- A pianist playing a chord, displaying the harmonies that the multiple notes create
- 1 Basic Description
- 2 A More Mathematical Explanation
- 3 Why It's Interesting
- 4 Teaching Materials
- 5 References
Harmony is the musical concept of adding tones that compliment the main melodic note being played. By layering these complementary notes, composers can provide a pleasing swell in sound to capture the audience. Well-composed harmonies support the melody of the song and add to its strength. The frequencies resonated by the notes control whether or not a harmony sounds pleasing to the human ear.
A More Mathematical Explanation
Ratios and FrequenciesIn musical notation, notes run from the letters A through G with middle C [...]
Ratios and Frequencies
In musical notation, notes run from the letters A through G with middle C on the piano as an important reference point. Middle C itself can also be called C4, with the following number signifying which octave the note falls into. The new numbering for the octave starts at C, so after G4 is A4. When referencing the keys of a piano, the white keys represent these lettered notes and the five black keys per 8 note octave represent the flats and sharps, which have differing names depending on the key the musician is playing in. Each of these keys are one "Semitone" or "Half Step/Tone" apart from the keys directly adjacent to it. Furthermore, each of the notes resonate at a certain frequency that makes their sound discernible to the human ear. The frequency ratios that these notes resonate at when played together determines whether or not the resulting harmonies are audibly pleasing to the listener (consonant) or if their sound invokes a feeling of discomfort of tension in the listener (dissonant).
Frequency and Ratios
Tuning to Specific Frequencies
There are multiple ways to tune and play different instruments in order to produce the desired sound and harmonies from the notes being played. The piano was historically tuned to the “just-toned” scale in the 16th century. In this method of tuning, the tonic, fundamental note, scale was "pure" in that the frequencies of its component notes were pure integer ratios of one another. However, the negative side to this method of tuning are very readily apparent when a musician changes keys, as the new harmonies sound impure.
This issue was later resolved with the “equal-tempered” or "even-tempered" scale that was popularized by Johann S. Bach. The name of the method comes from attempting to “even out” the problems present in the frequency ratios. By tuning the frequency ratios between all twelve notes on the chromatic scale. With this method, it became possible to change keys and still obtain audibly-pleasing harmonies – only an especially well-trained ear would notice that the frequencies on an equal-tempered scale piano are not exactly pure. This method itself was not perfected until the 20th Century due to the discrepancies in human hearing regarding the pitches of each note. With the aid of modern equipment, it is possible to tune instruments to a much more exact pitch frequency. However, for the most part, tuning is still done by ear to a reference note (i.e. Middle C) due to the greater part of the mathematical nature of tuning being lost in the even-tempered scale. 
There are other methods for tuning certain instruments, such as the organ, which which are harder to perfect. The system used before the current Kirnberger method of tuning for these instruments is called the Mean Tone system, which attempts to use an arithmetic mean in order to even out the commas present in the notes.
As for stringed instruments – the frequency that a string resonates at is controlled by the musician playing it and his finger placement along the fret board. For this reason, it is possible for a skilled musician to subtly change the frequency of the note to best fit the desired key. With this approach, he is able to create a pure sounding harmony independent of key.
In recent years, there has been an level of interest in new tuning techniques other than the standard even-tempered method. Methods like "just intonation" were developed for modern microtunable MIDI instruments in order to approach the classical pieces and have them played the way they were intended to be heard. However, the even-tempered tuning method is still the most widespread of tuning instruments and playing music. Furthermore, there is a wider study of ethnic music, which employs different notes and frequencies than western music. 
|Note||Frequency (Hz)||Wavelength (cm)|
Above is a table of notes ranging from C4 (middle C) to C6 (high C) along with their corresponding frequencies and wavelengths. By comparing frequencies and observing the resulting ratios, it is possible to identify the resulting harmony and determine whether it is consonant or dissonant. Please note that since these values are rounded and also based on the equal-tempered method of tuning, the resulting ratios that should be whole integers may not be exactly as such.
|Frequency Ratio||Common Name|
Below is a table containing the general titles that correspond to certain frequency ratios between a few specific harmonies. These ratios can be mathematically determined through the frequency values provided in the table above. A more complete list is can be found at the "Musical Intervals" page by Dale Pond .
The above listed harmonies and their corresponding ratios are all consonant -- through observation, it is apparent that the frequency ratio that each has contains only positive integers, all of which are small integers as well. These qualities usually signify the consonance of a harmony that corresponds to the ratio. If the integers in the frequency ratio are too big, then the resulting harmony created between the notes is noticeably dissonant.
However, it should be noted that while the listed harmonies are consonant, they do not sound as pleasing to the listener as Unison, a Perfect Fourth and a Perfect Fifth, which are all perfectly consonant. Major and Minor Thirds and Sixths are also classified as consonant, however they are imperfectly consonant.
What makes these notes and harmonies sound different between instruments, however, is the presence of integer mupltiples of the base frequencies being played. These "overtones" define the specific sound an instrument has. 
Below in a Flash application that allows you to select two notes at a time and play them together, providing information on the interval ratio and the name of the selected harmony. If the application freezes and only allows you to play one note or no notes at all, please refresh the page. To prevent freezing, please do not click too quickly, especially when deselecting notes:
Music Theory and Math Components
The Circle of Fifths
The Circle of Fifths, as shown above, is a visual representation of notes and musical keys. The outer circle of letters in red signifies the name of the root note of the major scale with the given key, and the inner circle of green lower-cased letters represents the name of the root note of the equivalent minor scale with the same key. The labels in the gray outlined circle represents the number of tones above (sharp sign) or below (flat sign) the base note at the top of the circle in the 12 o'clock position.
The Circle of Fifths can be used to understand why two notes that are in perfect harmony in one key suddenly sound dissonant in another key. By going around the circle and multiplying each of the pitches by 3/2 twelve times (with octave shifts to remain in the same octave) until the starting note is reached again, the resulting frequency is not equal to the initial frequency. This ditonic comma is the difference between the expected note and the one created as a result of the multiplication.
It should be noted that the Circle of Fifths is thus not a cycle, but a sequence despite its representation. By going about the circle, one reaches the notes of the next octave instead of the initial note, this minute change in tone is audible between the octaves. Furthermore, there is a "syntonic comma" which occurs when going four fifths up the Circle of Fifths and not producing a true major third. These commas all represent frequency discrepancies regarding what a note's played frequency is and the mathematically calculated frequency that the note should be played at.
In order to resolve these commas, certain compromises must be made in the paying of fifths. Alternatively, the fifths could be sacrificed for the sake of acquiring purer-sounding thirds. A shift of approximately 24 cents allows for the harmonies to sound pleasing despite being slightly off from the mathematically calculated actual wavelength for the ideal frequency ratios. This reasoning is the basis behind even-tempered tuning, which includes these compromises to allow for more keys to sound consonant instead of the strictly mathematical tuning behind the "just-toned" method that does not sound as pleasing when there is a change in musical key. 
The Circle of Fifths also proves to be a very useful tool as a visual representation of the notes and where they are in relation to one another. Through this visualization, certain aspects of the circle can be used to identify different notes corresponding to the tone being played. By observing the corresponding consonant and dissonant harmonies, it is possible to arrange music according to the relation between each of the. Furthermore, the note related to a named harmony can be found by traveling around the Circle of Fifths the same fraction as the frequency ratio. For example, a Major Third to a certain note can be found by traveling 4/5 forwards or backwards along the Circle of Fifths. The circle also makes transposing music to a different key much easier, visually allowing the musician to see how many semitones up or down the notes should move in order to be properly played in this new key.
Commas, as briefly mentioned before, are the discrepancies present between a played note and its actual mathematically calculated frequency. The more commonly known musical commas are:
- Ditonic Comma -- 23.46 Cents: Also known as the Pythagorean Comma -- it occurs when navigating the Circle of Fifths mathematically. The difference represented by the comma is that between 12 Perfect Fifths and 1 Octave.
- Syntonic Comma -- 21.51 Cents: Occurs when the note represented on the Circle of Fifths 4/5 of a revolution from the root note and represents the discrepancy between the new played note and the actual frequency the Major Third of the base note should sound like. Alternatively: The difference between 4 Perfect Fifths and 2 Octaves plus 1 Major Third.
- Diaschisma -- 19.55 Cents : The difference between 3 Octaves and 4 Perfect Fifths plus 2 Major Thirds.
- Diesis -- 41.06 Cents : The difference between and Octave and 3 Major Thirds.
Chords are the result of three or more notes that are played at the same time -- allowing the given instrument to provide its own harmonizing tones. There are numerous chords that exist within music theory, all of which are created through the combination of different tones.
The Circle of Fifths above can further be used to observe the similarities between certain major and minor chords, which are mirror images across the circle. Some examples include F Major (IV) and E Minor (III) chords, C Major (I) and A Minor (VI) chords and the G Major (V) and D Minor (II) chords.
By playing multiple chords in succession, one creates a chord progression. These progressions provide the backbone for the main melody of the musical piece. Thus, by working with the appropriate chords, a full-sounding musical piece can be formed. It should be noted that it is possible to create a good chord progression with the appropriate use of a dissonant chords to break the flow or create tension.
In the context of post seventeenth century harmonic progressions, numerous chords that a composer could choose to utilize were accepted as common, including the chords for all the scale degrees (the one, two, three, four, five, six, and seven chords) even though some of these have a certain degree of dissonance, such as the seven chord . These additions to the norms of harmonization, along with inversions of chords and decorative non-chord tones allow for interesting compositions to come about.
- Inverted chords -- Beyond simplistic root position chords there are musical techniques for modification and variation of melodies and harmonies. A common method of modification on root position chords is to invert them, so that the notes present in the chord remain the same, but the lowest note (bass) of the chord is not the root of the chord. For example, a normal major triad is composed of the root of the chord (in the base), the third of the root, and the fifth of the root. If the normal triad is inverted, then the note that will appear in the bass will either be the third of the root, or the fifth of the root, in which case the chord is denoted as being in first or second inversion, respectively. Johann Sebastian Bach frequently used inverted chords when reinterpreting old pieces whose melody was instilled in popular culture by folk tradition. Through inverted chords and the manipulation of harmonic progression, Bach composed a number of variations on well-known traditional pieces.
- Transposition -- Another common method of adding zest to a musical excerpt. In Edward Elgar’s Pomp and Circumstance transposition is used to move a musical phrase up an interval in order to alter the piece. This can be done however many times the composer deems appropriate.
- Composition by chance -- Yet another example of a mathematical concept applied to music in order to achieve a brand new piece of music every time something is played. The Austrian composer Franz Joseph Haydn wrote a sixteen measure long trio, with six different versions of every measure (except the eighth and sixteenth measures, which have four and three versions, respectively). Thusly, in total, Haydn composed 91 measures (including all the variations) for the trio, however, there are 940,369,969,152 possible combinations of each of the measures. Given the astronomical number of possibilities, it is quite probable that a person playing a random combination of measures from this Haydn trio will play something that has never been played before . The number of possible combinations of all the trios is a simple calculation (614 x 41 x 31), but the fact that Haydn constrained the possible outcomes of the trio by designing measures so that they would harmonically work with one another independent of their order remains impressive, especially since so many possible outcomes came out of it.
From the perspective of physics, concepts such as dissonance and consonance can be explained fairly simply. In the seventeenth century, Galileo was one of the first to explain consonant intervals such as the perfect fifth in terms of waves. He concluded that if the frequencies of two notes are in a simple integer ratio, then their total waveform is periodic, and their individual sine waves will match up at certain points. What we as humans perceive as music can be modeled by mathematics describing the physics of sound waves. Movement of any object causes the gas molecules in air to vibrate and this motion gets ultimately transmitted to our ears and we perceive it as a sound or music. For example, the loudness we distinguish is simply the amplitude of the sound wave. If the vibration was made using a lot of energy, then the vibrations have greater amplitude and we perceive that sound wave as louder. The same relationship exists between the perceived pitch and the frequency of a wave. A high pitch sound can be described mathematically as a high frequency sound wave while low pitch sounds are low frequency sound waves.
- Simple Harmonic Motion -- Simple Harmonic Motion is a type of periodic motion that further connects the concepts of amplitude and frequency to loudness and pitch. It can be illustrated by the phase circle, a circular motion diagram with a screen (acting as a y axis of sorts) placed to the left of a circle. The shadow of a point moving along the circle appears on the screen. The tangential velocity of the circle vt is related to the angular velocity ω by the equation vt = Aω where A is the amplitude, and the y-axis component of the function is modeled by the velocity v of the shadow that is projected on the screen by v = vt cos θ. Theta can also be rewritten as ωt, thus the model equation can be simplified to v = Aω cos ωt. The equation shows that though an object is moving with uniform circular motion, the analogous simple harmonic motion is not uniform. This means that the object undergoing simple harmonic motion will have a higher velocity if the corresponding circular motion has a longer radius, or if the radius turns faster. An item moving in simple harmonic motion would therefore have greater momentum if its frequency or amplitude were to increase . This can be understood if one thinks about the strings on a guitar. When the length of the string is shortened, the tone played by the string has a higher frequency, and for that to happen, the string must vibrate more times (this is elaborated in the Standing Waves page).
(Diagrams taken from Simple Harmonic Motion site)
Why It's Interesting
Pythagoras and Harmonies
It was through Pythagoras’ work that it was discovered that the sound a string made was related to the proportion of the string in relation to its root note. For example, by placing your finger on the twelfth fret of a guitar string, you are effectively splitting the string in half. The resulting sound is that of a pitch that is exactly one octave higher than the original note the string played. Pythagoras took great interest in this phenomenon and sought to research the notes created by splitting the string into different proportions. To observe his theory, he use the lyre and monochord, which contain one string and frets at different points along the board that splits the string into different proportions .
In order to reflect his findings regarding the octave, Pythagoras added one more string to his lyre -- totaling in 8 notes with one full octave and the corresponding higher C. At the time, 7 was seen as a number containing mystic properties so the addition of this new string was not taken upon lightly. Through their research, the Pythagoreans were able to conclude that the speed of vibration and the size of the size of the instrument being played a great part in the sounds that these instruments played. Melanie Richards pointed out in her paper on Pythagoras and
Music that the stringed bass, with its towering size, is able to play the lowest notes because of its large size.
This concept is of proportion and sound is further explained in the Standing Waves page.
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- Su, Francis E., et al. "Music Math Harmony." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.
- Su, Francis E., et al. "Music Math Harmony." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.
- Michael Beer: http://www.pages.drexel.edu/~jjn27/mathandmusic.pdf
- Benson, D. J. Music: A Mathematical Offering. Cambridge: Cambridge UP, 2007. Print.
- Harkleroad, Leon. The Math behind the Music. Cambridge: Cambridge UP, 2006. Print.
- Loy, D. Gareth. Musimathics: The Mathematical Foundations of Music. Vol. 1. Cambridge, MA: MIT, 2006. Print.
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