How Did Music Theorists Decide the Pitch of Each Note?
By Jane Yang 楊靜悠
Introduction
Known as the temperament or scale in music theory, we learn to sing “do-re-mi-fa-sol-la-ti” as early as kindergarten. We also learn that all sounds are essentially produced by the vibrations that hit our eardrum, whose frequency decides the pitch. Then, have you ever thought about how music theorists chose a pitch for each note in “do-re-mi-fa-sol-la-ti” from an infinite number of options on a number line? In this article, we will introduce you to “Pythagorean temperament,” an early musical scale often attributed to an ancient Greek mathematician, Pythagoras [1, 2]. We will also delve into how music theorists and mathematicians later developed “equal temperament,” which has become the most widely used musical scale in western music since the 19th century [1].
Pythagorean Temperament
First of all, let's understand the concept of an "octave." Mathematically, two sounds are considered an octave apart if their frequencies have a ratio of 2:1. For example, the standardized “middle A” has a frequency of 440 Hz (Footnote 1), or vibrations per second, while the next A which is an octave higher has a frequency of 880 Hz. When they are played at the same time, they sound so consonant that the human brain perceives them as the “same” note but the latter in a higher pitch. This phenomenon is called “octave equivalence [1, 2].”
Therefore, to create a musical scale, we only need to consider an octave, or one cycle of “do-re-mi-fa-sol-la-ti”. We can then multiply or divide the frequencies of the notes in an octave by any power of two to obtain a higher or lower octave because of octave equivalence. Pythagoras also discovered two notes that are a "fifth" apart, meaning their frequencies have a ratio of 3:2, also sound pleasant when played together. Hence, he decided that the task was to create as many ratios of 3:2 and 2:1 as possible to provide convenience for composers.
Obviously, Pythagoras should have no access to the accurate frequency of each note, so the tuning was probably completed by hearing the pitch and comparing its relative distance to the base note. However, for a better understanding, let’s unveil the ancient method based on our modern understanding.
To decide a frequency to each of the notes in an octave, Pythagoras started with the note A at 440 Hz and multiplied its frequency by 3/2 to obtain the note at 660 Hz. By multiplying 3/2 again, he got 990 Hz. However, this exceeded the desired octave range (i.e. greater than 880 Hz), so he divided it by two to get the note equivalent to it at 495 Hz. Pythagoras repeated this process of multiplying by 3/2, and dividing by two if the resulting frequency exceeds 880Hz, until he obtained a musical scale consisting of seven nonequivalent notes which is enough to play simple melodies [1]. He rearranged those frequencies in order, creating a musical temperament very similar to the one we use today (Table 1).
| Ratio | 1 | 9/8 | 81/64 | 4/3 | 3/2 | 27/16 | 243/128 | 2 |
| Frequencies (Hz) | 440 | 495 | 557 | 587 | 660 | 743 | 835 | 880 |
Table 1 The frequencies of notes and their ratios with respect to the note A in Pythagorean temperament. The values are rounded off to the nearest integer.
Equal Temperament
However, the seven notes in Pythagorean temperament are just enough for playing simple melodies. Before we examine the problem of Pythagorean temperament, let’s look at the modern system called "equal temperament". This temperament divides an octave into 12 equal musical intervals. Keep in mind that our brain perceives the distance of musical interval by ratio instead of difference. Therefore, the frequencies of each note in a scale should have an exponential relation, with a ratio r between each pair of adjacent notes satisfying r12 = 2, i.e. r = 21/12. By multiplying the starting frequency by the ratio r = 21/12 for 12 times, we obtain the frequencies of all the notes within an octave (Table 2).
| Ratio | 1 | 21/12 | 22/12 | 23/12 | 24/12 | 25/12 | 26/12 |
| Frequencies (Hz) | 440 | 466 | 494 | 523 | 554 | 587 | 622 |
| Ratio | 27/12 | 28/12 | 29/12 | 210/12 | 211/12 | 2 | |
| Frequencies (Hz) | 659 | 698 | 740 | 784 | 831 | 880 |
Table 2 The frequencies of notes and their ratios with respect to the note A in equal temperament. The values are rounded off to the nearest integer. The frequencies in bold are played by the black keys of a piano.
Key Change
So, why is equal temperament preferred over Pythagorean temperament? You may have heard of a musical jargon called “key change” before. Actually, the mathematical implication of key change is to multiply the frequency of each note of a melody by a constant number. After performing this trick, human brains will still perceive the two melodies as the same since the musical interval (i.e. the frequency ratios) between any two adjacent notes are retained [1]. For example, a melody that plays 440Hz, 660Hz, and 733.3Hz in order is considered equivalent to a melody that plays 550Hz, 825Hz and 916.6Hz. Key change in music usually helps musicians express their feelings: Changing to a higher key in the midway of a piece of music can express excitement or encouragement, while lowering the key may convey sorrow or tranquility. In addition, by lowering the key of a song, a singer whose voice range is too low to cover the high pitch can now sing the song.
After understanding the concept of key change, you would discover that the equal temperament adapts to key change perfectly because the ratio of the frequencies between any adjacent notes is a constant [1, 2]. Pianists, for example, only need to move up every note for one key on a piano keyboard tuned with equal temperament to complete the key change, and the finite number of keys on the keyboard is sufficient to cover all notes required for any key changes.
On the other hand, the seven notes in Pythagorean temperament don’t suffice. Instead of having a constant ratio, adjacent notes in Pythagorean temperament have a ratio of either 9:8 or 256:243 [2]. We have to continue Pythagoras’ calculation to create more and more notes so that key changes can be performed perfectly from any note. By extending his calculation beyond the first octave, we wish the value will return to the starting point 440 Hz at some point, so that we can get a finite number of notes. Nevertheless, this has been proved impossible, due to the fact that (3/2)n is never a power of two, so we will need an infinite number of black keys for a musical instrument to perform key changes, which is simply not practical [1]. Although Pythagoras was able to get close to the desired frequency 440 Hz, there was still a small discrepancy known as the "Pythagorean comma" [2]. This slightly higher frequency ratio of 1.0136:1 posed challenges for musicians and mathematicians until the invention of equal temperament (Footnote 2).
Historical Controversies Over the Invention of Equal Temperament
One interesting coincidence is that the equal temperament was invented by the Chinese mathematician, physicist and music theorist Zhu Zaiyu in 1584, and given a mathematical definition by the Flemish mathematician Simon Stevin around the period between 1585–1608 [3]. There are still controversies on who should receive the credit and whether the development was independent [3, 4], but we may never know the truth.
Nevertheless, one thing you can take away is that anything we take for granted today may have been the outcome of the struggle of our predecessors for thousands of years, and there may actually be a scientific reason behind it. From Pythagoras' exploration to the invention of equal temperament, these mathematicians have shaped the music we enjoy. So next time you sing "do-re-mi-fa-sol-la-ti", remember the mathematical journey that led to these familiar notes.
1 Middle A: In the case of C major (one of the easiest modes in music), “do-re-mi-fa-sol-la-ti” corresponds to C, D, E, F, G, A, B respectively in representation. Chosen as a standard note for tuning musical instrument, the “middle A” corresponds to “la” in C major. Although it should have a frequency of 440 Hz by the ISO 16 standard [5], the tune is sometimes set at 442 Hz in some wind bands to cater the wind instruments.
2 Editor’s note: The number 1.0136 is given by 312 / 219, i.e. taking the fifths 12 times while reducing the octaves seven times.
References
[1] Formant. (2022, August 12). The Mathematical Problem with Music, and How to Solve It [Video]. YouTube. https://www.youtube.com/watch?v=nK2jYk37Rlg
[2] Benson, D. (2008, December 14). Music: A Mathematical Offering. Cambridge University Press. https://homepages.abdn.ac.uk/d.j.benson/pages/html/music.pdf
[3] Yung, B. (1981). A Critical Study of Chu Tsai-yü's Contribution to the Theory of Equal Temperament in Chinese Music. By Kenneth Robinson. Additional Notes by Erich F. W. Altwein; Preface by Joseph Needham. Wiesbaden: Franz Steiner Verlag (Sinologica Coloniensia Band 9), 1980. x, 136 pp. Figures, Appendixes, Bibliography. N.p. The Journal of Asian Studies, 40(4), 775–776.
[4] Kuttner, F. A. (1975). Prince Chu Tsai-Yü's life and work: A re-evaluation of his contribution to equal temperament theory. Ethnomusicology, 19(2), 163–206.
[5] International Organization for Standardization. (1975). ISO 16:1975 Acoustics — Standard tuning frequency (Standard musical pitch). https://www.iso.org/standard/3601.html