26 February 2019; Feast of Sts Porphyrius and Alexander
3. Transcendental Constant Music Scale Tables part 2
The second table of Transcendental Constant Music Scale starting with initial frequency A4 = 440.000Hz.
This fundamental frequency generates much bigger differences at the high end of the scale - at the note B8 , the difference is slightly over 50 Hz; However, the lower frequencies are with smaller differences. For example Tenor C = C5 key number 52, has the difference between Transcendental and Piano frequency only 2.04 Hz, and Soprano C = C6 , 64 number key, has the difference also 2.04 Hz, beyond recognition for the human hearing. These results prove that the initial frequency A4 = 440.000 [Hz] is not the best choice, it is just an approximation.
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).
It is really difficult to say now which scale is better - without actual experiments on some programmable music machine.
Figure 1. Table of Transcendental Constant Music scale starting with the initial frequency A4 = 440.000 Hz.
FUNDAMENTAL FREQUENCY IN [HZ]
0.44000000D+03
KEY TRANSC SCALE NOTE PIANO SCALE NOTE ABS. DIFF. [HZ] REL. DIFF.
-8 0.1623116776D+02 0.1635159783D+02 -0.12043007D+00 -0.73650339D-02
-7 0.1719855354D+02 0.1732391444D+02 -0.12536089D+00 -0.72362915D-02
-6 0.1822359601D+02 0.1835404799D+02 -0.13045199D+00 -0.71075323D-02
-5 0.1930973152D+02 0.1944543648D+02 -0.13570496D+00 -0.69787564D-02
-4 0.2046060125D+02 0.2060172231D+02 -0.14112105D+00 -0.68499638D-02
-3 0.2168006341D+02 0.2182676446D+02 -0.14670106D+00 -0.67211546D-02
-2 0.2297220612D+02 0.2312465142D+02 -0.15244530D+00 -0.65923286D-02
-1 0.2434136119D+02 0.2449971475D+02 -0.15835356D+00 -0.64634859D-02
0 0.2579211859D+02 0.2595654360D+02 -0.16442501D+00 -0.63346265D-02
1 0.2732934187D+02 0.2750000000D+02 -0.17065813D+00 -0.62057504D-02
2 0.2895818442D+02 0.2913523509D+02 -0.17705067D+00 -0.60768575D-02
3 0.3068410682D+02 0.3086770633D+02 -0.18359951D+00 -0.59479480D-02
4 0.3251289506D+02 0.3270319566D+02 -0.19030061D+00 -0.58190217D-02
5 0.3445068000D+02 0.3464782887D+02 -0.19714887D+00 -0.56900787D-02
6 0.3650395790D+02 0.3670809599D+02 -0.20413809D+00 -0.55611190D-02
7 0.3867961220D+02 0.3889087297D+02 -0.21126077D+00 -0.54321426D-02
8 0.4098493659D+02 0.4120344461D+02 -0.21850802D+00 -0.53031494D-02
9 0.4342765948D+02 0.4365352893D+02 -0.22586945D+00 -0.51741395D-02
10 0.4601596989D+02 0.4624930284D+02 -0.23333295D+00 -0.50451129D-02
11 0.4875854490D+02 0.4899942950D+02 -0.24088460D+00 -0.49160695D-02
12 0.5166457876D+02 0.5191308720D+02 -0.24850844D+00 -0.47870094D-02
13 0.5474381371D+02 0.5500000000D+02 -0.25618629D+00 -0.46579326D-02
14 0.5800657261D+02 0.5827047019D+02 -0.26389758D+00 -0.45288390D-02
15 0.6146379359D+02 0.6173541266D+02 -0.27161907D+00 -0.43997287D-02
16 0.6512706669D+02 0.6540639133D+02 -0.27932464D+00 -0.42706016D-02
17 0.6900867270D+02 0.6929565774D+02 -0.28698504D+00 -0.41414578D-02
18 0.7312162440D+02 0.7341619198D+02 -0.29456758D+00 -0.40122972D-02
19 0.7747971008D+02 0.7778174593D+02 -0.30203585D+00 -0.38831199D-02
20 0.8209753988D+02 0.8240688923D+02 -0.30934935D+00 -0.37539258D-02
21 0.8699059466D+02 0.8730705786D+02 -0.31646320D+00 -0.36247150D-02
22 0.9217527797D+02 0.9249860568D+02 -0.32332771D+00 -0.34954874D-02
23 0.9766897101D+02 0.9799885900D+02 -0.32988798D+00 -0.33662431D-02
24 0.1034900909D+03 0.1038261744D+03 -0.33608345D+00 -0.32369820D-02
25 0.1096581525D+03 0.1100000000D+03 -0.34184745D+00 -0.31077041D-02
26 0.1161938337D+03 0.1165409404D+03 -0.34710664D+00 -0.29784095D-02
27 0.1231190448D+03 0.1234708253D+03 -0.35178049D+00 -0.28490980D-02
28 0.1304570020D+03 0.1308127827D+03 -0.35578066D+00 -0.27197699D-02
29 0.1382323051D+03 0.1385913155D+03 -0.35901040D+00 -0.25904249D-02
30 0.1464710202D+03 0.1468323840D+03 -0.36136377D+00 -0.24610632D-02
31 0.1552007669D+03 0.1555634919D+03 -0.36272501D+00 -0.23316847D-02
32 0.1644508108D+03 0.1648137785D+03 -0.36296763D+00 -0.22022894D-02
33 0.1742521621D+03 0.1746141157D+03 -0.36195364D+00 -0.20728773D-02
34 0.1846376788D+03 0.1849972114D+03 -0.35953254D+00 -0.19434485D-02
35 0.1956421776D+03 0.1959977180D+03 -0.35554041D+00 -0.18140028D-02
36 0.2073025500D+03 0.2076523488D+03 -0.34979877D+00 -0.16845404D-02
37 0.2196578865D+03 0.2200000000D+03 -0.34211345D+00 -0.15550612D-02
38 0.2327496074D+03 0.2330818808D+03 -0.33227340D+00 -0.14255651D-02
39 0.2466216013D+03 0.2469416506D+03 -0.32004930D+00 -0.12960523D-02
40 0.2613203731D+03 0.2616255653D+03 -0.30519217D+00 -0.11665227D-02
41 0.2768951992D+03 0.2771826310D+03 -0.28743182D+00 -0.10369763D-02
42 0.2933982927D+03 0.2936647679D+03 -0.26647526D+00 -0.90741310D-03
43 0.3108849789D+03 0.3111269837D+03 -0.24200486D+00 -0.77783309D-03
44 0.3294138804D+03 0.3296275569D+03 -0.21367654D+00 -0.64823627D-03
45 0.3490471138D+03 0.3492282314D+03 -0.18111767D+00 -0.51862264D-03
46 0.3698504978D+03 0.3699944227D+03 -0.14392495D+00 -0.38899220D-03
47 0.3918937739D+03 0.3919954360D+03 -0.10166204D+00 -0.25934495D-03
48 0.4152508405D+03 0.4153046976D+03 -0.53857080D-01 -0.12968088D-03
49 0.4400000000D+03 0.4400000000D+03 0.00000000D+00 0.00000000D+00
50 0.4662242219D+03 0.4661637615D+03 0.60460369D-01 0.12969770D-03
51 0.4940114206D+03 0.4938833013D+03 0.12811937D+00 0.25941223D-03
52 0.5234547504D+03 0.5232511306D+03 0.20361982D+00 0.38914358D-03
53 0.5546529175D+03 0.5543652620D+03 0.28765556D+00 0.51889175D-03
54 0.5877105111D+03 0.5873295358D+03 0.38097527D+00 0.64865675D-03
55 0.6227383539D+03 0.6222539674D+03 0.48438650D+00 0.77843858D-03
56 0.6598538739D+03 0.6592551138D+03 0.59876005D+00 0.90823725D-03
57 0.6991814975D+03 0.6984564629D+03 0.72503465D+00 0.10380527D-02
58 0.7408530674D+03 0.7399888454D+03 0.86422194D+00 0.11678851D-02
59 0.7850082838D+03 0.7839908720D+03 0.10174118D+01 0.12977343D-02
60 0.8317951734D+03 0.8306093952D+03 0.11857782D+01 0.14276003D-02
61 0.8813705852D+03 0.8800000000D+03 0.13705852D+01 0.15574831D-02
62 0.9339007165D+03 0.9323275230D+03 0.15731935D+01 0.16873828D-02
63 0.9895616702D+03 0.9877666025D+03 0.17950676D+01 0.18172994D-02
64 0.1048540045D+04 0.1046502261D+04 0.20377835D+01 0.19472328D-02
65 0.1111033560D+04 0.1108730524D+04 0.23030363D+01 0.20771831D-02
66 0.1177251721D+04 0.1174659072D+04 0.25926490D+01 0.22071502D-02
67 0.1247416517D+04 0.1244507935D+04 0.29085820D+01 0.23371341D-02
68 0.1321763170D+04 0.1318510228D+04 0.32529427D+01 0.24671349D-02
69 0.1400540922D+04 0.1396912926D+04 0.36279961D+01 0.25971526D-02
70 0.1484013867D+04 0.1479977691D+04 0.40361762D+01 0.27271872D-02
71 0.1572461842D+04 0.1567981744D+04 0.44800980D+01 0.28572386D-02
72 0.1666181361D+04 0.1661218790D+04 0.49625703D+01 0.29873069D-02
73 0.1765486610D+04 0.1760000000D+04 0.54866100D+01 0.31173920D-02
74 0.1870710502D+04 0.1864655046D+04 0.60554562D+01 0.32474940D-02
75 0.1982205792D+04 0.1975533205D+04 0.66725865D+01 0.33776129D-02
76 0.2100346256D+04 0.2093004522D+04 0.73417339D+01 0.35077487D-02
77 0.2225527952D+04 0.2217461048D+04 0.80669046D+01 0.36379014D-02
78 0.2358170541D+04 0.2349318143D+04 0.88523973D+01 0.37680709D-02
79 0.2498718694D+04 0.2489015870D+04 0.97028243D+01 0.38982573D-02
80 0.2647643588D+04 0.2637020455D+04 0.10623133D+02 0.40284606D-02
81 0.2805444481D+04 0.2793825851D+04 0.11618630D+02 0.41586808D-02
82 0.2972650387D+04 0.2959955382D+04 0.12695006D+02 0.42889179D-02
83 0.3149821849D+04 0.3135963488D+04 0.13858362D+02 0.44191718D-02
84 0.3337552820D+04 0.3322437581D+04 0.15115239D+02 0.45494427D-02
85 0.3536472651D+04 0.3520000000D+04 0.16472651D+02 0.46797304D-02
86 0.3747248205D+04 0.3729310092D+04 0.17938112D+02 0.48100351D-02
87 0.3970586087D+04 0.3951066410D+04 0.19519677D+02 0.49403567D-02
88 0.4207235020D+04 0.4186009045D+04 0.21225976D+02 0.50706951D-02
89 0.4457988349D+04 0.4434922096D+04 0.23066254D+02 0.52010505D-02
90 0.4723686703D+04 0.4698636287D+04 0.25050416D+02 0.53314227D-02
91 0.5005220813D+04 0.4978031740D+04 0.27189073D+02 0.54618119D-02
92 0.5303534497D+04 0.5274040911D+04 0.29493587D+02 0.55922180D-02
93 0.5619627828D+04 0.5587651703D+04 0.31976125D+02 0.57226410D-02
94 0.5954560480D+04 0.5919910763D+04 0.34649717D+02 0.58530809D-02
95 0.6309455288D+04 0.6271926976D+04 0.37528312D+02 0.59835378D-02
96 0.6685502004D+04 0.6644875161D+04 0.40626843D+02 0.61140115D-02
97 0.7083961295D+04 0.7040000000D+04 0.43961295D+02 0.62445022D-02
98 0.7506168961D+04 0.7458620184D+04 0.47548777D+02 0.63750098D-02
99 0.7953540416D+04 0.7902132820D+04 0.51407596D+02 0.65055343D-02
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