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# Math In Music

Q4 Math Project

by

Tweet## Sara Kimble

on 6 June 2013#### Transcript of Math In Music

By Jasmine Falk, Katie Pinto, and Sara Kimble Math in Music Time Signatures The top number of a time signature represents the number of beats in the measure. Types of Notes Before we get started,

here are some basic music terms you need to know. Measure: the groupings that notes are divided by within a song

Bar line: the lines that divide the measures Dotted Notes A dotted note means that you take the note value and add an additional half of the note's value to the duration of its count. How is Math related to Music? In a 4/4 Time Signature: Math is found in music through: Fractions of Notes

Time Signatures

Pitch Intervals

Frequencies

The Harmonic Series The bottom number of a time signature represents which note gets the beat. The most common time signature is the one pictured above, 4/4. In a 4/4 time signature, or "common time," the quarter note gets the beat and there are 4 beats in a measure.

But what's a quarter note? Clef: the symbol at the beginning of a staff that tells the musician the pitch of the notes in that staff. Treble Clef Bass Clef 4 beats 2 beats 1 beat 1/2 beat 1/4 beat To find the length of a dotted note, the formula is 0.5n + n,

or 3n/2. Now that we have a basic understanding of some musical terms, let's take a look at how sound reaches our ears... Frequency is defined as the number of times per second sound waves hits our ears (measured in Hertz). What is a frequency? What is pitch? Pitch is the frequency of a note that determines how high or low it sounds. Music Staffs A staff consists of 5 lines and 4 spaces.

Each one represents a different note. The notes are either placed on a line or in a space. What do notes look like on an instrument? How does frequency relate to pitch? The greater the frequency is, the higher the pitch. This means that the faster a string on a piano or violin, for example, vibrates, the higher the note will sound to the human ear. For example: The note A has a frequency of 440, while E (a 5th above A) has a frequency of 660. This explains why we perceive A to sound lower than E. In fact, there is a formula to calculate the frequency (in Hz) of any note: MIDI Pitch Values MIDI is a technical standard used for computerized music. Every note is assigned a "MIDI pitch" value, which we can substitute into the formula. Using this formula, let's calculate the frequency of middle C. From the MIDI pitch value chart, we know that the MIDI value of C is 60. When we substitute this into the equation, we get: Thus, middle C has a frequency of 261.626 Hz. *This formula can be applied to any pitch using the MIDI pitch value chart. * Why do certain chords sound better than others? In short: ratios! The frequencies of two notes can be compared in a ratio.

This ratio tells us how well the waves of the notes will match up, and consequently how well the notes will sound when played together. For example... Let's first take the case of two notes that sound good together; C and G. Frequency of middle C ≈ 261.6 Hz

Frequency of G ≈ 392.0 Hz

Ratio of C/G = 261.6/392.0 ≈ 2/3

This tells us that every 2nd C wave matches up with every 3rd G wave,

which is why they sound good together. The Harmonic Series The Harmonic Series, an infinite series, is a naturally occurring progression of pitches that is present in every note we hear. To Sum it All Up... Math can be found everywhere

in music! Whether it is through fractions of notes, time signatures, pitch intervals, frequencies, or the Harmonic Series, math can be found and appreciated in music in many ways. The next time you hear your favorite song on the radio or sit down to play a piece on the piano, you're learning more than just music - you're doing math too! Sources: Thanks for Watching! •Ms. Berry and Ms. Notari! The Harmonic Series is a mathematical representation of how a string vibrates.

A string vibrates not only as a whole, which is the primary pitch (fundamental) we hear, but also in fractional segments (e.g. halves, thirds, fourths, etc.) known as overtones. This is the same as if there were many strings being played simultaneously, each one a fraction of the length of the fundamental. Each segment vibrates faster, creating a higher pitch. In every primary tone that we hear, such as middle C, we are also hearing all the overtones (the notes above it that are in its Harmonic Series) Applying this to the equation... Explaining the Equation The 1 represents the string's vibration as a whole

The fractions following the 1 represent the string's vibration in segments. These are the overtones

For instace, the second term in the equation is 1/2. This means that the overtone represented by 1/2 vibrates at a length half of the whole string. Thus, this overtone has a frequency 2 times that of the fundamental.

The numbers in the series are added together to represent a string's total vibrating motion (the whole vibration + all its overtones) Our Experiment

We created an experiment to test if a piano's strings lengths are correctly predicted by the harmonic series.

We measured the lengths of certain strings of a piano to show that they followed the pattern of terms in the harmonic series equation. For example... In contrast to the pleasing sound of C and G, the notes C and C# sound very dissonant together. Frequency of middle C ≈ 261.6 Hz

Frequency of C# ≈ 277.2 Hz

Ratio of C/C# = 261.6/277.2 ≈ 17/18

This tells us that every 17th C wave matches up with every 18th C# wave.

It takes too many waves for them to match up which is why they do not sound good together. Middle C (marked #1 in the picture) is the fundamental note of this series. All the other notes above it which are marked in blue (C, G, C, E, F, Bb...), are overtones. All of these notes played together represent the total vibration of middle C.

According to the harmonic series equation, the first overtone should vibrate at a length 1/2 of the string of the fundamental. The first overtone of middle C is the C an octave above it, which must mean that the string length of high C is 1/2 the string length of middle C.

Likewise, the next overtone in the middle C harmonic series is G. Thus, the length of G's string must be 1/3 the length of middle C's string. And so on for the remaining overtones. We decided to experiement with the harmonic series of middle C To tie all our research together,

we picked a song and found its sheet music, then listened carefully for the chords. Finally... The Experiment Explained The song we chose is

"We are Young" by Fun. We measured the lengths of the first 6 notes in the harmonic series of middle C (the fundamental + 5 overtones). Below is a chart of the strings' lengths: The Sheet Music: Breaking down the chords As you can see from the purple circles, the key of this piece is in F major, which means that there is one flat (Bb - a black key). There are several different chords in this song that are circled in the picture in yellow. Let's listen to the notes of the above portion of the song in more detail.

Pay attention to the chords and syncopation. Our measurements show that the strings of the piano do approximately follow the harmonic series. In conclusion, our experiment proved that the harmonic series predicts the string lengths of a piano because the pattern (1, 1/2, 1/3, 1/4...) was present in overtones of middle C. Triplets Three notes played in the same amount of time as one or two beats. A half note is 2 beats (in a 4/4 time signature). In a triplet, 3 quarter notes would be squeezed into only 2 beats, as opposed to 3. The same logic applies to a quarter note triplet. One beat of a measure would actually have 3 eighth notes squeezed into it. Mathematically... Half Note Triplet Quarter Note Triplet Half note = 2 beats Quarter note = 1 beat 3 notes 3 notes = each note is 2/3 beat = each note is 1/3 beat

Full transcripthere are some basic music terms you need to know. Measure: the groupings that notes are divided by within a song

Bar line: the lines that divide the measures Dotted Notes A dotted note means that you take the note value and add an additional half of the note's value to the duration of its count. How is Math related to Music? In a 4/4 Time Signature: Math is found in music through: Fractions of Notes

Time Signatures

Pitch Intervals

Frequencies

The Harmonic Series The bottom number of a time signature represents which note gets the beat. The most common time signature is the one pictured above, 4/4. In a 4/4 time signature, or "common time," the quarter note gets the beat and there are 4 beats in a measure.

But what's a quarter note? Clef: the symbol at the beginning of a staff that tells the musician the pitch of the notes in that staff. Treble Clef Bass Clef 4 beats 2 beats 1 beat 1/2 beat 1/4 beat To find the length of a dotted note, the formula is 0.5n + n,

or 3n/2. Now that we have a basic understanding of some musical terms, let's take a look at how sound reaches our ears... Frequency is defined as the number of times per second sound waves hits our ears (measured in Hertz). What is a frequency? What is pitch? Pitch is the frequency of a note that determines how high or low it sounds. Music Staffs A staff consists of 5 lines and 4 spaces.

Each one represents a different note. The notes are either placed on a line or in a space. What do notes look like on an instrument? How does frequency relate to pitch? The greater the frequency is, the higher the pitch. This means that the faster a string on a piano or violin, for example, vibrates, the higher the note will sound to the human ear. For example: The note A has a frequency of 440, while E (a 5th above A) has a frequency of 660. This explains why we perceive A to sound lower than E. In fact, there is a formula to calculate the frequency (in Hz) of any note: MIDI Pitch Values MIDI is a technical standard used for computerized music. Every note is assigned a "MIDI pitch" value, which we can substitute into the formula. Using this formula, let's calculate the frequency of middle C. From the MIDI pitch value chart, we know that the MIDI value of C is 60. When we substitute this into the equation, we get: Thus, middle C has a frequency of 261.626 Hz. *This formula can be applied to any pitch using the MIDI pitch value chart. * Why do certain chords sound better than others? In short: ratios! The frequencies of two notes can be compared in a ratio.

This ratio tells us how well the waves of the notes will match up, and consequently how well the notes will sound when played together. For example... Let's first take the case of two notes that sound good together; C and G. Frequency of middle C ≈ 261.6 Hz

Frequency of G ≈ 392.0 Hz

Ratio of C/G = 261.6/392.0 ≈ 2/3

This tells us that every 2nd C wave matches up with every 3rd G wave,

which is why they sound good together. The Harmonic Series The Harmonic Series, an infinite series, is a naturally occurring progression of pitches that is present in every note we hear. To Sum it All Up... Math can be found everywhere

in music! Whether it is through fractions of notes, time signatures, pitch intervals, frequencies, or the Harmonic Series, math can be found and appreciated in music in many ways. The next time you hear your favorite song on the radio or sit down to play a piece on the piano, you're learning more than just music - you're doing math too! Sources: Thanks for Watching! •Ms. Berry and Ms. Notari! The Harmonic Series is a mathematical representation of how a string vibrates.

A string vibrates not only as a whole, which is the primary pitch (fundamental) we hear, but also in fractional segments (e.g. halves, thirds, fourths, etc.) known as overtones. This is the same as if there were many strings being played simultaneously, each one a fraction of the length of the fundamental. Each segment vibrates faster, creating a higher pitch. In every primary tone that we hear, such as middle C, we are also hearing all the overtones (the notes above it that are in its Harmonic Series) Applying this to the equation... Explaining the Equation The 1 represents the string's vibration as a whole

The fractions following the 1 represent the string's vibration in segments. These are the overtones

For instace, the second term in the equation is 1/2. This means that the overtone represented by 1/2 vibrates at a length half of the whole string. Thus, this overtone has a frequency 2 times that of the fundamental.

The numbers in the series are added together to represent a string's total vibrating motion (the whole vibration + all its overtones) Our Experiment

We created an experiment to test if a piano's strings lengths are correctly predicted by the harmonic series.

We measured the lengths of certain strings of a piano to show that they followed the pattern of terms in the harmonic series equation. For example... In contrast to the pleasing sound of C and G, the notes C and C# sound very dissonant together. Frequency of middle C ≈ 261.6 Hz

Frequency of C# ≈ 277.2 Hz

Ratio of C/C# = 261.6/277.2 ≈ 17/18

This tells us that every 17th C wave matches up with every 18th C# wave.

It takes too many waves for them to match up which is why they do not sound good together. Middle C (marked #1 in the picture) is the fundamental note of this series. All the other notes above it which are marked in blue (C, G, C, E, F, Bb...), are overtones. All of these notes played together represent the total vibration of middle C.

According to the harmonic series equation, the first overtone should vibrate at a length 1/2 of the string of the fundamental. The first overtone of middle C is the C an octave above it, which must mean that the string length of high C is 1/2 the string length of middle C.

Likewise, the next overtone in the middle C harmonic series is G. Thus, the length of G's string must be 1/3 the length of middle C's string. And so on for the remaining overtones. We decided to experiement with the harmonic series of middle C To tie all our research together,

we picked a song and found its sheet music, then listened carefully for the chords. Finally... The Experiment Explained The song we chose is

"We are Young" by Fun. We measured the lengths of the first 6 notes in the harmonic series of middle C (the fundamental + 5 overtones). Below is a chart of the strings' lengths: The Sheet Music: Breaking down the chords As you can see from the purple circles, the key of this piece is in F major, which means that there is one flat (Bb - a black key). There are several different chords in this song that are circled in the picture in yellow. Let's listen to the notes of the above portion of the song in more detail.

Pay attention to the chords and syncopation. Our measurements show that the strings of the piano do approximately follow the harmonic series. In conclusion, our experiment proved that the harmonic series predicts the string lengths of a piano because the pattern (1, 1/2, 1/3, 1/4...) was present in overtones of middle C. Triplets Three notes played in the same amount of time as one or two beats. A half note is 2 beats (in a 4/4 time signature). In a triplet, 3 quarter notes would be squeezed into only 2 beats, as opposed to 3. The same logic applies to a quarter note triplet. One beat of a measure would actually have 3 eighth notes squeezed into it. Mathematically... Half Note Triplet Quarter Note Triplet Half note = 2 beats Quarter note = 1 beat 3 notes 3 notes = each note is 2/3 beat = each note is 1/3 beat