VIDEO DESCRIPTION / NOTES
There is a serious mathematical problem with the tuning of musical instruments. A problem that even Galileo, Newton, and Euler tried to solve. This video is about this problem and about some of the ways to tackle it. It starts from the basic physics of sound, proves mathematically why some musical instruments can never be perfectly in tune, and then introduces the main solutions that were proposed to solve this problem, along with their upsides and downsides: Pythagorean tuning, Just intonation, the Meantone temperament, and finally – the equal temperament, which is the tuning system almost everybody uses today in the West.
To learn more about the connections between music and mathematics, I highly recommend the book “Music – A Mathematical Offering” by David Benson. This terrific book is a treasure trove of information, extremely well written, and its thorough discussion of temperaments is just one of the many topics it covers. The book can be downloaded legally and for free from here: https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-music.html
Sevish is a master of electronic microtonal music. His compositions, despite their ominous genre, sound light and fun. Check him out. https://sevish.com
Paul Davis explaining how and why John Frusciante (of the Red Hot Chili Peppers) “mistuned” his guitar in the song “Scar Tissue”
A Madrigal by Nicola Vicentino (1555), played on a 24-tone harpsichord tuned in meantone temperament, by Johannes Keller.
A concise introduction to Arabic music. Pay attention especially to the Albayati, Alsaba, Alsard, and Ahuzaam maqams, with their intense microtonality.
The Lumatone Isomorphic Keyboard is a cool interface to microtonal music.
Yehezkel Raz, the Ableton wizard, for transforming me from a complete Ableton noob to a good-enough user in less than two hours. https://yehezkelraz.com
The intro and outro music is “Snowfall Butterflies” by Asher Fulero (via YouTube Audio Library). https://asherfulero.com
Photo of the Antegnati Organ in Santa Barbara, Mantua (1565), courtesy of the organist Simon Lloyd. https://simon-lloyd.com
=== Contents ===
07:00 – Choosing frequencies
11:56 – Pythagorean Tuning
18:36 – Meantone Temperament
24:34 – Equal Temperament
29:50 – Other temperaments
The sound samples were prepared with Ableton Live 11. To make them piano-like but still as accurate as possible, I used the physical-modelling Pianoteq plugin, with the unison width set to 0, octave stretching ratio set to 1, string length set to its maximum value (to minimize string inharmonicity), hammer noise set to 0.5, pedal noise set to 0, and the velocity-to-dynamics curve considerably lowered (ending at mezzo-piano).
Created by Yuval Nov for the 2022 “Summer of Math Exposition” (SoME2) competition, hosted by the one and only 3Blue1Brown (Grant Sanderson).
CHOICE COMMENTS TO VIDEO
Fun fact: violinists frequently use all three systems: pythagorean, just, and equal temperament. We tune the instrument’s strings according to the perfect 3:2 pythagorean fifth and usually play single lines without accompaniment wholly in the pythagorean system, as the whole steps are wider and the half steps are narrower, giving melodic lines more direction. When playing chords, or playing with other instruments which also use pythagorean tuning (like other stringed instruments), we often will adjust certain notes to just intonation to avoid clashing. We try to avoid adjusting melodic notes this way, instead preferring to adjust only the harmony notes. When playing with equal temperament instruments like piano, if there are any long sustained notes where the intonation difference and resultant clash will be clearly noticeable, we occasionally adjust to equal temperament for just a moment to avoid this. Performance is an art of compromise!
Equal temperament without any just interval permit to keep a permanent place for ALL notes. To try it you must avoid the just fifts-A/E can stay just because E is metalic and will go down itself. But D cord must be tuned maximum hight possible. The same for G string. The difference is very small. Each interval must be modified respecting follow rules: 1/unisson =just, 3minor=smaller, 3major=bigger, 4=bigger, 5=smaller, 6minor=smaller, 6major=bigger, 8=bigger. How much? It’s you who decide! The direction of modification is important only.
DISSONANCE OF BEAT FREQUENCIES When you mix frequencies together you end up with the cross-Cartesian product of all of the sums and differences of the frequencies being mixed. So, if I mix two frequencies A and B together, I end up with four different frequencies: A, B, A+B, A-B (or B-A if B should be larger). In time domain graph of the waveform, you would simply see one frequency superimposed over another similar to images depicting an AM modulated waveform.
This is one of the best outlines of scales, intervals and temperaments I have seen on line. Some historical perspective, just enough to explain the new demands due to harmony singing, or modulation, but not going into excessive and misleading detours that you get so often in explanations of musical scales. [Nice how] you point out clearly that equal temperament is a necessary compromise and not a perfect solution.
One thing that could have been explored a bit more (perhaps) is the reason why small integer ratios sound more harmonious, and how that’s literally connected to constructive and destructive interference in physical waves. For example, a brief audio & visual representation of how ‘beats’ form when two notes slightly off-tune from each other are played. 29:34 The transposition of the melody here actually sounds great past the first chord. The final dominant 7th chord is more in-tune with the harmonic series, being constructed of 5/4, 3/2, and ~7/4. The preceding chord is a minor chord with a lowered minor third based on that 7/4 interval. 11:50 — Equating a Temperament with a Tuning is a common mistake, but in fact not quite correct! Temperaments a subset of Tunings ; all Temperaments are Tunings but not all Tunings are Temperaments! A temperament is a scheme for adjusting pitches from their exact-integer-ratios. So, Pythagorean and Just Intonation are Tunings, but they are not Temperaments, because they use exact integer ratios. Equal-Temperaments, Meantone Temperaments, and Well-Temperaments are temperaments. They have deliberately and systematically adjusted their pitches away from exact whole-number ratios. 18:37 — Minor Historical nit: Meantone Temperaments were much more common in the mid-late Renaissance than in the Baroque, by which time Well-Temperaments began to take over (and persisted into the mid-late 1800s, BTW — longer than most people realize).
>> The ultimate problem is digitalia which removes all heart and soul and feeling from music so muddled and reproduced. It turns out that the micro-anomalies, errors, perturbations are in fact what makes music lovely and musical performance (and performers) indispensible. MIDI did not ‘oust all drummers’ as was initially predicted and in fact proved the opposite need. Autotune makes its hallmark awful mark routinely. “ProTools sound” has become denigrantly well-known amongst recording engineers. “Perfection” is always an aspiration and never an attainment, IMO.
Well, thanks for the informative video. The equal temperment has not only the octave as a geometrically sound interval but also the tritone that the dominant-seventh chord is the best tuned chord on the piano with its notes only a hair above what the seventh and leading-tone should be. This changed music drastically in that the dominant-seventh gained in popularity. The out-of-tunedness of the piano is compensated by the deeper and less harmonic notes of the bass overpowering the difference-tones of the above notes. Singing instruments and singers fluctuate their notes within a pitch range in order to accommodate the exigencies of the harmonic movement. Tones which occur simultaneously would otherwise form dissonances with the actual harmonies. Using your a=440hz a c# would need (on the flute for ex.) to be 550hz in order to produce a difference tone of 110hz (or an A’). at the same time, this c# combines with an e”=660hz in order to produce an A’=110hz. The a’ played together with the e” produces an E= 330hz. Should one of the harmony notes not have this geometry to the tonic, the chord would sound horrific. The higher the instruments are playing, the easier this is to hear this. The aformentioned c# changes its pitch as soon as a g” is introduced, bending higher to what’s called a leading tone to c#’= 556hz, while the g” is a geometric fourth (4/3) above the d’ which is also a 5th below the a’ (g = 391hz). The difference tone produced between these two notes g (391hz)and c# (556hz) is an E = 165hz, fitting with the 110hz = A’ perfectly consonant in the harmony. Walter Piston illustrated this phenomenon with a contrabass playing first the C# of an A-major chord and adding the septime and watching the visible raising of the bass-note C#. The Juilliard String Quartett with Bob Mann as the first violin experimented with playing tempered for about 10 years from the mid 50s (heard on many recordings) but stopped this practice due to intonational differences. The Guarneri (David Sawyer as the cellist) thought that they were playing “wide” thirds but I showed them that this wasn’t so – they were indeed playing in-tune using what I call “Integrated Intonation”.
>> One of JAN IRVIN (LOGOS MEDIA) many interviews with Dr Hans Utter notes 57 notes in the octave of Indian music played on Sitar.
So, I built a guitar about 55 years ago. I knew how long my fingerboard was and so I knew where I wanted to put the fret for the first octaves (which I know as the first harmonics). On the other hand I didn’t have any idea if I was going to tune the open strings to the right frequency – though I would of course tune the strings to each other – so I guess I was going to transpose just about everything until I got that right! That more or less forces me to go with 12 geometrically equal intervals and it seemed so obvious that I simply got out the log tables – yes, that long ago, big shout out to John Napier here – and worked out where to cut my fingerboard for the frets. It seemed so obvious at the time, I never thought there could be any (sensible) other way, so that’s very good to know now. This is the no-nonsense music theory! Many called themselves teachers in music theory are priggers, get your tuition and gave a lot of crappy terminologies…ruined many in music schools…
I think this could actually the best video on YouTube.
MUSICAL CULT CONTROL paper by Dr Leonard Horowitz apropos. See also youtubes by MARK DEVLIN and NEIL SANDERS which delve into the MK aspects of modern music. There is much more to this than realized by most.
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Michelle Gibson is a researcher and author known for her work on topics related to alternative history, ancient civilizations, sacred geometry, and energy phenomena. While she has discussed various subjects in her research, including star forts, cathedrals, resonance, and healing, it’s important to note that her work is often considered speculative and controversial within mainstream academic circles. Here are some key themes that Michelle Gibson has explored in her research: 1. **Star Forts**: Gibson has discussed the architectural design and purported symbolism of star forts, which are a type of fortification characterized by a star-shaped layout with protruding bastions or points. Some researchers, including Gibson, have proposed that star forts may have served purposes beyond military defense, such as energy manipulation, spiritual symbolism, or navigation aids. 2. **Cathedrals**: Gibson has examined the construction and symbolism of cathedrals, particularly Gothic cathedrals in Europe. She has suggested that these structures incorporate principles of sacred geometry and esoteric knowledge, and may have served as energetic centers or healing spaces in addition to their religious functions. 3. **Resonance**: Gibson has explored the concept of resonance and its potential effects on human consciousness, health, and well-being. She has suggested that certain architectural features, materials, or locations may resonate with specific frequencies or energies, influencing the environment and individuals within it. 4. **Healing**: Gibson has discussed various holistic and alternative healing modalities, including energy healing, sound therapy, and vibrational medicine. She has proposed that ancient structures, natural landscapes, and sacred sites may possess healing properties or energetic qualities that promote physical, emotional, and spiritual well-being. While Gibson’s research may offer alternative perspectives on history, architecture, and consciousness, it is important to approach her work critically and evaluate claims based on empirical evidence and scholarly consensus. Some of the ideas and interpretations put forth by Gibson and others in the alternative history community may lack scientific validation or rely on speculative assumptions. As with any interdisciplinary inquiry, it is essential to maintain skepticism and rigorously evaluate claims in light of available evidence and established principles of scholarship.