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.

Thoughts and Comments

Notes in a Melodic Context

While Indian music is tonal, it is also strongly dependent on the intervallic distance between successive notes. As a consequence, a melodic sequence like

minor tone (10:9) * semitone (16:15)

yields a rather flat minor-third (32/27) which is missing in Table 1 [Gill2009] because its percentage similarity value is 6.7%.

A very common note 45/32 does not appear in Table 1 [Gill2009]. While 45/32 may have a low percentage similarity value (5.3%), it attains a strong position in the scale due to the melodic context imposed by its neighbours (invariably notes like a rather flat third or flat sixth); e.g. राग् मियाँ की तोड़ी.

There is a whole host of Sharp Fourth notes that are missing in Table 1 [Gill2009]: 25/18, 45/32 (discussed above), 64/45, and 36/25. These notes can have strong positions in a scale depending on melodic context.

Intervals Smaller than Semitone
Some smaller intervals (like 25:24 which occurs between the 6/5 and 5/4 notes) are important in melodic music, even though their percentage similarity values are low (8% for 25:24). These do not show up in Table 1 [Gill2009] either.

In fact, choosing the top 60 intervals has caused quite a few intervals smaller than 100c to be dropped!

Percentage Similarity Metric vs. Melodic Music

Merely using the percentage similarity as a metric suggests that the 27/20 note (8.5%) is more important than, say, 45/32 (5.2%). However, in my opinion, no Indian scale would use 27/20 as a position of rest as it would be construed as a mistuned fourth!

Pentatonic राग्s and a Postulate for हिंदूस्तानी Music
Some very popular pentatonic राग्s don't end up in Table 2 [Gill2009]. For e.g., consider the simple but powerful राग् बैरागी भैरव् (Southern रेवती):

1/1 16/15 4/3 3/2 16/9 2/1

The percentage similarity value comes out to be 38.26% well below the first 50 scales of Table 2 [Gill2009].

Or consider राग् कलावती, another popular and immensely pleasing राग्:

1/1 5/4 3/2 5/3 16/9 2/1

with percentage similarity value 35.46% - once again not in the top 50.

More surprisingly, राग् माल्कौँस्:

1/1 6/5 4/3 8/5 16/9 2/1

has a percentage similarity value of 39%. As a result, it does not occur in Table 2 [Gill2009] either even though it's one of the more popular pentatonic राग्s in practice.

Similar thoughts apply for other pentatonic राग्s employed in Indian music.

It seems incongruous to me that a vast majority of the scales listed in Table 2 [Gill2009] are not really encountered in any musical system, while quite a few that are very popular (at least in हिंदूस्तानी music) fall below the 40% percentage similarity value.

The only way to explain this discrepancy may be that in melodic systems the overall percentage similarity of the scale is not as important as the immediate intervals that are used in transitioning from one note to another.

Given this, I'm tempted to posit that in हिंदूस्तानी music, when moving from one note to another (a pre-defined musical interval away), preference is given to that specific intervallic ratio which has a higher percentage similarity metric (per Table 1 [Gill2009]) - though this preference may be overridden by the overall melodic context.

Comment on South Indian राग्s
Finally, many South Indian Melakartas (generating scales) would automatically be disqualified under the percentage similarity criterion because often two or more semi-tonal intervals occur in sequence bringing down the overall percentage similarity of the scale. e.g. 6 Melakarthas each in the Indu Chakra, Rishi Chakra, Rutu Chakra, and Aditya Chakra (i.e. a total of 24 Melakartas) some of which are very popular even now (for instance, राग् नाट्टई).

---

References

---

[Gill2009] Gill, Purves, "A Biological Rationale for Musical Scales", PLoS ONE 4(12):e8144, 2009.
doi:10.1371/journal.pone.0008144

[Kamath2009] Kamath, "Comment from the Perspective of Indian Music", Comment on "A Biological Rationale for Musical Scales", PLoS ONE 4(12), 2009.

---

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.

A Cylindrical Minkowskian Universe

Imagine a flat two-dimensional universe. By flat, I mean a universe with an intrinsic Minkowskian metric. While the intrinsic metric is Minkowskian, SR doesn't place (nor can it place?) a restriction on the extrinsic topology of a Minkowskian universe. As a result, it is perfectly legitimate to consider the case where a Minkowskian universe is extrinsically curved onto itself. In particular, assume that the x coordinate axis of O wraps around to meet itself. The topology of this universe then is that of the surface of a cylinder. The y coordinate axis lies along the length of the cylinder while the x coordinate axis encircles the girth of the cylinder.

Re-framing the Twin Paradox

Now, reconsider the Twin Paradox in a cylindrical Minkowskian universe: The astronaut twin, who sets off in the x direction of the twin on Earth, will eventually 'circumnavigate' the universe and rendezvous with her Earth-bound sister. Under these circumstances, there is no switch of inertial frames nor is there any acceleration. The situation is completely symmetrical, and the usual explanations of the Twin Paradox no longer apply. Thus, under the conventional interpretation of time-dilation as a physical effect, we are once again faced with the paradox of determining which twin has aged more than the other.

Reinterpreting time-dilation

The Lorentz transforms have been verified empirically over and over again. Provisionally, one can then assume that the transforms are correct. Since we nevertheless arrive at a paradox above, the only conclusion is that the physical interpretation of time-dilation in SR is to blame [3].

Since the situation is perfectly symmetrical, the one thing we can be certain about is that neither twin is older than the other. Alternatively put, they've both aged the same amount of time physically.

In other words, the effect of time-dilation derived from the (rescaling) equations of the Lorentz transforms does not actually imply slow down of physical time in a moving inertial frame. The Lorentz transforms are, simply put, rescaling equations that achieve covariance of physical laws across inertial frames. In effect, both O and O' physically age equally. However, if we wish to maintain the form of the laws of physics as we move from frame O to frame O', we need to employ scaling in the units used to express the laws. This scaling is a purely kinematical (and not physical) artefact and it is incorrect for us to infer any physical results from it.

Other Notes

Since my initial explorations on this topic I've come across multiple other expositions that argue that time-dilation is not a physical effect: for instance Kracklauer [Kracklauer2002], and most notably multiple references to Sachs [Sachs1971] (which I've not been able to obtain a personal copy of). However, to the best of my knowledge, no one has used the formulation of the Twin Paradox in a cylindrical Minkowskian universe as stated above, and I believe it is my unique contribution to this problem.

Footnotes

[1] That such a physical effect should arise simply from kinematical considerations is untenable. Unfortunately, even Einstein initially believed time-dilation to be physically true: “... it follows that the time marked by the clock (viewed in the stationary frame) is slow ...” [Lorentz1952].

[2] The Ehrenfest paradox similarly arises from interpreting length-contraction as a manifest physical effect.

[3] Alternatively, it is possible that SR prohibits extrinsically curved universes whose intrinsic metric nevertheless remains Minkowskian. However, there is nothing in SR that compels us to assume this at this time.

---

Update [20Sep2009]: I've emailed a précis of these deliberations to Dr. Mendel Sachs via his website.

---

References

---

[Kracklauer2002] Kracklauer, Kracklauer, "Twins: never the twain shall part", Proceedings: Physical Interpretations of Relativity Theory VIII, 2002, pp. 248-255.

[Lorentz1952] Lorentz et al., "The Principle of Relativity", Dover Publications, 1952.

[Sachs1971] Sachs, "A resolution of the clock paradox", Physics Today, vol. 24(9), 1971, pp. 23-29. http://dx.doi.org/10.1063/1.3022927

[Schutz1985] Schutz, "A first course in general relativity", Cambridge University Press, 1985.

---

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'.

---

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

---

References

---

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

---

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.

2001

[Agrawal2001] Agrawal et al, "Distribution of HLA Class II Antigens in Three North Indian Populations", International Journal of Human Genetics, vol. 1(4), pp. 283-291, 2001.

[Bamshad2001] Bamshad et al, "Genetic Evidence on the Origins of Indian Caste Populations", Genome Research, vol. 11, pp. 994-1004, 2001.
doi:10.1101/gr.GR-1733RR

[Chattopadhyay2001] Chattopadhyay et al, "Gene Diversity at Three Tetrameric STR Loci Among Eight Ethnic Populations of West Bengal and Manipur, India", International Journal of Human Genetics, vol. 1(4), 255-262, 2001.

[Maca-Meyer2001] Maca-Meyer et al, "Major genomic mitochondrial lineages delineate early human expansions", BMC Genetics, vol. 2:13, 2001.
doi:10.1186/1471-2156-2-13

[Majumder2001a] Majumder, "Indian Caste Origins: Genomic Insights and Future Outlook", Genome Research, vol. 11, pp. 931-932, Jun. 2001.
doi:10.1101/gr.192401

[Majumder2001b] Majumder, "Ethnic populations of India as seen from an evolutionary perspective", Journal of Biosciences, vol. 26(4), pp. 533-545, Nov. 2001.

[Mukherjee2001] Mukherjee et al, "High-resolution analysis of Y-chromosomal polymorphisms reveals signatures of population movements from Central Asia and West Asia into India", Journal of Genetics, vol. 80(3), pp. 125-135, 2001.

[Quintana-Murci2001] Quintana-Murci et al, "Y-Chromosome Lineages Trace Diffusion of People and Languages in Southwestern Asia", American Journal of Human Genetics, vol. 68, pp. 537-542, 2001.
doi:10.1086/318200

[Ramana2001] Ramana et al, "Y-chromosome SNP haplotypes suggest evidence of gene flow among caste, tribe, and the migrant Siddi populations of Andhra Pradesh, South India", European Journal of Human Genetics, vol. 9, pp. 695-700, 2001.

[Reddy2001] Reddy et al, "Population Substructure and Patterns of Quantitative Variation among the Gollas of Southern Andhra Pradesh, India", Human Biology, vol. 73(2), pp. 291-306, 2001.
PMID: 11446430

[Shankarkumar2001] Shankarkumar et al, "HLA Antigen Distribution in Maratha Community from Mumbai, Maharastra, India", International Journal of Human Genetics, vol. 1(3), pp. 173-177, 2001.

[Veerraju2001] Veerraju et al, "Insertion / Deletion DNA Polymorphisms in two South Indian Tribal Populations", International Journal of Human Genetics, vol. 1(2), pp. 129-132, 2001.

[Venkatramana2001] Venkatramana et al, "Apolipoprotein E Polymorphism Among the Indian Populations and its Comparison with Other Asian Populations", International Journal of Human Genetics, vol. 1(2), pp. 123-128, 2001.

2002

[Bhasin2002] Bhasin, Walter, "Genetics of Castes and Tribes of India: Serum Protein Polymorphisms–KM and GM Systems", International Journal of Human Genetics, vol. 2(2), pp. 119-138, 2002.

[Chattopadhyay2002] Chattopadhyay et al, "Genetic Affinity Among Eight Ethnic Populations of West Bengal and Manipur, India: A Study Based on Six Polymorphic Functional Loci (HLADQA1, LDLR, GYPA, HBGG, D7S8 and GC)", International Journal of Human Genetics, vol. 2(4), pp. 233-242, 2002.

[Das2002] Das et al, "Genetic Variation of Serum Proteins Among the Koch Sub-populations of West Bengal, India", International Journal of Human Genetics, vol. 2(4), pp. 255-260, 2002.

[Edwin2002] Edwin et al, "Mitochondrial DNA diversity among five tribal population of southern India", Current Science, vol. 83(2), pp. 158-162, 2002.
ISSN 0011-3891

[Kivisild2002] Kivisild et al, "The Emerging Limbs and Twigs of the East Asian mtDNA Tree", Molecular Biology and Evolution, vol. 19(10), pp. 1737-1751, 2002.
PMID: 12270900

[Qamar2002] Qamar et al, "Y-Chromosomal DNA Variation in Pakistan", American Journal of Human Genetics, vol. 70, pp. 1107-1124, 2002.
PMCID: PMC447589

[Rajalingam2002] Rajalingam et al, "Distinctive KIR and HLA diversity in a panel of north Indian Hindus", Immunogenetics, vol. 53, pp. 1009-1019, 2002.
doi:10.1007/s00251-001-0425-5

2003

[Basu2003] Basu et al, "Ethnic India: A Genomic View, With Special Reference to Peopling and Structure", Genome Research, vol. 13, pp. 2277-2290, 2003.
doi:10.1101/gr.1413403

[Chhaya2003] Chhaya, Shankarkumar, "HLA Antigen Distribution in Selected Gujarati Subcaste from Mumbai, Maharastra, India", International Journal of Human Genetics, vol. 3(3), pp. 135-138, 2003.

[Cordaux2003a] Cordaux et al, "Mitochondrial DNA analysis reveals diverse histories of tribal populations from India", European Journal of Human Genetics, vol. 11, pp. 253-264, 2003.
doi:10.1038/sj.ejhg.5200949

[Cordaux2003b] Cordaux, Stoneking, "South Asia, the Andamanese, and the Genetic Evidence for an Early Human Dispersal out of Africa", American Journal of Human Genetics, vol. 72, pp. 1586-1590, 2003.
doi:10.1086/375407

[Endicott2003a] Endicott et al, "The Genetic Origins of the Andaman Islanders", American Journal of Human Genetics, vol. 72, pp. 178-184, 2003.
PMCID: PMC378623

[Endicott2003b] Endicott et al, "Reply to Cordaux and Stoneking [Cordaux2003b]", American Journal of Human Genetics, vol. 72, pp. 1590-1593, 2003.

[Kashyap2003] Kashyap et al, "Molecular Relatedness of The Aboriginal Groups of Andaman and Nicobar Islands with Similar Ethnic Populations", International Journal of Human Genetics, vol. 3(1), pp. 5-11, 2003.

[Khan2003] Khan et al, "ApoB 3’HVR Polymorphism a Genetic Variation in Indian Subcontinent", International Journal of Human Genetics, vol. 3(3), pp. 139-145, 2003.

[Kivisild2003a] Kivisild et al, "The Genetics of Language and Farming Spread in India", Chapter 17 in "Examing the farming/language dispersal hypothesis", McDonald Institute Monographs, 2003.

[Kivisild2003b] Kivisild et al, "The Genetic Heritage of the Earliest Settlers Persists Both in Indian Tribal and Caste Populations", American Journal of Human Genetics, vol. 72, pp. 313-332, 2003.
PMCID: PMC379225

[Kumar2003] Kumar, Reddy, "Status of Austro-Asiatic groups in the peopling of India - An exploratory study based on the available prehistoric, linguistic and biological evidence", Journal of Biosciences, vol. 28(4), pp. 507-522, 2003.

[Ranjan2003] Ranjan et al, "Genomic Diversity in North-East, India: Genetic Relationship Among Five Ethnic Populations From Manipur Based on VNTR Polymorphism", International Journal of Human Genetics, vol. 3(1), pp. 21-27, 2003.

[Shankarkumar2003] Shankarkumar et al, "HLA DRB1 and DQB1 Gene Diversity in Maratha Community from Mumbai India", International Journal of Human Genetics, vol. 3(1), pp. 39-43, 2003.

[Sidhu2003] Sidhu et al, "Study of Genetic Polymorphism at D21S11 and D21S215 Loci in the Jat Sikh Population of Punjab", International Journal of Human Genetics, vol. 3(1), pp. 45-50, 2003.

[Thangaraj2003] Thangaraj et al, "Genetic Affinities of the Andaman Islanders, a Vanishing Human Population", Current Biology, vol. 13, pp. 86-93, 2003.
doi: 10.1016/S0960982202013362

2004

[Babita2004] Babita, Usha, "HLA Antigen Distribution in Sikhs from Punjab, India", International Journal of Human Genetics, vol. 4(2), pp. 111-113, 2004.

[Baig2004] Baig et al, "Mitochondrial DNA Diversity in Tribal and Caste Groups of Maharashtra (India) and its Implication on Their Genetic Origins", Annals of Human Genetics, vol. 68, pp. 453-460, 2004.
doi: 10.1046/j.1529-8817.2004.00108.x

[Cordaux2004a] Cordaux et al, "Independent Origins of Indian Caste and Tribal Paternal Lineages", Current Biology, vol. 14, pp. 231-235, Feb. 2004.
doi:10.1016/j.cub.2004.01.024

[Cordaux2004b] Cordaux et al, "The Northeast Indian Passageway: A Barrier or Corridor for Human Migrations?", Molecular Biology and Evolution, vol. 21(8), pp. 1525-1533, May 2004.
doi:10.1093/molbev/msh151

[Das2004] Das et al, "Minimal Sharing of Y-Chromosome STR Haplotypes Among Five Endogamous Population Groups from Western and Southwestern India", Human Biology, vol. 76, no. 5, pp. 743-763, 2004.
doi:10.1353/hub.2005.0003

[Kankonkar2004] Kankonkar et al, "HLA Antigen Distribution in Selected Population Groups from Maharashtra", International Journal of Human Genetics, vol. 4(2), pp. 115-118, 2004.

[Khan2004] Khan et al, "Genetic variation of ApoB 3′ hypervariable region polymorphism among Brahmins of North India", Current Science, vol. 86(5), pp. 697-701, 2004.

[Lee2004] Lee et al, "Molecular evidence for absence of Y-linkage of the Hairy Ears trait", European Journal of Human Genetics, vol. 12, pp. 1077-1079, 2004.
doi:10.1038/sj.ejhg.5201271

[Mansoor2004] Mansoor et al, "Investigation of the Greek ancestry of populations from northern Pakistan", Human Genetics, vol. 114, pp. 484-490, 2004.
doi:10.1007/s00439-004-1094-x

[Metspalu2004] Metspalu et al, "Most of the extant mtDNA boundaries in South and Southwest Asia were likely shaped during the initial settlement of Eurasia by anatomically modern humans", BMC Genetics, vol. 5:26, 2004.
doi:10.1186/1471-2156-5-26

[Palanichamy2004] Palanichamy et al, "Phylogeny of Mitochondrial DNA Macrohaplogroup N in India, Based on Complete Sequencing: Implications for the Peopling of South Asia", American Journal of Human Genetics, vol 75., pp. 966-978, 2004.

[Quintana-Murci2004] Quintana-Murci et al, "Where West Meets East: The Complex mtDNA Landscape of the Southwest and Central Asian Corridor", American Journal of Human Genetics, vol. 74(5), pp. 827-845, 2004.

PMCID: PMC1181978

[Rajkumar2004] Rajkumar, Kashyap, "Genetic structure of four socio-culturally diversified caste populations of southwest India and their affinity with related Indian and global groups", BMC Genetics, vol. 5:23, 2004.
doi:10.1186/1471-2156-5-23

[Shankarkumar2004] Shankarkumar, "Diversity Among Selected Caste and Tribal Groups of Western India: Origin, HLA and Disease Associations", International Journal of Human Genetics, vol. 4(2), pp. 105-110, 2004.

[Sridharan2004] Sridharan, Shankarkumar, "HLA Antigens in Nadars a Native Dravidian Caste Group of Tamil Nadu, South India", International Journal of Human Genetics, vol. 4(2), pp. 119-124, 2004.

[Tandon2004] Tandon et al, "Genetic Affinity Between Two Ethnically Diverse Caste Groups of North India: A Study Based Upon 15 Microsatellite Loci", International Journal of Human Genetics, vol. 4(1), pp. 37-43, 2004.

[Vishwanathan2004] Vishwanathan et al, "Genetic structure and affinities among tribal populations of southern India: a study of 24 autosomal DNA markers", Annals of Human Genetics, vol. 68, pp. 128-138, 2004.
doi: 10.1046/j.1529-8817.2003.00083.x

[Wooding2004] Wooding et al, "Directional migration in the Hindu castes: inferences from mitochondrial, autosomal and Y-chromosomal data", Human Genetics, vol. 115, pp. 221-229, 2004.
doi:10.1007/s00439-004-1130-x

2005

[Agrawal2005] Agrawal et al, "YAP, signature of an African-Middle Eastern migration into Northern India", Current Science, vol. 88, no. 12, pp. 1977-1980, 2005.
ISSN 0011-3891

[Balgir2005] Balgir, "Detection of a Rare Blood Group “Bombay (Oh) Phenotype” Among the Kutia Kondh Primitive Tribe of Orissa, India", International Journal of Human Genetics, vol. 5(3), pp. 193-198, 2005.

[Kalaydjieva2005] Kalaydjieva et al, "A newly discovered founder population: the Roma/Gypsies", BioEssays, vol. 27.10, pp. 1084-1094, 2005.
doi:10.1002/bies.20287

[Kankonkar2005] Kankonkar, Shankarkumar, "HLA DRB1 Gene Study in Different Population Groups From Mumbai, Maharashtra, India", International Journal of Human Genetics, vol. 5(4), pp. 267-271, 2005.

[Metspalu2005] Metspalu, "Through the course of prehistory in India: tracing the mtDNA trail", Dissertation, Institute of Molecular and Cell Biology, University of Tartu, 2005.
ISBN 9949–11–202–8

[Rajkumar2005] Rajkumar et al, "Phylogeny and antiquity of M macrohaplogroup inferred from complete mt DNA sequence of Indian specific lineages", BMC Evolutionary Biology, vol. 5:26, 2005.
doi:10.1186/1471-2148-5-26

[Ravindranath2005] Ravindranath et al, "Alu Insertion / Deletion Polymorphisms in Yadava Population of Andhra Pradesh, South India", International Journal of Human Genetics, vol. 5(3), pp. 223-224, 2005.

[Thangaraj2005] Thangaraj et al, "Reconstructing the Origin of Andaman Islanders", Science, vol. 308, page 996, 2005.
doi:10.1126/science.1109987

[Thomas2005] Thomas, Banerjee, "HLA-A allele frequency and haplotype distribution in the Dravidian tribal communities of South India", Indian Journal of Human Genetics, vol 11, issue 3, pp. 140-144, 2005.

2006

[Barnabas2006] Barnabas et al, "High-Resolution mtDNA Studies of the Indian Population: Implications for Palaeolithic Settlement of the Indian Subcontinent", Annals of Human Genetics, vol. 70, pp. 42-58, 2006.
doi: 10.1111/j.1529-8817.2005.00207.x

[Biswas2006] Biswas, Pawar, "Phylogenetic tests of distribution patterns in South Asia: towards an integrative approach", Journal of Biosciences, vol. 31(1), pp. 95-113, 2006.

[Carvalho-Silva2006] Carvalho-Silva et al, "Ancient Indian roots?", Journal of Biosciences, vol. 31(1), pp. 1-2, 2006.

[Debnath2006] Debnath, Chaudhuri, "Study of Genetic Relationships of Indian Gurkha Population on the Basis of HLA - A and B Loci Antigens", International Journal of Human Genetics, vol. 6(2), pp. 159-162, 2006.

[Endicott2006] Endicott et al, "Multiplexed SNP Typing of Ancient DNA Clarifies the Origin of Andaman mtDNA Haplogroups amongst South Asian Tribal Populations", PLoS ONE, vol. 1(1):e81, pp. 1-8, 2006.
doi:10.1371/journal.pone.0000081

[Gutala2006] Gutala et al, "A shared Y-chromosomal heritage between Muslims and Hindus in India", Human Genetics, vol. 120, pp. 543-551, 2006.
doi:10.1007/s00439-006-0234-x

[Kashyap2006] Kashyap et al, "Genetic structure of Indian populations based on fifteen autosomal microsatellite loci", BMC Genetics, vol. 7:28, 2006.
doi:10.1186/1471-2156-7-28

[Palanichamy2006] Palanichamy et al, "Comment on 'Reconstructing the Origin of Andaman Islanders'", Science, vol. 311, page 470a, 2006.
doi:10.1126/science.1120176

[Rosenberg2006] Rosenberg et al, "Low Levels of Genetic Divergence across Geographically and Linguistically Diverse Populations from India", PLoS Genetics, vol. 2(12):e215, pp. 2052-2061, 2006.
doi:10.1371/journal.pgen.0020215

[Sahoo2006] Sahoo et al, "A prehistory of Indian Y chromosomes: Evaluating demic diffusion scenarios", Proceedings of the National Academy of Sciences, vol. 103, no. 4, pp. 843-848, 2006.
doi: 10.1073/pnas.0507714103

[Sengupta2006] Sengupta et al, "Polarity and Temporality of High-Resolution Y-Chromosome Distributions in India Identify Both Indigenous and Exogenous Expansions and Reveal Minor Genetic Influence of Central Asian Pastoralists", The American Journal of Human Genetics, vol. 78, pp. 202-221, 2006.

PMCID: PMC1380230

[Sun2006] Sun et al, "The Dazzling Array of Basal Branches in the mtDNA Macrohaplogroup M from India as Inferred from Complete Genomes", Molecular Biology and Evolution, vol. 23(3), pp. 683-690, 2006.
doi:10.1093/molbev/msj078

[Thangaraj2006a] Thangaraj et al, "Response to Comment on 'Reconstructing the Origin of Andaman Islanders'", Science, vol. 311, page 470b, Jan 2006.
doi:10.1126/science.1120670

[Thangaraj2006b] Thangaraj et al, "In situ origin of deep rooting lineages of mitochondrial Macrohaplogroup 'M' in India", BMC Genomics, vol. 7:151, Jun 2006.
doi:10.1186/1471-2164-7-151

[Thanseem2006] Thanseem et al, "Genetic affinities among the lower castes and tribal groups of India: inference from Y chromosome and mitochondrial DNA", BMC Genetics, vol. 7:42, 2006.
doi:10.1186/1471-2156-7-42

2007

[Balgir2007] Balgir, "Identification of a rare blood group, “Bombay (Oh) phenotype,” in Bhuyan tribe of Northwestern Orissa, India", Indian Journal of Human Genetics, vol. 13(3), pp. 109-113, 2007.

[Bhasin2007] Bhasin, "Genetics of Castes and Tribes of India: Indian Population Milieu", Chapter 16 in "Anthropology Today: Trends, Scope and Applications", Anthropological Special Volume No. 3, pp. 167-208, 2007.

[Chaubey2007] Chaubey et al, "Peopling of South Asia: investigating the caste–tribe continuum in India", BioEssays, vol. 29(1), pp. 91-100, 2007.
doi: 10.1002/bies.20525

[Firasat2007] Firasat et al, "Y-chromosomal evidence for a limited Greek contribution to the Pathan population of Pakistan", European Journal of Human Genetics, vol. 15, pp. 121-126, 2007.
doi:10.1038/sj.ejhg.5201726

[Gayden2007] Gayden et al, "The Himalayas as a Directional Barrier to Gene Flow", The American Journal of Human Genetics, vol. 80, pp. 884-894, 2007.

PMCID: PMC1852741

[Kanthimathi2007] Kanthimathi et al, "A Genetic Structure of the Early Immigrants (Mukkalathor) of Tamil Nadu as Inferred From Autosomal Loci", International Journal of Human Genetics, vol. 7(2), pp. 167-173, 2007.

[Khan2007] Khan et al, "Genetic affinities between endogamous and inbreeding populations of Uttar Pradesh", BMC Genetics, vol. 8:12, 2007.
doi:10.1186/1471-2156-8-12

[Krithika2007a] Krithika et al, "Geographic Contiguity, Patterns of Gene Flow and Genetic Affinity among the Tribes of Arunachal Pradesh, India", International Journal of Human Genetics, vol. 7(3), pp. 267-276, 2007.

[Krithika2007b] Krithika et al, "Molecular Genetic Perspective of Indian Populations: A Y-Chromosome Scenario", Chapter 30 in "Anthropology Today: Trends, Scope and Applications", Anthropological Special Volume No. 3, pp. 385-392, 2007.

[Kumar2007] Kumar et al, "Y-chromosome evidence suggests a common paternal heritage of Austro-Asiatic populations", BMC Evolutionary Biology, vol. 7:47, Mar 2007.
doi:10.1186/1471-2148-7-47

[Reddy2007] Reddy et al, "Austro-Asiatic Tribes of Northeast India Provide Hitherto Missing Genetic Link between South and Southeast Asia", PLoS ONE, vol. 2(11): e1141, 2007.
doi:10.1371/journal.pone.0001141

[Sharma2007] Sharma et al, "A novel subgroup Q5 of human Y-chromosomal haplogroup Q in India", BMC Evolutionary Biology, vol. 7:232, 2007.
doi:10.1186/1471-2148-7-232

[Trivedi2007] Trivedi et al, "High Resolution Phylogeographic Map of Y-Chromosomes Reveal the Genetic Signatures of Pleistocene Origin of Indian Populations", Chapter 31 in "Anthropology Today: Trends, Scope and Applications", Anthropologist Special Volume No. 3, pp. 393-414, 2007.

[Winters2007a] Winters, "Can parallel mutation and neutral genome selection explain Eastern African M1 consensus HVS-I motifs in Indian M haplogroups", Indian Journal of Human Genetics, vol. 13(3), pp. 93-96, 2007.
doi:10.4103/0971-6866.38982

[Winters2007b] Winters, "Did the Dravidian Speakers Originate in Africa?", BioEssays, vol. 29(5), pp. 497-498, 2007.

[Zerjal2007] Zerjal et al, "Y-chromosomal insights into the genetic impact of the caste system in India", Human Genetics, vol. 121, pp. 137-144, 2007.
doi:10.1007/s00439-006-0282-2

2008

[Badaruddoza2008] Badaruddoza et al, "Study of YAP Element among an Endogamous Human Isolate in Punjab", International Journal of Human Genetics, vol. 8(3), pp. 269-271, 2008.

[Barik2008] Barik et al, "Detailed mtDNA Genotypes Permit a Reassessment of the Settlement and Population Structure of the Andaman Islands", American Journal of Physical Anthropology, vol. 136, pp. 19-27, 2008.
doi:10.1002/ajpa.20773

[Carvalho-Silva2008] Carvalho-Silva, Tyler-Smith, "The Grandest Genetic Experiment Ever Performed on Man? – A Y-Chromosomal Perspective on Genetic Variation in India", International Journal of Human Genetics, vol. 8(1-2), 2008.

[Chaubey2008a] Chaubey et al, "Language Shift by Indigenous Population: A Model Genetic Study in South Asia", International Journal of Human Genetics, vol. 8(1-2), pp. 41-50, May 2008.

[Chaubey2008b] Chaubey et al, "Phylogeography of mtDNA haplogroup R7 in the Indian peninsula", BMC Evolutionary Biology, vol. 8:227, Aug 2008.
doi:10.1186/1471-2148-8-227

[Heyer2008] Heyer, "The Indian genetic landscape and disease-related genes", Journal of Biosciences, vol. 33(5), pp. 631-633, 2008.
doi:

[Kumar2008a] Kumar et al, "Molecular Genetic Study on the Status of Transitional Groups of Central India: Cultural Diffusion or Demic Diffusion?", International Journal of Human Genetics, vol. 8(1-2), pp. 31-39, 2008.

[Kumar2008b] Kumar et al, "The earliest settlers' antiquity and evolutionary history of Indian populations: evidence from M2 mtDNA lineage", BMC Evolutionary Biology, vol. 8:230, 2008.
doi:10.1186/1471-2148-8-230

[Maji2008a] Maji et al, "Distribution of Mitochondrial DNA Macrohaplogroup N in India with Special Reference to Haplogroup R and its Sub-Haplogroup U", International Journal of Human Genetics, vol. 8(1-2), pp. 85-96, March 2008.

[Maji2008b] Maji, Vasulu, "Genetic Structure of an Isolated Sub-Tribe of the Adi People of Arunachal Pradesh State in Northeast India: Isonymy Analysis and Selective Neutrality of Surname Distribution in Adi Panggi", Journal of Genetic Genealogy, vol. 4(1), pp. 1-11, Spring 2008.

[Thangaraj2008] Thangaraj et al, "Maternal Footprints of Southeast Asians in North India", Human Heredity, vol. 66(1), pp. 1-9, 2008.
doi:10.1159/000114160

[Tripathi2008] Tripathi et al, "Role of Alu Element in Detecting Population Diversity", International Journal of Human Genetics, vol. 8(1-2), pp. 61-74, 2008.

[Tripathy2008] Tripathy et al, "Trends in Molecular Anthropological Studies in India", International Journal of Human Genetics, vol. 8(1-2), pp. 1-20, 2008.

[Trivedi2008] Trivedi et al, "Genetic Imprints of Pleistocene Origin of Indian Populations: A Comprehensive Phylogeographic Sketch of Indian Y-Chromosomes", International Journal of Human Genetics, vol. 8(1-2), pp. 97-118, 2008.

[Veerraju2008] Veerraju et al, "Insertion/Deletion Polymorphisms in Indian Tribal Populations", International Journal of Human Genetics, vol. 8(1-2), pp. 75-83, 2008.

[Watkins2008] Watkins et al, "Genetic variation in South Indian castes: evidence from Y-chromosome, mitochondrial, and autosomal polymorphisms", BMC Genetics, vol. 9:86, 2008.
doi:10.1186/1471-2156-9-86

[Winters2008a] Winters, "Origin and Spread of Dravidian Speakers", International Journal of Human Genetics, vol. 8(4), pp. 325-329, 2008.

[Winters2008b] Winters, "High Levels of Genetic Divergence across Indian Populations", PLoS Genetics, Reader Response to Rosenberg et al ("Low Levels of Genetic Divergence across Linguistically Diverse Populations from India") [Rosenberg2006], 2008.

2009

[Chakravarti2009] Chakravarti, "Human Genetics: Tracing India's invisible threads", Nature, vol. 461, pp. 487-488, 2009.
doi:10.1038/461487a

[Chandrasekar2009] Chandrasekar et al, "Updating Phylogeny of Mitochondrial DNA Macrohaplogroup M in India: Dispersal of Modern Human in South Asian Corridor", PLoS ONE, vol. 4(10):e7447, 2009.
doi:10.1371/journal.pone.0007447

[Fornarino2009] Fornarino et al, "Mitochondrial and Y-chromosome diversity of the Tharus (Nepal): a reservoir of genetic variation", BMC Evolutionary Biology, vol. 9:154, 2009.
doi:10.1186/1471-2148-9-154

[Kumar2009] Kumar et al, "Reconstructing Indian-Australian phylogenetic link", BMC Evolutionary Biology, vol. 9:173, 2009.
doi:10.1186/1471-2148-9-173

[Maji2009] Maji et al, "Phylogeographic distribution of mitochondrial DNA macrohaplogroup M in India", Journal of Genetics, vol. 88(1), pp. 127-139, 2009.
doi:10.1007/s12041-009-0020-3

[Moiz2009] Moiz et al, "Population Study of 1311 C/T polymorphism of Glucose 6 Phosphate Dehydrogenase gene in Pakistan - an analysis of 715 X-chromosomes", BMC Genetics, vol. 10:41, 2009.
doi:10.1186/1471-2156-10-41

[Reich2009] Reich et al, "Reconstructing Indian population history", Nature, vol. 461, pp. 489-495, 2009.
doi:10.1038/nature08365

[Sharma2009] Sharma et al, "The Indian origin of paternal haplogroup R1a1* substantiates the autochthonous origin of Brahmins and the caste system", Journal of Human Genetics, vol. 54, pp. 47–55, 2009.
doi:10.1038/jhg.2008.2

[Thangaraj2009] Thangaraj et al, "Deep Rooting In-Situ Expansion of mtDNA Haplogroup R8 in South Asia", PloS ONE, vol. 4(8):e6545, 2009.
doi:10.1371/journal.pone.0006545

[Zhao2009] Zhao et al, "Presence of three different paternal lineages among North Indians: A study of 560 Y chromosomes", Annals of Human Biology, vol. 36(1), pp. 46-59, 2009.
doi:10.1080/03014460802558522

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.

Pre-1999

[Sarkar1961] Sarkar et al, "A Contribution to the Genetics of Hypertrichosis of the Ear Rims", American Journal of Human Genetics, vol. 13(2), pp. 214-223, 1961.

[Dronamraju1962] Dronamraju, Haldane, "Inheritance of Hairy Pinnae", American Journal of Human Genetics, vol. 14(1), pp. 102-103, 1962.

[Stern1964] Stern et al, "New Data on the Problem of Y-Linkage of Hairy Pinnae", American Journal of Human Genetics, vol. 16(4), pp. 455-471, 1964.

[Chattopadhyay1966] Chattopadhyay, "Inheritance of hairy pinna of the ear", Journal of Genetics, vol. 59(3), pp. 329-334, 1966.

[Karve1968] Karve et al, "A Biological Comparison of Eight Endogamous Groups of the Same Rank [and Comments and Replies]", Current Anthropology, vol. 9(2/3) pp. 109-124, 1968.

[Brega1986] Brega et al, "Genetic Studies on the Tharu Population of Nepal: Restriction Endonuclease Polymorphisms of Mitochondrial DNA", American Journal of Human Genetics, vol. 39, pp. 502-512, 1986.

[Pitchappan1988] Pitchappan, "Founder effects explain the distribution of the HLA A1-B17 but not the absence of the A1-B8 haplotypes in India", Journal of Genetics, vol. 67(2), pp. 101-111, 1988.

[Ballinger1992] Ballinger et al, "Southeast Asian Mitochondrial DNA Analysis Reveals Genetic Continuity of Ancient Mongoloid Migrations", Genetics, vol. 130, pp. 139-152, 1992.

[Passarino1993] Passarino et al, "COII/tRNA(Lys) Intergenic 9-bp Deletion and Other mtDNA Markers Clearly Reveal That the Tharus (Southern Nepal) Have Oriental Affinities", American Journal of Human Genetics, vol. 53, pp. 609-618, 1993.

[Melton1995] Melton et al, "Polynesian Genetic Affinities with Southeast Asian Populations as Identified by mtDNA Analysis", American Journal of Human Genetics, vol. 57, pp. 403-414, 1995.

[Mountain1995] Mountain et al, "Demographic History of India and mtDNA-Sequence Diversity", American Journal of Human Genetics, vol. 56, pp. 979-992, 1995.

[Passarino1996] Passarino et al, "Pre-Caucasoid and Caucasoid Genetic Features of the Indian Population, Revealed by mtDNA Polymorphisms", American Journal of Human Genetics, vol. 59, pp. 927-934, 1996.

[Bamshad1998] Bamshad et al, "Female gene flow stratifies Hindu castes", Nature, vol. 395, pp. 651-652, 1998.

1999

[Bhattacharyya1999] Bhattacharyya et al, "Negligible Male Gene Flow Across Ethnic Boundaries in India, Revealed by Analysis of Y-Chromosomal DNA Polymorphisms", Genome Research, vol. 9, pp. 711-719, 1999.
doi:10.1101/gr.9.8.711

[Kivisild1999a] Kivisild et al, "The Place of the Indian mtDNA Variants in the Global Network of Maternal Lineages and the Peopling of the Old World", in "Genome Diversity: Applications in Human Population Genetics", Kluwer Academic/Plenum Publishers, New York, pp. 135-152, 1999.

[Kivisild1999b] Kivisild et al, "Deep common ancestry of Indian and western-Eurasian mitochondrial DNA lineages", Current Biology, vol. 9, pp. 1331-1334, Nov. 1999.

[Majumder1999] Majumder et al, "Human-specific insertion/deletion polymorphisms in Indian populations and their possible evolutionary implications", European Journal of Human Genetics, vol. 7, pp. 435-446, 1999.

[Watkins1999] Watkins et al, "Multiple Origins of the mtDNA 9-bp Deletion in Populations of South India", American Journal of Physical Anthropology, vol. 109, pp. 147-158, 1999.

2000

[Kivisild2000a] Kivisild et al, "An Indian Ancestry: a Key for Understanding Human Diversity in Europe and Beyond", Chapter 31 in "Archaeogenetics: DNA and the population prehistory of Europe", Eds. Colin Renfrew & Katie Boyle, McDonald Institute Monographs, 2000.
ISBN: 1-902937-08-2
ISSN: 1363-1349

[Kivisild2000b] Kivisild, "The origins of southern and western Eurasian populations: an mtDNA study", Ph.D. Thesis, Department of Evolutionary Biology, Institute of Molecular and Cell Biology, Tartu University, Estonia, May 2000.

[Roychoudhury2000] Roychoudhury et al, "Fundamental genomic unity of ethnic India is revealed by analysis of mitochondrial DNA", Current Science, vol. 79(9), pp. 1182-1192, 2000.

[Su2000] Su et al, "Y chromosome haplotypes reveal prehistorical migrations to the Himalayas", Human Genetics, vol. 107, pp. 582-590, 2000.
doi:10.1007/s004390000406

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.


The Rise of Music in the Ancient World

This week I happened to stumble across Curt Sachs' "The Rise of Music in the Ancient World" [Sac1943] which has a section dealing with ancient Indian music. I was pleasantly surprised to find that Sachs had arrived at pretty much the same formulation as mine following an almost identical thought process. This post documents Sachs' major results and also indicates where they deviate from mine.

The following are verbatim quotes from Sachs' book (pages 166-167):

There has been much pondering over the puzzling problem of why and how the Hindus came to a division into twenty-two parts.
...
Actually, the srutis were not units but, on the contrary, of three different sizes necessitated by the very nature of Indian scales.
...
India's standard scales depended on the divisive principle; they had major whole tones of 204, minor whole tones of 182, and semitones of 112 Cents.
...
The incessant readaptation of the octave required facilities for changing semitones or major whole tones into minor whole tones, of adding and cutting off adequate portions.
All permutations in these 'give-and-take' were feasible with only three elements: (a) twenty-two Cents or a 'comma,' the difference between the major and the minor whole tone (204-182 Cents); (b) seventy Cents, the difference between the minor whole tone and the semitone; (c) ninety Cents , the difference between the semitone and the comma.
...
The give-and-take operation also indicates the exact sequence of the twenty-two shrutis:

D 112 70 E 22 90 F 22 70 22 90 G 22 90 70 22 A 90 22 70 B 22 90 C 22 70 112 D

The first and last steps of 112 Cents, minimum steps with which any model scale begins and ends, are not split in this operation.

Comparison with Sachs' Work

Similarities

The first similarity is that both Sachs and I identify the following three intervals as 1-शृति sized 'elements':

81:80 (Comma of Didymus, denoted c): ~021.51c
25:24 (Small Semitone, denoted s): ~070.67c
256:243 (Pythagorean Limma, denoted L): ~090.22c

The second similarity is that both Sachs and I partition the octave such that the notes at the 1st and the 21st शृति positions are indeterminate.

Differences

First of, Sachs doesn't clearly motivate how he arrived at the three different 'elements'. He states this as a 'matter-of-fact' with no caveats. (I am led to think that these were taken from Fox Strangways' earlier work [Str1914].)

Secondly, I identify the following interval as 0-शृति sized:

2048:2025 (Diaschisma, denoted d): ~019.55c

But there is no reference to this interval in Sachs' work (where it concerns ancient Indian music).

Finally, there is a slight difference in the exact partitioning of the octave. If the 16:15 semitone is denoted S, then Sachs' octave is partitioned into the interval sequence

"S s c L c s c L c L s c L c s c L c s S"

whereas my octave is partitioned into the (palindromic) interval sequence

"S s c L c s c L c s d s c L c s c L c s S"

Note that both sequences are identical in great measure. However, the interval between the 4/3 & 3/2 notes is partitioned by Sachs into the sequence "c L s c" with no obvious justification of why this particular order should be preferred over an alternative sequence, like "c s L c", for example, or indeed over any other combination of those four 'elements'.

On the other hand, my partitioning yields the (palindromic) sequence "c s d s c" between the same two notes. Now, s*d=L. Thus, by assimilating the d either into the preceding or the subsequent s, two different interval sequences can be arrived at: "c L s c" and "c s L c". Both are valid within the context of my model and there is no obvious reason to choose one over the other. It is possible that "c L s c" is favoured in ascending scales, while "c s L c" in descending scales. In any case, I've already argued that both these sequences may have been treated as 'equivalent' for all practical purposes since the d is 0-शृति sized interval (i.e. smaller than the प्रमाण शृति) [Kam2009a].


Similar Work By Other Authors

Fox Strangways too had followed a process similar to mine [Str1914]. He too had arrived at 20 notes in the octave; and in his scheme, 45/32 & 64/45 were two possibilities for the 11-शृति position. However, he had (artificially) introduced notes at the 1st & 21st शृतिs "by analogy" [Str1914, pg 114] even though those two शृतिs were never used in any जाती during मुनि भरत's time.


Final Thoughts

In retrospect, it is no surprise that multiple authors have arrived at conclusions similar to mine. Given the principle of a 'natural' scale (based on an overtone series) there are only a limited number of ways to generate a full gamut based on modulations of the parent scale.

---

References

---

[Kam2009a] Roshan Kamath, "An Acoustic Investigation into Ancient Indian Musical Tuning(s)", http://roshbaby.blogspot.com/2009/02/acoustic-investigation-into-ancient.html, Feb 2009.

[Sac1943] Curt Sachs, "The Rise of Music in the Ancient World", W. W. Norton & Company, Inc., 1943.

[Str1914] A. H. Fox Strangways, "The Music of Hindostan", Oxford University Press, 1914.

---

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.


An 'Acoustic Natural' Scale for the षड्ज-ग्राम

A 'natural' scale based on acoustic principles and which satisfies the necessary conditions of the नाट्यशास्त्र would be:

1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1

This scale employs three different intervals; namely, 9:8 (T), 10:9 (t), and 16:15 (S) much in the line of the scales documented by मुनि भरत. Specifically, the scale above is intended to represent the tuning known as षड्ज-ग्राम which was composed of two tetrachords internally bounded by the Just Fourth (4/3) and the Perfect Fifth (3/2). Note however that in NO way does this imply that मुनि भरत's षड्ज ग्राम was exactly the scale mentioned above; neither is it the point of this article.

The Sixth in the षड्ज-ग्राम is Pythagorean (27/16) and NOT the 'Just' Sixth (5/3) since tetrachordal symmetry was observed in the tuning of the षड्ज-ग्राम्. The scale that used the 'Just' Sixth instead was known as the मध्यम-ग्राम. More about it below.

The षड्ज-ग्राम can be represented in terms of intervallic values as "T t S T T t S".

(Open Question: As the वीणा was a bow-harp, it is hard to defend the stand that the वीणा was tuned to an 'acoustically natural' scale. Historically, instruments like these were tuned by a 'cyclic' process of ascending/descending fifths/fourths which would, without tempering, lead to only two interval sizes, not three as required by मुनि भरत's description. Clearly there's a discrepancy somewhere; but not if one allows the possibility that ancient Indian musicians may have tuned their harps by exploiting 'overtonal consonance' instead of following a 'cyclic' process.)


The मूर्छनs (Modes) of the षड्ज-ग्राम

By rooting the above 'Acoustic Natural' षड्ज-ग्राम on each note, different मूर्छन (modes) of the same basic scale are obtained.

Mode of the First

This is "T t S T T t S" and yields 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1
This mode was also a जाती named षड्जी.

Mode of the Second

This is "t S T T t S T" and yields 1/1, 10/9, 32/27, 4/3, 3/2, 5/3, 16/9, 2/1
This mode was also a जाती named अर्षभी.

Mode of the Third

This is "S T T t S T t" and yields 1/1, 16/15, 6/5, 27/20, 3/2, 8/5, 9/5, 2/1

Mode of the Fourth

This is "T T t S T t S" and yields 1/1, 9/8, 81/64, 45/32, 3/2, 27/16, 15/8, 2/1

Mode of the Fifth

This is "T t S T t S T" and yields 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 16/9, 2/1

Mode of the Sixth

This is "t S T t S T T" and yields 1/1, 10/9, 32/27, 4/3, 40/27, 128/81, 16/9, 2/1
This mode was also a जाती named धैवती.

Mode of the Seventh

This is "S T t S T T t" and yields 1/1, 16/15, 6/5, 4/3, 64/45, 8/5, 9/5, 2/1
This mode was also a जाती named निषादी.


The Modal Gamut

Collect the individual notes that result from the above मूर्छनs and the following gamut results. The notes of the (default) षड्ज-ग्राम are indicated with bold letters.

1/1, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 81/64, 4/3, 27/20, 45/32, 64/45, 40/27, 3/2, 128/81, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 2/1.

The S (16:15) interval

The S interval between the 5/4 and 4/3 notes is observed to be 'split' into two sub-intervals by the introduction of the 81/64 note. Hence, one is inclined to consider the S as a 2-step interval.

The t (10:9) interval

The t interval between the 9/8 and 5/4 notes is observed to be 'split' into three sub-intervals by the introduction of 32/27 and 6/5 notes. This is also true for the t interval between the 27/16 and 15/8 notes. Thus, the t may be considered as a 3-step interval.

The T (9:8) interval

The T interval between the 1/1 and 9/8 notes is 'split' into three sub-intervals by the 16/15 and 10/9 notes. However, since the S (16:15) is already determined to be a 2-step interval or since the t (10:9) is already determined to be a 3-step interval, this makes the T a 4-step interval. An identical reasoning applies to the T interval between the 3/2 and 27/16 notes.

What about the T between the 4/3 and 3/2 notes?

At first glance this interval seems to be split into five sub-intervals. However, this shall be addressed later below once we assign 'weights' (शृतिs) to each note in the complete gamut.


शृति values for each note in the gamut

'Weights' can be assigned to each note in the gamut by adding up the weights (i.e. step-sizes) of the individual intervals that make up the note. For e.g., the 5/4 composed of a T and t shall get assigned a weight of 4+3=7.

These weights are termed शृतिs following मुनि भरत's terminology.

00: 1/1
01: -/-
02: 16/15
03: 10/9
04: 9/8
05: 32/27
06: 6/5
07: 5/4
08: 81/64
09: 4/3
10: 27/20
11: 45/32
11: 64/45
12: 40/27
13: 3/2
14: 128/81
15: 8/5
16: 5/3
17: 27/16
18: 16/9
19: 9/5
20: 15/8
21: -/-
22: 2/1

(Note the conspicuous absence of notes at शृति positions 1 and 21. This may be the underlying reason why none of the जातीs defined by मुनि भरत employed notes at those positions. Also note that both the 45/32 and 64/45 notes end up with the same शृति value 11.)

The first observation is that we end up with a 22 शृति octave. मुनि भरत termed the interval between the 16th and 17th शृति positions as the प्रमाण शृति. This comes out to be the Comma of Didymus (81:80, ~21.51c). It is thought that मुनि भरत considered this as the smallest musically relevant interval, and any interval smaller that this could very easily have been identified with the unison (1:1). Now, the various intervallic distances in the above gamut are:

2048:2025 (Diaschisma): ~019.55c, शृति value 0(?).
81:80 (Comma of Didymus): ~021.51c, शृति value 1.
25:24 (Small Semitone): ~070.67c, शृति value 1.
256:243 (Pythagorean Limma): ~090.22c, शृति value 1.
16:15 (Just Semitone): ~111.73c, शृति value 2.

The only interval smaller than the Comma of Didymus is the Diaschisma which occurs exactly once in the gamut as the intervallic distance between the 45/32 and 64/45 notes. Thus, it is possible that the 45/32 and 64/45 notes were considered to be identical given the state of measurement and accuracy that must have been available to मुनि भरत. This is also obliquely reflected in the same शृति value that gets assigned to both notes. This in itself may have been sufficient justification for मुनि भरत to identify those two notes. On the other hand, the 45/32 note did not occur in any जाती of the षड्ज-ग्राम while the 64/45 note did not occur in any जाती of the मध्यम-ग्राम (see below) so it is conceivable that this confusion never arose.

Note that three different intervals (81:80, 25:24, and 256:243) are considered to be a 1-शृति interval.

Back to the T between the 4/3 and 3/2 notes

If the 45/32 and 64/45 notes truly get identified with each other, it leaves 4 sub-intervals between the 4/3 and 3/2 notes. Even without this identification, we have only 13-9=4 शृति values between the 4/3 and 3/2 notes anyway; and both the 45/32 and 64/45 notes occupy the same शृति position. Thus, the T interval between the 4/3 and 3/2 notes too can be argued to be a 4-step interval consistent with its step-size in other positions.


The मूर्छनs (Modes) of the मध्यम-ग्राम

A complementary tuning, the मध्यम-ग्राम, was also in vogue during मुनि भरत's time. The primary difference between the two tunings was that the Sixth in the मध्यम-ग्राम was 'Just' (5/3) compared to the Pythagorean (27/16) of the षड्ज-ग्राम.

1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1

In terms of intervallic values, this scale can be represented as "T t S T t T S". Following the same procedure as done above for the षड्ज-ग्राम and collecting the various notes yields a subset of the gamut that has already been determined.

1/1, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 4/3, 27/20, 45/32, 64/45, 40/27, 3/2, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 2/1.

No new notes are 'discovered'.

As an aside, of the seven modes of the मध्यम-ग्राम, the following three were also जातीs in use:

गाँधारी (mode of the Third) - This is "S T t T S T t" and yields 1/1, 16/15, 6/5, 4/3, 3/2, 8/5, 9/5, 2/1.

मध्यम (mode of the Fourth) - This is "T t T S T t S" and yields 1/1, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1.

पंचमी (mode of the Fifth) - This is "t T S T t S T" and yields 1/1, 10/9, 5/4, 4/3, 3/2, 5/3, 16/9, 2/1.


Conclusions

An octave with 22 शृति divisions naturally arises from the modulations of an 'acoustically natural' scale. However, these 22 शृतिs provide only 20 notes since the notes at the 1 & 21 positions are indeterminate. Inspite of the 22 शृतिs, it must not be forgotten that (ancient) Indian music itself was based only on 7 स्वरs (notes). Thus, the 22 शृतिs that arise from the modes of the two ग्रामs are of theoretical interest only and serve no practical purpose otherwise. However, a knowledge of these शृतिs may be handy in comprehending the subsequent evolution of Indian music.

---

References

---

[Bha1984] Pt. Vishnu Narayan Bhatkhande, "Music Systems in India: A Comparative Study of Some of the Leading Music Systems of the 15th, 16th, 17th, & 18th Centuries", South Asia Books, 1984.

[Dan1995] Alain Danielou, "Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness", Inner Traditions International, 1995.

[Day1891] ^^ ... the historian Strabo says that among the Greeks those who regard all Asia as far as India as a country sacred to Dionysius, "attribute to that country the invention of nearly all the science of music."^^, Capt. Charles Russell Day, "The Music and Musical Instruments of Southern India and the Deccan", Novello, Ewer & Co., 1891, page 19.

[Jai1975] Nazir Ali Jairazbhoy, "An Interpretation of the 22 Srutis", Asian Music, Vol. 6, No. 1/2 (Perspectives on Asian Music: Essays in Honor of Dr. Laurence E. R. Picken), 1975, pp. 38-59.

[Kam2008] Roshan Kamath, "A Model for Implied Intonations in Classical हिन्दूस्तानी (Hindustani) Music", http://roshbaby.blogspot.com/2008/07/model-for-implied-intonations-in.html, 2008.

---