MATHEMATICS and MUSIC
MUSICAL DOW In 1987, there was an article in the Hartford Courant about Myron Schwager, a cellist, who teaches at the Hartt School of Music, part of the University of Hartford. He has a stock market charting system based on scales, and I wrote him a letter. See 1987 Musical Dow. A month or so later, as I was a senior officer, Mary and I were invited to a black tie ITT/Hartford Board dinner at the Hartford Club. A small group played music during the dinner. I recognized the cellist from his picture in the Hartford Courant article. After dinner I went up to Myron Schwager and asked him what instrument he was playing. He said "It's a cello." I asked if I could look at it. He handed it to me. I held it up to my ear and exclaimed: "Its playing the Dow!" He said: "You must be Don Sondergeld."
Symmetry and Music The six symmetries of music refers to a set of transformations that can be applied to music while leaving a fundamental essence of the music unchanged. The six symmetries are: pitch translation invariance, time scaling invariance, octave translation invariance, time translation invariance, amplitude scaling invariance, and pitch reflection invariance. Also see: http://orion.math.iastate.edu/mathnight/activities/modules/music/
Patterns by Natasha Glydon:
Musical pieces are read much like you would read math symbols. The symbols represent some bit of information about the piece. Musical pieces are divided into sections called measures or bars. Each measure embodies an equal amount of time. Furthermore, each measure is divided into equal portions called beats. These are all mathematical divisions of time.
Fractions are used in music to indicate lengths of notes. In a musical piece, the time signature tells the musician information about the rhythm of the piece. A time signature is generally written as two integers, one above the other. The number on the bottom tells the musician which note in the piece gets a single beat (count). The top number tells the musician how many of this note is in each measure. Numbers can tell us a lot about musical pieces.
Each note has a different shape to indicate its beat length or time. Notes are classified in terms of numbers as well. There are whole notes (one note per measure), half notes (two notes per measure), quarter notes (four notes per measure), eighth notes (eight notes per measure), and sixteenth notes (sixteen notes per measure). These numbers signify how long the notes last. That is, a whole note would last through the entire measure whereas a quarter note would only last one quarter of the measure and thus there is enough time for four quarter notes in one measure. This can be expressed mathematically since 4 x 1/4 = 1. A note with a dot after it lengthens the note by half. For example, a quarter note with a dot after it would be held for 3/8 of a measure, since: 1/4 + 1/2(1/4) = 3/8.
Three eigths of a measure is midway between a quarter note and a half note. It is important for musicians to understand the relationships and values of fractions in order to correctly hold a note.
Fibonacci: The Fibonacci sequence is a famous and well-known sequence that follows as: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on, adding each term to the one before it to create the next term. That is, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34, and continuing infinitely. In music, the Fibonacci sequence can be seen in piano scales. For example, the C scale on the piano consists of 13 keys from C to C; eight white keys and five black keys, with black keys arranged in groups of three and two. In the Fibonacci sequence, the ratio between each term is very close to 0.618, which is known as the golden ratio.
Pythagoras and Frequency: It was Pythagoras who realized that different sounds can be made with different weights and vibrations. This led to his discovery that the pitch of a vibrating string is proportional to and can be controlled by its length. Strings that are halved in length are one octave higher than the original. In essence, the shorter the string, the higher the pitch. He also realized that notes of certain frequencies sound best with multiple frequencies of that note. For example, a note of 220Hz sounds best with notes of 440Hz, 660Hz, and so on.
The closest tie between music and math is patterns. Musical pieces often have repeating choruses or bars, similar to patterns. In mathematics, we look for patterns to explain and predict the unknown. Music uses similar strategies. When looking at a musical piece, musicians look for notes they recognize to find notes that are rare (high or low) and less familiar. In this way, notes relate to each other. Relationships are fundamental to mathematics and create an interesting link between music and math. See http://www.italmeds.com/index.php?option=com_content&view=article&id=110:music-and-mathematics&catid=87:music-philosophy
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems". It may be derived from the harmonic series as the interval between the first and second harmonics.
The frequency of middle C on a piano is often set at 261.6 Hz. There are twelve semitones in an octave. A piano keyboard has 7 white keys and 5 black keys to play notes within any octave. A trumpet has 3 valves that can be all open, two closed, etc. with 7 combinations of fingering, fingering, so a trumpet can only play 7 tones within an octave.
Also the trumpet reference is mathematically correct about the valve openings, but trumpet players use different techniques and can play a full range of notes within an octave, and also over several octaves."
The treble clef and bass clef are most commonly used in sheet music. The numbers below indicate which octave the note is in. There are 12 semitones in an octave. There are five lines and four spaces in a clef.. In the treble clef the spaces contain F, A, C, E and the lines contain E, G, B, D, and F. In an octave, not counting E and F twice, the lines and spaces handle 7 of the semitones, and special sharp or flat symbols are used to designate the other 5 semitones in the octave.
Don Francis writes: "The ratio of the interval ending on F sharp is sometimes called the devil's interval, probably because there is no good ratio to define it. It was rarely used in pre 20th century music because of perceived dissonance. It came to typify some types of jazz (the famous flatted fifth of bebop). A good reference for remembering the interval is the first two notes of Maria."
A fourth is a musical interval encompassing four staff positions in sheet music. For example, the ascending interval from C to F is a perfect fourth, as there are four staff positions including C to F (counting C, D, E, and F). F lies five semitones above C (counting C sharp, D, D sharp, E, and F).
A fifth is a musical interval encompassing five staff positions in sheet music. For example, the ascending interval from C to G is a perfect fifth, as there are five staff positions including C to G (counting C, D, E, F, and G). G lies seven semitones above C (counting C sharp, D, D sharp, E, F, F sharp, and G).
Heinrich Rudolf Hertz (1857-1894): A Hertz, or Hz, is named after this German physicist. It is a measure of the frequency, the number of vibrations of a string per second. People in different musical traditions have different ideas about which notes they think sound good together. If you double the frequency, the human ear tends to hear both notes as the same. This is called "octave equivalency". The doubled frequency is called a higher octave. This "octave" is two times higher, not eight times higher. In the "diatonic scale", there are 8 notes counting both ends of the octave hence the term "octave". In the "chromatic scale" there are 13 notes counting both ends, and the "Arab classical scale" has 17, 19, or even 24 notes in its "octave.
Scales can be classified as "Just" (See http://en.wikipedia.org/wiki/Just_intonation (if the ratios between the frequencies are ratios of integers), "tempered" (if the just scale is tempered), and "practice-based" (if it reflects musical practice) In the Even Tempered Scale going from one semitone to the next is the 12th root of 2, or 1.05946... (Pythagoras discovered that frequencies whose ratio is equal to the ratio of two simple whole numbers yield "harmonious" and pleasing sounds. The ratio of 3:2 is 1.5. A "perfect fifth" corresponds to a separation of seven semitones, as the seventh power of 1.05946 is close to 1.5. A "perfect fourth" corresponds to a frequency ratio of 4/3 and five semitones.)
A harmonic series is the sequence of all multiples of a base frequency. Pitched musical instruments often use a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves. Interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the typical spacing of the resonances, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest frequency, and such multiples form a harmonic series
The harmonic series is an arithmetic series (1×f, 2×f, 3×f, 4×f, 5×f, ...). In terms of frequency (measured in cycles per second, or hertz (Hz) where f is the fundamental frequency), the difference between consecutive harmonics is therefore constant and equal to the fundamental. But because our ears respond to sound nonlinearly, we perceive higher harmonics as "closer together" than lower ones. On the other hand, the octave series is a geometric progression (2×f, 4×f, 8×f, 16×f, ...), and we hear these distances as "the same" in the sense of musical interval. In terms of what we hear, each octave in the harmonic series is divided into increasingly "smaller" and more numerous intervals.
The second harmonic (or first overtone), twice the frequency of the fundamental, sounds an octave higher; the third harmonic, three times the frequency of the fundamental, sounds a perfect fifth above the second. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher).
Pitches are labeled using: i) Letters, as in Helmholtz pitch notation, ii) a combination of letters and numbers—as in scientific pitch notation, where notes are labelled upwards from C0, the 16 Hz C or iii) Number that represent the frequency in hertz (Hz), the number of cycles per second.
For example, one might refer to the A above middle C as A4, or 440 Hz. In standard Western equal temperament, the notion of pitch is insensitive to "spelling": the description "G4 double sharp" refers to the same pitch as A4; in other temperaments, these may be distinct pitches. Human perception of musical intervals is approximately logarithmic with respect to fundamental frequency: the perceived interval between the pitches "A220" and "A440" is the same as the perceived interval between the pitches A440 and A880.
Motivated by this logarithmic perception, music theorists sometimes represent pitches using a numerical scale based on the logarithm of fundamental frequency. For example, one can adopt the widely used MIDI standard to map fundamental frequency, f, to a real number, d, as follows
If ƒ is a frequency, then the corresponding frequency data value d may be computed by
The quantity log2 (ƒ / 440 Hz) is the number of octaves above the 440-Hz concert A (it is negative if the frequency is below that pitch). Multiplying it by 12 gives the number of semitones above that frequency. Adding 69 gives the number of semitones above the C five octaves below middle C.
Since 440 Hz is a widely used standard concert A (e.g. USA, UK), and since that is represented in MIDI terms by the integer 69 (nine semitones above middle C, which is 60), this gives a real number which expresses pitch in a manner consistent with MIDI and integer notation, known as the midi note number.
Converting from midi note number (d) to frequency (f) is given by the following formula:
This creates a linear pitch space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1.
MIDI Note Numbers for Different Octaves
The relative pitches of individual notes in a scale may be determined by one of a number of tuning systems. In the west, the twelve-note chromatic scale is the most common method of organization, with equal temperament now the most widely used method of tuning that scale. In it, the pitch ratio between any two successive notes of the scale is exactly the twelfth root of two (or about 1.05946). In well-tempered systems (as used in the time of Johann Sebastian Bach, for example), different methods of musical tuning were used. Almost all of these systems have one interval in common, the octave, where the pitch of one note is double the frequency of another. For example, if the A above middle C is 440 Hz, the A an octave above that is 880 Hz .
Below are the Hertz values for the octave beginning with middle C
You will note in the table below, the log of 1.05946 in base 2 is .08333. Check the calculation at http://logbase2.blogspot.com/2008/08/log-calculator.html
The following table (from http://en.wikipedia.org/wiki/Music_and_mathematics) reveals how accurately various equal-tempered scales approximate three important harmonic identities: the major third (5th harmonic), the perfect fifth (3rd harmonic), and the "harmonic seventh" (7th harmonic).
The Chromatic Scale in C has 12 notes, and uses every half-tone / semitone position.
BLACK KEYS ON A PIANO
At Carnegie Hall, gospel singer Wintley Phipps delivers perhaps the most powerful rendition of Amazing Grace ever recorded. He says, "A lot of people don't realize that just about all Negro spirituals are written on the black notes of the piano. Probably the most famous on this slave scale was written by John Newton, who used to be the captain of a slave ship, and many believe he heard this melody that sounds very much like a West African sorrow chant. And it has a haunting, haunting plaintive quality to it that reaches past your arrogance, past your pride, and it speaks to that part of you that's in bondage. And we feel it. We feel it. It's just one of the most amazing melodies in all of human history." After sharing the noteworthy history of the song, Mr. Phipps delivers a stirring performance that brings the audience to its feet!
"Amazing Grace" is a Christian hymn published in 1779, with words written by the English poet and clergyman John Newton (1725–1807). Newton wrote the words from personal experience. He grew up without any particular religious conviction, but his life's path was formed by a variety of twists and coincidences that were often put into motion by his recalcitrant insubordination. He was pressed (conscripted) into service in the Royal Navy, and after leaving the service, he became involved in the Atlantic slave trade. In 1748, a violent storm battered his vessel off the coast of County Donegal, Ireland, so severely that he called out to God for mercy, a moment that marked his spiritual conversion. He continued his slave trading career until 1754 or 1755, when he ended his seafaring altogether and began studying Christian theology.Amazing grace!
How sweet the sound
I once was lost, but now am found,
'Twas grace that taught my heart to fear,
Through many dangers, toils and snares
The Lord has promised good to me
Yea, when this flesh and heart shall fail,
When we've been there ten thousand years
Probably the most marvelous fugue was the one between the Hatfields and the McCoys.
When electric currents go through them, guitars start making sounds. So would anybody.
"If a thing
isn't worth saying, you sing it." -- Pierre Beaumarchais, The Barber of
possibly hear the last movement of Beethoven's Seventh and go slow." --
Oscar Levant, explaining his way out of a speeding ticket.
"I would rather play Chiquita Banana and have my swimming pool than play Bach and starve." -- Xavier Cugat
Augmented fifth: A 36-ounce bottle.
English horn: A woodwind that got its name because it's neither English nor a horn. Not to be confused with French horn, which is German.
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Perfect Pitch: When you throw a viola into the toilet and it doesn't hit the sides.
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you run over first?
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Q: What do you call a musician with a college degree?
This site was last updated 04/08/17