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