27 February 2019; the feast of Sts Gabriel of Our Lady of Sorrows and Leander
4. Transcendental Constant Music Scale Tables part 3
An attempt at getting the exact value of the key A4 = 440.00 Hz (in piano frequency).
While playing with the numbers I noticed that Ln ( frequency D0/ Fundamental frequency fF) = - ( constant C16/ constant C8 i.e. pi ).
The equation looks like this:
Ln ( 18.2235960 Hz / ffundamental ) = - ( 9.99983879780488 / 3.14159265358979312)
It simplifies as follows:
Ln ( 18.2235960 Hz / ffundamental ) = -3.18304755
e-3.18304755 = ( 18.2235960 Hz / ffundamental ) = 0.0414591137
Giving finally the fundamental value of the frequency ffundamental:
ffundamental= 439.555851 [Hz]
Having fundamental frequency we can easily determine base of power in general formula for key notes:
( D0 = 18.2235960 Hz ) = [ X (-6-49)/12] * [ffundamental= 439.555851 [Hz] ]
This simplifies to:
X(-55 / 12 )= 0.0414591137
(-55 / 12 ) Ln X = Ln 0.0414591137
Ln X = 0.694483102
X = e0.694483102 = 2.00267363
and final equation for calculating the Transcendental Constant Notes is:
f ( n ) = [ ( 2.00267363 )( n - 49 ) / 12 ] * 439.555851 [Hz] [ Eqn. 1 ]
This fundamental frequency, fFundamental = 439.555851 [Hz] calculated from the constants, not by assigning some arbitrary number ( A4 = 440.00 Hz) looks much more interesting. It is time for experiments to check all these notes if they sound harmonious. I am pretty sure they do.
The minimum absolute difference in [Hz] equals to -0.12 at D0
The maximum difference is 36.09 [Hz] at B8
The relative difference varies from 0.002 to 0.0074
The tables contain five columns: 1st column- key number and the name of the key; 2nd column - Transcendental Scale Note frequency in Hz; 3rd column - Piano Scale Note frequencies in Hz; 4th column - Absolute difference between those frequencies in Hz, 5th column - The relative difference between these frequencies from column 4 ( absolute value divided over the frequency from second column - so it is similar to relative error)
Figure 1. Table of Transcendental Constant Frequencies with fundamental frequency 439.555851 [Hz]
FUNDAMENTAL FREQUENCY IN [HZ]
0.43955585D+03
KEY TRANSC SCALE NOTE PIANO SCALE NOTE ABS. DIFF. [HZ] REL. DIFF.
-8 0.1623176385D+02 0.1635159783D+02 -0.11983398D+00 -0.73285790D-02
-7 0.1719886935D+02 0.1732391444D+02 -0.12504509D+00 -0.72180620D-02
-6 0.1822359600D+02 0.1835404799D+02 -0.13045199D+00 -0.71075326D-02
-5 0.1930937694D+02 0.1944543648D+02 -0.13605954D+00 -0.69969910D-02
-4 0.2045984984D+02 0.2060172231D+02 -0.14187246D+00 -0.68864370D-02
-3 0.2167886913D+02 0.2182676446D+02 -0.14789533D+00 -0.67758707D-02
-2 0.2297051886D+02 0.2312465142D+02 -0.15413256D+00 -0.66652921D-02
-1 0.2433912644D+02 0.2449971475D+02 -0.16058831D+00 -0.65547012D-02
0 0.2578927709D+02 0.2595654360D+02 -0.16726651D+00 -0.64440980D-02
1 0.2732582923D+02 0.2750000000D+02 -0.17417077D+00 -0.63334825D-02
2 0.2895393076D+02 0.2913523509D+02 -0.18130433D+00 -0.62228547D-02
3 0.3067903629D+02 0.3086770633D+02 -0.18867004D+00 -0.61122145D-02
4 0.3250692540D+02 0.3270319566D+02 -0.19627026D+00 -0.60015621D-02
5 0.3444372207D+02 0.3464782887D+02 -0.20410680D+00 -0.58908973D-02
6 0.3649591511D+02 0.3670809599D+02 -0.21218088D+00 -0.57802202D-02
7 0.3867037997D+02 0.3889087297D+02 -0.22049300D+00 -0.56695307D-02
8 0.4097440171D+02 0.4120344461D+02 -0.22904290D+00 -0.55588290D-02
9 0.4341569949D+02 0.4365352893D+02 -0.23782944D+00 -0.54481149D-02
10 0.4600245234D+02 0.4624930284D+02 -0.24685050D+00 -0.53373885D-02
11 0.4874332664D+02 0.4899942950D+02 -0.25610286D+00 -0.52266498D-02
12 0.5164750510D+02 0.5191308720D+02 -0.26558209D+00 -0.51158987D-02
13 0.5472471756D+02 0.5500000000D+02 -0.27528244D+00 -0.50051353D-02
14 0.5798527356D+02 0.5827047019D+02 -0.28519663D+00 -0.48943596D-02
15 0.6144009690D+02 0.6173541266D+02 -0.29531576D+00 -0.47835715D-02
16 0.6510076223D+02 0.6540639133D+02 -0.30562910D+00 -0.46727711D-02
17 0.6897953384D+02 0.6929565774D+02 -0.31612391D+00 -0.45619584D-02
18 0.7308940672D+02 0.7341619198D+02 -0.32678526D+00 -0.44511333D-02
19 0.7744415014D+02 0.7778174593D+02 -0.33759580D+00 -0.43402959D-02
20 0.8205835372D+02 0.8240688923D+02 -0.34853550D+00 -0.42294462D-02
21 0.8694747640D+02 0.8730705786D+02 -0.35958146D+00 -0.41185841D-02
22 0.9212789812D+02 0.9249860568D+02 -0.37070756D+00 -0.40077097D-02
23 0.9761697480D+02 0.9799885900D+02 -0.38188420D+00 -0.38968229D-02
24 0.1034330964D+03 0.1038261744D+03 -0.39307798D+00 -0.37859238D-02
25 0.1095957486D+03 0.1100000000D+03 -0.40425136D+00 -0.36750123D-02
26 0.1161255782D+03 0.1165409404D+03 -0.41536223D+00 -0.35640885D-02
27 0.1230444617D+03 0.1234708253D+03 -0.42636357D+00 -0.34531524D-02
28 0.1303755797D+03 0.1308127827D+03 -0.43720298D+00 -0.33422038D-02
29 0.1381434933D+03 0.1385913155D+03 -0.44782221D+00 -0.32312430D-02
30 0.1463742273D+03 0.1468323840D+03 -0.45815665D+00 -0.31202698D-02
31 0.1550953571D+03 0.1555634919D+03 -0.46813476D+00 -0.30092842D-02
32 0.1643361009D+03 0.1648137785D+03 -0.47767751D+00 -0.28982863D-02
33 0.1741274180D+03 0.1746141157D+03 -0.48669773D+00 -0.27872760D-02
34 0.1845021120D+03 0.1849972114D+03 -0.49509940D+00 -0.26762533D-02
35 0.1954949411D+03 0.1959977180D+03 -0.50277694D+00 -0.25652183D-02
36 0.2071427344D+03 0.2076523488D+03 -0.50961436D+00 -0.24541710D-02
37 0.2194845155D+03 0.2200000000D+03 -0.51548447D+00 -0.23431112D-02
38 0.2325616329D+03 0.2330818808D+03 -0.52024788D+00 -0.22320391D-02
39 0.2464178986D+03 0.2469416506D+03 -0.52375205D+00 -0.21209547D-02
40 0.2610997351D+03 0.2616255653D+03 -0.52583019D+00 -0.20098578D-02
41 0.2766563308D+03 0.2771826310D+03 -0.52630014D+00 -0.18987486D-02
42 0.2931398048D+03 0.2936647679D+03 -0.52496309D+00 -0.17876271D-02
43 0.3106053815D+03 0.3111269837D+03 -0.52160225D+00 -0.16764931D-02
44 0.3291115755D+03 0.3296275569D+03 -0.51598145D+00 -0.15653468D-02
45 0.3487203879D+03 0.3492282314D+03 -0.50784355D+00 -0.14541881D-02
46 0.3694975139D+03 0.3699944227D+03 -0.49690883D+00 -0.13430171D-02
47 0.3915125628D+03 0.3919954360D+03 -0.48287316D+00 -0.12318336D-02
48 0.4148392914D+03 0.4153046976D+03 -0.46540615D+00 -0.11206378D-02
49 0.4395558510D+03 0.4400000000D+03 -0.44414903D+00 -0.10094296D-02
50 0.4657450490D+03 0.4661637615D+03 -0.41871251D+00 -0.89820904D-03
51 0.4934946269D+03 0.4938833013D+03 -0.38867434D+00 -0.78697608D-03
52 0.5228975537D+03 0.5232511306D+03 -0.35357687D+00 -0.67573074D-03
53 0.5540523377D+03 0.5543652620D+03 -0.31292423D+00 -0.56447301D-03
54 0.5870633564D+03 0.5873295358D+03 -0.26617944D+00 -0.45320289D-03
55 0.6220412061D+03 0.6222539674D+03 -0.21276132D+00 -0.34192038D-03
56 0.6591030728D+03 0.6592551138D+03 -0.15204103D+00 -0.23062549D-03
57 0.6983731243D+03 0.6984564629D+03 -0.83338572D-01 -0.11931821D-03
58 0.7399829266D+03 0.7399888454D+03 -0.59188221D-02 -0.79985288D-05
59 0.7840718845D+03 0.7839908720D+03 0.81012554D-01 0.10333354D-03
60 0.8307877087D+03 0.8306093952D+03 0.17831357D+00 0.21467801D-03
61 0.8802869107D+03 0.8800000000D+03 0.28691068D+00 0.32603487D-03
62 0.9327353269D+03 0.9323275230D+03 0.40780391D+00 0.43740413D-03
63 0.9883086748D+03 0.9877666025D+03 0.54207227D+00 0.54878579D-03
64 0.1047193141D+04 0.1046502261D+04 0.69087970D+00 0.66017985D-03
65 0.1109586005D+04 0.1108730524D+04 0.85548129D+00 0.77158631D-03
66 0.1175696302D+04 0.1174659072D+04 0.10372300D+01 0.88300517D-03
67 0.1245745519D+04 0.1244507935D+04 0.12375840D+01 0.99443644D-03
68 0.1319968342D+04 0.1318510228D+04 0.14581142D+01 0.11058801D-02
69 0.1398613438D+04 0.1396912926D+04 0.17005127D+01 0.12173362D-02
70 0.1481944292D+04 0.1479977691D+04 0.19666013D+01 0.13288047D-02
71 0.1570240085D+04 0.1567981744D+04 0.22583415D+01 0.14402856D-02
72 0.1663796635D+04 0.1661218790D+04 0.25778443D+01 0.15517789D-02
73 0.1762927381D+04 0.1760000000D+04 0.29273809D+01 0.16632846D-02
74 0.1867964441D+04 0.1864655046D+04 0.33093949D+01 0.17748028D-02
75 0.1979259719D+04 0.1975533205D+04 0.37265141D+01 0.18863333D-02
76 0.2097186087D+04 0.2093004522D+04 0.41815641D+01 0.19978763D-02
77 0.2222138630D+04 0.2217461048D+04 0.46775826D+01 0.21094317D-02
78 0.2354535978D+04 0.2349318143D+04 0.52178344D+01 0.22209995D-02
79 0.2494821698D+04 0.2489015870D+04 0.58058280D+01 0.23325797D-02
80 0.2643465788D+04 0.2637020455D+04 0.64453326D+01 0.24441724D-02
81 0.2800966249D+04 0.2793825851D+04 0.71403971D+01 0.25557775D-02
82 0.2967850752D+04 0.2959955382D+04 0.78953701D+01 0.26673950D-02
83 0.3144678408D+04 0.3135963488D+04 0.87149206D+01 0.27790249D-02
84 0.3332041642D+04 0.3322437581D+04 0.96040615D+01 0.28906672D-02
85 0.3530568174D+04 0.3520000000D+04 0.10568174D+02 0.30023220D-02
86 0.3740923124D+04 0.3729310092D+04 0.11613032D+02 0.31139893D-02
87 0.3963811242D+04 0.3951066410D+04 0.12744832D+02 0.32256689D-02
88 0.4199979268D+04 0.4186009045D+04 0.13970223D+02 0.33373610D-02
89 0.4450218432D+04 0.4434922096D+04 0.15296337D+02 0.34490655D-02
90 0.4715367108D+04 0.4698636287D+04 0.16730822D+02 0.35607825D-02
91 0.4996313620D+04 0.4978031740D+04 0.18281881D+02 0.36725119D-02
92 0.5293999219D+04 0.5274040911D+04 0.19958309D+02 0.37842537D-02
93 0.5609421238D+04 0.5587651703D+04 0.21769536D+02 0.38960080D-02
94 0.5943636432D+04 0.5919910763D+04 0.23725668D+02 0.40077747D-02
95 0.6297764517D+04 0.6271926976D+04 0.25837541D+02 0.41195538D-02
96 0.6672991923D+04 0.6644875161D+04 0.28116762D+02 0.42313454D-02
97 0.7070575772D+04 0.7040000000D+04 0.30575772D+02 0.43431495D-02
98 0.7491848084D+04 0.7458620184D+04 0.33227899D+02 0.44549660D-02
99 0.7938220240D+04 0.7902132820D+04 0.36087420D+02 0.45667949D-02
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