There are integer multiples of a certain frequency (fundamental), that are called harmonics, partial tones (partials) or overtones. It is important to note that the term 'overtones' does not include the fundamental frequency. The first overtone is therefore already the second harmonic or the second partial. The term overtone should never be mixed with the other terms, as the counting is unequal.
The term harmonic has a precise meaning - that of an integer (whole number) multiple of the fundamental frequency of a vibrating object.
A harmonic frequency is a multiple of a fundamental frequency, also called "harmonic". Especially when it comes to counting, do not say: "overtones are harmonics".
Musicians prefer the term overtones and physicists prefer the term harmonics.
Sound engineers are somewhat uncertain between these two terms.
Harmonics and overtones are also called resonant frequencies.
http://www.sengpielaudio.com/calculator-harmonics.htm
"Overtones" = Harmonics minus 1, or "Harmonics" = Overtones + 1
Frequency ratio | Tone example | Frequency in Hz | ||
fundamental frequency | fundamental | 1:1 | C | 65 |
double frequency | octave | 2:1 | c | 130 |
trifold frequency | fifth | 3:2 | g | 195 |
fourfold frequency | fourth | 4:3 | c' | 260 |
Why is that? Because the Perfect Fourth is NOT an overtone but is derived from doubling the UNDERTONE as noncommutative phase of the Perfect Fifth! And so the concept of "harmonic" as an "integer" is designed to coverup the Fundamental Frequency as the Fundamental Force of noncommutative phase.
So then the Fundamental Frequency is defined as a GEOMETRIC RATIO of 1:1 and NOT a natural number ratio and the concept of Integer then already assumes a symmetric geometric irrational magnitude ratio of the Octave as the 2 number as the SQUARE.
This is explained as the Geometric Mean Squared as the Octave or 2 from the Harmonic Mean (4/3) x the Arithmetic Mean (3/2)....
Conceptual structuralism is illustrated in (Feferman 2008) by discussion of two constellations of structural notions, first of the positive integer sequence and second of the power set conception of the continuum. For comparison with what I say below about the latter and other conceptions of the continuum, let me review briefly what I have to say there about the former.https://math.stanford.edu/~feferman/papers/Continuum-I.pdf
From the structural point of view, our conception is that of a structure (N+, 1, Sc, <), where N+ is generated from the initial unit 1 by closure under the successor operation Sc, and for which m < n if m precedes n in the generation procedure.So if 3 is "after" 2 as an integer then it can't be a ratio of 2/3 in the opposite time direction or undertone. Another way to say this is that the Overtone has to have a denominator that is an octave pitch equivalence of the fundamental tone. Therefore 4/3 is NOT an overtone yet it is a "frequency ratio."
The division is a method of distributing a group of things into equal parts.
So the Octave and the Perfect Fifth can NOT be distributed equally because two does not go into three. And so if you start at the "one" as the fundamental frequency then there can NOT be the Perfect Fifth as the UNDERtone at the same time as the overtone because that creates a NEW octave as the "2" since 3 does not go into 2 evenly and therefore Two can not stay as the same octave PITCH of the one.
A division ring is generally a noncommutative ring. It is commutative if and only if it is a field, in which case the term "division ring" is rarely used, except for properties of division rings that are true even if they are commutative or in the proof that a specific division ring is commutative.So the algebraic definition of division is different than the geometric definition of division!! Fascinating indeed.
Noncommutative version of natural numbers - pdf
The hybrid term subharmonic is used in music in a few different ways. In its pure sense, the term subharmonic refers strictly to any member of the subharmonic series (1⁄1, 1⁄2, 1⁄3, 1⁄4, etc.). When the subharmonic series is used to refer to frequency relationships, it is written with f representing some highest known reference frequency (f⁄1, f⁄2, f⁄3, f⁄4, etc.). ...
The human voice can also be forced into a similar driven resonance, also called “undertone singing” (which similarly has nothing to do with the undertone series), to extend the range of the voice below what is normally available.
https://en.wikipedia.org/wiki/Undertone_series
why not related?
In the German theory by or derived from Hugo Riemann, the minor mode is considered the inverse of the major mode, an upside down major scale based on (theoretical) undertones rather than (actual) overtones (harmonics) (See also: Utonality).https://arxiv.org/html/1202.4212v1
undertones must be produced in unusual ways. While the overtone series is based upon arithmetic multiplication of frequencies, resulting in a harmonic series, the undertone series is based on arithmetic division.https://en.wikipedia.org/wiki/Undertone_series
Fields Medal math professor Alain Connes:
a non-commutative algebra naturally engenders a one-parameter group of automorphisms that makes it evolve, that makes it rotate in such a way that the passage from xy to yx corresponds to what evolution yields for the purely imaginary value t = sqrt(-1) of the parameter t of the evolution group. We must ... introduce evolution only after choosing a state in the algebra, but the evolution obtained is modified only by inner automorphisms and these are in a certain sense invisible. Here, I believe, is the key link that Hamilton sought between time and algebra. ...
Music, Math, and Time - In the book Triangle of Thoughts (American Mathematical Society, 2001) Alain Connes, Andre Lichnerowicz, and Marcel Paul Schutzenberger discuss the interrelationships among Mathematics, Physics, and Philosophy.
And now
Music of Quantum Circles October 2017 DOI:10.1007/978-3-319-47337-6_11 In book: The Musical-Mathematical Mind (pp.99-110)
However, even in this case there is a highly non-commutative world of higher order collectivity algebras B(n). This can be used to capture the geometry of rotations, like those appearing in the classical Pythagorean octave versus perfect fifth considerations.
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