A2,m3

M3,d4

A3,P4

A4,d5

P5,d6

A5,m6

M6,d7

A6,m7

M7,d8

A7,P8 A8

Intervals in order of chromatic sizeThe table below lists all possible basic intervals in order of number of semitones, these are also presented in notationwith their enharmonic equivalents, with c as reference tone (remember, all intervals can occur on -above or below- alltones!).

Enharmonic equivalenceThe possibility of writing one and the same sound with two different diatonic spellings is called enharmonicequivalence; such two intervals are enharmonically the same. When listened to separately, the number of semitoneswill determine the sound, and in ear-training usually the simpler spelling is heard, interpreted and given. Within acertain musical context however, more complex phenomena may occur, requiring specific writing, spelling and

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nature type

P P 2 7

M m 3 6

A d 4 5

nomenclature, giving prevalence to one of the two possibilities for contextual reasons.

In one case, a special name was assigned to such a pair of enharmically equivalent intervals: the name tritone can beused to designate both the augmented fourth and the diminished fifth; this unique interval acoustically equals its owninversion. The name was derived from the fact that this interval equals exactly three whole tone distances (sixsemitones).

Inversions of the intervalsThe inversions describe close relationships between intervals in pairs: an interval and its inversion use the same notes,but in different order or position. The two inversions together span a perfect octave, 12 semitones, thus the inversionof a Major third is the minor sixth, calculated as M3 (4 semitones) and m6 (8 semitones), together 4 + 8 = 12semitones (P8).

The close relationship between inversions can be clearly heard, as well as the fundamental difference in sound. Nointerval should be mixed up with its inversion, they are not identical, nor is one derived from the other[17].

The inversions are below presented in notation, with c as reference tone (remember, all intervals can occur on -aboveor below- all tones!).

The inversion of a perfect interval is always a perfect interval, the inversion of a major interval always a minorinterval, the inversion of a diminished interval always an augmented interval, and all vice versa as well. The inversionof a second is a seventh, of a third a sixth, of a fourth a fifth, and vice versa. The tritone is the only interval which isenharmonically the same as its inversion, such as can be heard by clicking and playing the above example.

A table to help quickly find the names of the inversions is given here:

Further study

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Special:Browse/Interval - browse to find whiteboards treating intervals.

TriadsDefinition of triadA triad is a special type of chord which consists of 3 different tones, in consecutive thirds.

The triad thus consists of a minimum of three intervals: 2 thirds and 1 fifth, or their respective inversions.

Basic notationThe basic notation of a triad, is by virtually taking tones 1, 3 and 5 of a scale and writing them together.

The examples below use the basic tones only and can be clicked for playback.

Basic scale and triad of c:

Basic scale and triad of d:

Basic scale and triad of e:

Basic scale and triad of f:

Basic scale and triad of g:

Basic scale and triad of a:

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Triad Symbols

Major Maj M

Minor min m -

Diminished dim d

Augmented Aug A +

Flat 5 5 -5

Doublediminished (no symbol)

Basic scale and triad of b:

Types of triadsConsisting of 2 thirds and one fifth, the thirds can be major or minor, and the fifth can be perfect, diminished oraugmented, there are 6 types of triads.

Major triad: consisting of a major third, a minor third and a perfect fifth;Minor triad: consisting of a minor third, a major third and a perfect fifth;Augmented triad: consisting of a major third, a major third and an augmented fifth;Diminished triad: consisting of a minor third, a minor third and a diminished fifth;Flat 5 triad: consisting of a major third, a diminished third and a diminished fifth;Double Diminished triad: consisting of a diminished third, a major third and a diminished fifth.

These triads can occur on any tone, the examples below are the 6 triads of c (alterations are needed to obtain thetones):

Symbols for triadsFor most music in which improvising plays an important role, symbols are used for triads and chords.

There is no unity in the systems used for these symbols, as different symbols occur in different books.

Below is a table summarizing the most commonly used chord-symbols for triads:

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Inversions of triadsA triad is still considered to be the same triad when the 3 tones it consists of are presented in another position than thebasic 1-3-5 described so far. There are three basic positions for a triad:

The 1 is the lowest tone, this is called root position, short: root1.The 3 is the lowest tone, this is called first inversion or sixth position, short: 62.The 5 is the lowest tone, this is called second inversion or six-four position, short: 6-43.

The acoustic properties are different, yet these positions can be perceived as derived from one basic structure: thetriad. This perception evolved by the end of the 15th century in Europe, and the triads (and their inversions) quicklyentered Western classical music to dominate it for centuries to come, at the cost of the polyphony which prevailedbefore. Harmony thus replaced melody as the structuring principle of musical logic.

The examples below are the positions of the triads of c:

The names of the inversions are derived from the intervallic structure: the inversions of the thirds become sixths(major, minor or augmented), and the inversions of the fifths become fourths (perfect, augmented or diminished).

Other positions of triadsAny chord with more than just three tones, but consisting of just three tones making up a triad, is also perceived as,and considered to be, a triad. In this way, triads can be performed with more pitches, yet three basic tones only, as isthe case in many compositions for piano, band, choir, ensemble or orchestra.

Other designations for positions of triads are:

closed position - short CP, when no triad-tones fit between the tones played, the position of the lowest tone, thebass, is disregarded in this respectopen position - short OP, when exactly one triad tone would fit between the tones played, the position of thelowest tone, the bass, is disregarded in this respectmixed position - short MP, when the position has a mix of "open" and "closed" intervals, the position of thelowest tone, the bass, is disregarded in this respectoctave position - short 8P, when the highest tone played is the 1third position - short 3P, when the highest tone played is the 3fifth position - short 5P, when the highest tone played is the 5

Some examples, demonstrating these positions of the triad of c-major, for piano:

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Use of triadsAny scale or mode can be harmonized by forming triads on its steps, these triads (or, when expanded: seventh chords)are called degrees.

The harmony thus obtained is proper to that particular scale itself and can be modal or tonal, depending on its use.

Further studySpecial:Browse/Triad - browse to find more whiteboards treating triads.

Seventh chordsDefinition of seventh chordA seventh chord is a special type ofchord which consists of 4 differenttones, in consecutive thirds.

The seventh chord thus consists of aminimum of six intervals: 3 thirds, 2fifths and 1 seventh, or theirrespective inversions.

A seventh chord can be regarded asen expanded triad; each seventhchord itself consists of 2 triads.

The relationships of all types ofseventh chords with their constituenttriads and intervals are shown in thewhiteboard picture.

Basic notationThe basic notation of a seventh chord, is by virtually taking tones 1, 3, 5 and 7 of a scale and writing them together,sounding as one.

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The example below uses the basic tones only and can be clicked for playback.

Types of seventh chordsConsisting of 3 thirds, 2 fifths and 1 seventh, the thirds can be major, minor or diminished, fifths can be perfect,diminished or augmented, the sevenths can be major, minor or diminished. There are no seventh chords which containan augmented third, as the enharmonic equivalence of this interval, the perfect fourth, is such a strong consonant thatits top note would acoustically assume the position of the root of the whole chord, thus leading to the perception of anextended triad. Thus the number of possible seventh chords is limited to just 11:

Major seventh: consisting of a major third, a minor third, a major third, two perfect fifths and a major seventh1.Dominant seventh: consisting of a major third, a minor third, a minor third; one perfect fifth, one diminishedfifth and a minor seventh

2.

Minor seventh: consisting of a minor third, a major third, a minor third; two perfect fifths and a minor seventh(enharmonically equivalent to a Major triad with added 6 in third inversion)

3.

Minor major seventh: consisting of a minor third, a major third, a major third; one perfect fifth, one augmentedfifth and a major seventh

4.

Half diminished: consisting of two minor thirds, a major third; one diminished fifth, a perfect fifth and a minorseventh (enharmonically equivalent to a Minor triad with added 6 in third inversion)

5.

Diminished seventh: consisting of three minor thirds; two diminished fifth and a diminished seventh(enharmonically equivalent to itself in all inversions)

6.

Augmented seventh: consisting of two major thirds, a minor third; an augmented fifth, a perfect fifth and amajor seventh

7.

Augmented dominant: consisting of two major thirds, a diminished third; an augmented fifth, a diminishedfifth and a minor seventh

8.

Flat 5 dominant: consisting of a major third, a diminished third, a major third; two diminished fifths and aminor seventh (enharmonically equivalent to itself in second inversion)

9.

Double diminished minor seventh: consisting of a diminished third, a major third, a major third, a diminishedfifth, an augmented fifth and a minor seventh (enharmonically equivalent to an Augmented dominant in thirdinversion)

10.

Double diminished seventh: consisting of a diminished third, a major third, a minor third, a diminished fifth, aperfect fifth and a diminished seventh (enharmonically equivalent to a Dominant seventh in third inversion)

11.

The whiteboard image shows the relative mutual proximity and similarity of the 9 basic seventh chords. The arrowsindicate how an alteration of one chord tone will change the chord into a neighbouring, different, but closely related,related seventh chord.

All seventh chords can occur on any tone, the notated example below shows all 11 theoretically possible seventhchords of c (alterations are needed to obtain the correct tones). In practice however, only 9 seventh chords have actualchord-symbols: the last two seventh chords have no chord-symbol of themselves. These two would rather be notatedas third inversions, slash chords of d (D7#5/C and D7/C, as notated in small black notes), of which they areenharmonic equivalents:

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Seventh chord Symbols

Major seventh maj7 M7 7

Dominant seventh Dom7 7

Minor major seventh minmaj7 mM7 -7

Minor seventh min7 m7 -7

Half diminished min75 m75

Diminished seventh dim7 d7 7

Augmented seventh maj7#5 M7#5 7+5

Augmented dominant 7#5 7+5 +7

Flat 5 dominant 75 7-5

Symbols for seventh chordsFor most music in which improvisation plays an important role, symbols are used for seventh chords.

There is no unity in the systems used for these symbols, as different symbols occur in different books.

Below is a table summarizing the most commonly used chord-symbols for seventh chords:

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Double diminished types(notated as

thirdinversions onanother root)

Inversions of seventh chordsA seventh chord is still considered to be the same seventh chord when the 4 tones it consists of are presented inanother position than the basic 1-3-5-7 described so far. There are four basic positions for a seventh chord:

The 1 is the lowest tone, this is called root position, short: root1.The 3 is the lowest tone, this is called first inversion or six-five position, short: 6-52.The 5 is the lowest tone, this is called second inversion or four-three position, short: 4-33.The 7 is the lowest tone. this is called third inversion or second position, short: 24.

The acoustic properties are different, yet these positions can be perceived as derived from one basic structure: theseventh chord.

The examples below are basic positions of seventh chords of c and on c:

The names of the inversions arederived from the intervallicstructure: the inversions of the thirdsbecome sixths (major, minor oraugmented), the inversions of thefifths become fourths (perfect,augmented or diminished) and theinversions of the sevenths becomeseconds (major, minor oraugmented).

The whiteboard image shows a fulltable of the nine seventh chords thathave chord symbols, in allinversions on c (not "of" c, as onlythe root positions are C chords!).

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perfect consonants imperfect consonants dissonants

P1, P8, P5, P4 M3, m3, M6, m6M2, m2, M7, m7, tritoneplus all augmented and diminished intervals(no matter if they are enharmonicallyconsonant)

Use of seventh chordsAny scale or mode can be harmonized by forming seventh chords on its steps. These seventh chords are then calleddegrees.

The harmony thus obtained is proper to that particular scale itself and can be modal or tonal, depending on its use.

Further studySpecial:Browse/Seventh chord - browse to find more whiteboards treating seventh chords.

Consonance and dissonanceDefinitions of consonance and dissonanceConsonant literally means: sounding together

Dissonant literally means: sounding apart

Let us stop a moment and consider these concepts more closely. Especially the concept dissonant may seem strange inthe context of music, which after all consists of sounds combined together into one piece. When playing simultaneoussounds, these will "sound together" by the very definition of their simultaneousness. The whole purpose of introducingthe concepts of consonance and dissonance however, is to make distinctions in the nature and quality of theinnummerable possibilities of combining sounds. But then, why just make this distinction seemingly oversimple andmerely dual, almost resembling the opposites "good" vs "bad"?[18]

The quality of the "sound together" is in fact strongly depending on the musically formal and harmonical contexts inwhich it occurs. Correspondingly, according to Louis & Thuilles "Harmonielehre" one should distinguish between thephenomena of acoustic dissonance and interpretational dissonance (German: Auffassungsdissonant)[19]. I considerthis phenomenon as an integral part of the concept of harmony in a wider sense, and distinguish further betweenformally and harmonically contextual dissonances.

Before going into finer detail, we shall consider the traditional classical theoretical view on consonance versusdissonance first.

Traditional classification of intervals in consonant and dissonantTraditionally, only the basic the intervals are classified, based on their use in the type of harmony used in classicalmusic.

This classification, though all too often still taught as factual, is actually rather outdated and of little practical value formost contemporary music, such as Jazz, Pop and World music, as well as contemporary composition; hence we willtake a closer look at their actual acoustic properties next.

Acoustic order of consonance and dissonance in intervals

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An attempt has been made here to establish an acoustic order of the basic and wide intervals, starting with the mostconsonant, gradually evolving to the more dissonant, somewhat like Hindemith's Reihe 2[20]. In this order it can beobserved how the wide intervals are in fact found in perhaps unexpected places for the traditionally educated musictheoreticians, but as "the proof of the pudding is in the eating", in sound "the proof of the theory is in the hearing", asone can observe oneself.

This order may be true for these intervals exactly as notated, on c one octave below central c, and yet it may differsomewhat when gradually transposed to other tones, more so even in different registers. Also, this order can basicallybe regarded to be true for the instrument on which it is perceived, here: the piano. But consonance and dissonancedepend on the harmonics-structure of the instrument used, as well as on the medium for the sound waves: the air itself,so far taken for granted. Dolphins, porpoises and whales, living and using an enormous variety of sounds under water,in many more octaves than we humans can perceive, should have a quite different perception of these intervals. Goodexamples of completely different sets of intervals in world music are offered by Indonesian gamelan: the bells usedproduce totally different overtones than the piano.

When composing, arranging or improvising, great care has to be taken to judge the sounds intended not on meretheoretical grounds, but on the actual perception intended, clearly and precisely defined, taking exact register positionsinto account, or else one may end up with music that looks good, but sounds bad, as is all too often the case whenrelying too much on general theoretical knowledge, superficially studied, only partly digested and not fullyunderstood.[21]

Consonance and dissonance in scalesConsonance and dissonance in scales is generated from the presence and position of contextual harmonicallydissonant intervals embedded within a particular scale. Thus all diatonic scales contain at least one tritone, and mostpentatonic scales have no dissonant intervals at all.

Dissonance in a scale adds a harmonic element of tension to its melodic structure, certain tones always stand out in aparticular scale, exactly because of this.

Let us consider this in the church modes.

The most remarkable tones by which we recognize the church modes are:

the sharp 4 in lydian1.the flat 7 in mixolydian2.the natural 6 in dorian3.the flat 2 in phrygian4.the flat 5 in locrian5.

Which does make one wonder about the two remaining modes: by what special characteristics do we recognize ionianand aeolian, also known as the basic major and minor scales, or do they stand out in a negative sense, by having nospecial characteristics at all?

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chord type: consonants dissonants

triads maj and min all others: , #5 , 5 and double dim

seventh chords maj7 and min7 all others: 7, minmaj7, , 7, maj7#5, 7#5 and75

Now, as the image presented hereattempts to show:

all remarkable tones are partof an embedded tritone;

1.

in all 5 remarkable cases thistritone is connected to thebasic triad 1-3-5 supportingthe scale, the remarkable tonebeing outside the triad on 2, 4,6 or 7 (except in locrian whichhas the tritone on 1-5);

2.

only in ionian (4-7) andaeolian (2-6) the completeembedded tritone isdisconnected from this basictriad.

3.

So we can conclude that the majorand natural minor scales are wellsuited for tonal music, whichdepends on the use of triads 1-3-5throughout, as the tritones embeddedin these scales do not burden itsbasic sound with dissonance,reserving its embedded tension to beused on degrees other than its propertonic I.

The accuracy of this phenomenological observation and line of reasoning are strongly confirmed by the fact that fortonal use, the minor harmonic scale is customary, which does not flatten the 7th tone, and which like natural minor hasa tritone between 2 and 6. In this scale a second tritone is introduced by the natural 7, adding more tension to thisscale. The position of this additional tritone now is remarkably enough between 4 and 7: exactly the same as in ionian(major). Therefore we can conclude that by embedding both tritones the minor harmonic scale combines the tensionproperties of major and minor into one scale, which sheds quite a new light on the concepts of molldur and durmoll,which are customary in functional harmony.[22]

Finally, this procedure gives some clues as to how one can go about harmonizing makams and ragas, and in fact to thedifferent phenomenology of harmony itself in music based on modes such as these, many of which are notharmonically connected to 3 and 5 by their performance traditions.

Traditional classification of chords in consonant and dissonantTraditionally, only chords with perfect fifths between 1 and 5, and in the case of seventh chords also between 3 and 7,are considered consonant.

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Noteworthy is that the second inversion of the consonant chords is considered to be a typical contextual dissonant.

Use of consonance and dissonanceConsonance is a term that describes the phenomenon of several sounds or pitches blending together, almost as ifbecoming one, a unity of sound. Dissonance on the other hand, is a term that describes tension in sound, as if soundsor pitches do not blend together, and remain separate auditive entities. The apperception of whether sounds do or donot blend is highly subjective however, and strongly dependent on musical context, whereas tension is a moreobjectively measurable property in sound itself.

In functional harmony, that is: harmony employing degrees and thus harmonic functions, the usual order is that of anincrease of tension followed by a moment relaxation, so using more and increasing dissonance, to be followed and thetension released by consonance. The consonance is always chosen in relation to the musical context, the dissonantsemployed are style- and context-related as well. As a general rule however, it is the dissonants which represent thestrongest expressional means in harmony.

Circle of fifthsMajor scales in order of accidentalsIt is possible to construct a major scale on every tone, and different accidentals are needed to induce the proper orderof steps: whole, whole, half in both tetrachords (4 tone scale part). Only the scale of c major yields this without use ofaccidentals. Both c d e f and g a b c share this structure: 1 1 . These tetrachords are both connected to thefundamental c, the first has it as lowest, the second as highest note. The remaining interval f g is a consequence of thistetrachordal construction, and in itself a whole step (major second). The total scale thus shows 1 1 1 1 1 .

When constructing this on different tones, we can set the scales we find into the following order by number ofaccidentals, those we find having sharps:

And those we find having flats:

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If we allow for the use of double sharps and flats, more scales are possible, but for practical reasons we limited the listof scales to those employing a maximum of 7 accidentals.

Minor scales in order of accidentalsIt is also possible to construct a similar series when studying minor. In order to obtain similar accidentals, we use theminor-parallel scale, or aeolian mode, which can be found on the VI of any major scale. VI of c being the a we findthe following order by number of accidentals, with sharps:

And another series yet, employing flats instead, starting with the parallel of f, d:

Key signatures in order of accidentalsNote that both orders are based upon this diatonical series of fifths:

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either, for the sharps

f c g d a e b

or the reverse, for the flats

b e a d g c f

which form a circle as shown in theimage, that also mentions the churchmodes. Not all these fifths areperfect however, this diatonic circleof fifths contains one diminishedfifth: f - b.

The order of appearance of sharps is fixed as: f# c# g# d# a# e# b#, the example shows thecorrection notation of the key-signatures with sharps on two clefs:

The order of appearance of flats is also fixed and the exact reverse as to the use of basic tones: b e a d g c f, the example shows the correction notation of the key-signatures with flats on two clefs:

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Geometry of the full circle of fifthsWe can now summarize thisstructure and knowledge in thecircle of fifths.

The image of the circle of fifthssummarizes and visually representsthe observations on major and minorscales and keys and the keysignatures with number and order ofsharps and flats, but it also allows afurther geometrical look into theintervals, and their place within thisrepresentation.

Here are some relevant observationswhich help in understanding andmemorizing this visualrepresentation of "musicalgeometry":

There are the enharmonicequivalences of the majorkeys C=B (7 flats equals 5sharps enharmonically),G=F# (6 flats equals 6 sharpsenharmonically), D=C# (5flats equals 7 sharpsenharmonically);

1.

There are the enharmonic equivalences of the minor keys a=g# (7 flats equals 5 sharps enharmonically), e=d#(6 flats equals 6 sharps enharmonically), and b=a# (5 flats equals 7 sharps enharmonically);

2.

The enharmonic equivalences are true for the tempered tuning system alone, but this is the contemporary tuningsystem which we actually use in this basic music theory;

3.

The circle of fifths is not valid in other tuning systems;4.In the outer black ring the major keys are represented in white capitals;5.In the inner white ring the minor keys are represented in black minuscules;6.In the inner gray circle all possible interconnections between points are shown, these represent intervals;7.Outside the rings, the number of accidentals is shown, from zero up to seven;8.On the top and bottom left and right are red accidentals, to show the left side contains flats, and the right sidecontains sharps;

9.

The fact that the starting point C - a is represented on top, is arbitrary but customary;10.Intervals can also be visualized in this geometrical representation, as the innermost gray circle contains allinterconnections between all 2 points, these represent the following intervals:

neighbours are a perfect fifth (or its inversion the perfect fourth) apart - completing this as a pattern onegets a dodecagon or the full circle of fifths;

1.

neighbours to neighbours, skipping one in between, are a major second apart (or its inversion the minorseventh) - completing this as a pattern one gets a hexagon or half the circle as a whole tone scale(acoustically there are only two such scales);

2.

skipping two one finds a minor third (or its inversion the major sixth) - completing this as a pattern onegets a square or a diminished seventh chord (acoustically there are only 3 such chords);

3.

skipping three one finds a major third (or its inversion the minor sixth) - completing this as a pattern onegets an equilateral triangle or an augmented triad (acoustically there are only 4 such triads);

4.

11.

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opposites are in a tritone (augmented fourth or its inversion diminished fifth) relationship (acousticallythere are only 6 such intervals);

5.

neigbours to opposites are a minor second apart (or its inversion the major seventh) - completing this as apattern one gets the full circle as a star representing the chromatic scale (acoustically there is only 1 suchscale);

6.

A still fuller circle of fifths is possible when wrapping the full sequence of flats, naturals and sharps of basictones in order onto itself, 21 tones wrapped in the form of a 12 tone circle, with only g, d and a having noenharmonic equivalents with a single accidental:

f - c - g - d - a - e - b - f - c - g - d - a - e - b - f# - c# - g# - d# - a# - e# - b#;

12.

where all fifths are perfect. Doing this however would lead to needlessly encumbering the image withhighly unusual keys and scales, as well as to overflooding the eye with too much information, but one canand should attempt to imagine this overcomplete image nevertheless.

The whiteboard below shows the gradual development of the circle of fifths out of a series of basic tones, curled ontoitself as a spiral:

step 0: Original ancient deduction of basic tones from a as [23] using perfect fifths, creating pentatonicand diatonic scales, finding the tritone;step 1: Diatonic circle of fifths, also the basis for the order of sharps and flats;step 2: Diatonic spiral of fifths, charting all flat and sharp alterations of basic tones;step 3: Chromatic spiral of fifths including Pythagorean comma,[24] a dilemma for tuning;step 4: Full chromatic circle of fifths with major and minor keys and geometry of intervals.

Making sure to check and gradually understand fully all of these steps towards, and observations on, the circle of

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fifths, will enable one to memorize it as more than a mere photographic image, but rather as a dynamic "Gestalt" or"Mandala", implied by and underlying the tone-structure of the tempered tuning system.

TranspositionDefinition of transpositionTransposition is the shifting of musical notation by a certain specific interval.

Use of transpositionTransposition may be used in order for a certain musical instrument to play the correct tones, or to change the key of acomposition, song, or part of any music.

Commonly occurring reasons to transpose music are:

to correctly write the staff or