Course:PHYS341/2022/Project9

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The Controversial Topic of 432 Hz vs 440 Hz

We all understand for musical instruments to be musical, they need to be correctly tuned to a frequency and temperament in order to reduce dissonance. Until 1955, it is likely that the frequency of A tone used by different countries and musical movements fell within the large range of 400 Hz to 460 Hz before the international benchmark was established for 440 Hz[1]. Then, the notes were scaled to equal temperament, where the interval between the two adjacent notes were always the same regardless of the note (i.e frequency ratio being a consistent 12th root of 2) and the rest of the frequencies were derived from the base frequency. Thus, this means previously, music in different countries had different standards, and may also be tuned at a different temperament, and frequency[2].

An interesting claim claims that tuning a musical instrument lower than the international standard of 440 Hz will result in a calmer, more stable heart.This frequency is said to be 432 Hz. And recent scientific studies have confirmed this specific healing frequency could indeed improve human health [3][4][5][6][7][8]. If these are all true, what are the differences between tuning to A=432 Hz, compared to A=440 Hz for such a benefit to occur? What are the costs of tuning to A=432 Hz compared to the standard A=440 Hz? Why had the International Organization for Standardization (ISO) set the International Standard Pitch (i.e., Concert Pitch) to A=440 Hz? Does the number of the frequency actually matter in music-making?  If not, what actually matters?

History

Frequency and Temperament Prior to Standardization

In the History of Musical Pitch by musicologist Alexander J. Ellis[9], he clearly stated that before the standardization of concert pitch at A=440Hz, the frequency of A tone used in different countries and different musical movements should be within the large range of 400Hz to 460Hz. This was measured and documented by ancient pipe organs and tuning forks in France, England, Germany, Austria, Italy, Belgium, America, Russia and Spain during the years 1600 to 1900[9][10]. Thus, it is credible to agree upon that is past generations, music in different countries had different standards in accordance with the musician’s preference/style or piece’s performance. This meant an instrument from America may be tuned to a different note than an instrument from France. In the similar way, tuning temperaments from the base frequency would also have been adjusted accordingly, varying from country to country, village to village, and with no standardized pitch or temperament.

Purpose of Standardization A=440 Hz

Given no standardization between instruments across the globe, this makes it inefficient for musicians as there was no set quality of instruments being “in tune.” This means one musical piece in one country may sound different in another country, even if the same instrument was used because the tuning pitch and temperament may be different[11]. Composition and reproduction of the movements would not simply be challenging but also, imagine playing in an orchestra and all instruments were tuned differently. Not only would this make transposition and replication more tedious, but this variation would pose as a burden with time as musicians and scores become more complex and is travelled around for performances of large venues, concert halls, or orchestra opera houses. Thus, A=440 Hz was compromised in the international conference held in London for many reasons such that it follows tendencies and pitch standards of that time since it was an averaged frequency and was a round number. It was broadcasted by the B.B.C. as follow:

“The B.B.C. tuning-note is derived from an oscillator controlled by a piezo-electric crystal that vibrates with a frequency of one million Hz. This is reduced to a frequency of 1,000 Hz by electronic dividers; it is then multiplied eleven times and divided by twenty-five, so producing the required frequency of 440 Hz. As 439 Hz is a prime number a frequency of 439 Hz could not be broadcast by such means as this[12]."

Definitions

Sound (and sound waves)

The general principle of sound is that sound vibrates through the air, generating sound waves that are picked up by the nerves in the human ear, in which they are then transmitted to the brain so humans can hear the sound. These sound waves are basically a frequency in this case, where they could only be heard by the human ear (where the frequency range of hearing is from 20 Hz to 20,000 Hz) and it’s measured by Hertz, which is the number of complete “waves” generated in 1 second.

Frequency

The measurement of the number of cycles per unit of time. In music, frequency is measured in Hertz (Hz) and 1 unit of this equals one cycle per second. Thus, for example, 440 Hz is equivalent to “vibrating” 440 waves per second. And the wave frequency of music is within the audible frequency. It is worth noting that frequency could also be seen as a unit of time and space since Hertz describes how many waves vibrates in one second, and for how long it vibrates. Using 440 Hz as an example again, this means the wave vibrates 440 times in 1 second, and after 440 times, a full length of 440 meters is generated.

Temperament

The different kinds of tuning system in music where pure intervals of just intonation may be slightly altered through a different frequency ratio to satisfy certain requirements.

Twelve-Tone Equal Temperament (i.e., Scientific Tuning)

The most mainstream music tuning system, where twelve-tone originates from the 12 notes (7 white keys and 5 black keys on the piano) and meets the frequency ratio where each semitone step goes up by a factor of the 12th root of 2. Thus, equal temperament is tuning where the ratio between a musical note and its next semitone is to be the exact same for all notes.

Stravidari/Verdi Tuning

The kind of tuning system that advocates for A=432Hz as the tuning standard. [13]

Pythagorean Tuning

The tuning system of basing all notes and intervals from series of pure perfect fifths on the frequency ratio of 3:2. [13]

Schumann Resonance Tuning

The tuning system to resonant with Earth’s natural frequency of 7.83 Hz – 8 Hz, where tuning C-1 to 8 Hz, thereby C4 would be 256 Hz.[14]

"Factor 9 Grid"

The tuning system that revolves around the number 9, where the number could be found on every note, but each scale is separated by 9 Hz. [15]

"Chakras of Human Body Frequencies"

The tuning system that “resonates” with human chakras frequency with its own “vibration, sound, colour and symbol.” [16]

Discussion

Tuning to A=432 Hz vs A=440 Hz

440Hz and 432Hz sine formula graph

The typical musical scales, chords, and intervals (pair of notes with specific relationship with one another, where mathematically, the frequencies have a specific simple, small integer ratio) use specific combinations of frequencies notwithstanding temperaments on top. Tuning A=432 Hz means to tune the A above middle C at 432 Hz on the piano, where in this case, the octave above’s A is 864 Hz and an octave below is 216 Hz. This follows the same nature for A=440 Hz, where instead of A above middle C at 432 Hz, it is tuned back up to 440 Hz. In visualization, this means there are 8 additional cycles per second (Hz). The period of the wave can be derived from the angular frequency (T=2πω)[17], also known as an alternative unit of measurement in describing oscillation of sine waves.

Subjective commentary on the respective frequencies

Music, like art, is always subject to the eye (or ears) of the beholder— and the tuning down from A=440 Hz to A=432 Hz is no different. Some say A=440 Hz is more engaging, alert, and perking to the ears whereas A=432 Hz is more somber, settling, and relaxing. Furthermore, Rosenberg’s results in the Journal of Popular Music Studies[8] from conducting a survey of qualitative data from 118 individuals claimed profound results when listening to music tuned to A=432 Hz. Although these are all subjective commentary and there is no “right” or “wrong” to which tuning is used, novelty wears off once tuning is integrated into modern production. Though, it is worth mentioning that as far as human resonance goes, it is posing to be an interesting conversation.

Health Benefits

Many recent studies have indicated music tuned to 432 Hz has interestingly positive health effects such as decreasing heart rate, improving sleep, and decreasing clinical anxiety levels. It would be worthwhile to dig deeper should further funding and research be conducted involving autistic individuals or individuals in need. [3][4][5][6][7][8]

Conveniency and Efficiency

In terms of modern music production, because of the standardization of A=440Hz, it is generally a mess to work with music and instruments playing in anything but 440 Hz. Though, it is noted that many orchestras may tune a couple frequencies higher or lower depending on specialization. However, some instruments when tuned to A=432 Hz and when played with some chords, just sounds off, and sometimes sound really weird[15]. Additionally, some instruments are specifically built to be played in a specific temperament like how the guitar is built for equal temperament. The instrument would sound like a brutal mess if tuned otherwise.

Human Resonance

In music therapy or sound healing, it is said that 432 Hz is the “healing frequency.” There are many theories behind this, yet the reason behind why it is exactly 432 Hz is unknown, or whether there is a specific range, for example, between 430 Hz to 435 Hz, is also unknown.

Verdi Tuning—different kinds of tuning, under the "Verdi Tuning" branch

There are many different tunings all called under the “Verdi Tuning” branch which consists of different temperaments on top of the base frequency when the kind of intonation is not specified[4]. As different temperaments/frequency ratio is applied to different base frequencies, the tunings of the following notes would be different. Some examples, along with a table that compares the frequencies of the notes using three different tuning methods can be viewed in this cited link or copied below as it is on public domain and considered under fair dealing exemption in Copyright laws, in which I do not own the copyright for table image below[18]. It is worth noting however, that if you calculate the frequencies with based on A4=432 Hz, C4 does not give you 256 Hz; Rather, you have to be based on C4=256 Hz in order to get A4=432 Hz (with the exception of using the Scale of Twelve Fifths tuning)[13][18].

This table image from Sacred Geometry by Jordi Solà-Soler compares the frequencies of notes using three different tuning systems (Just, equal and Scale of Twelve-fifths), highlighting 432 Hz changes depending on the ratios.
A4 = 432Hz + equal temperament[13]
C4 = 256 Hz + equal temperament[13]
C4 = 256 Hz + pythagorean tuning (A4 = 432Hz)[13]
A4 = 432 Hz + equal temperament/ pythagorean tuning/ 5-limit[13]

WHAT DIGITAL CALIBRATION CANNOT CHANGE: TEMPERAMENT RATIO

A musician, instrumentalist or composer can do anything to tuning. Setting it to A=440 Hz, or 442, 443, 434… all these do not really matter as with the technology of this generation, all digital stuff can be calibrated and adjusted. All these does not make as big of a difference as the temperament, which can make a big difference in the really small details. What digital calibration cannot change is the ratio after setting the base note frequency. Perhaps the specific frequencies of the notes could be manually customized, however, these temperament ratios such as the twelve-tone equal tempered system, the ratio itself, cannot be changed to tune our musical instruments. Though the controversy debate may be worthwhile in terms of human resonance and health benefits, music production wise, learning the history, invention, and function of tuning systems (i.e., just intonation/true tuning compared to equal temperament) may be of greater importance as at what digit the base pitch is for a note, is not that significant as it is convenient.

Debate

Fiction pertaining to using A=432 Hz as base pitch

There has been much debate that 432 Hz is more superior to 440 Hz. And pertaining to that, a lot of fiction has been built such as ancient instruments being tuned to 432 Hz, 432 Hz was the original tuning system, the frequency being the one of the heart/brain/earth/sun/waters among some other really far-fetched ones with chakras and cosmic energies[10].

Cymatics (Resonance Pattern)

The cymatics is the aesthetics of resonance patterns in water or plates, in which the visualization of specific frequencies is characterized as “good/pretty” or “bad/ugly.” The resonance pattern is created by introducing vibration frequencies into a container with water, or a plate with sand, and the vibrations would cause the water/sand to form different patterns at different frequencies. This pattern could be calculated mathematically, thus, in a way is scientific[13]. However, the subjective conclusion of a pattern from frequencies as nice-looking or not in determining its effect on humans, is quite premature in itself as other factors such as the medium’s surface, or material sound travels may play in part to affect the frequency’s resonance pattern. Open-close/open-close duct/bar can change the resonance as the wave frequency and the points of antinodes’ displacement, and node accumulation would differ[13]. And, by changing the dimensions of the plate or water container, patterns subjectively known to be “pretty” could be achieved with “bad” frequencies (i.e., non-432 Hz). In terms of materials, since the speed of sound (channel that frequencies travel in) is different in all sorts of air, liquid and metal, this would thereby change the wavelength of the frequency, which would then produce a different pattern.

Frequency of heart/brain/earth/sun/water

It was claimed that 432 Hz is the frequency of the heart/brain/earth/sun/waters. Yet, as Sagol well concluded in “Music Theory: 432 Hz Tuning – Separating Fact From Fiction,” heart and brainwaves fluctuates in a range, earth’s resonance an average of 7.83 Hz, the sun oscillates at approx. 5.964 GHz and water molecules does not vibrate at any specific frequency[10].

Ending Thoughts

Though certain frequencies and tunings for music would sound a mess in actual music production, numerology in tuning frequencies associates integer numbers to being more harmonious and more pure. Despite this favouritism, as previously mentioned, pertaining to physics of music, the ratio between the frequencies is perceptually more important than the frequency values (whether music is tuned to 440 Hz, or 441, 442, etc.) as the ratio from the base pitch is what produces harmonies.

432 Hz Numerology

It is fascinating to find numbers and calculations of the universe relating to the “432” number. As mentioned earlier, if the A above middle C is tuned to 432 Hz on the piano, then an octave above A would be 864 Hz and an octave below would be 216 Hz. This interestingly correlates to the diameter of the moon which is 2160 miles, at 216Hz in C3 (following ISO’s register designation[19]) and the diameter of the sun is approx. 864,000 miles, which is 864 Hz of the A in C5[20][21]. Another interesting calculation when time is involved is in that there are 86,400 seconds in a day (24*60*60), which equal to 432 x 200. Following one modern-day calculation of the precessional cycle, calculates a total of 25,920 year per cycle, which equals 432 x 60[22][23][24]. We could also spiral this into further mathematical relationships or the Fibonacci sequence metaphysicians are proud of such that the speed of light is 186,282 miles per second ((432/2)*864) = 216*864=186,282[25].

Sound absorption—Evelyn Glennie

When music, in the form of sound waves, are projected, it likely gets dissipated in the body somehow. An example of this is how percussionist Evelyn Glennie could “hear” pitches fairly accurate despite being totally deaf as she does it through feeling the resonance in her bones[13][26]. Thus, perhaps there is a specific (possible range) of frequencies that resonate well directly with our human bodies yet not enough of research is there to back it up yet.

DEMO

These are extracts from CC Youtube videos of songs tuned to A=440 Hz or A=432 Hz when played for ease of comparison[27][28]. I do not own the copyright for any of these audio and all sources are cited as linked, and considered under fair dealing exemption in Copyright laws. It would be optimal to open the two comparison audios and listen to them side by side for best comparison results.

A=440 Hz[27][28]:
Fur Elise guitar played in A=440Hz, by Paul Davids
I See Fire guitar played in A=440Hz, by Paul Davids
Harp played in A=440Hz, by Lilian Chan
A=432 Hz[27][28]:
Fur Elise guitar played in A=432Hz, by Paul Davids
I See Fire guitar played in A=432Hz, by Paul Davids
Harp played in A=432Hz, by Lilian Chan

References

  1. ↑ Ellis, Alexander J. The History of Musical Pitch. Nature 21, 550–554 (April 8, 1880). https://doi.org/10.1038/021550a0
  2. ↑ Musical Scales of Various Nations. Nature 31, 488–490 (March 26, 1885). https://doi.org/10.1038/031488a0
  3. ↑ 3.0 3.1 Calamassi, Diletta, and Gian Paolo Pomponi. “Music Tuned to 440 Hz Versus 432 Hz and the Health Effects: A Double-blind Cross-over Pilot Study.” Explore (New York, N.Y.), vol. 15, no. 4, 2019, pp. 283-290. doi:10.1016/j.explore.2019.04.001; https://pubmed.ncbi.nlm.nih.gov/31031095/
  4. ↑ 4.0 4.1 4.2 Calamassi, Diletta et al. “Music tuned to 432 Hz versus music tuned to 440 Hz for improving sleep in patients with spinal cord injuries: a double-blind cross-over pilot study.” Acta bio-medica : Atenei Parmensis vol. 91,12-S e2020008. 30 Nov. 2020, doi:10.23750/abm.v91i12-S.10755; https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8023109/
  5. ↑ 5.0 5.1 Aravena, Pedro Christian, et al. “Effect of Music at 432 Hz and 440 Hz on Dental Anxiety and Salivary Cortisol Levels in Patients Undergoing Tooth Extraction: A Randomized Clinical Trial.” Journal of Applied Oral Science: Revista FOB, vol. 28, 11, May 2020, p. e20190601. EBSCOhost, https://doi.org/10.1590/1678-7757-2019-0601; https://pubmed.ncbi.nlm.nih.gov/32401941/
  6. ↑ 6.0 6.1 Di Nasso, Luca, et al. “Influences of 432 Hz Music on the Perception of Anxiety during Endodontic Treatment: A Randomized Controlled Clinical Trial.” Journal of Endodontics, vol. 42, no. 9, Sept. 2016, pp. 1338–43. EBSCOhost, https://doi.org/10.1016/j.joen.2016.05.015
  7. ↑ 7.0 7.1 Halbert, James D., et al. “Low Frequency Music Slows Heart Rate and Decreases Sympathetic Activity.” Music and Medicine, vol. 10, no. 4, Oct 2018, pp. 180-185. doi:10.47513/mmd.v10i4.614; https://www.researchgate.net/publication/328792293_Low_Frequency_Music_Slows_Heart_Rate_and_Decreases_Sympathetic_Activity
  8. ↑ 8.0 8.1 8.2 Rosenberg, Ruth E. “Perfect Pitch: 432 Hz Music and the Promise of Frequency.” Journal of Popular Music Studies (University of California Press), vol. 33, no. 1, Mar. 2021, pp. 137–54. EBSCOhost, https://doi.org/10.1525/jpms.2021.33.1.137
  9. ↑ 9.0 9.1 Ellis, Alexander J. The History of Musical Pitch. Nature 21, 550–554 (April 8, 1880). https://books.google.ca/books?id=hnkWAAAAYAAJ&pg=PA333%23v%3Donepage&q=&redir_esc=y#v=onepage&q&f=false
  10. ↑ 10.0 10.1 10.2 Sagol, Assaf D. "Music Theory: 432 Hz Tuning - Separating Fact From Fiction." https://ask.audio/articles/music-theory-432-hz-tuning-separating-fact-from-fiction
  11. ↑ Walk That Bass. "7. Why A 440 Hz?" https://www.youtube.com/watch?v=GwWXDwqgWjo
  12. ↑ Cavanagh, Lynn. "A brief history of the establishment of international standard pitch a=440 hertz."https://www.wam.hr/sadrzaj/us/Cavanagh_440Hz.pdf
  13. ↑ 13.0 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 AwesomeAcoustics in English. "The real physics and math of 432Hz vs 440Hz and Schumann resonance." https://www.youtube.com/watch?v=lMS30ybk1_w
  14. ↑ Brown, Peter. "432 Hz—The Truth Behind Nature's Healing Frequency." https://www.binauralbeatsmeditation.com/432-hz-truth-behind-natures-frequency/#The_Schumann_Resonance_The_Earth_s_Heartbeat
  15. ↑ 15.0 15.1 Neely, Adam. "Testing 432 Hz Frequencies (and temperaments)." https://www.youtube.com/watch?v=ghUs-84NAAU&t=645s
  16. ↑ Davis, Faith. "Chakra Frequencies: What They Are & How to Achieve Resonance." https://cosmiccuts.com/en-ca/blogs/healing-stones-blog/chakra-frequenciess
  17. ↑ Elster, Allen D. "Angular Frequency (w)." https://mriquestions.com/angular-frequency-omega.html
  18. ↑ 18.0 18.1 Solà-Soler, Jordi. "On the concert pitch A=432 and C=128." https://www.sacred-geometry.es/?q=en/content/concert-pitch-a432-and-c128
  19. ↑ Open Music Theory. "Pitches and octave designations." http://openmusictheory.com/pitches.html.
  20. ↑ Benson, Tom. “Moon.” NASA, 13 May 2021, https://www.grc.nasa.gov/www/k-12/rocket/moon.html.
  21. ↑ Dunbar, Brian. “The Sun.” NASA, 6 May 2015, https://www.nasa.gov/sun.
  22. ↑ Benjamin, Roy. “What Era’s O’ering? The Precession of the Equinoxes in ‘Finnegans Wake.’” James Joyce Quarterly, vol. 48, no. 1, University of Tulsa, 2010, pp. 111–28, http://www.jstor.org/stable/41429839.
  23. ↑ Hoadley, Renee. “Article 257: Geometry of the Solar System - Part 4 - Precession of the Equinoxes.” Cosmic Core, 22 Feb. 2021, https://www.cosmic-core.org/free/article-257-geometry-of-the-solar-system-part-4-precession-of-the-equinoxes/.
  24. ↑ Jones, Scott, and Elaine Jones. “ The Precession of the Equinoxes.” Precession of the Equinoxes, http://www.ancient-wisdom.com/precession.htm.
  25. ↑ Khan, Lucien. “Fibonacci Sequence & 60 Number Fibonacci Repeat Cycle.” Facebook, 11 Sept. 2017, https://www.facebook.com/FibonacciSequence60/photos/the-diameter-of-the-moon-is-2159-miles-2160the-diameter-of-the-sun-is-8645759-mi/705123093030790/.
  26. ↑ Ted Radio Hour. "How Do We Listen When We're Unable to Hear?" https://www.npr.org/2015/06/05/411730683/how-do-we-listen-when-we-re-unable-to-hear
  27. ↑ 27.0 27.1 27.2 Davids, Paul. "The Ultimate 432Hz VS 440Hz | CONSPIRACY + Comparison." https://www.youtube.com/watch?v=Rt3EAPDn-Ug
  28. ↑ 28.0 28.1 28.2 Chan, Lilian. "A Fun Test! “432Hz vs 440Hz” | 那一部豎琴的頻率較療癒?" https://www.youtube.com/watch?v=jLEHI6qDZh0
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