WikiJournal Preprints/Phase periodicity and the mystery of musical consonance

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Consonant just intervals

The frequency ratios (6/5, 5/4, 4/3, 3/2, 8/5, 5/3) approximate the piano's (E♭, E, F, G, A♭, A) played above the C.
Simple explanation of beats: Amplitude vanishes when waves cancel.

Musical consonance is defined by a collection of frequency ratios defined by ratios of small whole numbers. Exactly which ratios create consonance is a matter of opinion, but our primary interest is in fractions between ${\displaystyle 1}$ and ${\displaystyle 2}$, where the coprime integer pairs range from from ${\displaystyle 2}$ through ${\displaystyle 8}$. Temporarily removing ${\displaystyle 7}$ from the list leaves us with the six consonant intervals shown in Figure 1. Six piano keys (E♭ E, ..., A) make the frequency ratio with the lower C shown in the figure. The question mark on D informs us that the "correct" ratio for that note is ambiguous, because two paths to D yield different results: Starting at C, one could move up by a factor of ${\displaystyle 3/2}$ (fifth) and down by ${\displaystyle 4/3}$ (fourth). Instead, one could obtain, ${\displaystyle 10/9=(5/3)\cdot (2/3)}$ by moving up a sixth and down a fifth. This explains the need for a tempered scale way that does burden a student in a research group who has an aversion to higher mathematics.[1]

This hypothetical math-adverse student might play an essential role in an effort to sort out a mystery dates back many years before Helmholtz proposed an explanation for the consonance of just intervals back in 1877.[2] An important component of supervised student research involves students learning from students. Students begin higher education by developing a broad vocabulary, but only mastering one discipline. But most progress is interdisciplinary, and therefore demands communication between experts. Traditional courses (two midterms and a final) cannot tolerate the chaos of students always teaching each other. But that chaos might transform to joy if students are in a research group trying to understand acoustical beats. Of course, that joy is tempered when students are told that one group member's ability to detect rhythmic irregularities will carry absolutely no weight in the scientific community. Now we need an expert in experimental psychology.

Amplitude beats

It is well known that acoustical beats can be heard when a just interval is slightly detuned.[3] An explanation by Helmholtz explanation is a common starting point for investigating this phenomenon, even by authors who concede that this model is less than convincing.[4][5][6][7][8][9] What is easily taught is often misleading,[10] yet there are reasons for introducing at topic with misleading simplicity. For example, student research that uncovers the flaws in a simple explanation has practical value in the teaching of that subject. In our case, the formulas and ideas presented below will prepare the reader for the more sophisticated calculations that follow.

Characteristic periods for the tritone and fifth.

Characteristic frequencies

Consider a perfectly tuned just interval with frequencies ${\displaystyle (f_{p},f_{q})}$, where:

Eq. 1 | ${\displaystyle \quad f_{p}=pf_{0}>qf_{0}=f_{q}\,,}$

and ${\displaystyle p>q}$ are relatively prime integers. Adopting the usual symbols for period and frequency, we define ${\displaystyle T_{0}=1/f_{0}}$, is the longest of the four characteristic periods. The highest of these four frequencies is inspired by the fact that the smallest number divisible by both ${\displaystyle p}$ and ${\displaystyle q}$ is their product, ${\displaystyle pq}$. Hence we define:

Eq. 2 | ${\displaystyle \quad f_{x}=pqf_{0}}$

This allows us to write the period of the two pitches in the interval as multiples of ${\displaystyle T_{x}=1/f_{x}}$:

Eq. 3 | ${\displaystyle \quad T_{q}=pT_{x}}$ and ${\displaystyle T_{p}=qT_{x}}$

Figure 3 illustrates this for the (3/2) fifth, the high note's period is two units of ${\displaystyle T_{x}}$, while the lower note's period is three units long. A summary of relationships between these frequencies and periods is shown in Table 1.

Beats between matching harmonics

This depiction of a violin's waveform appears to have a "harmonic" structure, i.e., it is free of inharmonicity.
Comparing harmonic (top) and inharmonic (bottom) waveforms

The history of recognizing importance of ratios of small numbers in defining consonant intervals is so ancient that it is almost a matter of convenience to associate it with Helmholtz (1821-1894),[2][4] (who incidentally made great contributions to the field of musical acoustics.) The amplitude beats depicted in Figure 2 (above) can be shown to occur at the frequency,

Eq. ? | ${\displaystyle \quad f_{\text{beat}}=|\Delta f|\equiv |f_{2}-f_{1}|}$

where ${\displaystyle \Delta f}$ is the difference between the two frequencies. Since we are seeking beats with frequencies of a few Hertz, this equation stipulates that we must consider beats between frequencies that are nearly equal.

It was Fourier (1768–1830) who demonstrated that periodic signals can be represented as a sum of sine and cosine wave whose frequencies follow a harmonic series. In other words, suppose a sound wave is periodic with period ${\displaystyle T}$, or equivalently, has a fundamental frequency of ${\displaystyle f=1/T}$. Subject to constraints beyond the scope of this article, any such function time can be expressed as a sum over sine and cosine waves, all with frequencies that are a multiples of ${\displaystyle f}$ (i.e. the frequencies are ${\displaystyle f}$, ${\displaystyle 2f}$, ${\displaystyle 3f}$, ${\displaystyle 4f}$, et cetera.)

We may neglect the cosine waves if the fluctuating pressure, ${\displaystyle p(t)}$ happens to be zero time, ${\displaystyle t=0,}$ and write this sum in the in what is called a Fourier series:

Eq. ? | ${\displaystyle \quad p(t)=C_{1}\sin(\omega t)+C_{2}\sin(2\omega t)+C_{3}\sin(3\omega t)+\ldots \,,}$

where ${\displaystyle (C_{1},C_{2},\dots )}$ are arbitrary constants, and "omega" is defined as, ${\displaystyle \omega =2\pi f}$. The top graph in Figure ? shows an example with the following choices for these constants:[11]

Eq. ? | ${\displaystyle \quad C_{1}=2}$, ${\displaystyle \;C_{2}=-1}$, ${\displaystyle \;C_{3}={\tfrac {2}{3}}}$, and ${\displaystyle C_{j}=0}$ if ${\displaystyle j>3}$.

The bottom graph in Figure ? depicts the waveform for a case that is inharmonic because ${\displaystyle 3}$ was replaced by ${\displaystyle 3.1}$.

Formula

Table 2 shows the 3/2 "fifth" interval with the higher frequency, 301Hz, sharp by one Hertz. The harmonics, 602, 903, ..., are shown in that column. The harmonics of the lower frequency (200Hz) are also shown. The cases when the frequencies are match are shaded yellow, and accompanied by the frequency difference, which we call ${\displaystyle \Delta f}$.[12] It can be shown that all the beat frequencies are given by:

Eq. ? | ${\displaystyle \quad {}^{i}\!f_{B}=i\cdot \left|q\Delta f_{p}-p\Delta f_{q}\right|\,,}$

where the prefix ${\displaystyle i}$ is a positive integer that can identify the pair of harmonics invovled: The ${\displaystyle ip^{\text{th}}}$ harmonic of ${\displaystyle f_{q}}$ matches the ${\displaystyle iq^{\text{th}}}$ harmonic of ${\displaystyle f_{p}}$. If the just interval's tuning is exact, this pair of harmonics have the same frequency, which is ${\displaystyle f_{x}=ipqf_{0}}$. In other words, ${\displaystyle f_{x}}$ as described in Table 1 is the frequency of the shared harmonic that is responsible for beats, according Helmholtz's model of consonance.

Equation (?) would lead one to believe that beats only occur when p-wave and h-wave each contain the corresponding harmonic in sufficient strength to cause the amplitude variations to be noticeable to the human ear. Yet, beats can be heard between pure sine waves that deviate slightly from just intonation.[2][3][4][5]

Phase beats

Equation ? also represents the frequency of what we shall call phase beats, except that the integer ${\displaystyle i}$ is restricted to only two values:

• ${\displaystyle i=1}$ for all just intervals.
• ${\displaystyle i=2}$ if either ${\displaystyle p}$ or ${\displaystyle q}$ is even (verified for ${\displaystyle \max(p,q)\leq 8}$ only).

Two important caveats must be mentioned:

1. While there is evidence that people can distinguish phase differences between two pure sinusoidal waves, I was unable to find any mention of these "phase beats" on the internet.[13]
2. The rule regarding phase beats for ${\displaystyle i=2}$ has only been established for cases where ${\displaystyle p}$ and ${\displaystyle q}$ are both less than 8.
The waveform for a fifth (P5) that was detuned by 4.3 cents. The centers of the shaded regions A and C are separated by one beat period (${\displaystyle i=1}$). The temporal separation between B and the two outer regions (A, C) identify the shorter (${\displaystyle i=2}$) beat period. Note how the shape at B is "upside down" in comparison with A and C.

Figure 6 shows the waveform for P5 (perfect fifth) that has been detuned by about twice the amount associated with equal temperament. With a 3/2 ratio, we expect to hear the ${\displaystyle i=2}$ beat, which occurs at twice the rate of the ${\displaystyle i=1}$ beat. This extra beat is shown in the center of the graph, where two cycles of the ${\displaystyle T_{0}}$ interval as shown (shaded cyan). Note how the central shaded region is the inverse of the two yellow shaded regions at each end. If humans perceive the central shaded section to be identical to those on the ends, the perception is that this is an ${\displaystyle i=2}$. But if the central section is perceived differently, humans will perceive two simultaneous beats: The ${\displaystyle i=1}$ occurs at frequency ${\displaystyle {}^{1}\!f_{B}^{\phi }}$, with an additional beat at twice that frequency.

Hearing the beats

All files play the beat at 90 beats per minute and are arranged in 3/4 time:

1. Four bars with no metronome: Listen for the beats
2. Four bars with metronome. Think: click-2-3 | 2-2-3 | 3-2-3 | 4-2-3
3. Four bars with no metronome. Count: rest-2-3 | 2-2-3 | 3-2-3 | 4-2-3
4. One bar with metronome. Listen: click-2-3

The challenge occurs at step 3: Try to count the beats as if they were a four measure rest in a waltz. If you are the clicker, you would come on the first beat of the fifth bar after that rest. After a while you might be able to start counting at the beginning. Think of yourself as a clicker, who has four bars of rest, four bars of clicks, four bars of rest, and one bar of clicks. The frequency of each note is indicated (measured in Hz.)

MENTION NEED TO LISTEN SEVERAL TIMES ALSO THE FAKE WALTZ EFFECT

OGG files are compressed, but often easier to access online:[14]

WAV files for these intervals are shown below:

The parity condition

Periodicity of the phase between two notes in a musical interval can be ascertained by time shifting one note.

Phase beats will occur even if both notes in the interval are pure tones (i.e., have no harmonics.) If humans can detect phase beats, they would mimic amplitude beats among the lowest matching harmonic (${\displaystyle i=1}$) if both ${\displaystyle p}$ and ${\displaystyle q}$ have odd parity. For most familiar intervals, one of the integers have even parity. It might not be not known whether humans would perceive these beats as due to only the second matching harmonic (${\displaystyle i=2}$) both harmonics (${\displaystyle i=1}$ and ${\displaystyle i=2}$.) The question is whether humans perceive a signal to be identical to its inverse.

Biological Synchronization

Human hearing from eardrum to cochlea to inner ear to brain. For a more accurate sketch see File:Ear-anatomy-text-small-en.png
• W:Auditory cortex: The right auditory cortex has long been shown to be more sensitive to tonality (high spectral resolution), while the left auditory cortex has been shown to be more sensitive to minute sequential differences (rapid temporal changes) in sound, such as in speech
• Lits&Stone:[2] Devil's staircase figure. Neurons cannot fire at rates much beyond a kilohertz.
• w:Neurotransmission#Convergence_and_divergence. wikt:white lie. The brain is neither analog nor digital, but works using a signal processing paradigm that has some properties in common with both. ... the signals sent around the brain are "either-or" states that are similar to binary. A neuron fires or it does not. These all-or-nothing pulses are the basic language of the brain. So in this sense, the brain is computing using something like binary signals.forbes.com (quora)

THEMES: Neurological spikes not sine waves ... trianguar waves.. Two analogs: Metronomes and circuits. Both are odes. Reference Beats (acoustics) on Wikiversity.

My humnan ear pictuer: Brain as spectral analyzer. Both sides listen to both ears. Does Wikipedia separate the left and right sides? Nobel prize?

Purpose of physics is to predict, not explain (half truth).

Does math explain the 7/5 and 10/7 effect?

For modelling simplicity, firing frequencies may be the same as the driving frequencies, but in reality may be scaled-down versions of them, since neurons cannot fire at rates much beyond a kilohertz.[2]

Acknowledgements

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Competing interests

Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.

Ethics statement

The following passage was lifted from Institutional review board (Wikipedia). It emphasizes policy in the United States. For a worldwide perspective, see Ethics committee (Wikipedia).

An Institutional Review Board (IRB) is a committee that applies research ethics by reviewing the methods proposed for research to ensure that they are ethical. Such boards are formally designated to approve (or reject) behavioral research involving humans. Certain research categories are considered exempt from IRB oversight. One such category is research in conventional educational settings. Generally, human research ethics guidelines require that decisions about exemption are made by an IRB representative, not by the investigators themselves.

Editorial scrapbook

references and websites
• Coombes, S., and Gabriel James Lord. "Intrinsic modulation of pulse-coupled integrate-and-fire neurons." Physical Review E 56.5 (1997): 5809. (link) ... rhythmic motor behavior... w:Hodgkin–Huxley model (4 nonlinear odes).
Tadpole quote: "For example, in the mollusc Tritonia and the tadpole Xenopus the escape swim behavior is generated in this fashion ... The study of coupled oscillators has applications in understanding CPG (central pattern generators) neuronal circuits ... "
Three of the 41 citing articles involve music:
1. LotsStone[2] coupled neural oscillators ... PROBLEMS WITH HELMHOLTZ'S THEORY ...
2. Heffernan, B., and A. Longtin. "Pulse-coupled neuron models as investigative tools for musical consonance." Journal of Neuroscience Methods 183.1 (2009): 95-106. (link)
3. Hadrava, Michal, and Jaroslav Hlinka. "A Dynamical Systems Approach to Spectral Music: Modeling the Role of Roughness and Inharmonicity in Perception of Musical Tension." Frontiers in Applied Mathematics and Statistics 6 (2020): 18.(link)
• I found the references in Lots&Stone:
1. Schellenberg, E. Glenn, and Sandra E. Trehub. "Frequency ratios and the discrimination of pure tone sequences." Perception & Psychophysics 56.4 (1994): 472-478.(link) They studied sequential intervals and found people recognize them. Even infants do. Goes against Helmholtz model... but what does this have to do with mode locking?
2. Mirollo, Renato E., and Steven H. Strogatz. "Synchronization of pulse-coupled biological oscillators." SIAM Journal on Applied Mathematics 50.6 (1990): 1645-1662. (link) mentioned in Lits&Stone. Fireflies and women's menstrual cycles!
• Hodgkin–Huxley differential equations as simple as I can find them (link)

References

1. Student's a working knowledge of the musical scale can also appreciate the need for a tempered scale looking at a more "perfect keyboard" at w:Pythagorean tuning
2. Shapira Lots, Inbal, and Lewi Stone. "Perception of musical consonance and dissonance: an outcome of neural synchronization." Journal of the Royal Society Interface 5.29 (2008): 1429-1434. link
3. See reference 11 on w:special:permalink/1130834192#Uses
4. Plomp, Reinier, and Willem Johannes Maria Levelt. "Tonal consonance and critical bandwidth." The journal of the Acoustical Society of America 38.4 (1965): 548-560. link
5. Interference beats and Tartini tones (UNSW)
6. Steedman, Mark. "Helmholtz’and Longuet-Higgins’ Theories of Consonance and Harmony." Unpublished Tutorial Paper (2002).link
7. https://knilt.arcc.albany.edu/4-4_Beat_Theory
8. [THE PHYSICS AND PSYCHOPHYSICS OF SOUND & MUSIC (David Worrall) http://www.avatar.com.au/courses/PPofM/consonance/con1.html]
9. The following references found on the internet but were not seen by the author because they are not freely available: ✶ Hartmann, William M. Signals, sound, and sensation. Springer Science & Business Media, 2004. ✶ Tenney, James. "A history of consonance and dissonance." (1988). ✶ Roederer, Juan G. Introduction to the Physics and Psychophysics of Music. Springer Science & Business Media, 2012.
10. Two examples of necessary but misleading introductions to science are (1) the Bohr model is introduce long before students are ready for quantum mechanics, and (2) a simple relationship between frequency, wavelength and (phase) velocity is taught, with no mention of essential concept of group velocity.
11. The ${\displaystyle \pm 4}$ is not part of the waveform but simply added to offset the two waves above and below the axis.
12. The delta ${\displaystyle \Delta }$ is the Greek "D" and stands for "difference"
13. My only access to the scientific literature is through the open internet, which limits the articles that I can read.
14. A good student project would investigate whether OGG compression changes the phase structure in a way that renders them useless for investigating phase beats.