Comments on "A Biological Rationale for Musical Scales"

On 03 December 2009, PLoS ONE published an enlightening new study [Gill2009] which attempts to provide a biological rationale for musical scales that are used across various cultures. The gist of its conclusions is

... the component intervals of the most widely used scales throughout history and across cultures are those with the greatest overall spectral similarity to a harmonic series.

However, an initial cursory look suggests that this conclusion doesn't hold for many of the राग्s commonly employed in हिंदूस्तानी music. I submitted my initial opinion and other observations as comments on the paper [Kamath2009].

This post is a broader treatment based on those original comments.

Demolishing "Paradoxes" in Special Relativity: Twins in a Cylindrical Minkowskian Universe

Introduction

The kinematics of Special Relativity (SR) is predicated on the Lorentz transforms [Lorentz1952]. Although these equations are merely (rescaling) transforms that conserve the covariance of physical laws across relatively moving inertial frames, it is the general perception that the effect of time-dilation (derived from the Lorentz transforms) is a manifest physical effect [1]. As any student of SR knows, this interpretation leads to various paradoxes; notably, the Twin (or Clock) Paradox which is the focus of discussion here [2].

The Twin Paradox

Very simply, the Twin Paradox involves two identical twins, one of whom stays on Earth (frame O) while the other, an astronaut (frame O'), goes on a space-faring journey in a high-speed rocket. Upon her return, the astronaut twin ostensibly finds that she has aged less than her sister who stayed back on Earth - presumably due to effects of time-dilation. This is clearly a paradox because the astronaut twin could very well argue that it's her sister on Earth who should age less, due to the very same effects of time-dilation.

The paradox has been explained away in multiple ways, but mostly by invoking the fact that the situation is not completely symmetrical. For instance, only the astronaut twin actually experiences acceleration when she turns around. Alternatively, the astronaut twin uses two inertial frames - one in each direction of travel - and it's the switch between the frames that causes the asymmetry [Schutz1985].

I submit that all the explanations of the Twin Paradox are faulty; primarily because they seek to explain something that is not a paradox in the first place. And to demonstrate this, I have to re-frame the problem in a way that eliminates the underlying basis for asymmetry.

Anisotropic Wavefront in Special Relativity - Comment on Vankov's "On Controversies in Special Relativity"

Prologue
I recently read Anatoli Andrei Vankov's paper "On Controversies in Special Relativity" [Van2006] with great interest. I've always been fascinated by discussions of paradoxes in Relativity (especially, the Ehrenfest Paradox), but I wasn't in the least aware that the Spherical Wave-front example of Einstein was controversial.

Vankov's Analysis
See Section 3 (Shape of light front, and constancy of the speed of light) of Vankov's paper [Van2006] for

1. Specifics on why the example is controversial.
2. An analysis of the physics underlying the example.
   
3. Subsequent interpretation towards the resolution of the controversy.

Specifically, the analysis concludes that the shape of an isotropic (spherical) wave-front in frame S' becomes an ellipsoid in frame S. I do not disagree with the analysis itself. However, I submit that the eventual interpretation is untenable. [There are a few typos in equations (11), (13), and (14). However, the typos don't affect the analysis perse; so I don't dwell on them here.]

Note that we start off with a spherical wave-front in frame S' which is a surface of constant t'. This surface is then parametrized using theta'. Subsequently, using the Lorentz transforms, this spherical surface is determined to be transformed into an ellipsoid in frame S. And the analysis is absolutely correct.

My Comments
However, at this point it is also asserted that an ellipsoid is exactly what is perceived by frame S. But, it must be realized that this ellipsoid surface is not a surface of constant t in frame S (because of relativity of simultaneity). [This is also evident in the analysis: The expression for the time coordinate in frame S is a function of theta/theta'.] As a result, the conclusion that the ellipsoid surface is perceived as-is in frame S, is, philosophically speaking, untenable: If frame S were to specify the shape of the wave-front, it would be based on a surface of constant t, and not based on a surface that corresponds to space-time events belonging to varying, or even arbitrary, t. Thus, the ellipsoid surface has no significance in frame S.

From a physical perspective, frame S would assert that the wave-front is spherical because frame S will make the shape determination at a specific time t which would give an obviously spherical wave-front (just as Einstein had originally remarked). Note that the space-time events that make up this spherical wave-front in frame S similarly carry no significance when transformed to frame S'.

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Update [15Sep2009]: I've emailed a one-page write-up to Anatoli Vankov soliciting his comments.

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References

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[Van2006] Vankov, "On Controversies in Special Relativity", arXiv.org, 2006.

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Bibliography of Indic Genetics (2001-2010)

This is a list of (mostly open access) papers in my collection that pertains to the genetics of the Indian Subcontinent.

Bibliography of Indic Genetics (2000 and prior)

This is a list of (mostly open access) papers in my collection that pertains to the genetics of the Indian Subcontinent.

An Acoustic Investigation into Ancient Indian Musical Tuning(s) - II

Some time ago I'd pondered the question of why the ancient Indian musical octave was considered to be a 22-step (22-शृति) interval [Kam2009a]. In my investigation I had followed primarily two cardinal principles:

(1) Ancient Indian musicians must have tuned their वीणाs (bow-harps) acoustically, using their ears as a guide (and without using any 'cyclic' principles). Hence their scales must have been largely reflective of our contemporary understanding of acoustic/harmonic principles.

(2) Only those processes (e.g. मूर्छनs) that were described by मुनि भरत could be used for further investigation.

Thus, I had postulated a 'acoustic natural' scale as a plausible model of the षड्ज-ग्राम and inferred that its मूर्छनs (modulations) automatically generated a 22-शृति gamut.

An Acoustic Investigation into Ancient Indian Musical Tuning(s)

Ancient Indian music theory was based primarily on the tuning of the (ancient) वीणा, a Bow-Harp [Bha1984]. The topic of tuning of the वीणा leads to much discussion and disagreement in musicological circles even today. Based on the information which has reached us via Greek physicists who 'reverse-engineered' Greek tuning from Greek musical scales, and presumably Greek music being inherited from an Eastern musical practice [Dan1995, Day1891], it is quite possible that Indian tunings were also founded on acoustic/harmonic principles.

मुनि भरत's treatise, the नाट्यशास्त्र, which became the basis for subsequent musicological works, identifies three different interval sizes in the two parent tunings (षड्ज-ग्राम and मध्यम-ग्राम) and assigns them certain weights or शृति values [Jai1975]. More importantly, the scales of मुनि भरत's time were heptatonic (i.e. employing 7 स्वरs) and based on मूर्छनs (modes) of a given ग्राम (parent tuning). This is central to what follows.

(Note: It is more correct to translate ग्राम as 'pitch collection'; however, the phrase 'parent tuning' is used in this article since tuning is the topic under discussion.)

The purpose of this article is to start with an acoustically determined scale, and then see if the basis of the नाट्यशास्त्र cannot be recovered using our current understanding of the state-of-the-art in those days. Primarily, it seeks to understand why the ancient Indian octave was presumably divided into 22 steps (शृतिs). This article is also motivated somewhat by earlier work on contemporary Indian scales which ended up with a non-coarse grouping of 22 equivalence note classes [Kam2008]. However, it is in no way a defense of the claim that contemporary Indian intonation too is based upon a set of 22 fixed शृतिs.