Yahoo Tuning Groups Ultimate Backup tuning Accommodating a rank two temperament to 1200edo

Chris remarked: "V-Vocal which is a Roland version of autotune (which requires more work) which came in Sonar 8.5 lets me adjust on a level of a cent."

This suggests it might be useful to look at what rank two temperaments we can accommodate to 1200edo. This is not always possible, and when it is the results might not be good. On the other hand, they can be excellent, as eg with meantone.

<1200.000 1901.955 2786.314 3368.826 4151.318|

🔗Graham Breed <gbreed@...>

🔗genewardsmith <genewardsmith@...>

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> The magic mapping I listed is a 400edo version;
> 400edo can also be used for unidec and that really works better.
> We also have 200edo, for garibaldi, valentine or guiron...

Hi Gene,
fully agreed, that both approximations do work so fine and well,
because 200 is the next convergent after 53
out of the well-known series:

http://www.research.att.com/~njas/sequences/A060528
"1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, ..."

hence 200edo has especially excellent approxs of 3-limit:
....
2^(7/12) = ~1.498307076... 700 Cents exactly
2^(17/29) = ~1.501294382... (703+13/29) = ~703.448275...
2^(31/53) = ~1.499940903... (701+47/53) = ~701.886792...
2^(24/41) = ~1.500419433... (702+18/41) = ~702.439024...
2^(117/200) = ~1.500038989... just 702 Cents exactly
2^(148/253) = ~1.500018441... (701+247/253) = ~701.97628...
2^(179/306) = ~1.500005010... (701+49/51) = ~701.96078...
...

but why not stepping direct the even much better

2^(389/665) = ~1.4999999015... (701+127/133) = ~701.954887...?

that is even more narrow closer to 1.5 ~701.954887218...

as suggested in:
http://www.eurotrib.com/story/2008/5/13/20499/5787
quote:
"
To approximate the major fifth 3:2 ratio, approximations of log[2](3/2) are needed. The partial quotients of the continuous fraction are

[0, 1, 1, 2, 2, 3, 1, 5, 2, 23, 2, 2, 1, 1, 55, 1, ... ]

the first few approximants are

0, 1, 1/2, 3/5, 7/12, 24/41, 31/53, 179/306, 389/665, 9126/15601, ...

so the 5, 12, 41, 53, 306, 665 tone scales as good for the 3:2, 4:3, 8:3... harmonics as you can cope for. Especially 12, 53 and 665 tone scales are good, because the next partial quotients 3, 5 or 23 are large. (The devil must be using the 666-scale.)
"

Not to mention the "trans-satanic" accuracy in percision

2^(9126/15601) = ~1.500000001749... (701+14899/15601) ~701.95500288...

that can be found in
http://www.informaworld.com/smpp/content~db=all~content=a772617196
too.

Attend that re-discovery of 200edo:
/tuning/topicId_39499.html#39504
"...because i just found out that 200edo (6 cents per step)..."
/tuning/topicId_78311.html#78341

But why using all that "EDO"-stuff approximation instead
of simply the true original: 3/2=1.5 JI?
What is all that ado about EDOs for,
in order to express the plain ratio 3/2 in such an
compilcated cirumstandial roundabout way,
without nay chance to fit it as exact match.

bye
Andy

🔗genewardsmith <genewardsmith@...>

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>... 385/384, 441/440, 325/324 and 364/363.
> These are all useful properties in some connections.

In deed Gene,
fully agrred that they can be applied very useful as already showed in
/tuning/topicId_91520.html#91552

especially when arrange them into an chain of a dozen 5ths
in the following sequential order

Eb 384/385 Bb F C 539/540 G 384/385 D 440/441 A...
...A E 539/540 B 384/385 F# C# G# 4375/4374 Eb

alternatively that's when expressed in [Monzo's> prime-vectors

Eb [7 1,-1 -1 -1> Bb F C [-2 -3, -1 2 1> G [7 1,-1 -1 -1> D
D [3 -2,1 -2 1> A E [-2 -3, -1 2 1> B [7 1,-1 -1 -1> F# C# G#
G# [-1 -7, 4 1> Eb

or approximate that ratios in yours preferred Cents-units

Eb ~-4.50cents Bb F C ~-3.21... G ~-4.50... D ~-3.93... A
A E ~-3.21... B ~-4.50... F# C# G# ~+0.396... Eb

[please: Use the FIXED-with font option]

The above arrengement yields for the 3rds-progression an intensionally mild emergencing 'key-characteristics'
That consists in only three different characteristic
graduations in seize of sharpnesses above 5/4.
That epimoric partition of the PC gains by that
an appreciable preference for the most frequently used tierces
F-A, C-E, G-B and D-F#.
That reasonable smaller bias in the common keys turn out to be significant lower in offness than the constant conformly equal,
but wrecked 12edo:

Eb 126/125 G [1 2, -3 1> ~13.79... rel. Cents off from 5/4
Bb 126/125 D [1 2, -3 1> ~13.79...
F 176/175 A [4 0, -2 -1 1> ~9.86...
C 176/175 E [4 0, -2 -1 1> ~9.86...
G 176/175 B [4 0, -2 -1 1> ~9.86...
D 176/175 F# [4 0, -2 -1 1> ~9.86...
A 126/125 C# [1 2, -3 1> ~13.79...
E 126/125 G# [1 2, -3 1> ~13.79...
B 100/99 Eb [2 -2, 2 0 -1> ~17.39...
F# 100/99 Bb [2 -2, 2 0 -1> ~17.39...
C# 100/99 F [2 -2, 2 0 -1> ~17.39...
G# 100/99 C [2 -2, 2 0 -1> ~17.39...

because I think that the remote, less common
3rds B-Eb, F#-Bb, C#-F and G#-C
may allow the tolerance to be charged
with somewhat more strong deviation from just pure
Here in that case, my choice of bearing concrete: ~17~c
should rather easily be accepted,
when considering against that most historically 'well'-tunings
drop out in that remote region even as Pythagoren ditones 81/64.
That Baroque practice offends us modern ears with at least with
the stress of an full SC out of tune when the modulations do lead
into the remote accidential keys, as already Mozart and Beethoven often did so.

Quest:
What do you think about my choice of dareing 100/99,
which amounts at least almost a tinly little bit more than 1%,
that it could be regarded as still passable compromise,
but should barely endurable, even for an die-hard 12-EDOist?

! SpDoubEpi11lim.scl
Sparschuh's [2010] double (5ths & 3rds) epimoric 11-lim. dodecatonics
12
132/125 ! C# [ 2 1, -3 0 1>
28/25 ! D [ 2 0, -2 1>
385/324 ! Eb [-2 -4, 1 1 1> = (D# = 297/250)*(4375/4374 Ragisma)
44/35 ! E [-2 0, -1 -1 1>
4/3 ! F [-2 1>
176/125 ! F# [ 4 0, -3 0 1>
539/360 ! G [-3 -2, -1 1 1>
198/125 ! G# [ 1 2, -3 0 1> or (Ab = 385/243)*(4374/4375 Ragisma)
176/105 ! A [ 4 -1, -1 -1 1>
16/9 ! Bb [ 4 -2>
847/450 ! B [-1 -2, -2 1 2>
2/1
!
![eof]

Attend the Ragismatic enharmonics @ Eb=D# and G#=Ab !

At last here the absolute frequencies in Hertzians [Hz = cps]
for implementing that particular well-temperament
on modern pianos and organs at todays standard pitch:
Gain the finaly list simply multiplying
all the 'scala'-ratios by the common factor 525/2 = [-1 1,2 1>

c' 262.5 middle_C4
#' 277.2
d' 294
#' 311+199/216
e' 330
f' 350
#' 369.6
g' 393+1/16
#' 415.8
a' 440 Hz
#' 466+2/3
b' 494+1/12
c" 525 tenor_C5

Conclusion
Scheibler's made an clever choice when overtakeing Werckmeister's
'septenarian' ratio of 441/440 and defineing by that his previsional
normal-pitch standard 440Hz, that got later more and more accepted:
http://en.wikipedia.org/wiki/Johann_Scheibler
http://de.wikipedia.org/wiki/Johann_Heinrich_Scheibler_%28Krefeld%29

By the way arises the quest:
In what "connections" do you employ all that actually exploited:
"...useful properties of 385/384, 441/440, 325/324 and 364/363..." ?
Knowing more the better about that further purposes,
how to apply them elswhere, would be very interesting for me!

bye
Andy