Inbetween K. Wegscheider 416Hz (June 2003) and T. Dent 419Hz (September 2006)
- --- In email@example.com, Brad Lehman <bpl@...> wrote:
> All righty...this exercise is principally for Johnny, but also for...
> Andreas and anyone else who wants to listen along.
> For Andreas: in the altered version, where three notes have been
> off-spot from W-III, please tell us *exactly* where you hear theseveral
> 707-cent 5ths or 4ths. Where are these wide ones within the circle ofyour
> 5ths? (Or, if you can't pick them out at all reliably here, what's
> objection to the sound of 704-cent 5ths elsewhere?)Hi Brad & all others lovers of his wide"French"-5ths hypothesis,
Johnny also disagrees with Brad's hypothesis in:
"My instincts agree with Andreas's;
Bach's fifths were flatted in only one direction from just."
So thinks also too
M.Zapf's & K.Brigs in their's reinterpretation (May 2003):
But here my new proposal inbetween 2 others modern "Bach"-tunings:
1. Tom Dent's A4=419Hz reinterpretation of "squiggle"(Sept. 1999)
2. Kristian's A4=416Hz
Presented as double inequality for the 12 absolute-pitches inbetween:
Tom Dent's 419Hz "squiggle" >= new-proposal 418Hz >= Wegscheider 416Hz
compare that 3 ones in chromatically order:
c' 250 > 249.5 < 248.7 at 'middle-C4'
C# 264 = 264 < 262.279688...
D: 280 = 280 < 278
Eb 297 > 296 > 295.111111...
E: 314 > 312 > (311,25 = 311+1/4)
F: 333.5> 333 > 332
F# 352 > 351 > 349,70625
G: 374 = 374 > 372.05
G# 396 > 394.75> 393,419531...
A: 419 > 418 > 416 > neo-"Baroque" modern 'Cammer-Thone' ~415Hz
Bb 445 > 444 > (442.66666... = 442+2/3)
B: 470 > 468 > 466.275
c" 500 > 499 > 497.4 at 'tenor-C5'
Inbetween K.Wegscheider416Hz(June2003) and T.Dent419Hz(September2006)
1059/998 ! C# 529.5'tenor-C#5'/499
560/499 ! D
592/499 ! Eb
624/499 ! E (5:4)*(2496:2495) ~0.7 Cents wider than an 5/4 JI 3rd
666/499 ! F (4:3)*(999:998) ~1.7 Cents wider than an 4/3 JI 4th
702/499 ! F#
748/499 ! G (3:2)*(1496:1497)~-1.2 Cents narrower than an 3/2 JI 5th
1579/998 ! G# 789.5/499
836/499 ! A5 that's an octave above the absolute A4=418Hz reference
888/499 ! Bb
936/499 ! B
2/1 ! C6 = 998 'sopran-C6'
That proposal meets even Johnny demands
due to satisfying Sorge's condition:
All 5ths in that become barely tempered down in one direction
by the following 7 epimoric ratios lowered inbetween the 5ths,
that do amout totally an PC=3^12/2^19
499Hz=C5 1496:1497 G 560:561 D 209:210 A 208:209 E B F# C#...
C# 3158:3159 G# 4736:4737 Eb Bb F 998:999 C5=499Hz
or as expanded cycle of a dozen duodecimes 3:2 and
19-times octaves 2:1 down:
C5 = 499 'tenor-C5'
G2 = 187 374 748 1496 (<1497 := 3*C5)
D2 = 70 140 280 560 (<561 := 3*G2)
A3 = (13 26 52 104 208<) 209 (<210 := 3*D2)
E1 = 39 := 3*13
B2 = 117 := 9*13
F#4 = 351 :=27*13
C#6 = 1059 :=81*13
G#6 = 1579 3158 (<3159 := 243*13)
Eb1 = 37 74 148 296 592 1184 2368 4736 (<4737 := 3*G#6)
Bb2 = 111 := 3*Eb1
F4 = 333 := 3*Bb2
C5 = 499 998 (<999 := 3*F4)
if you posseses by change an todays modern
that meets even:
"Tuning Pitch in nowadays' practice is taken often at a=415 Hz, a
semitone below modern standard concert pitch of a=440 Hz."
or more precisely:
# "48 g♯′/a♭′ G♯4/A♭4 ~415.305..."
when calculated from an theoretically 12-EDO step downwards:
440 / 2^(1/12) = ~415.304698...
with even 2 different keyboards for 2 independent 8-foot stops
then try to tune in practice on the one hand:
versus on the other manual
After that unify that both versions in one instrument
by synchonizing them to the above procedure into
the new intermediate tuning, in order to get rid of
'objections' against Wegscheider's broade-"French"5th.
Who in that group here dares to tune that new
another 'Bach' on his/hers own 415Hz instument?
No warranty garanteed for what happens then!
- --- In firstname.lastname@example.org, "Andreas Sparschuh" <a_sparschuh@...>
>I don't know abot others, here, Andreas, but I am not even gonna try
> --- In email@example.com, Brad Lehman <bpl@> wrote:
to unravel what you might trying to say until you stop doing several
(1) Peppering you posts with endless wiki links for really stupid
things, like middle C or piano keyboard frequencies.
(2) Start using just ONE clear and easy to understand method for
indicating temperaments. All that scala mismash and wierd stuff like
multiplying frequencies by 3 instead of 1,5 just makes it all not
worth the time.
Try being simple and clear for just once. Maybe you've really got
something to say, who knows? At the moment it just looks like the
ravings of a madman.
--- In firstname.lastname@example.org, "Paul Poletti" <paul@...> wrote:
> Start using just ONE clear and easy to understand method for
> indicating temperaments.
Simply consider all given values there
as frequencies in Hz of absolute pitches,
that are subjects of 3 possible sequential operations:
Algorithm for synchroneous well-temperaments:
1. Step 19-times an octaves down, by halfing the pitch-frequency
2. Go 12 times to partial 3:1, by multipying with facor 3.
3. Decrement frequncy by -1Hz down, when intend tempering flattend.
but only if you insist in "wide-5ths" then allow also too:
(4. Increment by +1Hz upwards, for an sharper "French"-5th.)
Choose the chain of flow in the operation sequence
so that the circle of a dozen 5ths returns back to the initial
start after 12times 3:1 and 19times 1:2 while fitting the
decrements so, that they yield an distribution of the PC=3^12:2^19
> All that scala mismash and wierd stuff like
> multiplying frequencies by 3 instead of 1,5 just makes it all not
> worth the time.
That ratio of 3/2 = 1.5 arises operationally from taking the
quotient of the 3rd partial (3:1) over an octve (2:1),
when realting that both overtones #2 and #3 to theirs fundamental
"...allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, etc. times of the
but on strings there never appear 2/3 = 1:(3/2) due to the lack of
"Subharmonics do not normally occur in natural sounds, although the
subharmonic f/2 may be generated by the cone of a LOUDSPEAKER."
That makes an 5th (3:2 = 1.5) less fundamental than the ratio
inbetween the overtones 3:1 and 2:1 within the harmonic series.
Hence an 5th is composed by an
division of an 12th (3:1) as nominator
over an octave (2:1) as denominator by the calculation
(3:2) := (3:1):(2:1)
In other words:
any 5th (3:2) consists terms of
as composed of the difference of an '12th'-'8th'.
"3 just perfect fifth P8 + P5 1902.0 702.0"
when both do refer to the same (1:1) base or
hope that helps,
why i do prefer the multiplication by the "harmoic" factor 3
in order to stay wihin the partial-series.
Even Brad understood that in his:
"...in the line of fifths A-E-B-F#-C#-G#-D#-Bb-F-C-G-D-A to reduce the
next note by 1 Hz, i.e. introducing a beat rate of 1 per second
against the preceding fifth. The fifths F#-C#-G#-D# and D-A are kept
pure. The other eight are adjusted by different geometric amounts,
based on the superparticular ratios described in his algorithm.
(Arithmetically, it amounts to subtracting 1 Hz from the top of most
of the columns, in his chart, wherever there are values in
Brad contiues or the experts:
"Sparschuh's mathematical algorithm resembles the classic unproven
"Collatz Conjecture" from 1937, except that Sparschuh's iterated
function uses (3n-1) rather than (3n+1). [And see Eric Roosendaal's
3x+1 web site, along with this page by Frits Beukers demonstrating and
comparing the numerical sequences....]"