26 February 2019, Feast of Sts Porphyrius and Alexander

These two articles contain two tables of notes. First calculated from the base frequency 49th key 437.3333 Hz and second with a base frequency A4 = 440.0000Hz.

The first table with adjusted base frequency to 437.3333 Hz gives the maximum difference between the piano scale used now in music at value 5.44 Hz at key 79 = D#7. And the second table contains notes calculated from base frequency 440.0000 Hz, i.e. the same as for the piano. This time, however, the maximum difference is 51.4 Hz at the key 99 = B. 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). There are of course many ways of setting up the initial (fundamental ) frequency but the fundamental frequency equal to (440.000 Hz - 2.6666 Hz) = 437.3333 Hz seems to give the best fit.

Figure 1. The Table of Transcendental Constant Frequencies starting with initial frequency = (440.000 - 2.666) Hz = 437.333 Hz.

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FUNDAMENTAL FREQUENCY IN [HZ]

0.43733333D+03

KEY    TRANSC SCALE NOTE    PIANO SCALE NOTE    ABS. DIFF. [HZ]     REL. DIFF.

C0 -8    0.1613279704D+02    0.1635159783D+02   -0.21880079D+00   -0.13381003D-01
C#0 -7    0.1709431988D+02    0.1732391444D+02   -0.22959455D+00   -0.13253041D-01
D0 -6    0.1811314997D+02    0.1835404799D+02   -0.24089803D+00   -0.13125062D-01
D#0 -5    0.1919270284D+02    0.1944543648D+02   -0.25273364D+00   -0.12997067D-01
E0 -4    0.2033659761D+02    0.2060172231D+02   -0.26512470D+00   -0.12869055D-01
F0 -3    0.2154866908D+02    0.2182676446D+02   -0.27809538D+00   -0.12741026D-01
F#0 -2    0.2283298063D+02    0.2312465142D+02   -0.29167079D+00   -0.12612981D-01
G0 -1    0.2419383779D+02    0.2449971475D+02   -0.30587696D+00   -0.12484919D-01
G#0  0    0.2563580272D+02    0.2595654360D+02   -0.32074088D+00   -0.12356841D-01
A0  1    0.2716370949D+02    0.2750000000D+02   -0.33629051D+00   -0.12228746D-01
A#0  2    0.2878268027D+02    0.2913523509D+02   -0.35255482D+00   -0.12100634D-01
B0  3    0.3049814253D+02    0.3086770633D+02   -0.36956380D+00   -0.11972506D-01

C1  4    0.3231584721D+02    0.3270319566D+02   -0.38734845D+00   -0.11844361D-01
C#1 5    0.3424188800D+02    0.3464782887D+02   -0.40594087D+00   -0.11716199D-01
D1  6    0.3628272179D+02    0.3670809599D+02   -0.42537420D+00   -0.11588021D-01
D#1 7    0.3844519031D+02    0.3889087297D+02   -0.44568266D+00   -0.11459827D-01
E1  8    0.4073654304D+02    0.4120344461D+02   -0.46690158D+00   -0.11331615D-01
F1  9    0.4316446154D+02    0.4365352893D+02   -0.48906739D+00   -0.11203387D-01
F#1 10    0.4573708522D+02    0.4624930284D+02   -0.51221762D+00   -0.11075143D-01
G1 11    0.4846303856D+02    0.4899942950D+02   -0.53639093D+00   -0.10946881D-01
G#1 12    0.5135146010D+02    0.5191308720D+02   -0.56162710D+00   -0.10818603D-01
A1 13    0.5441203302D+02    0.5500000000D+02   -0.58796698D+00   -0.10690309D-01
A#1 14    0.5765501763D+02    0.5827047019D+02   -0.61545256D+00   -0.10561998D-01
B1 15    0.6109128575D+02    0.6173541266D+02   -0.64412691D+00   -0.10433670D-01

C2 16    0.6473235719D+02    0.6540639133D+02   -0.67403414D+00   -0.10305325D-01
C#2 17    0.6859043832D+02    0.6929565774D+02   -0.70521942D+00   -0.10176964D-01
D2 18    0.7267846304D+02    0.7341619198D+02   -0.73772894D+00   -0.10048586D-01
D#2 19    0.7701013608D+02    0.7778174593D+02   -0.77160985D+00   -0.99201919D-02
E2 20    0.8159997903D+02    0.8240688923D+02   -0.80691020D+00   -0.97917808D-02
F2 21    0.8646337893D+02    0.8730705786D+02   -0.84367893D+00   -0.96633531D-02
F#2 22    0.9161663992D+02    0.9249860568D+02   -0.88196576D+00   -0.95349087D-02
G2 23    0.9707703786D+02    0.9799885900D+02   -0.92182114D+00   -0.94064477D-02
G#2 24    0.1028628783D+03    0.1038261744D+03   -0.96329613D+00   -0.92779700D-02
A2 25    0.1089935577D+03    0.1100000000D+03   -0.10064423D+01   -0.91494756D-02
A#2 26    0.1154896287D+03    0.1165409404D+03   -0.10513117D+01   -0.90209645D-02
B2 27    0.1223728688D+03    0.1234708253D+03   -0.10979565D+01   -0.88924368D-02

C3 28    0.1296663535D+03    0.1308127827D+03   -0.11464292D+01   -0.87638925D-02
C#3 29    0.1373945335D+03    0.1385913155D+03   -0.11967819D+01   -0.86353314D-02
D3 30    0.1455833170D+03    0.1468323840D+03   -0.12490669D+01   -0.85067537D-02
D#3 31    0.1542601561D+03    0.1555634919D+03   -0.13033357D+01   -0.83781593D-02
E3 32    0.1634541392D+03    0.1648137785D+03   -0.13596392D+01   -0.82495482D-02
F3 33    0.1731960884D+03    0.1746141157D+03   -0.14180273D+01   -0.81209205D-02
F#3 34    0.1835186626D+03    0.1849972114D+03   -0.14785488D+01   -0.79922760D-02
G3 35    0.1944564674D+03    0.1959977180D+03   -0.15412506D+01   -0.78636149D-02
G#3 36    0.2060461709D+03    0.2076523488D+03   -0.16061779D+01   -0.77349371D-02
A3 37    0.2183266266D+03    0.2200000000D+03   -0.16733734D+01   -0.76062426D-02
A#3 38    0.2313390037D+03    0.2330818808D+03   -0.17428771D+01   -0.74775314D-02
B3 39    0.2451269250D+03    0.2469416506D+03   -0.18147257D+01   -0.73488035D-02

C4 40    0.2597366133D+03    0.2616255653D+03   -0.18889520D+01   -0.72200589D-02
C#4 41    0.2752170464D+03    0.2771826310D+03   -0.19655845D+01   -0.70912977D-02
D4 42    0.2916201212D+03    0.2936647679D+03   -0.20446467D+01   -0.69625197D-02
D#4 43    0.3090008275D+03    0.3111269837D+03   -0.21261563D+01   -0.68337250D-02
E4 44    0.3274174326D+03    0.3296275569D+03   -0.22101243D+01   -0.67049136D-02
F4 45    0.3469316767D+03    0.3492282314D+03   -0.22965547D+01   -0.65760855D-02
F#4 46    0.3676089796D+03    0.3699944227D+03   -0.23854431D+01   -0.64472407D-02
G4 47    0.3895186602D+03    0.3919954360D+03   -0.24767758D+01   -0.63183792D-02
G#4 48    0.4127341687D+03    0.4153046976D+03   -0.25705288D+01   -0.61895010D-02
A4 49    0.4373333333D+03    0.4400000000D+03   -0.26666667D+01   -0.60606061D-02
A#4 50    0.4633986205D+03    0.4661637615D+03   -0.27651410D+01   -0.59316944D-02
B4 51    0.4910174120D+03    0.4938833013D+03   -0.28658892D+01   -0.58027660D-02

C5 52    0.5202822974D+03    0.5232511306D+03   -0.29688332D+01   -0.56738209D-02
C#5 53    0.5512913847D+03    0.5543652620D+03   -0.30738773D+01   -0.55448591D-02
D5 54    0.5841486292D+03    0.5873295358D+03   -0.31809066D+01   -0.54158806D-02
D#5 55    0.6189641821D+03    0.6222539674D+03   -0.32897853D+01   -0.52868853D-02
E5 56    0.6558547595D+03    0.6592551138D+03   -0.34003543D+01   -0.51578733D-02
F5 57    0.6949440339D+03    0.6984564629D+03   -0.35124290D+01   -0.50288445D-02
F#5 58    0.7363630488D+03    0.7399888454D+03   -0.36257967D+01   -0.48997991D-02
G5 59    0.7802506578D+03    0.7839908720D+03   -0.37402141D+01   -0.47707369D-02
G#5 60    0.8267539905D+03    0.8306093952D+03   -0.38554047D+01   -0.46416579D-02
A5 61    0.8760289452D+03    0.8800000000D+03   -0.39710548D+01   -0.45125622D-02
A#5 62    0.9282407122D+03    0.9323275230D+03   -0.40868109D+01   -0.43834498D-02
B5 63    0.9835643267D+03    0.9877666025D+03   -0.42022758D+01   -0.42543206D-02

C6 64    0.1042185257D+04    0.1046502261D+04   -0.43170046D+01   -0.41251747D-02
C#6 65    0.1104300023D+04    0.1108730524D+04   -0.44305005D+01   -0.39960120D-02
D6 66    0.1170116862D+04    0.1174659072D+04   -0.45422100D+01   -0.38668326D-02
D#6 67    0.1239856417D+04    0.1244507935D+04   -0.46515181D+01   -0.37376364D-02
E6 68    0.1313752484D+04    0.1318510228D+04   -0.47577432D+01   -0.36084234D-02
F6 69    0.1392052795D+04    0.1396912926D+04   -0.48601307D+01   -0.34791937D-02
F#6 70    0.1475019844D+04    0.1479977691D+04   -0.49578472D+01   -0.33499473D-02
G6 71    0.1562931770D+04    0.1567981744D+04   -0.50499738D+01   -0.32206841D-02
G#6 72    0.1656083292D+04    0.1661218790D+04   -0.51354985D+01   -0.30914041D-02
A6 73    0.1754786691D+04    0.1760000000D+04   -0.52133089D+01   -0.29621073D-02
A#6 74    0.1859372863D+04    0.1864655046D+04   -0.52821832D+01   -0.28327938D-02
B6 75    0.1970192423D+04    0.1975533205D+04   -0.53407819D+01   -0.27034635D-02

C7 76    0.2087616885D+04    0.2093004522D+04   -0.53876373D+01   -0.25741164D-02
C#7 77    0.2212039904D+04    0.2217461048D+04   -0.54211436D+01   -0.24447526D-02
D7 78    0.2343878598D+04    0.2349318143D+04   -0.54395454D+01   -0.23153720D-02
D#7 79    0.2483574944D+04    0.2489015870D+04   -0.54409254D+01   -0.21859746D-02
E7 80    0.2631597264D+04    0.2637020455D+04   -0.54231918D+01   -0.20565604D-02
F7 81    0.2788441788D+04    0.2793825851D+04   -0.53840639D+01   -0.19271294D-02
F#7 82    0.2954634324D+04    0.2959955382D+04   -0.53210574D+01   -0.17976816D-02
G7 83    0.3130732020D+04    0.3135963488D+04   -0.52314679D+01   -0.16682171D-02
G#7 84    0.3317325227D+04    0.3322437581D+04   -0.51123535D+01   -0.15387357D-02
A7 85    0.3515039484D+04    0.3520000000D+04   -0.49605164D+01   -0.14092376D-02
A#7 86    0.3724537609D+04    0.3729310092D+04   -0.47724827D+01   -0.12797227D-02
B7 87    0.3946521929D+04    0.3951066410D+04   -0.45444809D+01   -0.11501910D-02

C8 88    0.4181736626D+04    0.4186009045D+04   -0.42724185D+01   -0.10206424D-02
C#8 89    0.4430970238D+04    0.4434922096D+04   -0.39518576D+01   -0.89107711D-03
D8 90    0.4695058299D+04    0.4698636287D+04   -0.35779879D+01   -0.76149498D-03
D#8 91    0.4974886141D+04    0.4978031740D+04   -0.31455985D+01   -0.63189604D-03
E8 92    0.5271391864D+04    0.5274040911D+04   -0.26490468D+01   -0.50228029D-03
F8 93    0.5585569477D+04    0.5587651703D+04   -0.20822257D+01   -0.37264773D-03
F#8 94    0.5918472235D+04    0.5919910763D+04   -0.14385286D+01   -0.24299836D-03
G8 95    0.6271216165D+04    0.6271926976D+04   -0.71081111D+00   -0.11333217D-03
G#8 96    0.6644983811D+04    0.6644875161D+04    0.10864924D+00    0.16350832D-04
A8 97    0.7041028197D+04    0.7040000000D+04    0.10281966D+01    0.14605066D-03
A#8 98    0.7460677028D+04    0.7458620184D+04    0.20568436D+01    0.27576730D-03
B8 99    0.7905337141D+04    0.7902132820D+04    0.32043210D+01    0.40550077D-03```