shruti determination
[This is a draft article]
मला स्वरांच्या उंचीतला फरक गाता गाता अनेकदा जाणवला आहे. पण ह्याचा थोडाही खोलात अभ्यास करायचा असेल तर त्याला थोड्या विश्लेषणाची जोड देण्याची गरज भासते.
ह्या लेखाचे ३ भाग:
१. एक Hierarchical Frequency Determination Algorithm (HFDA)
२. २२ श्रुती सिद्धांत asis
३. better proposals
Hierarchical Frequency Determination Algorithm
If one takes the Hexagon shown on the right (note: this hexagon itself is locally consistent, related by simple rations) as a Building Block of the Tonnetz diagram and tries to expand from there to construct an extended Tonnetz diagram, we very quickly run into inconsistencies. We need to approximately stitch together some edges.
The HDH in short states that if one takes only the inRaag notes from the Tonnetz diagram, the Vaadi/Samvaadi ones will have a higher amount of local consistency around them, and the approximate stitching will happen around less important notes.
Put differently, one arranges the notes & चलनs of a Raag in hierarchical importance and goes down that chart to start assigning frequency values consistent with those already determined.
The 22 Shruti Theory
This theory only recognizes powers of 3 (up to the 5th power!) and powers of 5 (only 1st power!), as integers that can make up “simple” ratios.
[insert extended Tonnetz diagram here]
Computations by some other theorists: https://22shruti.com/research_topic_6.asp
Computations by Dr Oak:
स्वर  Pure Ratio  Decimal Ratio 

S  1  100 
r1  256/243  105.3497942 
r2  16/15  106.666667 
R1  10/9  111.111111 
R2  9/8  112.5 
g1  32/27  118.51851851 
g2  6/5  120 
G1  5/4  125 
G2  81/64  126.5625 
M1  4/3  133.333333 
M2  27/20  135 
m1  45/32  140.625 
m2  729/512  142.3828125 
P1  40/27  148.148148148 
P2  3/2  150 
d1  128/81  158.0246913 
d2  8/5  160 
D1  5/3  166.666667 
D2  27/16  168.75 
n1  16/9  177.77778 
n2  9/5  180 
N1  15/8  187.5 
N2  243/128  189.84375 
S2 😞  81/80  202.5 
As mentioned above, this is sort of a hybrid 5primelimit ∩ 729oddlimit system.

noddlimit = {p/q : there does not exist odd m > n such that m p or m q} 
nprimelimit = {p/q : there does not exist prime m > n such that m p or m q}
e.g. ratios with a factor of 9 are prohibited in a 7oddlimit system but allowed in a in a 7primelimit system.
A Better Proposal
Set no oddlimits and no primelimits. Choose the first P prime numbers where P is significantly large (say 810) and construct a Pdimensional Tonnetz net.
We will be assigning lower preferences to higher prime axes.*
Now, given the chalan of the raag, choose nodes that minimize path length (or some such metric) while traversing over the net. (*This computation is the place where.) Essentially, travelling along largeprime axes will cost us more. The further we do away from the root as well, it will incur more path cost. In the way, we can enforce a soft primelimit and oddlimit.
Here’s a roadmap:
References
 https://22shruti.com/research_topic_6.asp
 https://puretones.sadharani.com/learn/musicalscales
 https://en.wikipedia.org/wiki/Fokker_periodicity_block
 https://www.anaphoria.com/wilson.html (works of https://en.wikipedia.org/wiki/Erv_Wilson)
Categories: music