A rude response to the theses of a "study" aimed at our understanding of Pythagorean values.
It is my choice not to address the authors privately before publishing the following text, as I believe that their already published position is an act of shameless insult, touching the pillar of Science upon which Art rests. And since already none of the artists seem determined to speak out in the face of this avalanche, I choose this response to be fierce, balancing all the brutal silence that "studies" like this one have taken advantage of.
The following lines are my immediate response after reading the following documents. What follows is not a proper critic to the scientific results this paper is supposed to deliver. It is an outrage against the lack of research inspiration.
The article advertising the paper is this one:
But the above article is nothing compared to the study and the bibliography upon which it is based, listed here:
For those of you who don’t understand, or don’t want to understand, the content of my protest, please consider my text as an act of advertisement to their work. The authors attempt to disassociate the sentiment of “pleasantness” with the Pythagorean fundamental proportions of tetractys (1,1/2,2/3,3/4) by collecting opinions of perception. Because how can you even think of measuring “pleasantness” when listening to the 5th (2/3) or the fourth (3/4)? By putting it in public vote? And who are the people who answered? Really now?
Of course, a gamelan instrument produces frequencies of its harmonic stele that are “out of tune”! This is part of the instrument’s nature! But whether Cambridge acknowledges that or not, the overall timbre obeys to the fundamental principles of “τετρακτύς” and whether you like it or not, a sound consonance is the sonification of mathematics!
Thanks to Pythagoras, tetractys will always be there to prove your stupidity!
So “λετ με σταρτ” :-)
“The phenomenon of musical consonance is an essential feature in diverse musical styles.”
It is certainly not essential! “Musical” consonance or its “essence” is literally unknown to many musical traditions. Ask my perceptions on the traditional music of my country. I’m a Greek. The way authors of this study conceive “musical consonance” comes from the experience of hearing at least three tones that sound simultaneously - which, by the way, is how the average listener understands music harmony today. In the world of Pythagoras the harmony of today’s “consonance” did not exist. (In his time, three natural sounds, in their entire harmonic stele, could not ever musically consist of a consonance! In our days, or in reality - or in mathematics, I have news for you: this is absolutely true, because, literally, they cannot!)
“The traditional belief, supported by centuries of Western music theory and psychological studies, is that consonance derives from simple (harmonic) frequency ratios between tones and is insensitive to timbre.”
Actually, this is true. But in order to fully understand this connection (consonance deriving from simple frequency ratios insensitive to timbre) outside mathematics, you need to analyze sound using electronic devices. And that would have been a good start of your thoughts, although there is no such “traditional” belief. There is evidence of historic “Western Music Theory” that certainly experimented with consonance out of the expected “traditional” music theory - and I leave “psychological” studies out of this. Artists who made harmony their profession were also scholars -they don’t exist today, don’t bother searching. It is they who I think you should ask to measure their “pleasantness” in consonance. Unlike us, they were true masters of harmony as defined in the Enlightenment. Therefore I ask you, have you ever heard of Charles De Lusse[1] (not to mention numerous scholars before him or even before the Fall of Constantinople)?
Picture 1, The fingering table of special notation for the performance of the "genii", in the transverse flute with one key. De Lusse, Paris 1763
But, don’t bother answering. You are ruining it in your next phrase:
“Here we show through five large-scale behavioral studies, comprising 235,440 human judgments from US and South Korean populations, that harmonic consonance preferences can be reshaped by timbral manipulations, even as far as to induce preferences for inharmonic intervals.”
Really? “Harmonic consonance preferences…reshaped”? “Preferences?” There are no preferences in mathematical results…(!). As I said, consonance of the 5th (Re - La in your example) will always be a consonance coming out of the proportion 2/3 whether that derives from a sine wave or from a sound of a complex timbre with thousands of frequencies (if it wouldn’t then it wouldn’t have been “Re - La”). The evidence of (not the “essence” of) this connection will always be there to stick its tongue out to your “research”!
And let me ask: Do you authors believe that the actual world in which we live in - either that being US or South Corea- has sufficiently inherited us the necessary skills, means or resources, to measure the perception of 235,440 people who can judge the “consonance” or “dissonance” of two tones sounding together according (or, even better, “discording”) to their own perception of “pleasantness”? Or do you -the authors- feel trained enough considering yourself equipped with the proper audio perceptions to do that? Are your electronic devices or analyzing methods sophisticated enough to quantitatively measure the connection of psychic pleasentness with frequency?
Really now? 235,440 you consider people? And what about the rest of “the people”? Are they monkeys? Because I can give you hundreds of millions of “people” jumping up and down like monkeys in whatever s#!t-“concert” the “whatever” s#†t “artist” gives in stadiums all around the world. Should the authors measure the deviations in timbre-consonance during these s#!t-“concerts” ? Or should they measure the “pleasantness” of these jumping monkeys to prove Pythagoras was wrong? (I think they should. Because that would have turned your study into something even more ridiculous).
And what about the “inharmonic” intervals? There is no such term as «αναρμονία». Instead, in the Greek language we use the word «δυσαρμονία». This word describes a situation when nothing can get along (or bond) with anything else (in other words: no two things can bond with each other). Although you don’t speak (or study) Greek, I know that you are arrogant enough not to seek for the meaning of the word «δυσαρμονία» (“dis-harmony”). You know of course that in our world, by definition, two or more octaves are «δυσαρμονικές» -“disharmonic” (or “inharmonic” as you like to call them) in pure measurements, depending on the tuning they follow. In the world of Pythagoras, this is a «δυστοπία» (“dystopia”). I’m sure you know that. But this “dystopia” is actually the blessing of western polyphony we all accept and enjoy.
“We show how such effects may suggest perceptual origins for diverse scale systems ranging from the gamelan’s slendro scale to the tuning of Western meantone and equal-tempered scales.”
Again, gamelan and all other percussion instruments, include a certain percentage of harmonic disruption or even “noise” in their spectrum. Therefore this study is so wrong in many aspects:
1) Gamelan intervals consist of sounding bodies that produce distorted harmonic steles. Therefore it should be expected that the guinea pigs (sorry, the people) who took part in this “study” would define the consonance of two timbres (not tones) in positions non-proportional to the pythagorean intervals. That’s because each timbre of the interval produces distorted harmonics.
2) Gamelan scales, as a result of intervals combined, are by definition not compatible with meantone or equal-tempered scales, not only because the scales upon which they are tuned are traditional, but also because the timbre of their sounding body is not harmonic! It cannot be! It is self-evident that the interval proportions from the scale would have been different from meantone and equal tempered scales.
3) It is the above fact that should guide you, the researchers, not to analyze measurements with precise measuring electronic devices as a whole, but to examine the results taking in account the deviation of harmonics in your experiment - assuming the fact that all people taking part in your experiment, are capable of feeling the “pleasantness” of fine-tuned 5th or 4th interval (because this is not a given). Of course, you didn’t do any of these. Why? Because you wouldn’t have any paper if you did.
Do you really think that Pythagoras didn’t understand anything about the difference between metal percussion and a string when listening to the blacksmiths’ hammering? I think it is you who don’t understand that his contribution to science is that he distinguished a certain relation between two sounds. The fact that he proved that there is a kind of mathematical connection between the harmony of the two sounds, a mathematical connection that can be confirmed with sound and the fact that it was always there before him or any other being, is awesome.
The actual paper begins with the following sentence, which bear citations to existing institutes that prove its scientific stability:
"Many musical styles involve multiple performers playing or singing
simultaneously [1–3]."
N O K I D D I N G !
Now that I keep on reading this, I’m afraid I’m not able to find a single line that fails to prove its pseudoscientific stupidity. I stand in awe, reading every single line after that. I find it difficult to keep on writing.
“…4272 online participants and 235,440 human judgments…” (!)
For those who might believe that the paper is about widening the audience of consonance perception without losing respect to the 12-note scale, I think they are wrong. Authors clearly reject mean-tone, just and any kind of intonation. Their perception of pleasantness does not include any kind of the 12-tone scale:|
“First: continuous treatment of pitch intervals. Previous consonance research has used stimuli drawn solely from discrete scales, most commonly the Western 12-tone chromatic scale [citations 12,27,29,34,35,38,39]. This is problematic because it neglects potentially interesting structure in between the scale degrees of the chromatic scale, and because the resulting paradigm is inherently Western-centric."...
It is clear that you investigate the “potentially interesting structure”, outside the compass of historical tunings, since every result of this investigation, taking in account historical tunings, would be “inherently Western-centric” and “problematic”. That’s of course contradictory to your own mention of Zarlino - and of course many other scholars that you omit, but you don’t seem to care much…
Truth is that the perception of consonance will always include timbre -even if the sound is constructed with parts of the harmonic column removed- because you cannot conduct experiments of sound appreciation with the brains of 235,440 people connected to sound source with electrodes bypassing their ears. The bias on the results of dichotic experiments would also have been even more problematic (good for them they avoided that).
You cannot measure perception without the ears (unfortunately). And, believe me, that’s inclusive because it includes all of us with working ears.
You seem to avoid evidence of the actual samples the participants were given. Instead you prefer to give us a rough description:
“In the dense rating paradigm, participants are played chords that are randomly and uniformly sampled from continuous intervallic space. For dyads (chords comprising two tones), we typically study intervals in the range [0, 15] semitones; in successive trials we might therefore see dyads such as 4.87 semitones, 12.32 semitones, or 1.83 semitones.”
And then a rating with a “pleasantness” scale between 1-7…
The very heart of your experiment, which is the actual audio presented to the participants, is described in less than 13 words and in parenthesis! :
“(Study 1: harmonic dyads; Study 4B: major 3rd with 3 dB/octave roll-off).”
For study 1: Which harmonic dyads? In which tuning? In which scale? In what durations? With what sounds/instruments?
For study4b: In which tuning? In which register? With what sounds/instruments? Why 3dB down as the thirds scale up the registers (I assume this is the explanation of the inexplicable: “3 dB/octave spectral roll-off”)?
And above all for both studies: in which sequence? If you think it doesn’t matter, you know you are wrong. It matters, and it matters big time! A group of tones (or consonances) form very different perceptions if you change the sequence they are presented.
Information like “1.3 seconds in duration” which is short, or “we played dyads… over the range G3-F4” which is vague, does not answer the above questions and I think either you don’t understand their significance or you deliberately didn’t bother dealing with them. Despite the fact that most parts of your study are uploaded to a repository, I was unable to locate a video of the sessions or even a sample of an online session-recording the participant’s online experience.
Every mathematical method for scaling you use on top of that to produce your “results” with all the statistical mambo-jumbo -even with… bootstrapping (!)- cannot diminish the fact that since you don’t care to answer the above fundamental questions in your study, your “ground” is invalid.
“Second: systematic exploration of timbral features. Several recent consonance studies have included timbral manipulations, but generally only explored a limited number of manipulations [12,34,35] or used manipulations designed to demonstrate generalizability rather than to test particular hypotheses[27].Here we take a more systematic approach. We focus in particular on spectral manipulations, because (as we show later) these yield particularly clear hypothetical effects in computational modeling. In a series of studies, we address the three main ways of manipulating a harmonic spectrum: (1) changing the frequencies of the harmonics (Studies 2 and 5), (2) changing the amplitudes of the harmonics (Study 3), and (3) deleting individual harmonics entirely (Study 4). In Study 1 we establish an experimental baseline for harmonic dyads (two-note chords), in Studies 2–4 we explore timbral features in the context of dyads, and in Study 5 we generalize our results to triads (three-note chords).”
All of a sudden we end-up speaking about three-note chords! Pythagorean approach is not based in “three-note” consonance. Actually the term “three-note” is wrong but English language does not help you understand the difference. The correct term is «τρίφωνος συγχορδία» (“three-voice chord”) from «τρία» (“three”), «φωνή» (“voice”) and «συγχορδία» which comes from «συν» (which means “plus” suggesting “the incusion of” something) and «χορδή» (“string”). But, unfortunately, English, as always, messes-up with the right terms: “Chord” is the transliteration of «χορδή» (“chordi”) becoming /xorˈði/ in the phonetic alphabet.
So, one “chord” aka «χορδή» (which is “chordi”, the singular of one “string”) producing (multiple “chordes” aka «χορδές» (many “chords”, /xorˈðes/, the plural of one «χορδή») resulting to a «συγχορδία» (“synchordia” /siɣxorˈði.a/) which means “multiple” + “strings”) is a pure paradox! In other words, an «οξύμωρον» (“oxymoron”). No “χορδή” can produce many “χορδές”. Every “χορδή” consists of its own independent harmonic structure. Now, one natural «χορδή» contains the spectral content of its sounding body which is connected to the properties of its material. The more you derive from the fundamental proportions of tetractys (1,1/2,2/3,3/4) the more distorted the result of tuning is against the fundamental.
In other words, it is absolutely expected to have deviations in frequency that does not comply with either the mean-tone or the equal-tempered scales, since the testing sound is a «συγχορδία».
“Together, these 23 experiments characterize the relationship between timbre and consonance in great detail, shedding light on the psychological mechanisms underpinning consonance perception, as well as the close connection between musical instruments and the cultural evolution of musical styles.”
It is impossible to go through all the studies to which you refer before making this post. Nevertheless, I consider the reading of them unnecessary. The idea of dismantling «διακρότημα» (search in libraries for the Greek term and its ancient-language properties, not the encyclopedia) or even its ingredients into their elements, or the «διακροτηματικές» frequencies by manipulating parts of their harmonic stele, will not ever lead you to safe conclusions. And the reason for that is that humans will always have the necessity for understanding the aesthetics of «διαστηματική» (/djas.ti.maˈti.ci/, google the term) and its connection to music, with their bodies.
Now, regarding the sounds used, I should have expected nothing less. You use cheap MIDI soundfont libraries to produce your experiments that don’t correspond, not even near to the harmonic truth of real sounds. The constant artificial vibrato, or the poor sound envelope of the soundfont itself, or the unreal low registers used to produce your examples are leading only to artificial results.
A crucial element of psychoacoustic approach beyond the mathematical analysis and processing of frequencies should be the “pronunciation” of sound. Again English language is poor. The appropriate word is «εκφορά» (/ek.foˈra/) which intensifies the meaning of sound articulation in musical sounds as a characteristic of the human language.
There can be no “scientific” explanation to the human preferences on harmonic consonances based on the analysis and manipulation of timbre, because the concept of human speech is connected to the perception of timbre from both our ears right at birth along with the rapid understanding of the world, still using our ears, years and decades after that. This perceptual notion of sound defines us from the very beginning. You should connect your brains with electrodes instead of having sounds vibrating your ears in order for your “research” to gain ground. But whether you like it or not, Music and its aesthetics will always be the altera pars of the Human voice and the perception of Human voice will always be considered as a moving notion between these two: «Ταῦτα δὲ θεωρεῖται ἐκ φωνῆς ποιότητι, ἧς κινήσεις εἰσὶ δύο. Ἡ μὲν συνεχής τε καὶ λογικὴ καλουμένη· ἡ δὲ διαστηματική τε καὶ μελωδική. ἡ μὲν οὖν συνεχὴς κίνησις τῆς φωνῆς, τάς τε ἐπιτάσεις καὶ τὰς ἀνέσεις ἀφανῶς ποιεῖται, μηδαμοῦ ἱσταμένη ἢ μέχρι σιωπῆς. ἡ δὲ διαστηματικὴ κίνησις τῆς φωνῆς, ἐναντίως κινεῖται τῇ συνεχεῖ̇̇ ̇ μονάς τε γὰρ ποιεῖ καὶ τὰς μεταξὺ τούτων διαστάσεις· ἐναλλὰξ αὐτῶν ἑκάτερον τιθεῖσα.» [Anecdota Graeca e codd. manuscriptis bibliothecarum Oxoniensium, I. “Περὶ φθόγγων. περὶ διαστημάτων. περὶ γενῶν. περὶ συστημάτων. περὶ τόνων. περὶ μεταβόλων. περὶ μελοποιίας”]
There can be no “scientific” explanation to the human preferences on the deviations of harmonic consonances either. And this is because we evolve as human beings, we experiment with the properties of our bodies and that certainly include our ears along with their interaction with our brain. It is the bending of sounds upon which we have the ability to experiment. “What would happen if…” is the main question that circles around the mind of an infant. To get reliable results, following the rational of your study, upon the perception of consonance, you should ask 235,440 babies before becoming infants.
This leads us to write a few things regarding the musicians participated in the study. I would say they already have a timbral concept. Again the best term is «ηχοχρωματική αντίληψη» from «ηχόχρωμα» (/ˈixoxroma/, meaning rather “sound color” than “tone color”) which focuses on the final aesthetic perception from humans and not the “tymbre” (French word, coming from “tambourin”/ “tympanum”/ «τύμπανο») of the sounding body itself. Therefore the «ηχοχρωματική αντίληψη» of the participants is biased -to say the least- by their sound experience in today’s way of life which is incredibly different in comparison to the ancient. You assume by default that we share today’s notion of “pleasantness” with the ancient world - responsible also for the fragments of Pythagorean values to which you focus - as a common field of reference. «Η αισθητική της διαστηματικής» (“the aesthetics of the science of intervals”) and its projection in today’s world is the aeternal mathematical proof of the Pythagorean legacy and its connection with Music.
The giant leap you are silently trying to establish as a given, betrays your attempt to question that relation described in an arrogant argument of pure “presentism” that falls to the assumption that the aesthetics of sound-perception connected to the Pythagorean values is a misconception as it is “inherently Western-centric”.
Therefore the ground upon which your statistical analysis is based is shaky, to say the least.
Whether you are feeling “pleasant” listening to your voice, or not, allow me to enlighten you or the readers of this post, that ‘tetractys’, lies in the very foundation of your own voice. Since you, as a team of humans, wrote this study and you refer to humans please feel free to see for yourself the triumph of Pythagoras by analyzing your own voice, or any other human voice!
As I said, sound will always be there to prove your stupidity!
[1] L'Art de la flûte traversière par M. de Lusse, Paris 1763