Basic Description Harmony is the musical concept of adding tones that compliment the main melodic note being played. In musical notation, notes run from the letters A through G with middle C on the piano as an important reference point. There are multiple ways to tune and play different instruments in order to produce the desired sound and harmonies from the notes being played.
This issue was later resolved with the “equal-tempered” or "even-tempered" scale that was popularized by Johann S.
There are other methods for tuning certain instruments, such as the organ, which which are harder to perfect. 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. In recent years, there has been an level of interest in new tuning techniques other than the standard even-tempered method. Above is a table of notes ranging from C4 (middle C) to C6 (high C) along with their corresponding frequencies and wavelengths. Below is a table containing the general titles that correspond to certain frequency ratios between a few specific harmonies.
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. 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.
What makes these notes and harmonies sound different between instruments, however, is the presence of integer mupltiples of the base frequencies being played. 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.
The Circle of Fifths, as shown above, is a visual representation of notes and musical keys. 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.
It should be noted that the Circle of Fifths is thus not a cycle, but a sequence despite its representation.
In order to resolve these commas, certain compromises must be made in the paying of fifths. 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.
Commas, as briefly mentioned before, are the discrepancies present between a played note and its actual mathematically calculated frequency. 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.
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. 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 [13]. Inverted chords -- Beyond simplistic root position chords there are musical techniques for modification and variation of melodies and harmonies.
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.
From the perspective of physics, concepts such as dissonance and consonance can be explained fairly simply. 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 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.
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.
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. Music that the stringed bass, with its towering size, is able to play the lowest notes because of its large size[16].
Bartolomeo Cristoferi had intended to construct a harpsichord, but when his bag of tools fell into the wooden body of the half-finished instrument he was building, he instead changed the course of musical history.
Each weight on a key translates to the strike of a hammer on a string, and each strike conjures art. A row of black and white ivory becomes a horizon of possibilities that can transport us to any place, any time, any feeling.
By layering these complementary notes, composers can provide a pleasing swell in sound to capture the audience.
Middle C itself can also be called C4, with the following number signifying which octave the note falls into.

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. For this reason, it is possible for a skilled musician to subtly change the frequency of the note to best fit the desired key. 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.
By comparing frequencies and observing the resulting ratios, it is possible to identify the resulting harmony and determine whether it is consonant or dissonant. These ratios can be mathematically determined through the frequency values provided in the table above. Major and Minor Thirds and Sixths are also classified as consonant, however they are imperfectly consonant. If the application freezes and only allows you to play one note or no notes at all, please refresh the page. 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. 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.
Alternatively, the fifths could be sacrificed for the sake of acquiring purer-sounding thirds. Through this visualization, certain aspects of the circle can be used to identify different notes corresponding to the tone being played. There are numerous chords that exist within music theory, all of which are created through the combination of different tones.
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. These additions to the norms of harmonization, along with inversions of chords and decorative non-chord tones allow for interesting compositions to come about. 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.
In Edward Elgar’s Pomp and Circumstance transposition is used to move a musical phrase up an interval in order to alter the piece. 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).
In the seventeenth century, Galileo was one of the first to explain consonant intervals such as the perfect fifth in terms of waves. 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 resulting sound is that of a pitch that is exactly one octave higher than the original note the string played. At the time, 7 was seen as a number containing mystic properties so the addition of this new string was not taken upon lightly. 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.
The name of the method comes from attempting to “even out” the problems present in the frequency ratios.
However, the even-tempered tuning method is still the most widespread of tuning instruments and playing music. 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.
If the integers in the frequency ratio are too big, then the resulting harmony created between the notes is noticeably dissonant.
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.
This ditonic comma is the difference between the expected note and the one created as a result of the multiplication. Furthermore, there is a "syntonic comma" which occurs when going four fifths up the Circle of Fifths and not producing a true major third.
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. By observing the corresponding consonant and dissonant harmonies, it is possible to arrange music according to the relation between each of the. 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. 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. 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. Pythagoras took great interest in this phenomenon and sought to research the notes created by splitting the string into different proportions.

Indeed, the sound struck a chord in Cristofori, and it inspired him to substitute the plucking mechanism of the harpsichord with the hammers in his tool bag.
The frequencies resonated by the notes control whether or not a harmony sounds pleasing to the human ear. 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. However, the negative side to this method of tuning are very readily apparent when a musician changes keys, as the new harmonies sound impure. Furthermore, there is a wider study of ethnic music, which employs different notes and frequencies than western music.
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. 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. 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. 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.
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. 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 [14].
What we as humans perceive as music can be modeled by mathematics describing the physics of sound waves. The tangential velocity of the circle vt is related to the angular velocity ? by the equation vt = A? 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 ?.
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 .
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.[1] This method itself was not perfected until the 20th Century due to the discrepancies in human hearing regarding the pitches of each note.
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. 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. Theta can also be rewritten as ?t, thus the model equation can be simplified to v = A? cos ?t. The interdependence of musician and instrument allows souls to be touched and emotions to be stirred. Furthermore, each of the notes resonate at a certain frequency that makes their sound discernible to the human ear. With the aid of modern equipment, it is possible to tune instruments to a much more exact pitch frequency. 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. The equation shows that though an object is moving with uniform circular motion, the analogous simple harmonic motion is not uniform. 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). If the vibration was made using a lot of energy, then the vibrations have greater amplitude and we perceive that sound wave as louder.
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. Middle C) due to the greater part of the mathematical nature of tuning being lost in the even-tempered scale.
An item moving in simple harmonic motion would therefore have greater momentum if its frequency or amplitude were to increase [15]. The questions, just as much as the answers, breathe life into the wood, metal and varnish, and become a part of the piano’s story, a story that grows richer with each piece that is played on the piano.
A high pitch sound can be described mathematically as a high frequency sound wave while low pitch sounds are low frequency sound waves. 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).

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