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What are Musical Scales? A scale is a sequence of musical notes in an ascending or a descending order. They are organised into categories such as diatonic, major, minor, and others, with a specific group of notes thus being described as a C-major scale, D-minor scale. Scales are typically listed from low to high. For instance, the increasing C major scale is C-D-E-F-G-A-B-[C], with the bracket indicating that the last note is an octave higher than the first note. Scales may be described according to the intervals they contain: for example: diatonic, chromatic, whole tone or they may be described according by the number of different pitch classes they contain: - Ditonic, tritonic, tetratonic scales having two, three or four tones - used in prehistoric music. Pentatonic, hexatonic, heptatonic scales, having five, six, and seven tones - are most common. Octatonic or eight tones - used in jazz and modern classical music. One or more scales may be used in a composition, such as in Claude Debussy's composition L'Isle Joyeuse. Scales in traditional Western music generally consist of seven notes and repeat at the octave. Notes in the commonly used scales are separated by whole and half step intervals of tones and semitones. Scales through History 1100–1600 Western music in the Medieval and Renaissance periods tends used the white-note diatonic scale C-D-E-F-G-A-B. 1600–1900 Music of this period used three types of scale: * The diatonic scale (seven notes) * The melodic and harmonic minor scales (seven notes) These scales are used in all of their transpositions. The music of this period introduces modulation, which involves systematic changes from one scale to another. Modulation occurs in relatively conventionalised ways. For example, major-mode pieces typically begin in a "tonic" diatonic scale and modulate to the "dominant" scale a fifth above. 1900 Onwards In the nineteenth and twentieth century, additional types of scales were explored: * The chromatic scale (twelve notes) * The whole tone scale (six notes) * The pentatonic scale (five notes) * The octatonic or diminished scales (eight notes)

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What is a Major Scale? The major scale is the first of the diatonic scales. Another name for the major scale is the Ionian Mode. A major scale has 7 notes. The first and simplest major scale is the C major scale:

C Major Scale The C major scale is the only major scale that doesn't have sharps (#) or flats (b). If you play the C major scale on the piano, only white notes are used (all the white notes). In solfege, the notes of the major scale are named like this: Do, Re, Mi, Fa, Sol, La, Ti, Do Another notation that is used is the following: - C D E F G A B C Do Re Mi Fa Sol La Ti Do All major scales have a typical structure. This is the C major scale: C D E F G A B C The C at the end of the scale is one octave higher (12 half tones) compared to the first C. Here are the intervals between the notes of the C major scale. An interval is the distance between two notes. * C-D: D is 2 half steps higher than C * D-E: 2 half steps * E-F: 1 half step * F-G: 2 half steps * G-A: 2 half steps * A-B: 2 half steps * B-C: 1 half step Every note in the C major scale is 2 notes higher than the previous note, except for the F and the C. C D E F G A B C 2 2 1 2 2 2 1 Major Scale Formula: 2 - 2 - 1 - 2 - 2 - 2 - 1 We can use this formula to construct other major scales: - e.g. D Major Scale D E F# G A B C# D 2 2 1 2 2 2 1 * # the sharp sign makes a note a half step higher. * b the flat sign makes a note a half step lower. List of all the Major Scales: C Major Scale: C D E F G A B

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D Major Scale: D E F# G A B C# E Major Scale: E F# G# A B C# D# F Major Scale: F G A Bb C D E G Major Scale: G A B C D E F# A Major Scale: A B C# D E F# G# B Major Scale: B C# D# E F# G# A# C# Major Scale: C# D# E# F# G# A# B# Eb Major Scale: Eb F G Ab Bb C D F# Major Scale: F# G# A# B C# D# E# Ab Major Scale: Ab Bb C Db Eb F G Bb Major Scale: Bb C D Eb F G A Minor Scales The notes on a major scale sound bright and cheerful while the notes of the minor scale sound solemn and sad. List of all the Minor Scales: C = C - D - Eb - F - G - Ab - Bb - C D = D - E - F - G - A - Bb - C - D E = E - F# - G - A - B - C - D - E F = F - G - Ab - Bb - C - Db - Eb - F G = G - A - Bb - C - D - Eb - F - G A = A - B - C - D - E - F - G - A B = B - C# - D - E - F# - G - A - B C# = C# - D# - E - F# - G# - A - B - C# Eb = Eb - F - Gb - Ab - Bb - Cb - Db - Eb F# = F# - G# - A - B - C# - D - E - F# G# = G# - A# - B - C# - D# - E - F# - G# Bb = Bb - C - Db - Eb - F - Gb - Ab - Bb The formula to create a minor scale is: – whole step - half step - whole step - whole step - half step - whole step - whole step w - h - w - w - h - w - w. 2 -1 - 2 - 2 - 1 - 2 – 2 Pentatonic Scale The word "pentatonic" comes from the Greek word ‘pente’ meaning five and ‘tonic’ meaning tone. The pentatonic scale consists of five notes within one octave. The pentatonic scale is a very ancient scale. Pentatonic scales are very common and are found all over the world. They are found in Celtic folk music, Hungarian folk music, West African music, African-American spirituals, American folk music, Jazz, American blues music and rock music, Sami joik singing, children's songs and in the music of ancient Greece. Pentatonic scales are also found in the music of Indonesia, the Philippines, Korea, Malaysia, Japan, China, India and Vietnam. The pentatonic scale is also found in Andean music, Afro-Caribbean music in the Great Highland Bagpipe. Western Classical composers such as the French composer Claude Debussy also used this scale.

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Pentatonic scales are either hemitonic or anhemitonic. Hemitonic scales contain one or more semitones and anhemitonic scales do not contain semitones. The two basic forms of pentatonic scales are: Major Pentatonic Scale - Consists of the 1st - 2nd - 3rd - 5th - 6th notes of a major scale. Minor Pentatonic Scale - Consists of the same 5 notes of a major pentatonic scale but its tonic (first note of the scale} is 3 semitones below the tonic of the major pentatonic scale. For example, the C major pentatonic (C - D - E - G – A) has the same notes as the A minor pentatonic (A - C - D - E – G) but arranged differently. The first note or tonic of the A minor pentatonic scale is 3 semitones (half steps) lower than the first note of the C major pentatonic scale. It uses the 1st - minor 3rd - 4th - 5th - minor 7th notes of a scale. Major Pentatonic Scale Anhemitonic pentatonic scales can be constructed in many ways. One construction takes five consecutive pitches from the circle of fifths; starting on C, these are C, G, D, A, and E. Transposing the pitches to fit into one octave rearranges the pitches into the major pentatonic scale: C, D, E, G, A, C. This common scale is found in the opening bars of "My Girl" by The Temptations. Another construction works backward: It omits two pitches from a diatonic scale. If we were to begin with a C major scale, for example, we might omit the fourth and the seventh notes of the scale F and B. The remaining notes, C, D, E, G, and A are transpositionally equivalent to the black keys on a piano keyboard: G-flat, A-flat, B-flat, D-flat and E-flat. Omitting the third (E) and seventh note (B) of the C major scale obtains the notes for another anhemitonic pentatonic scale: (F,G,A,C,D). Omitting the first and fourth notes of the C major scale gives a third an hemitonic pentatonic scale: (G,A,B,D,E). Minor Pentatonic Scale Although various hemitonic pentatonic scales might be called minor, the term is most commonly applied to the relative minor pentatonic scale derived from the major pentatonic, using scale tones 1, 3, 4, 5, and 7 of the natural minor scale. The C minor pentatonic would be C, E-flat, F, G, B-flat. The A minor pentatonic, the relative minor of C, would be the same tones as C major pentatonic, starting on A, giving A, C, D, E, G. This minor pentatonic contains all three tones of an A minor triad. The major pentatonic scale is the basic scale of the music of China and the music of Mongolia. The fundamental tones rendered by the 5 holes of the Japanese shakuhachi flute play a minor pentatonic scale. The minor pentatonic scale can be heard as the first notes of the verse of Don McLean's hit song ‘Vincent’ (Starry, Starry Night). The pentatonic scale is very common in Scottish music. The Great Highland bagpipe scale is made of three interlaced pentatonic scales. The pentatonic scale also features in Irish traditional music. The minor pentatonic scale is also used in Appalachian folk music. Both the major and the minor pentatonic scales are commonly used in jazz. Jazz pianists such as Art Tatum, Chick Corea and Herbie Hancock use this scale. It is also used in blues and rock. Pentatonic scales are useful for musicians who improvise in modern jazz, pop, and rock contexts because they work well over several chords. The blues scale is predominantly derived from the minor pentatonic scale, a very popular scale for improvisation in the realm of blues and rock alike.

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The pentatonic scale occurs in the melodies of popular music: for example in "Old Man River". It is also used in film music, where it is used to create a primitive or exotic atmosphere. The pentatonic scale also occurs in hymns and other religious music. The hymn "Amazing Grace", has a melody set to the notes of the pentatonic major scale. Antonín Dvořák, inspired by the native American music and African-American spirituals he heard in America, made extensive use of pentatonic themes in his "New World" Symphony and his "American" Quartet. Giacomo Puccini's operas Madam Butterfly and Turandot used the pentatonic scale to create a Japanese and Chinese atmosphere.

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The Ancient Greek Modes The Greek modes were named after the tribes who lived in Ancient Greece. The Ionians were from Asia Minor, the Aeolians were from the island of Lesbos and the Dorians were from the Peloponnesus. The other tribes were the Phrygians, Lydians, Lesbians and Locrians. "The Phrygian mode is found in the flamenco music of Spain and Spanish speaking countries in Latin America. The Lesbian (Mixolydian) mode still survives in the music of Scotland and Ireland as well as the American Appalachia (the mountain dulcimer is often tuned to this scale. The Lesbian mode is also found in the Blues scale which was developed by black American and is of African origin." (Gardner, K. 1990 p132) "The earliest Greek modes are attributed to Terpander of Lesbos (750BC), Sappo of Lesbos (610BC) and Pythagoras of Samos (500BC)." "These musicians devised their particular tone sequence on the seven stringed lyre. All seven strings on the lyre were controlled by one tuning peg, therefore the mode was determined by the length of the strings." (Gardner, K. 1990 p133) "Each string of the lyre was thought to echo one of the seven planets, the sun being the middle string or key note (Mese) of the mode, and the moon (nete) at the top string, and the Chronos (Saturn) at the lowest string (hypate), which was kept at a constant pitch." “The modes were built on one or two ‘tetrachords’.” "One tetrachord descended a fourth from the keynote (mese), and the other ascended a fourth, making a heptatonic, or seven note scale." (Gardner, K. 1990 p133) A tetrachord is a mini-scale of four notes, in descending pitch order, that are contained within a perfect fourth (five half steps) instead of an octave (twelve half steps). According to Catherine Schmidt-Jones, we have very detailed descriptions of these tetrachords, in ‘Harmonics’, written by Aristoxenus in the fourth century BC, but there is no way of knowing exactly what pitches were being used in them. (Schmidt-Jones, C 2010) We can only guess what music from ancient Greek times really sounded like. The Ancient Greeks didn’t write down their music but they did write about their music. References in Ancient Greek texts to "shading" suggest that they probably did not use the same intervals we do. It is more likely that ancient Greek music sounded more like other traditional Mediterranean and Middle Eastern music. (Schmidt-Jones, C 2010) "The Lesbian mode was used for open and joyful music. Often used in religious chants of the day, it was extroverted and happy." The Church changed its name to Mixolydian from Lesbian!” (Gardner, K. 1990 p134) "For melancholy, quiet and dreamy moods, the Aeolian mode was used. Intimate and poetic, this mode brought a simple, sweet quality to its melodies. The Aeolian mode still survives today as the natural minor scale." (Gardner, K. 1990 p134) "The Phrygian mode was dramatic and passionate, emphasising inner sadness and sensitivity in its expression." (Gardner, K. 1990 p134) "The strangest mode was Lydian, strange to medieval times because it had a raised fourth (tritone), and was thought to the Asian in origin. In the Republic (572 BC) Plato banned the Lydian mode because it was too feminine." (Gardner, K. 1990 p135)

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"Monique Witting, in her book ‘Lesbian Peoples’, writes that the Lydians were a tribe of Amazons that settled on Crete." (Gardner, K. 1990 p136) In his book ‘The Republic’ Plato says that playing music in a particular ‘harmonia’ (mode) would incline one towards specific behaviours associated with it. He suggested that soldiers should listen to music in the Dorian or Phrygian modes to help make them stronger, but avoid music in Lydian, Mixolydian or Ionian modes. (Mathiesen, T. J. 1999) Plato believed that boys should be allowed to hear music only in the Dorian and Phrygian modes. This would make them imitate the actions of a brave man defending himself against fortune steadily with endurance. Plato believed that a change in the musical modes of the state would cause a wide-scale social revolution. Plato believed that the strongest of all life’s influences is music. Plato forbade music in the Mixolydian and Lydian modes for his boys. He said that these modes were "useless even for women if they are to be decent". He said that music in the Ionian mode is "soft, lazy, and fit for drunkenness". (Gardner, K. 1990 p136) In ‘Laches’, Plato calls the Dorian the real Greek mode, and says that it creates a feeling of sincerity. He contrasts it with the Ionian, Phrygian, and Lydian. The Dorian would remain legal for use in war and crisis, and the Phrygian for peace, dignity, temperance, and worship. These comments indicate that lyres were manufactured in various modes. Plato wanted the government to control popular music by allowing only the manufacture of lyres built for the approved modes. In particular, Plato wanted to ban lyres with lots of strings ("a multiplicity of notes") able to play several different modes. Aristotle in the "Politics" said that the Mixolydian mode can make you sad and grave. He said that the Dorian mode settles the mind and is gravest and manliest and "avoids extremes". The Phrygian mode inspires enthusiasm and is exciting and emotional and the best for expressing "Bacchic fury". In contrast the "relaxed modes", i.e., Lydian and Ionian make people stupid. Complaints about styles of music corrupting young people aren't new! Clement of Alexandria knew the Dorian and Phrygian "harmonies" (tunings) and thought that David probably tuned his lyre to the Phrygian mode. From web site - www.pathguy.com/modes.htm Christian Church "The modes continued to be used into the times of the early Christian Church, their Greek names were maintained. Hundreds of years after their invention, when music notation began to be used in the monasteries of Europe, the modes came to be called Church Modes or Ecclesiastical Modes. The seven modes as we know them today are the Ionian, Dorian, Phrygian, Lydian, Lesbian (Mixolydian), Aeolian and Locrian. In the Middle Ages (400 AD to 1450 AD) each mode had its own set of ethics." (Gardner, K. 1990 p135) The Dorian mode was used for religious rituals and was the mode most used in Gregorian chant. Its magical, serene character brought a sense of exaltation to melodies. In the 1960s songwriter Donavan used the Dorian mode as did the Lennon and McCartney in the Beatles song 'Norwegian Wood'. The modes fell out of use when the character of melody changed. Ancient Greek music began as monophonic expression, meaning that it consisted only of one vocal or instrumental melody, often doubled at the octave and ornamented, but only consisting of a single melodic line. It was purely melodic and had no chords or harmony. The modes power was determined by the relationship of each of its tones to the central tone.

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In Western music ‘monophony’ was the main form of musical expression up until about 1000 AD when polyphony, meaning 'more than one voice' began to make musical inroads." (Gardner, K. 1990 p135) At first voices sang the melodies only in unison or octaves. In Gregorian chant they began to sing at distances of a fourth or a fifth from each other, still in modes and moving along in perfect parallels. These fourth and fifths were consonant intervals and sounded very pure to the ear. In twelfth century sacred music the principle melody, called 'plain chant,' was slowed down, and upper voices, moving more quickly, provided counter melodies. By the fourteenth century, because of corruption and economic problems, music moved away from the Church. In Western secular music, parallel thirds and sixths began to occur more often. More diverse intervals were used as well as clashing melodies and dissonance. "I believe that by the fifteenth century with the introduction of the bass voice line, harmony became more important than melody. The lowest voice was now the foundation for everything above it, usually three other voices. Rather than flowing in a horizontal, lyrical fashion, music, because of chords, took on a vertical character. Melodies now longer existed as powerful entities; they came to be constructed on the basis of the base line and harmony." (Gardner, K. 1990 p138) By the sixteenth century melody had become subservient to and dependent upon harmony. The modes disappeared from use because of the change brought on by harmony. The modes became condensed onto the two new scales - major and minor. Later the major and minor scales dissolved into the chromatic twelve-note scale. Melody began to move from semi-tone to semi-tone, half-step to half-step, all dictated by the harmony system devise within the more convenient equal temperament tuning system. (Gardner, K. 1990 p142) According to Don Campbell the different Greek modes were merged into the Major scale (viewed as happy) and the Minor scale (seen as sad). This development meant the loss of the range and subtlety of emotions available to the ancients. Music was used as a form of medicine up until medieval times. Each of the Greek modes had its own character and healing power. After the fifteenth century music was used less for healing and more for aesthetic purposes. (Campbell, D (ed) 1991) The modern modes that are playable in the well-tempered 12-tone system are named using the ancient Greek names, and can be described as the diatonic (white key) sequences: Ionian

The Ionian scale is the major scale. Is the root scale of a key. C Ionian scale is C major scale. It starts on the first note of the major scale and you play it just like the major scale. Dorian

The Dorian mode is a scale that starts and ends on the second note of the major scale. So for example, in the key of C, instead of starting the scale on C, you start on D.

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Phrygian Scale

The Phrygian scale is similar to the Dorian scale only it starts a note higher. It starts on the 3rd note of the major scale. In the key of C, this scale starts and ends on E. Lydian

The Lydian scale starts and ends on the 4th note of the major scale. In the key of C, that would be F. Mixolydian

The Mixolydian scale starts and ends on the 5th note of the major scale. In C major, that would be G. Aeolian

The Aeolian scale is also known as the natural minor scale. It starts and ends on the 6th note of the major scale. In the key of C, it starts and ends on A. Locrian

The Locrian scale begins and ends on the 7th (and final) note of the major scale. In C major, that would be B. From the web site - http://www.piano-lessons-info.com/music-scale-modes.html

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The Ancient Greek Modes: Familiar Tunes Dorian C-D-E-F-G-A-A# Yankee Doodle (verse) Man on the Flying Trapeze My Grandfather's Clock Hi Ho ("Snow White") Ionian C-E-F-G-A-A#-C My Darling Clementine Streets of Loredo I Saw Three Ships Locrain C-C#-D#-F-F#-G#-A# My Country 'Tis of Thee It's a Wonderful Life Blowin' in the Wind (Bob Dylan) Lydian C-D-D#-F-G-A#-C Danny Boy (Verse) Mixolydian C-D#-F-G-G#-A#-C When Johnny Comes Marching Home Rock of Ages Phrygian C-D-E-F-G-A-C Ten Little Indians Where Have You Been Billy Boy Oh! Susannah Aeolian C-E-F-G-A-B-C Popeye the Sailor Man Sailing, Sailing Jingle Bells Rock Harry Potter theme The Ancient Musical Modes: What Were They? Ed Friedlander MD From the web site - http://www.pathguy.com/modes.htm

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What are Harmonics? Pythagorus Pythagoras lived on the Greek island of Samos from 560-480 BC. He is credited with the invention of the monochord and the discovery of the harmonic ratios in sound. Pythagoras said "study the monochord and you will know the secrets of the universe." The monochord is a long wooden box with one long string attached to two raised pieces of wood to permit it to vibrate. When the string is plucked it produces a sound that is called the fundamental tone of the string. Using a wooden bridge we can divide the string in two. When either half of the now divided string is plucked it will produce the same note as the fundamental except it will be an octave (eight notes) higher, since it is vibrating twice as fast as the vibration of the fundamental. If the string were divided equally into three, the note produced would be different from the fundamental note. If the fundamental note was 'C' the note produced would be 'G', an octave above the fundamental. If the string were divided equally into four the note produced would be the same as the fundamental, but it would be two octaves higher. Pythagoras found that whenever the whole string was plucked, higher sounds would be created at the same time as the fundamental note. These higher sounds or 'harmonics' were mathematically related in frequency to the fundamental sound through whole number ratios of 2:1, 3:1, 4:1. These harmonics were related to each other in ratios of 2:3, 4:3, 5:8. Pythagoras discovered that these ratios were found in nature and the proportions used in art and architecture. Pythagoras saw the universe as a giant monochord, an instrument that stretched between heaven and earth. The higher sounds were those of pure spirit and the lower ones were those of the earth. Pythagoras applied the law of harmonic intervals to all phenomena in nature, the elements, the planets and constellations. How are Harmonics Produced? If you were to attach a guitar string tightly between two posts and sharply pluck it, you would observe the string moving up and down in a wave motion. This movement would produce its fundamental pitch. As the sound waves travel to the ends of the string they bounce back along the string like a wave hitting a rock and rebounding. Waves travelling in opposite directions collide to create smaller waves. This effect produces the higher sounds that Pythagoras described. FIGURE THREE - Waves formed when a string is plucked

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As the waves pass back and forth along the string, the string will divide into proportions mathematically related to the overall length of the string. Each part of the string will produce individual tones called a 'partials' or 'overtones' because they form only part of the overall sound produced. The overall length of the string will create the lowest pitch sound with the longest wavelength. This is called the fundamental tone of the string. This sound wave has the longest wavelength and the greatest amplitude and is the easiest to hear of all the tones produced. The other tones or partials will have shorter wavelengths and higher pitches than the fundamental. They will be more difficult to hear but will add colour and richness to the fundamental sound. A harmonic is a sound frequency that has a special relationship to the fundamental sound. If we go back to our original example of the string tied between two posts, the lesser tones created when the string is plucked will all be related to the fundamental tone through whole number ratios. If we pluck a middle 'C' string with a frequency of 256Hz the string will divide itself into a series of proportions based on whole number ratios creating a series of harmonics. The first harmonic produced will be twice the frequency of the fundamental as the string divides itself into two equal halves. This will resonate at 512Hz and will be a 'C' an octave above the fundamental. The Waveforms of the Harmonics Created when a Middle 'C' string is Plucked

The next division that occurs will be a division of the string into thirds, with each third of the string producing a tone three times the frequency of the fundamental at 768 Hz, which will produce the note 'G' an octave above the fundamental. The next division will be a quarter, the frequency will be 1,044 Hz and the note will be a 'C', two octaves above the fundamental note. This sequence carries on to infinity. These harmonics are sometimes called 'unstruck' sounds because they are formed as a consequence of creating a fundamental sound. What reaches the ear is a series of pure tones; a series of waves, where each wave produces lesser waves, and so on to infinity.

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When the upper partials vibrate in whole number ratios to the fundamental tone these are called 'harmonics'. All partials are not harmonics, whereas all harmonics are partials. Some partials are 'inharmonic' due to the nature of the instrument producing them. Harmonics in Air Columns Many musical instruments consist of an air column enclosed inside of a hollow metal tube. Though the metal tube may be more than a meter in length, it is often curved upon itself one or more times in order to conserve space. If the end of the tube is uncovered such that the air at the end of the tube can freely vibrate when the sound wave reaches it, then the end is referred to as an open end. If both ends of the tube are uncovered or open, the musical instrument is said to contain an open-end air column. A variety of instruments operate on the basis of open-end air columns; examples include the flute and the recorder. Even some organ pipes serve as open-end air columns. A musical instrument has a set of natural frequencies at which it vibrates at when it is played. These natural frequencies are known as the harmonics of the instrument; each harmonic is associated with a standing wave pattern. A standing wave pattern is as a vibrational pattern that is created when reflected sound waves from one end of instrument to interfere with sound waves from the sound source in such a manner that specific points in the air column appear to be standing still. In the case of stringed instruments standing wave patterns appear as nodes, points of no displacement or movement along the string. In the case of air columns, a closed end in a column of air is analogous to the fixed end on a vibrating string. At the closed end of an air column, air is not free to undergo movement and thus is forced into assuming the nodal positions of the standing wave pattern. Conversely, air is free to undergo its back-and-forth longitudinal motion at the open end of an air column; and as such, the standing wave patterns will depict antinodes at the open ends of air columns. An antinode in a standing wave is the region where the medium is in motion. This is the point of maximum movement between the nodes. In the diagram below of the first harmonic the antinodes are at the end of the tube while the node is in the middle of the tube. So the basis for drawing the standing wave patterns for air columns is that vibrational antinodes will be present at any open end and vibrational nodes will be present at any closed end. If this principle is applied to open-end air columns, then the pattern for the fundamental frequency (the lowest frequency and longest wavelength pattern) will have antinodes at the two open ends and a single node in between. For this reason, the standing wave pattern for first two harmonics for an open-end air column looks like the diagram below.

Inharmonics The strings of a piano are made of thick steel and are constructed to be very taut to withstand the striking of the felt tipped hammers in loud sections of music. The piano is heavy because of its iron frame. This frame is necessary to keep the strings tight. Piano strings are much thicker than the stings of a guitar, harp or violin. Their thickness and taughtness means they behave very differently from the strings of other instruments.

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The stiffness of a piano string causes its vibration to accelerate rapidly when struck and then to die down quickly. The piano does not create a steady waveform like other string instruments. Its partials are inharmonic rather than harmonic. John Backus says that the partials in the tone of a piano string are not integral multiples of the fundamental frequency, and so they are not strictly harmonics. The higher partials become progressively sharp with respect to the harmonic frequencies. When one piano was measured, the frequency of the fifteenth partial was nearly sixteen times the fundamental note. (Backus, J.1977) Standard of Pitch For many centuries in the West it has been customary to take the frequency of the note 'A' above middle 'C' as the standard reference for pitch. In the early days of music there was no standard frequency for 'A', each village would take its own 'A' from the local church organ. As a result each town's 'A' could be semi-tones apart from the next town. It was not until John Shore, a contemporary of Handel, invented the tuning fork in 1711 that a standard pitch for 'A' was available. Handel's 'A' tuning fork resonated at a frequency of 422.5 Hz. John Backus says that by the middle of the eighteenth century 'A' was established around the frequency of 415-428 Hz. Over the years the frequency of the standard 'A' has crept upwards as the string sections of orchestras tune up their instruments to get a brighter sound. (Backus, J.1977) The rest of the orchestra would then tune up to this higher ‘A’ and the whole process would begin again leading to ‘A’ getting higher and higher. By the end of nineteenth century orchestra's in the UK were playing 'A' at the frequency of 455Hz and in the USA 'A's had crept up to the frequency of 461Hz. In 1953 the International Standards Organisation recommended the adoption of 440Hz for 'A' as the frequency around the world. However there is still pressure to increase the frequency of 'A'. At present the Los Angeles Philharmonic Orchestra tunes to 'A' at 442Hz! John Backus says as a result of the rise in the frequency of 'A', singers are now singing arias in operas by Mozart and Beethoven at least a semi-tone (18-20Hz) higher than the pitch for which they were written. (Backus, J.1977) Harmonic Scale At one time it was proposed that there should be a 'scientific' musical scale for which the various 'C's were powers of the number 2: - 2, 4, 8, 16, 32, 64, 128, 256Hz, etc. In this scale middle 'C' would have a frequency of 256 Hz. Unfortunately this scale was never adopted by musicians and according to John Backus is now obsolete. However ear, nose and throat specialists still test hearing using middle 'C' tunings forks at 256Hz. (Backus, J.1977) Instruments All instruments produce partials. The partials are what gives an instrument its unique tonal quality or timbre. The partials of a flute will be different from a clarinet. The partials produced will be influenced by the size and the shape of the instrument, the materials used in its construction, as well as by the way the person plays the instrument. Apart from the piano most non-electronic instruments will produce true harmonics. The flute has fewer harmonics and produces the simplest waveform. The clarinet emphasises the uneven numbered harmonics, particularly the 7th and 9th harmonics because of the shape of its mouthpiece and bell shaped end. An oboe has a complete range of harmonics because its tube is conical. The string instruments produce more high harmonics compared to woodwind instruments which produce more middle range harmonics.

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Instruments that produce a drone like the didgeridu, the tamboura and the bagpipes all produce a sound that is rich in harmonics. With these instruments the fundamental note is generally sustained for a long time allowing the listener more opportunity to hear the rich mixture of harmonics produced. Jonathan Goldman says that in an experiment harmonics were electronically removed from the sounds of three instruments, a violin, a trumpet and a piano. When the sounds were played to listeners they were unable to recognise each instrument. (Goldman, J. 1992) Electric organs produce pure tones without harmonics. Harmonics need to be added by a special process. If we take the note 'C' for example the harmonics 'C' an octave above and 'G' an octave above are added to the fundamental tone to make the organ sound more like a piano or church organ. According to John Backus the partials of the tone produced by electric organs cannot be called harmonic in the strict sense. The fifth harmonic is added by using the tone two octaves and a major third above the lowest tone, and since this third is in the well-tempered scale its frequency will not be exactly five times the fundamental frequency. The seventh harmonics will be missing since there are no notes in the tempered scale close enough to this sound frequency that can be used. (Backus, J.1977) Human Voice Each human voice also has its unique harmonic pattern that helps us to distinguish one voice from another when speaking on the telephone or listening to the radio. Each person has a unique voice pattern, as unique as a fingerprint. Harmonics Over Time Nadia Boulanger, the great music teacher has studied the way the harmonics and intervals have been used in music over time. She said that today's dissonance is the tomorrow's consonance. The intervals and chords used in music have become increasingly complicated over time: - Antiquity Harmonics 1-3 Middle Ages and Renaissance Harmonics 1-5 Modern Era (1600-1850) Harmonics 1-9 Contemporary Period Harmonics 1-13 (Campbell, D.1991) Gregorian Chant The early Gregorian scales were written down in the seventh century AD. They were based on the Greek modes: Dorian, Phrygian, Lydian and Mixolydian. In the eleventh century after centuries of unison singing the interval of the fifth was introduced in a style called Organum. The interval of the fifth is the basis of most of the music of the world. The earliest example of harmony in Western music is Gregorian Chant. Here one group of singers is singing a melody while another group sings the same melody five notes above. It is believed that the Gregorian chant we are familiar today developed when the monks who were used to singing plainchant started to hear the fifth as a harmonic when they sung in the highly resonant cathedrals and monasteries. Some of the monks began to sing what they were hearing, creating this natural interval. The building of the great Gothic cathedrals coincided with the flowering of liturgical music that took place around the tenth and eleventh centuries in Europe. There was an enormous increase in the number of monasteries at this time as wealthy landowners left their estates to the church in the hope of obtaining redemption for their sins.

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The height of the ceilings in these great cathedrals made it possible to hear the very high harmonics for the first time. Listening to these high harmonics would transport listeners to a higher state of consciousness. The Masons who built the Gothic cathedrals had a deep understanding of sacred geometry and the connection between sound and architecture. In the fourteenth century the church considered intervals like the fourth and the tri-tone (e.g. C-F#) evil, the tri-tone was called 'the devil in music.' The 'evil' sounds of the 14th century are now quite acceptable to our ears. Debussy and Ravel would have been burned in the Inquisition if they had lived at that time.

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Harmonics and Intervals There is a relationship between the harmonics and intervals. If we take the sound frequency 100Hz for example it will produce the following harmonics and intervals. One note will produce all these harmonics and intervals. Fundamental 100 Hz \ / 200 = 2.1 Octave 100 1st Harmonic 200 Hz \ / 300 200 = 3.2 Fifth 2nd Harmonic 300 Hz \ / 400 300 = 4.3 Fourth 3rd Harmonic 400 Hz \ / 500 400 = 5.4 Third 4th Harmonic 500 Hz \ / 600 500 = 6.5 Minor Third 5th Harmonic 600 Hz 6th Harmonic 700 Hz 7th Harmonic 800 Hz \ / 900 800 = 9.8 Major Second 8th Harmonic 900 Hz Other Intervals Created 800 500 = 8.5 Minor Sixth 500 300 = 5.3 Sixth 1500 800 = 15.8 Seventh