Tuning System

Tuning is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones. Tuning is usually based on a fixed reference, such as A = 440 Hz. The term “out of tune” refers to a pitch/tone that is either too high (sharp) or too low (flat) in relation to a given reference pitch.

A tuning system is the system used to define which tones, or pitches, to use when playing music. In other words, it is the choice of number and spacing of frequency values used.

Just intonation or pure intonation is tuning system where intervals are whole number ratios of frequencies. One example is the Pythagorean tuning system.

However, in Western musical practice, instruments are rarely tuned using only pure intervals – the desire for different keys have identical intervals makes this impractical. The equal temperament system is mostly used in Western music.

Pythagorean Tuning

Pythagoreans were obsessed with simple ratios, i.e., ratios formed by small prime numbers. They build a just-tuning by using these ratios, which lead to a strong dissonance for some combinations of notes.

As mentioned in section Intervals, the consonant frequency ratio is 2:1. The next most consonant frequency ratio is 3:1, then 4:1, and so on. When building a scale, we desire the absence of beating when two sounds are played together. Since a frequency ratio of 2:1 appears when we play the same note one octave apart, we know that our scale should span no more than twice the frequency of the root frequency.

Therefore, we use the 3:1 ratio to build our scale. Let us start with 440 Hz. Using the 3:1 ratio we add one frequency below \(440 \cdot 4/3 \approx 586.66\) and one above \(440 \cdot 3 / 2 = 660\). We can repeat this process to create even more frequencies: \(586.66 \cdot 2/3 \approx 391.11\) and \(660 \cdot 3/4 = 495\).

Overall we end up with the following five notes.

Frequency	391.11	440.00	495.00	586.66	660.00
Note	G	A	B	D	E
Ratio	1:1	9:8	81:64	3:2	27:16

This forms the pentatonic scale of G major. The jump from 391.11 to 440 Hz equates to 2 semitones. The interval between 391.11 to 495.00 equates to 4 semitones, and going from the root to 586.66 spans 7 semitones. 391.11 to 660 gives us 9 semitones.

440.cpsmidi-391.11.cpsmidi;     //  2.0391492001631
495.00.cpsmidi-391.11.cpsmidi;  //  4.0782492174708
586.66.cpsmidi-391.11.cpsmidi;  //  7.0194024592495
660.00.cpsmidi-391.11.cpsmidi;  //  9.058699208817

(
Pbind(
  \scale, Scale.majorPentatonic,
  \degree, Pseq((0..5), 1),
  \dur, 0.5
).play
)

From there, we can add two more notes by dividing 391.11 by 3/4 and multiplying 495 by 3/2, i.e., \(391.11 \cdot 4/3 = 521.48\) and \(495 \cdot 3 / 2 = 742.5\). We end up with the scale of G major.

521.48.cpsmidi-391.11.cpsmidi; //  4.9804499913461
742.5.cpsmidi-391.11.cpsmidi;  //  11.097799226125

Frequency	391.11	440.00	495.00	521.48	586.66	660.00	742.5
Note	G	A	B	C	D	E	F#
Ratio	1:1	9:8	81:64	4:3	3:2	27:16	243:128

The technique we just used to build a scale is called Pythagorean tuning or Pythagorean just intonation. It is close to the 12-TET but uses whole-number ratios. Overall it does not offer much advantage for tonal harmony. We get all the intervals by consecutively multiplying by \(3/2\) (ascending fifth), \(2/3\) (descending fifth), or their inversion \(4/3\) or \(3/4\), respectively.

The Pythagorean tuning involves ratios of very large numbers, corresponding to natural harmonics very high in the harmonic series that do not occur widely in physical phenomena. The primary reason for its use is that it is extremely easy to tune, as its building block, the perfect fifth, is the simplest and consequently the most consonant interval after the octave and unison.

Equal Temperament

An equal temperament is a tuning system that approximates just intervals, such as the Pythagorean tuning system, by dividing an octave (or another interval) into equal steps. It was first introduced in 1584 by Zhu Zaiyu who achieved an exact calculation of the well-known twelve-tone equal temperament tuning (12-TET) in China. One year later, the Flemish mathematician Simon Stevin was also able to develop this flexible tuning. Instead of using simple ratios, equal temperament tunings use real numbers such that an octave is divided into equally tempered (equally spaced) on a logarithmic scale.

In the case of the twelve-tone, the ratio is equal to the 12th root of 2, i.e.,

(94)\[\begin{equation} \sqrt[12]{2} = 2^{1/12} \approx 1.05946. \end{equation}\]

In modern times, 12-TET is usually tuned relative to a standard pitch of 440 Hz, called A440. Note A is tuned to 440 hertz, and all other notes are defined as some multiple of semitones apart from it, either higher or lower in frequency. However, the standard pitch has sometimes been 440 Hz, and there is still a lot of dispute and discussion. This development is comparatively recent in the musical community, and the agreement is still fragile among musicians [Loy06].

The significant advantage of an equal system is that we can change the key of a piece without re-tuning the instrument. Furthermore, moving from one key to the next or from one mode to the next is much more musically pleasant. The substantial disadvantage is imperfection! The equal system is the most “wrong” system since every perfect ratio is detuned a little bit. However, since everyone uses equal temperament, our culturally trained ears got used to it – most of us can not hear the imperfection.