This Kurt Ellenberger articles cites a fascinating melody study of the overtone series. In a sequence of notes, a listener always prefers the melody to end on a octave pitch equivalence of the root tonic, aka the "binary sequence" from the exponential series.
So this is assuming then that the octave is inherently a square from a geometric mean based on symmetric, as Kurt Ellenberger emphasizes this inherent symmetric of the wavelength or 0 to 1 fundamental tone root tonic. The problem with this derivation is it ignores the order of time based on the ratios. So 3:4 is mentioned as a Perfect Fourth pitch from a ratio that does not include the order of the ratio:
When he used adjacent members on the harmonic series, because of the peculiar relationship of the pitch's numerical position and its string length, the ratio of their wavelengths to each other is the same as their positions in the series. Thus, the string length ratio of the second overtone to the third overtone is 2:3, and the third and fourth is 3:4 and so on.
So that's a wavelength of 2/3 with a frequency of 3/2 for the Perfect Fifth. And so the wavelength of 3/4 is the frequency of 4/3 as the Perfect Fourth. In terms of Pitch though the 3 is now "noncommutative" to the root tonic octave pitch symmetric. So in the first case of the Perfect Fifth the 3 is a Pitch of G. In the second case the 3 is a Pitch of F to the root tonic of C.
This is what I call the "Bait and Switch" that hides the noncommutativity. By excluding the "third" parameter of the pitch then the Gurdejiff "Law of Three" is lost. I have found this "bait and switch" to be the consistent approach to the overtone series that in fact is treating it as the "harmonic ratio" series assuming a symmetric mathematics. And so in fact 4/3 as a frequency can never been an overtone of the root tonic since 3 is in the denominator.
https://newmusicusa.org/nmbx/iv-the-phantom-tonic/
Nicolas Slonimsky once pointed out, in an effort to dissuade readers from the idea that Western tonality is the inevitable result of how we hear (as opposed to a largely artificial invention), that no matter how high one goes in the harmonic series, a fundamental pitch will not produce a perfect fourth above the fundamental....Just as Slonimsky opined, the perfect fourth above the tonic is nowhere to be found....
One other dominant does exist, however: the one built on the tonic itself. It resolves, not to any note within the scale, but to a foreign pitch—the so-called subdominant. Thus the perfect fourth above the tonic enters the scene, not as part of a stable major scale, but as a tempter, a seducer, a built-in modulation away from the true tonic. The perfect fourth, and not the tritone, is the true “devil in music.” It’s no “subdominant.” It’s the phantom tonic.
This is from the book Music and Sound, published in 1937 by the English organist, composer, and theorist Llewelyn Southworth Lloyd:
All the evidence shows that, in the early stages of a scale developed in the attempt to sing melody, one of two intervals, the fourth as an interval approached downwards, or the fifth, would almost certainly provide its first essential note other than the octave. (Emphasis added.)
So here we have the key noncommutativity of the Perfect Fourth interval with the order of time reversed.
Later in the same paragraph:
Our subdominant is a true fifth below its (the fundamental’s) octave.
Not a fourth above, but a fifth below: the phantom tonic.
So how does this fit into the binary sequence of the octave? As Alain Connes, the Fields Medal math professor points out, the Perfect Fifth is noncommutative when reversed back into the same octave symmetry for the scale! Thus the Pythagorean Comma is 3 to the 12th against 2 to the 19th while back into the scale it is 3 to the (1/19th) against 2 to the (1/12th). Another way to state this is the inverse of the Perfect Fifth as 3/2 is not commutative since 2/3 is C to F and 3/2 is C to G and therefore 3/2 x 2/3 does not equal one as the root tonic! https://en.wikipedia.org/wiki/Pythagorean_tuning
So in terms of Pitch if we assume "C" again then the Perfect Fifth is "G" and the Perfect Fourth is "F" clearly showing a reversal of time in direction as noncommutative since the "F" is derived from the octave as a reversal against the root tonic, i.e. a Perfect Fifth in the other direction as an undertone.
The only way this can be justified in terms of the overtone series assuming a "root tonic" as the starting point in time is to use a DIFFERENT root tonic as Philolaus did. So therefore the true "wavelength" of the root tonic has to assume a double octave with the ratios doubled as the common denominator. So therefore 4/3 is now derived from a 0 to 8 wavelength with a 6/8 wavelength for a 8/6 frequency. So the ratio of wavelength is indeed the same as 3:4 but the root tonic or "fundamental tone" is different therefore creating noncommutative geometry instead of symmetric as the fundamental "ordering" process of listening to frequency via time. There is an inherent asymmetric shift in time inherent to pitch processing. The first logarithmic equation was created from music theory based on this "Liar of the Lyre" Philolaus flipping his lyre around so that 0 to 12 fundamental tone root tonic is 3/2 frequency as 8/12 wavelength and thus the Perfect Fourth (8/6) PLUS the Perfect Fifth (12/8) = the Geometric Mean Squared as 12/6 (again assuming geometric ratio as symmetry is definitive and NOT the ordering of time via listening!)
Fascinatingly this asymmetry in time-frequency processing is inherent to our biology as well! Contrary to the Western religion of science as symmetric mathematics, our biology is also inherently noncommutative. This is demonstrated in solving the mystery of Perfect Pitch.
Experiments with musicians showed that those with absolute pitch had faster reaction times and more accuracy for tones when they were played in the right ear only [2]. As people with absolute pitch have also displayed higher neuronal activation, the activation and processing sounds is related to absolute pitch [2]. This neurological difference can be explained in the
differences in auditory processing, as the left side processes rapidly-changing and fine-grained sound information,
https://journals.le.ac.uk/ojs1/index.php/jist/article/view/4326/3680
In other words the conversion of frequency via time is faster than the time-frequency uncertainty principle in people with Perfect Pitch, indicating a quantum coherent nonlocal noncommutative phase.
Having a tonal language, like Mandarin Chinese, Vietnamese, and Japanese,
where the tone differences in speech have different implications, as a first language may predispose someone to identifying subtle differences in notes, and could lead to an increased affinity for perfect pitch [1-3]. However, other research indicates that, while learning tonal languages may help to form an association between pitches and labels, it does not appear to improve one’s ability to discriminate between them [2]. Furthermore, the differences in distribution could be due to altered cultural attitudes towards musical education [8] rather than the language itself....Earlier musical training is also associated with absolute pitch being more prevalent, with the majority of possessors having begun their musical studies below the age of seven [1, 2, 8]. There is also the phenomenon of the Levitin effect, where people without absolute pitch can recall tunes and sing them back in the original key without a reference note, like nursery rhymes or popular songs on the radio or TikTok, indicating some capacity for learning [6, 7].
The Levitin Effect is subconscious though without any knowledge of what the particular frequency or pitch is relative to the root tonic. Since Absolute Pitch is more prevalent in early music learners that also have been shown to have an increased corpus callosum that integrates the right and left sides of the brain, this demonstrates again the key secret of nonlocal noncommutativity in developing Perfect Pitch as a conscious skill, knowledgeable of the frequency and/or pitch relative to the fundamental tone of a song or melody.
Furthermore, relative pitch, where a reference note is used, is often more useful, particularly when transposing music live [3, 4]. Having or lacking absolute pitch does not dictate the success of a musical career, and relative pitch can be just as useful, if not more so in practice.
This was the same point alluded to by Haley Reinhart in her recent podcast interview - as the person interviewing her bragged of having "absolute pitch" and she said she also has it, well actually "relative pitch" in the absolute sense! She then proceeded to perfectly repeat a complex melody that the interviewer sang to her, as the two began to improvise together, wondering who would do the timing as rhythm and who would improvise on the melody!
We should clarify what we mean by nonlocal noncommutativity, as per Fields Medal math professor Alain Connes definitive lecture, "Music of Shapes." The growth rate being exponential is the limit of the Hilbert Space as an infinite discrete noncommutative matrix. So the noncommutativity is the inherent imaginary phase in time that changes the order of the discrete invariant 2:3 ratio in music theory, thereby labeled by Alain Connes as, "two, three, infinity." The Heisenberg Uncertainty of position and momentum is due to this subtraction of noncommutative exponentials, via the imaginary noncommutative time phase. So you have a matrix of 0, 1, -1, 0 that is the commutator to restore the symmetric for classical physics via the 2 x 2 matrix of the exponentials and inverse ratios as logarithms.
Moulton, C. (2014). Perfect pitch reconsidered. Clinical Medicine, 14(5), 517–519. doi:10.7861/clinmedicine.14-5-517
The correlation between early training and development of AP is corroborated by numerous studies. A survey of over 600 music college students found that 40% of respondents who had begun their musical training before the age of 4 years developed AP, whereas only 3% of those beginning after the age of 9 years acquired AP.4
Clearly the AGE of music training is the key to developing Absolute Pitch aka Perfect Pitch.
A later study by the same authors negates the environmental component more effectively by discounting anyone who started training after the age of 6 years.
Quite fascinating indeed - HARD-wired for Perfect Pitch due to early music training before age 4.
Schlaug et al found that AP possessors exhibited a greater leftward asymmetry of the planum temporale (PT) in the temporal lobe, a region central to speech processing.9 The authors subsequently found that early beginning non-AP musicians did not exhibit the exaggerated leftward PT asymmetry of their AP counterparts, arguing that this asymmetry may be a determinant of AP acquisition rather than a consequence.10
So here is the key factor of the noncommutative asymmetric time shift since the left brain processes time shifts.
Western languages such as English use pitch syntactically to convey expression, such as raising the tone of the voice at the end of the sentence to indicate a question. The idea of a fixed tone then seems somewhat incongruous in a species that relies on RP for communication. Some animals, by contrast, do use AP to communicate with others. Experiments on songbirds found that RP perception, although within the birds’ capacity, was subordinate to stimulus processing based on AP.18
Only recently has credible evidence emerged that a
fundamental role for AP may not be confined to the animal
kingdom. Diana Deutsch showed that speakers of tonal
languages, such as Mandarin and Vietnamese, when asked to
recite words on different days, do so at very similar pitches,
a pattern not demonstrated among English speakers.19,20
Moreover, recent studies have shown that Cantonese speakers
outperform their English-speaking non-musician counterparts
on various measures of pitch and music perception. 21
Studies have demonstrated significantly higher prevalence
of AP among Asian students than their white counterparts,
despite no difference in frequency of early music exposure.7
Asian students, however, were significantly more likely to
have received ‘fi xed-do’ training than white students. Deutsch
hypothesises that AP for music might then be acquired by
tone language speakers in the same way as they would acquire
the pitches of a second tonal language. 22 A more recent study
demonstrated significantly higher AP prevalence among
fluent tonal language speakers than non-fluent tonal language
speakers, thereby questioning the role of ethnicity as a
predisposing factor in AP acquisition.
AP possessors showed stronger activations in the posterior part of the middle
temporal gyrus, an area involved in higher-order language
processing, as well as stronger left-lateralised activation of the
superior temporal sulcus during processing of segmental speech
information. 24 The authors propose, therefore, that the auditory
acuity of AP may extend beyond musical processing to a more
general notion of acoustic segmentation by fully integrating
left-hemispheric, speech-relevant networks.
The hypothesis that tonal language speakers use AP to aid
communication is potentially exciting, because it could provide
the missing link as to why AP should provide any fundamental
value to humans at all.
https://www.mdpi.com/1422-0067/22/10/5397
Despite this, humans are the only [species] which possess a motor synchronization based on temporal anticipation with a mental model of time [59–61]....As a confirmation of this hypothesis, the motor cortico-basal ganglia-thalamo-cortical (mCBGT) circuit, which is involved in beat perception [70 ], shows to be more developed in humans than non-human primates [71,72].
Interestingly, music is not only associated with increased speech
perception, but recent twin studies seem to provide evidence for heritability and a positive relationship between sight-reading music ability and reading passage comprehension, suggesting an influence of decoding mechanisms that are independent of working verbal memory [76]....
Therefore, the clear involvement of FOXP2 in music activities in humans, and in song production in songbirds, further supports a coevolution of music and language.
Frontiers in Psychology 01 frontiersin.org
TYPE Hypothesis and Theory
PUBLISHED 13 February 2023
DOI 10.3389/fpsyg.2023.1055827
Testosterone, oxytocin and
co-operation: A hypothesis for the
origin and function of music
Hajime Fukui 1
* and Kumiko Toyoshima
Particularly, music listening decreases T levels in men and increases them in women.
listening to and playing music promotes empathy (e.g., Fukui and Toyoshima, 2014). Furthermore, studies have found that music increases trust in others and promotes altruistic behaviour (Fukui and Toyoshima, 2014).
Infant-Directed Vocalization as Prodosy (rhythm and melody)...
I’m aware of the issues with the 4th (5th below?),
The Fifth is a compound tone in which the second partial is the third partial of the fundamental compound tone; the Fourth is a compound tone in which the third partial is the same as the second of the Octave.p. 255 On the Sensations of Tone as a Physiological Basis for the Theory of Music
The thesis can be expressed in the following way: If two drones either a fourth or fifth apart are sounded, one of these will 'naturally' sound like the primary drone. It is not always the lower of the two which will sound primary, but the one which initiates the overtone series to which the other note (or one of its octaves) belongs. By amplifying a prominent overtone the secondary drone lends support to the primary and intensifies its 'primary' character. Ma [Perfect Fourth as 4/3], although consonant to Sa (root tonic), is alien to the overtone series and is not evoked in the sound of Sa. On the other hand, Sa is evoked in the sound of Ma, since Sa is a fifth above Ma and is its second overtone.
For this reason it can be argued that the tendency to view Ma [the Phantom Tonic] as the ground-note has a 'natural' basis. The same cannot be said for Pa [Perfect Fifth] as Sa is not part of its overtone series. (Jairazbhoy,
Jairazbhoy N. A. (1995). The Rags of North Indian Music: Their Structure and Evolution, Popular Prakashan.
However, even in this case there is a highly
non-commutative world of higher-order collectivity algebras Bn . This can be used
to capture the geometry of rotations, like those appearing in classical Pythagorean
octave versus perfect fifth considerations."
"The 12 comes from the fact that there are 12 notes when you make the chromatic
range. And the 19 comes from the fact that 19 is 12 + 7 and that the seventh note
in the chromatic scale, this is the scale that allows you to transpose. So what does
it mean ? It means that going to the range above is multiplying by 2 and the ear is
very sensitive to that. And transpose is multiplication by 3, except that it returns to
the range before, i.e. so it is to multiply by 3 / 2, that agrees.
Well, that’s the music, well known now, to which the ear is sensitive, etc. Okay.
But... there is an obvious question ! It is "is there a geometrical object which range
gives us the range we use in music ?". This is an absolutely obvious question."
"It is precisely the irrationality of log(3)/ log(2) which is responsible for the noncommutative nature of the quotient corresponding to the three places {2, 3,∞}. "
Colloquium 7: Philolaus on NumberIn: Proceedings of the Boston Area Colloquium in Ancient Philosophy
So what Philolaus did is CHANGE the root tonic since 2/3 was not allowed to create the FIRST logarithmic equation in western science! So Philolaus flipped his Lyre around and then used 6/8 as the wavelength for an 8/6 frequency or 4/3.
So to get the 4/3 frequency the root tonic was changed to 8 instead of the previous root tonic of 12 for 2/3 as 8/12.
So then the logarithmic equation was then 12 to 8 as 3/2 PLUS 8 to 6 as 4/3 = 2 as the octave.
(1, 4) = (7, 5) ... In the present case this means that regardless if you go from string 1 to string 7 via string 4 or string 5, the result is the same: (1, 7) = 0.
What are the observables for gravitation? Who can we say where we are? The answer is spectral [frequency]. ...It's not enough to know the spectral operators... ...Two noncommutative shapes that are Isospectral [i.e. both Perfect Fifth]...They have the same spectrum but they do not have the same second invariant. [Second-order tensors may be described in terms of shape and orientation.]
You find three types of notes of the spectrum. Integers plus 1/4 [Perfect Fourth], Integers plus 1/2 [Perfect Fifth] and Integers in the square of the spectrum [Octave] there are three kinds of NOTES. When you look at the possible chords - this is like the piano in which you can play...because they are three kinds of notes. The chords of two notes are possible for some shapes [Perfect Fifth] but not other shapes [Perfect Fourth]. The point is spectral, given by correlations between the eigenvalues (frequencies) of the Dirac operator.
So https://blogs.scientificamerican.com/critical-opalescence/the-wholeness-of-quantum-reality-an-interview-with-physicist-basil-hiley/George Musser: How does this [nonlocal entanglement] enter into quantum mechanics?
Basil j. Hiley: In noncommutativity. Every day in our life, we always have to be careful of the order. You’ve got a cup in the cupboard. You’ve got to open the cupboard door before you can the cup out. All our experience is doing things in the right order, so our activity is noncommutative. It comes into quantum mechanics because Heisenberg sought to explain atomic energy levels and what he found was he had to make his objects into things that didn’t commute with each other. The order was vital. There was a difference between first measuring the momentum and then measuring the position, from measuring the position and then measuring the momentum. That became the basis of his Uncertainty Principle.
For their analysis, Sommerfeld and Brillouin examined the propagation of ‘square’ pulses, for which, “in almost any case the beginning of the signal is a discontinuity in the signal envelope or in a higher derivative.” It was to be in this step-like front velocity, vF, - which may not exceed c in order to preserve causality - that the fiducial role of c as the limit of the front velocity of a wave packet “is deduced from the Lorentz invariance of the Maxwell equations.” (Heitmann & Nimtz 1994, p. 155)....
the group velocity, vGr, - and analogously described as the ‘centre of gravity’ of a wave packet. Insofar as “the group velocity was then believed to be identical to the energy propagation,” careful definition became important. (Jackson, Lande & Lautrup 2001, p. 1).
Central to the ensuing inquest was disquiet as to whether such observed pulses were, in fact, signals as carriers of information within the ambit of information theory and the prevailing interpretations of special relativity. Such contentions evoked the Italian researchers’ concession that “the question as to whether a wave packet can be considered a signal is a much debated and complicated one.” (Mugnai, Ranfagni & Ruggeri 2000, p. 4830).
Any realistic wave packet, or envelope employed as a signal, is of necessity frequency limited and in consequence cannot exhibit a sharp and accurately step-like front.......in the now ubiquitous world of electronic data handling, information often derives from the half-height detection of the practical approximation of a square wave pulse. Heitmann and Nimtz point out that application of such practicalities to evanescent and tunneling phenomena would validate claims of information superluminality in their cases.
Dr. Guenter Nimtz sends superluminal sound signals through the 5th dimension:
"the superluminal acoustic effect we have described is likely a ubiquitous but imperceptible phenomenon in the everyday world."
https://www.youtube.com/watch?v=e-iy3_CnznA
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