26 February 2019; Feast of Sts Porphyrius and Alexander

 

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 = C 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 A = 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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