Yahoo Tuning Groups Ultimate Backup tuning-math Re: Piano tuning and "BODE'S LAW EXPLAINED" II

I, Bill Arnold, will put my remarks at the end of these inserts,
used here for scholarship and educational purposes, only,
from sources online, separated by double-lines, as follows:
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Mark Kesti wrote,
"This table assumes A = 440 and equal temperment. The ratio of adjacent notes
is the 12th root of 2, or about 1.05946.

NOTE FREQUENCIES (Hz)
+----------------------------------------------------------------------+
| OCTAVE |
+-----+----------------------------------------------------------------------+
|NOTE | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+-----+----------------------------------------------------------------------+
| A |13.7|27.5| 55.0|110.0|220.0|440.0| 880.0|1760.0|3520.0| 7040.0|14080.0|
|A#/Bb|14.6|29.1| 58.3|116.5|233.1|466.2| 932.4|1864.7|3729.4| 7458.9|14917.8|
| B |15.4|30.9| 61.7|123.5|247.0|493.9| 987.8|1975.7|3951.3| 7902.7|15805.3|
| C |16.4|32.7| 65.4|130.8|261.6|523.3|1046.6|2093.2|4186.5| 8372.9|16745.8|
|C#/Db|17.3|34.6| 69.3|138.6|277.2|554.4|1108.8|2217.7|4435.5| 8871.1|17742.1|
| D |18.4|36.7| 73.4|146.8|293.7|587.4|1174.8|2349.7|4699.5| 9398.9|18797.8|
|D#/Eb|19.4|38.9| 77.8|155.6|311.2|622.4|1244.8|2489.5|4979.1| 9958.1|19161.3|
| E |20.6|41.2| 82.4|164.9|329.7|659.4|1318.8|2637.7|5275.3|10550.6|21101.3|
| F |21.8|43.7| 87.3|174.7|349.3|698.7|1397.3|2794.6|5589.2|11178.4|22356.8|
|F#/Gb|23.1|46.2| 92.5|185.1|370.1|740.2|1480.4|2960.8|5921.8|11843.5|23687.1|
| G |24.4|49.0| 98.0|196.1|392.1|784.3|1568.2|3137.1|6274.1|12548.2|25096.4|
|G#/Ab|26.0|51.9|103.9|207.7|415.5|830.9|1661.9|3323.7|6647.4|13294.8|29589.7|
+-----+----------------------------------------------------------------------+

+----------------------------------------------------------------------+
| OCTAVE |
+-----+----------------------------------------------------------------------+
|NOTE | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+-----+----------------------------------------------------------------------+
| C |16.4|32.7| 65.4|130.8|261.6|523.3|1046.6|2093.2|4186.5| 8372.9|16745.8|
|+-----+----------------------------------------------------------------------+"

http://oyt.oulu.fi/notfreq.html
Michael Kesti Grass Valley Group, Inc. |
mrk@gvgspd.GVG.TEK.COM |
!tektronix!gvgpsa!gvgspd!mrk |
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http://www.phy.mtu.edu/~suits/notefreqs.html

Frequencies for equal-tempered scale
A4 = 440 Hz
Speed of sound = 345 m/s
("Middle C" is C4 )

Note Frequency (Hz) Wavelength (cm)

C0 16.35 2100.

C1 32.70 1050.

C2 65.41 527.

C3 130.81 264.

C4 261.63 132.

C5 523.25 65.9

C6 1046.50 33.0

C7 2093.00 16.5

C8 4186.01 8.2

http://www.phy.mtu.edu/~suits/notefreqs.html
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Frequencies and Ranges
Note Frequency (Hz) Comments Lowest note for: Highest note* for:
C-2
4.09
128'C: C""; CCCCC
Gregg Bailey's 64' PVC subcontrabass clarinet

C-1 8.18 64' C: C'''; CCCC; MIDI#0; lowest organ note Hill organ, Sydney Town Hall,
Sydney AU
C#-1/ Db-1 8.66 C#'''; DDDDb; MIDI#1
D-1 9.18 D'''; DDDD; MIDI#2
D#-1/ Eb-1 9.73 D#'''; EEEEb; MIDI#3
E-1 10.30 E'''; EEEE; MIDI#4
F-1 10.92 F'''; FFFF; MIDI#5
F#-1/ Gb-1 11.56 F#'''; GGGGb; MIDI#6
G-1 12.25 G'''; GGGG; MIDI#7
G#-1/ Ab-1 12.98 G#'''; AAAAb; MIDI#8
A-1 13.75 A'''; AAAA; MIDI#9
Bb-1 14.57 A#'''; BBBBb; MIDI#10 BBBb octocontrabass clarinet
B-1 15.44 B'''; BBBB; MIDI#11
C0 16.35 32' C; C"; CCC; MIDI#12; lowest note written for tuba ("Encounters II" by
William Kraft) large pipe organs, B�sendorfer Imperial Grand Piano
C#0/ Db0 17.32 C#"; DDDb; MIDI#13 Lowest bass guitar strings made
D0 18.35 D"; DDD; MIDI#14
D#0/ Eb0 19.45 D#"; EEEb; MIDI#15 EEEb octocontralto clarinet , slide reed subcontrabass

E0 20.60 E"; EEE; MIDI#16
F0 21.83 F"; FFF; MIDI#17 B�sendorfer Grand Pianos
F#0/ Gb0 23.12 F#"; GGGb; MIDI#18 7/8/9-string bass guitars (additional link)
G0 24.50 G"; GGG; MIDI#19 BBb tuba* , contrabass trombone
G#0/ Ab0 25.95 G#"; AAAb; MIDI#20 BBb contrabass sarrusophone
A0 27.50 A"; AAA; MIDI#21; lowest A on piano piano, extended contrabassoon , Wolfe contra
forte , string octocontrabass
A#0/ Bb0 29.14 A#"; BBBb; MIDI#22 contrabassoon , extended Bb contrabass clarinet , C
contrabass sarrusophone
B0 30.87 B"; BBB; MIDI#23 double-contrabass flute , 5 & 6 string bass
C1 32.70 C'; CC; MIDI#24; 16' C contrabass rackett , string bass with extension, Chapman
Stick� (standard tuning), contrabassophone
C#1/ Db1 34.65 C#'; DDb; MIDI#25 Bb contrabass clarinet (not extended) , Eb contrabass
saxophone , Eb contrabass sarrusophone , tubax
D1 36.71 D'; DD; MIDI#26 reed contrabass , Eb contrabass ophicleide
D#1/ Eb1 38.89 D#'; EEb; MIDI#27 extended Eb contralto clarinet slide reed
subcontrabass
E1 41.20 E'; EE; MIDI#28 string bass, bass guitar, bass harmonica , F contrabass
ophicleide
F1 43.65 F'; FF; MIDI#29 F sub-subcontrabass recorder , harpsichord
F#1/ Gb1 46.25 F#'; GGb; MIDI#30 Eb contralto clarinet (not extended)
G1 49.00 G'; GG; MIDI#31 Great Bass Sordune , Great Bass Shawm
G#1/ Ab1 51.91 G#'; AAb; MIDI#32 Bb bass saxophone, Bb bass sarrusophone
A1 55.00 A'; AA; MIDI#33 Bb bass Ophicleide
A#1/ Bb1 58.27 A#'; BBb; MIDI#34 bassoon, extended Bb bass clarinet
B1 61.74 B'; BB; MIDI#35 contrabass flute , contrabass oboe , C bass ophicleide
C2 65.41 C; MIDI#36; 8' C cello, mandocello, bass shawm (extended), baritone sax (with
low A), alto/tenor rackett , C subcontrabass recorder
C#2/ Db2 69.30 C#; MIDI#37 Bb bass clarinet (not extended), A bass clarinet, baritone
sax (with low Bb), baritone sarrusophone
D2 73.42 D; MIDI#38 F contrabass trumpet *
D#2/ Eb2 77.78 D#; MIDI#39
E2 82.41 E; MIDI#40 Guitar (standard tuning)
F2 87.31 F; MIDI#41 F basset horn, bass crumhorn, F contrabass recorder
F#2/ Gb2 92.50 F#; MIDI#42 Eb alto clarinet
G2 98.00 G; MIDI#43; lowest line of bass clef octave mandolin, euphonium * , bass trumpet
* , tenor shawm (extended range), bass octavin
Gb2/ Ab2 103.83 G#: MIDI#44 Bb tenor sax, Bb tenor sarrusophone, F alto clarinet
A2 110.00 A; MIDI#45; lowest space of bass clef Heckelphone
A#2/ Bb2 116.54 A#; MIDI#46
B2 123.47 B; MIDI#47 bass oboe, bass flute (with low B)
C3 130.81 c; MIDI#48; 4' C viola, mandola, bass flute , tenor crumhorn, tenor shawm
(without extension), great bass recorder , Eb alto sax with low A (rare)
C#3/ Db3 138.59 c#; MIDI#49 Eb alto saxophone, Eb alto sarrusophone
D3 146.83 d; MIDI#50 Bb soprano clarinet, Eb alto horn * , F mezzo Conn-O-Sax,
Heckel-Clarinet
D#3/ Eb3 155.56 d#; MIDI#51 F mezzo-soprano saxophone, English horn (with low Bb) Bb
contrabass sarrusophone*
E3 164.81 e; MIDI#52 English horn, C soprano clarinet bass harmonica
F3 174.61 f; MIDI#53 bass recorder, alto crumhorn, alto shawm
F#3/ Gb3 185.00 f#; MIDI#54 D soprano clarinet
G3 196.00 g; MIDI#55; top space of bass clef violin, alto flute, Eb soprano clarinet, Bb
trumpet, flugelhorn, cornet, mandolin C contrabass sarrusophone*
G#3/ Ab3 207.65 g#; MIDI#56 Bb soprano saxophone, Bb soprano sarrusophone, oboe d'amore,
octavin Eb contrabass saxophone * , tubax
A3 220.00 a; MIDI#57; top line of bass clef flute d'amore in A, oboe (with A extension),
Bb Heckel-clarina
A#3/ Bb3 233.88 a#; MIDI#58 oboe, Bb flute d'amore Eb contrabass sarrusophone *
B3 246.94 b; MIDI#59 C flute (with B foot)
C4 261.63 middle C, 2' C; c'; MIDI#60 C flute (with C foot), Ab sopranino clarinet, tenor
recorder, soprano crumhorn, soprano shawm
C#4/ Db4 277.18 c#'; MIDI#61 Eb sopranino saxophone, Eb sopranino sarrusophone
D4 293.66 d'; MIDI#62 Eb oboe musette, Eb piccolo Heckel-clarina
D#4/ Eb4 311.13 d#'; MIDI#63 Eb soprano flute bass saxophone *
E4 329.63 bottom line of treble clef; e'; MIDI#64 F piccolo heckelphone
F4 349.23 bottom space of treble clef, f', MIDI#65 F treble flute, alto (treble) recorder
bass sarrusophone *
F#4/ Gb4 369.99 f#', MIDI#66
G4 392.00 g', MIDI#67 G treble flute
G#4/ Ab4 415.30 g#', MIDI#68 Bb sopranissimo saxophone Baritone saxophone *
A4 440.00 a', MIDI#69, "tuning A"
A#4/ Bb4 466.16 a#'; MIDI#70 Ab piccolo flute baritone sarrusophone *
B4 493.88 b'; MIDI#71
C5 523.25 1' C; c"; MIDI#72 soprano (descant) recorder
C#5/ Db5 554.36 c#"; MIDI#73
D5 587.32 d"; MIDI#74 C piccolo flute
D#5/ Eb5 622.26 d#"; MIDI#75 Db piccolo flute Tenor Saxophone*
E5 659.26 e"; MIDI#76; top space of treble clef
F5 698.46 f"; MIDI#77; top line of treble clef sopranino recorder Tenor Sarrusophone*
F#5/ Gb5 739.99 f#"; MIDI#78
G5 783.99 g"; MIDI#79
G#5/ Ab5 830.61 g#"; MIDI#80 Alto Saxophone*
A5 880.00 a"; MIDI#81
http://www.contrabass.com/pages/frequency.html

A#5/ Bb5 932.33 a#"; MIDI#82 Alto Sarrusophone*
B5 987.77 b"; MIDI#83
C6 1046.50 c'''; MIDI#84 garklein recorder

* "Nominal range - the range commonly written. Skilled players frequently exceed this
range.

Copyright � 2000-2002 by Grant Green
Last Modified: 10/02/2002 19:39:06
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Here begins my, Bill Arnold's, remarks:

I published Arnold's Law in 1979, as follows:

Bodies_Proportion___Degreed Arcs___Fraction___Ideal Mean**
Or Perimeter

Sun__________0___________0________0____________0
Mercury______1___________3_____1/120______3.14 X10(7th)miles
Venus________2___________6______1/60______6.28
Earth________3___________9______1/40______9.42
Mars_________4__________12______1/30_____12.56
Ceres*_______8__________24______1/15_____25.13
Jupiter_____15__________45______1/8______47.12
Saturn______30__________90______1/4______94.24
Uranus______60_________180______1/2_____188.49
Neptune_____90_________270______3/4_____282.74
Pluto______120_________360______4/4_____376.99

*Ceres: prime representative of so-called "asteroids"

**means: adjusted for diameters of both bodies, sun and planet
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We can supplement my solar-system planetary data with the C scale note data
from the sources above, accordingly, adding:

Frequencies and Ranges
Note Frequency (Hz) Comments Lowest note for: Highest note* for:
C-2
4.09
128'C: C""; CCCCC
Gregg Bailey's 64' PVC subcontrabass clarinet

C-1 8.18 64' C: C'''; CCCC; MIDI#0; lowest organ note Hill organ, Sydney Town Hall,
Sydney AU

*Source:
Copyright � 2000-2002 by Grant Green
Last Modified: 10/02/2002 19:39:06
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*Source:
http://oyt.oulu.fi/notfreq.html
Michael Kesti Grass Valley Group, Inc. |
mrk@gvgspd.GVG.TEK.COM |
!tektronix!gvgpsa!gvgspd!mrk |
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Hertz [Hz], according to The American Heritage Dictionary of the English
Language: "symbol Hz, a unit of frequency equal to one cycle per second"

and

Hertzian Wave, same source, "former name for a radio wave"

and

Hertz, same source, "worked on 'electromagnetic phenomena'"
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I, Bill Arnold would add, here, that in my paper published in Cycles Bulletin,
Vol. XXX, No. 4, 1979, I pointed out that any "cycle" is a IDEAL linear phenomenon
created by a mathematician based upon a REAL observed phenomenon of an oscillating
nature which inherently is "circular," hence the mathematical expression thereof
contains the value of pi [3.14etc]. More anon.
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I published Arnold's Law in 1979, as follows:
however, note I have added the Harmonic C Note [bodies]

C Notes_____Bodies_Proportion___Degreed Arcs___Fraction___Ideal Mean**
Octaves Or Perimeter
Or Harmonics

0***________Sun__________0___________0________0____________0
1___________Mercury______1___________3_____1/120______3.14 X10(7th)miles
2___________Venus________2___________6______1/60______6.28
?___________Earth________3___________9______1/40______9.42
4.09________Mars_________4__________12______1/30_____12.56
8.18________Ceres*_______8__________24______1/15_____25.13
16.4________Jupiter_____15__________45______1/8______47.12
32.7________Saturn______30__________90______1/4______94.24
65.4________Uranus______60_________180______1/2_____188.49
?___________Neptune_____90_________270______3/4_____282.74
130.8_______Pluto______120_________360______4/4_____376.99

*Ceres: prime representative of so-called "asteroids"

**means: adjusted for diameters of both bodies, sun and planet

***the NO SOUND point, from which the C Scale originates in the 1 C Note,
expressed at a perimeter of pi [3.14] when it so oscillates and sounds,
audibly to perception, however perceived: [to me, a scale is really a
series of concentric spheres of sound, with each higher note created
by another sphere surrounding the inner spheres in the same way the
Music of the Spheres appears to us, visually and mathematically, with
the planetary orbitals as means of their distances, expressed as spheres.]

Thus, I, Bill Arnold, seek assistance from musicians who can explain
why the solar-planetary system music is basically octaval, and in the
C Note Scale, except the system IS bodies in curved space, in a perceived
vacuum, and WHAT are Earth at circa 3 and Neptune circa 90, musically expressed
as notes? They are at "fretted" midpoints between their respective-adjacent
bodies, in a planetary sense, and I wonder what they would be, musically
expressed in the C Scale? And, any guess, WHY they are there, musically?
They make perfect physical [in the sense of physics, as expressed mathematically]
sense to me, as I see them: physically and mathematically, and I expressed
in ON THE SPECIAL THEORY OF ORDER.
Bill Arnold

Bill Arnold
billarnoldfla@yahoo.com
http://www.cwru.edu/affil/edis/scholars/arnold.htm
Independent Scholar
Independent Scholar, Modern Language Association
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Re: Piano tuning and "BODE'S LAW EXPLAINED" II

🔗Bill Arnold <billarnoldfla@yahoo.com>