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 represents 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 is 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).

Two Octaves represented as a Piano keyboard. Each key is one semitone apart from the adjacent keys.

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 their frequencies were pure integer ratios of one another. However, the consequences to this method of tuning were very readily apparent when the player changed keys due to the new harmonies sounding 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 obtaining 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 very recently in 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 gerater part of the mathematical nature of tuning being lost in the even-tempered scale. ^[2]

Other methods for tuning and playing notes are also used for different instruments. As for instruments like the organ,, which are harder to tune, the system used to tune these instruments before the current Kirnberger method of tuning was the Mean Tone system which attempted 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 fretboard. Thus, it is possible for a skilled musician to subtly change the frequency of the note to best fit the key that he is playing in. With this approach, he is able to create a pure sounding harmony no matter what key he performing in ^[3].

In recent years, there has been an level of interest in new tuning techniques other than the now-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. ^[4]

Frequency Table

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 off of the equal-tempered method of tuning, the resulting ratios that should be whole integers may not be exactly as such.

Ratios

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 ^[5].

The above listed harmonies and their corresponding ratios are all consonant -- through observation, it should be 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 all imperfectly consonant, but still clearly classified as 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. ^[6]

Flash Application

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:

[show more][hide]

The following references were used for the application: Interval Ratios ^[7], Compound Interval Names ^[8], Corresponding Notes and Harmonies ^[9], Additional ^[10]

Music Theory and Math Components

The Circle of Fifths

The Circle of Fifths

The Circle of Fifths, as show above, is a visual representation of notes and music keys. The outer circle of letters in red signifies the name of the note if it is considered major and the inner circle of green lower-cased letters represents the name of the note if it is considered minor. 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 employed to understand why two notes that are in perfect harmony in one key suddenly sound out of tune in another key. By going around the circle and multiply 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 the same frequency that you started with. This ditonic comma created 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 should considered as a sequence and not a cycle despite how it is represented. By going about the circle, you reach the notes of the next octave instead of the note you started on. However, this minute change in tone is still audible between the octaves. Furthermore, there is the "syntonic comma" which occurs when going four fifths up the Circle of Fifths and not creating a true producing a true harmonic 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 most pleasing frequency ratios. This reasoning is the basis behind even-tempered tuning, which allows for these compromises in order 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 shift in the music's key. ^[11]

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 each of the notes have. 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.

^[12]

Commas

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

Chords are the playing of three or more notes at the same time -- allowing the instrument being played to provide its own harmonizing tones. There are numerous chords that exist within music, created through this simultaneous play of numerous notes.

By playing multiple chords in succession, one creates a chord progression. These progressions provide the backbone for the melody that the song works around. Thus, by working with these chords, a full-sounding song will form. However, when playing these chords, it is still possible to create a good progression with the clever use of a dissonant chords to break the flow or create tension.

The Circle of Fifths above can further be used in order 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.