Equal Tempered Tuning & Flaws in Just Tuning

Just tuning presents an interesting method of tuning the different notes of a scale based on specific frequency ratios, and ultimately ends up with twelve notes of unique frequency intervals that are perfectly harmonic. While this sounds like the best possible tuning system in theory, it runs into some substantial problems in practice. For example, when an instrument is tuned to C, the minor third of D-F ends up having a different frequency ratio than the normal minor third, C-Eb. There are many other cases in which these harmonic inconsistencies occur, such as changing the key of the song, and this leads to dissonance and a lack of flexibility in playing music.

Equal tempered tuning, also known as equal temperament tuning, aims to resolve the problems created by just tuning. It does so by making the twelve semitones of an octave equally spaced in terms of the relationship of their frequencies. This process adds consistency to the tuning process that just tuning lacks, but loses some of the harmonic purity of the fractional intervals. The relationship between the frequencies of notes in equal tempered tuning is given by the equation, frequency ratio = 2^(n/12), where n is the number of semitones (The Physics of Music and Color). To hear a comparison between equal temperament and just tuning, check out this video.

The general consensus is that the flexibility given by equal temperament tuning is essential and makes up for the lack of purity that is achieved through just tuning.

Works Cited:

1.  Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.

Cents and Musical Intervals

While the western musical scale is broken into twelve notes, we often need more specific ways of describing the pitch of a sound, especially during the tuning process. For example, let's say that you are tuning one of your guitar's strings to the E note played by a piano. As you twist your tuning peg and strike the corresponding string, you get closer the frequency of the E and eventually reach the point where the frequency of your string is lower than F but higher than E. How can we quantify this difference? The piano player may tell you that you are 50 cents sharp of E, but what does this mean?

The chromatic scale is broken into twelve notes, but the cents system allows us to further break this up. The basic definition of this is that the interval between two semitones consists of 100 cents, evenly spaced frequency values between the two notes. Since an octave consists of twelve notes in a chromatic scale, we also know that an octave is made up of 1200 cents. Given this understanding, we now know when the pianist tells us that we are 50 cents sharp of E when tuning, our note is tuned exactly in between E and F. Though this frequency does not have a specific letter name, it can easily be quantified by using the cents system.

Many electronic tuning devices (or tuning apps) can help you tune your instrument to standard pitches. With remarkable precision, these devices are often able to show how many cents sharp or flat your detuned note is in order to help you reach the ideal frequency.

Works Cited:

1.  Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.

The Just Chromatic Scale

The western musical scale consists of twelve notes, each a semitone (half step) apart, and the scale consisting of all twelve notes is called the chromatic scale. Tuning an instrument to be able to play all twelve half steps of the chromatic scale adds a bit more difficulty to the process of tuning a pentatonic or diatonic scale.

Starting with the eight frequencies of a just diatonic scale, major third intervals are used to calculate the needed sharp/flat notes to complete the chromatic scale. Both descending and ascending major thirds can be used to find the ratios of the new notes, and the process of reducing the octave, as mentioned in my previous post, can be used to ultimately end up with the twelve ascending ratios of a chromatic scale. The following graphic is very useful in understanding the just chromatic tuning process:

(The Physics of Music and Color)

Works Cited:

1.  Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.