Yahoo Tuning Groups Ultimate Backup tuning A new scale

To the tuning list,

The 12 tones of a new scale produce eleven perfect fifths and work in the geometric progression of musical cells with exactness. The frequency tones are found within its 624 consonant frequencies comprised between C = 1 and (9/8)^6.

The tone frequencies and their position in the progression follows:

C = 1 (Cell # 0), major third: 1.2542

C# = 1.05349794236, (Cell # 46), 90.225 cents, major third: 1.265625

D = 1.11488414335, (Cell # 96), 188.272, major third: 1.2599

Eb = 1.185185185, (Cell # 150), 294.135, major third: 1.265625

E = 1.25424466127, (Cell # 200), 392.182, major third: 1.2599

F = 1.33333333333, (Cell # 254), 498.045, major third: 1.2542

F# = 1.40466392312, (Cell # 300), 588.27, major third: 1.265625

G = 1.5, (Cell # 358), 701.955, major third: 1.2542

Ab = 1.58024691358, (Cell # 404), 792.18, major third: 1.265625

A = 1.67232621503, (Cell # 454), 890.227, major third: 1.2599

Bb = 1.77777777777, (Cell # 508), 996, major third: 1.2542

B = 1.88136699191, (Cell # 558), 1094.137, major third: 1.2599

2C =2

Since I do not have experience on evaluations, I would appreciate it if you could inform me about the negative and positive properties of this scale. I´ll try to detect other similar scales in the progression.

Thanks

Mario Pizarro

Lima, August 09

piagui@...

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🔗Michael <djtrancendance@...>

Mario>"The tone frequencies and their position in the progression follows:
C = 1 (Cell # 0), major third: 1.2542
C# = 1.05349794236, (Cell # 46), 90.225 cents, major third: 1.265625
D = 1.11488414335, (Cell # 96), 188.272, major third: 1.2599
Eb = 1.185185185, (Cell # 150), 294.135, major third: 1.265625
E = 1.25424466127, (Cell # 200), 392.182, major third: 1.2599

F = 1.33333333333, (Cell # 254), 498.045, major third: 1.2542
F# = 1.40466392312, (Cell # 300), 588.27, major third: 1.265625
G = 1.5, (Cell # 358), 701.955, major third: 1.2542
Ab = 1.58024691358, (Cell # 404), 792.18, major third: 1.265625
A = 1.67232621503, (Cell # 454), 890.227, major third: 1.2599
Bb = 1.77777777777, (Cell # 508), 996, major third: 1.2542
B = 1.88136699191, (Cell # 558), 1094.137, major third: 1.2599
2C=2"

Pretty good scale. A few things, though:

>"Eb = 1.185185185, (Cell # 150), 294.135, major third: 1.265625"
A) More than 13 cents from 7/6...could use a lot of improvement.
>"Ab = 1.58024691358"
B) More than 8 cents from 11/7...could use some improvement.
Also......
C) The 12th fifth is about 55/37 or 1.486...that's over 15 cents flat from a
perfect fifth, which is so sour it is virtually un-usable in my opinion. The
5th, of course, is much more easily hurt by mis-tuning than most other notes.

To fix the fifth issue without making anything else worse you might just want
to make a scale with the 11 perfect 5th's of about 1.497304851267718.

That would form the scale:
1
1.0545122744419693 (near 1.05555)
1.1209609088149217 (near 1.125)
1.18206703751496----(fairly out of tune from 1.166666666)
1.2565533590911753 (near 1.25)
1.3250509406529316 (near 1.3333)
1.4085471953812865 (near 1.4)
1.497304851267718 (near 1.5)
1.578926344243316 (near 1.5714)
1.6784202068500524---(fairly out of tune from 1.66666666)
1.7699147097948094 (near 1.7777777)
1.8814434404439637 (near 1.875)
2

Note that the 12th 5th would be 1.50937...much more usable than the 1.4864
5th is your original scale.
********************************************

Now, the next step would be to tweak the 1.678 to make it closer to 1.6666 and
1.16666 (within 1.005 or smaller difference, preferably) and then make sure the
dyads it forms with all other notes to make sure they are all good.
The value you gave of 1.67232 from your scale seems pretty good here (right
between 1.67842 and 1.6666)...so we can use that and get a scale of.

1
1.0545122744419693
1.1209609088149217
1.18206703751496----(still fairly out of tune)
1.2565533590911753
1.3250509406529316
1.4085471953812865
1.497304851267718
1.578926344243316
1.67232 (now within about 8 cents of 1.66666)
1.7699147097948094
1.8814434404439637
2

Now if you really want to get crafty you can try and solve the last
out-of-tune dyad and make the 1.182 a value somewhere around 1.171. But since
that is a fairly large change, although this solves the root dyad, it will
likely make other dyads sour.

Best of luck!

🔗Mario Pizarro <piagui@...>

Mario>"The tone frequencies and their position in the progression follows:

C = 1 (Cell # 0), major third: 1.2542

C# = 1.05349794236, (Cell # 46), 90.225 cents, major third: 1.265625

D = 1.11488414335, (Cell # 96), 188.272, major third: 1.2599

Eb = 1.185185185, (Cell # 150), 294.135, major third: 1.265625

E = 1.25424466127, (Cell # 200), 392.182, major third: 1.2599

F = 1.33333333333, (Cell # 254), 498.045, major third: 1.2542

F# = 1.40466392312, (Cell # 300), 588.27, major third: 1.265625

G = 1.5, (Cell # 358), 701.955, major third: 1.2542

Ab = 1.58024691358, (Cell # 404), 792.18, major third: 1.265625

A = 1.67232621503, (Cell # 454), 890.227, major third: 1.2599

Bb = 1.77777777777, (Cell # 508), 996, major third: 1.2542

B = 1.88136699191, (Cell # 558), 1094.137, major third: 1.2599

2C =2"

Pretty good scale. A few things, though:

>"Eb = 1.185185185, (Cell # 150), 294.135, major third: 1.265625"
A) More than 13 cents from 7/6...could use a lot of improvement.
>"Ab = 1.58024691358"
B) More than 8 cents from 11/7...could use some improvement.
Also......
C) The 12th fifth is about 55/37 or 1.486...that's over 15 cents flat from a perfect fifth, which is so sour it is virtually un-usable in my opinion. The 5th, of course, is much more easily hurt by mis-tuning than most other notes.

To fix the fifth issue without making anything else worse you might just want to make a scale with the 11 perfect 5th's of about 1.497304851267718.

Note that the 12th 5th would be 1.50937...much more usable than the 1.4864 5th is your original scale.
********************************************

Now, the next step would be to tweak the 1.678 to make it closer to 1.6666 and 1.16666 (within 1.005 or smaller difference, preferably) and then make sure the dyads it forms with all other notes to make sure they are all good.
The value you gave of 1.67232 from your scale seems pretty good here (right between 1.67842 and 1.6666)...so we can use that and get a scale of.

Now if you really want to get crafty you can try and solve the last out-of-tune dyad and make the 1.182 a value somewhere around 1.171. But since that is a fairly large change, although this solves the root dyad, it will likely make other dyads sour.

Best of luck!

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🔗Michael <djtrancendance@...>

>"Thanks for your comments, now I will be busy for the rest of the year studying
>what on the earth you wrote about the 10 perfect fifth scale (it is not 11)."
Thank you. But wait, now you have me confused. :-D
If it were only 10 perfect 5ths...wouldn't there be only 1 imperfect fifth?
I only counted one imperfect fifth between 1.5 and 1.114 on the next octave,
which would be 1.5 and 1.125 if it were pure.

________________________________
From: Mario Pizarro <piagui@...>
To: tuning@yahoogroups.com
Sent: Tue, August 10, 2010 3:42:18 PM
Subject: Re: [tuning] A new scale

Michael,
Thanks for your comments, now I will be busy for the rest of the year studying
what on the earth you wrote about the 10 perfect fifth scale (it is not 11).
The progression of cells is so fantastic that I promise you next time I will
send you a 14 perfect fifths scale.
Regards
Mario
Lima, August 10
----- Original Message -----
>From: Michael
>To: tuning@yahoogroups. com
>Sent: Monday, August 09, 2010 10:43 PM
>Subject: Re: [tuning] A new scale
>
>
>Mario>"The tone frequencies and their position in the progression follows:
>C = 1 (Cell # 0), major third: 1.2542
>C# = 1.05349794236, (Cell # 46), 90.225 cents, major third: 1.265625
>D = 1.11488414335, (Cell # 96), 188.272, major third: 1.2599
>Eb = 1.185185185, (Cell # 150), 294.135, major third: 1.265625
>E = 1.25424466127, (Cell # 200), 392.182, major third:
>1.2599
>
>F = 1.33333333333, (Cell # 254), 498.045, major third: 1.2542
>F# = 1.40466392312, (Cell # 300), 588.27, major third: 1.265625
>G = 1.5, (Cell # 358), 701.955, major third: 1.2542
>Ab = 1.58024691358, (Cell # 404), 792.18, major third: 1.265625
>A = 1.67232621503, (Cell # 454), 890.227, major third: 1.2599
>Bb = 1.77777777777, (Cell # 508), 996, major third: 1.2542
>B = 1.88136699191, (Cell # 558), 1094.137, major third: 1.2599
>2C=2"
>
>
>
>
> Pretty good scale. A few things, though:
>
>>"Eb = 1.185185185, (Cell # 150), 294.135, major third: 1.265625"
>A) More than 13 cents from 7/6...could use a lot of improvement.
>>"Ab = 1.58024691358"
>B) More than 8 cents from 11/7...could use some improvement.
>Also......
>C) The 12th fifth is about 55/37 or 1.486...that' s over 15 cents flat from a
>perfect fifth, which is so sour it is virtually un-usable in my opinion. The
>5th, of course, is much more easily hurt by mis-tuning than most other notes.
>
>
>
> To fix the fifth issue without making anything else worse you might just
>want to make a scale with the 11 perfect 5th's of about 1.497304851267718.
>
> That would form the scale:
>1
>1.0545122744419693 (near 1.05555)
>1.1209609088149217 (near 1.125)
>1.18206703751496- ---(fairly out of tune from 1.166666666)
>1.2565533590911753 (near 1.25)
>1.3250509406529316 (near 1.3333)
>1.4085471953812865 (near 1.4)
>1.497304851267718 (near 1.5)
>1.578926344243316 (near 1.5714)
>1.6784202068500524- --(fairly out of tune from 1.66666666)
>1.7699147097948094 (near 1.7777777)
>1.8814434404439637 (near 1.875)
>2
>
> Note that the 12th 5th would be 1.50937...much more usable than the 1.4864
>5th is your original scale.
>************ ********* ********* ********* *****
>
> Now, the next step would be to tweak the 1.678 to make it closer to 1.6666
>and 1.16666 (within 1.005 or smaller difference, preferably) and then make
>sure the dyads it forms with all other notes to make sure they are all good.
> The value you gave of 1.67232 from your scale seems pretty good here (right
>between 1.67842 and 1.6666)...so we can use that and get a scale of.
>
>1
>1.0545122744419693
>1.1209609088149217
>1.18206703751496- ---(still fairly out of tune)
>1.2565533590911753
>1.3250509406529316
>1.4085471953812865
>1.497304851267718
>1.578926344243316
>1.67232 (now within about 8 cents of 1.66666)
>1.7699147097948094
>1.8814434404439637
>2
>
> Now if you really want to get crafty you can try and solve the last
>out-of-tune dyad and make the 1.182 a value somewhere around 1.171. But
>since that is a fairly large change, although this solves the root dyad, it
>will likely make other dyads sour.
>
>Best of luck!
>
>
>
>
>__________ Información de ESET NOD32 Antivirus, versión de la base de firmas
>de virus 5353 (20100809) __________
>
>ESET NOD32 Antivirus ha comprobado este mensaje.
>
>http://www.eset. com
>

🔗martinsj013 <martinsj@...>

🔗Michael <djtrancendance@...>

Michael/Me> To fix the fifth issue without making anything else worse you might
just want to make a scale with the 11 perfect 5th's of about 1.497304851267718.

Martin>"Michael, I meant to ask, how did you choose that value for the fifth?"

I have a program that creates scales based on circles of near-5ths. I
started plugging in values IE each note in the series (near 5th)^11 and seeing
the remainder of 2 over the total of that series. Note the program lists all
the dyads from the root for each case...creating data much like what Mario
presented for his tuning.

When I looked at dyads for several values ranging from about 1.4932 to 1.50675
(about 7.5 cents flat and sharp of a fifth)...I noticed 1.4973, unlike the other
values

A) Only formed a couple of sour (IE over 8 cents from pure) dyads (IE the dyads
around 1.18 and 1.678...the minor third and major sixth)

B) Left a remainder/"comma" of more like 10 cents to the octave.
That 12th "commatic fifth" is near 1.509, which sounds much better to me than,
say, 10 cents below the perfect fifth: in fact I've even tried 1.515 IE 50/33
(much sharper) and it still sounds pretty relaxed to me.

So that's only two sour dyads from the root and one only slightly sour fifth
in an almost completely equal temperament (and the slightly sour fifth only
appears once IE regardless of transposition).
Thus you could expect very few (maybe 10 or so) sour dyads in the entire
tuning given any note as a root.

Compare that to 12TET's rather lousy (IMVHO) minor third, major third,
tritone, minor sixth, and major sixth (4 notes "off" per root instead of 3) (all
over 10 cents or so off lowest-limit nearby values like 7/5,11/7, 8/5,5/3, and
10/7).

the fact those keep appearing for each transposition...and I think it should be
obvious why a circle of 11 1.4973 5ths and one 1.509 fifth makes a highly
competitive scale vs. 12TET.

🔗Michael <djtrancendance@...>

🔗genewardsmith <genewardsmith@...>

🔗Michael <djtrancendance@...>

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:

Hola Mario,

try to find the hidden ratios within yours decimal data.
Use for that purpose some helpful mathematical tools from the web,
alike for instance the aids:

http://superspace.epfl.ch/approximator/
http://www.wolframalpha.com/

in order to gain more understanding about the properties
of yours proposal:

> C = 1
1/1
^^^

> C# = ~1.05349794236...
256/243 = 1.053497942386831275720164609^_(period 27)
^^^^^^^

> D = ~1.11488414335...
29/26 = 1.1153846^_(period 6)
^^^^^
no to mention the next more complicated convergent:
262/235=1.11489361702127659574468085106382978723404255319^_(period 46)
^^^^^^^

> Eb = ~1.185185185...
32/27 = 1.185^_(period 3)
^^^^^

> E = ~1.25424466127...
74/59 = 1.254237288135593220338983050847457627118644067796610169^_\
^^^^^ 4915^_(period 58)

> F = ~1.33333333333...
4/3 = 1.3^_(period 1)

> F# = ~1.40466392312...
1024/729 =1.404663923182441700960219478737997256515775034293552812^_\
^^^^^^^^^ 071330589849108367626886145^_(period 81)

> G = 1.5
3/2
^^^

> Ab = ~1.58024691358...
128/81 = 1.580246913^_(period 9)
^^^^^^

> A = ~1.67232621503...
296/177 = 1.672316384180790960451977401129943502824858757062146892^_\
^^^^^^^ 6553^_(period 58)

> Bb = ~1.77777777777...
16/9 = 1.7^_(period 1)
^^^^

> B = ~1.88136699191...
111/59 = 1.881355932203389830508474576271186440677966101694915254^_\
^^^^^^ 2372^_(period 58)

> 2c' = 2
2/1
^^^

Analysis ready done.
>
> Since I do not have experience on evaluations,....

Express the above evaluated ratios in the
http://www.huygens-fokker.org/scala/scl_format.html

so far as I do understand yours original data at the moment
after that suitable reconditioning:

Attention:
The following file is barely intended as preliminary draft,
because maybe Mario wants some revison?

!Pizarro_ratio.scl
Mario Pizarro's [2010] rationalized and compiled by A.Sparschuh
!
12
!
256/243 ! C# the 'limma'
29/26 ! D = (9/8)(116/117 {~-14.9c}) = (10/9)(261/260 {~+6.7c})
74/59 ! E = (5/4)(296/295 {~+5.9c}) = (81/80)(116/117 {~-14.9c}
4/3 ! F
1024/729 ! F#
3/2 ! G
128/81 ! G#=Ab
296/177 ! A = (5/3)(296/295 {~+5.9c}) = (27/16)(116/117 {~-14.9c})
16/9 ! Bb
111/59 ! B = (15/8)(296/295 {~+5.9c}) = (243/128)(116/117 {~-14.9c})
2/1
!
![eof]

? Do the epimoric ratios 262/261 indicate for the middel_C4 ~261 Hz
an absolute pich D4 = ~295 Hz and the range for A4= ~441-444 Hz ?

For recapitulation here comes again once more concise the chain of 5ths in ratios and {Cent units} in relative view as my final summary synopsis

Ab-Eb-Bb-F-C-G 116/117 ~-14.9c D 15392/15399 {~-0.8c} A-D...
...D-A-E 241664/242757 {~-7.8c} B-F#-C#-G#

For control check if really all higher primes do cancel out each others:
(116/117)(15392/15399)(241664/242757) = 2^19/3^12
test succeded!

or was it really arbitrarily intended to invole another 5th D~A
chargeing with such an tiny amount of less than ONE cent?

Some further comments:
1.) That bisection of the PC sounds somehow alike "Baron von Wiese"
http://groenewald-berlin.de/tabellen/TAB-006.html
(sorry that resource is sadly only in German-language available)

2.) Maybe the small residue of less than 1Cent among D~A
might arise from roundings while the rationalization process,
probably due to the choice of all in all to low raw convergents

> I would appreciate it if you could inform me about
> the negative and positive properties of this scale.

I.m.h.o. yours nice tuning is still passable,
even the harsh 15 Cents inbetween the 5th G-D
can be tolerated, as central distinctive desired feature.

> Iï¿½ll try to detect other similar scales in the progression.

Here the frequently occuring "59" remembers me about the
second possible bisection of the schisma in:
/tuning-math/message/17405

Combine that '59' with the well documented historically absolute-pitch back in the times of Vienna-Classics:

http://drjazz.ca/musicians/pitchhistory.html
excerpt quote from Alexander Ellis:
"
Year:
1780 A= 421.3 Hz, Vienna. Tuning fork of the Saxon organ builder Schulz who lived in Vienna during Mozart's lifetime.
1780 A=421.6 Hz Vienna. Tuning fork used by the piano builder Stein. The fork was inherited by his son-in-law Streicher who Ellis calls "the present great pianoforte maker." A= 421.6 Hz is probably the pitch which Mozart used to tune his fortepianos and clavichords.
"

hence find and invent an corresponding chain of 5ths in order to meet
just somewhere the middle inbetween that both ancient tuning-forks:

Bb 7 14 28 56 112 224 448Hz
F 21 := Bb*3
C 63 := F*3 126 252Hz := middle_C4 frequency
G (D/3=47 94 188 < ) 189 := C*3
D (E/9=35 70 140 < A/3=140.5 <) 141 := 47*3
A (E/3=105 210 420 <) 421.5 Hz coeval Vienna-classics absolute-pitch
E 315
B (F#/3 = 59 118 236 472 944 <) 945 := E*3
F# 177 := 59*3
C# 537
G# 1593
Eb 4779
Bb 7 14 28 56 112 224 448 896 1792 3584 7168 14336 (<14337:=3^8*5*59)

Consider barely the relative representation, 5-fold partition of the PC in terms of ratios:

Ab-Bb 14336/14337 FCG 188/189 D 281/282 A 280/281 E 944/945 B-F#-C#-G#

or the same epimorics in logarithmic 'cent'-units approximation:

Ab-Bb ~-0.12c F-C-G ~-9.18c D -6.15c A -6.17c E ~-1.83c B-F#-C#-G#

then arrange that dozen pitches in chromatically ascending order,
here given as absolute frequencies and finally in normalizied ratios:

!SpOldVienna.scl
Sparschuh's 'old-Vienna' classics for abs.pitch A4=421.5Hz [2008]
!
! c' 252 middle_C4
! #' 268.5
! d' 282
! #' 298.6875
! e' 315
! f' 336
! #' 354
! g' 378
! #' 398.25
! a' 421.5 Hz ~absolute pitch of coeval Vienna [~1800] tunig-forks
! #' 448
! b' 472.5
! c" 504 tenor_C5
!
12
!
59/56 ! C# = (256/243)(14337/14336 {~+0.12c}) = (135/128)(944/945)
47/42 ! D = (9/8)(188/189 {~-9.18c}) = (10/9)(141/140 {~+12.23c})
531/448 ! Eb = (32/27)(14337/14336 {~+0.12c}) = (1215/1024)(944/945)
5/4 ! E
4/3 ! F
59/42 ! F# = (1024/729)(14337/14336 {~+0.12c}) = (45/32)(944/945)
3/2 ! G
177/112 ! G# = (128/125)(14337/14336 {~+0.12c}) = (405/256)(944/945)
281/168 ! A = (5/3)(281/280 {~+6.17c})= (27/16)(562/567 {-15.33c})
16/9 ! Bb = (3645/2048)/schisma
15/8 ! B
2/1
!
! [eof]

Remarks:
Attend here the epimoric BIsection of the schisma 3^8*5/2^15 within
the both 5ths: Bb~F and E~B using

59,7-limit, in more precisely accuracy:

(945/944)*(14337/14336) := (7*5*3^3/59/2^4)*(59*3^5/7/2^11)
~1.8329637...Cents + ~0.120757092...Cents

and the corresponding TRIsection of the SC=81/80 within the four
empty-strings of the violin G~D~A~E

81/80 = (189/188)(282/281)(281/280) = (189/188)(141/140)

Due to the many JI intervals it works also fine at other pitches.
But attend that that for A4=440Hz the speed of beatings among
the tempered invervals get accelerated by the factor 880/843
That arisen frequency ratio corresponds for all originally
@ 60 Bpm intended Metronome speed an discernible quicken of
the pulse-rate up to ~63 beats/min against modern 440Hz reference.

Conclusion:
But now I prefer the probably older,
but more comfortable leisurely pace of a'=421.5
It did so only after a few problems to get used to this delicate deep sounding tone.
It took my almost over an year to accept it as an
second alternative standard, compareable to bilingualism.
Gotcha!

bye
Andy

🔗genewardsmith <genewardsmith@...>

🔗Ozan Yarman <ozanyarman@...>

🔗martinsj013 <martinsj@...>

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
> Mario's D-A seemed to me to be 701.955 cents ...?
Hi Martin,
surely you are right,
fully agreed, because if

D = ~1.11488414335...
A = ~1.67232621503....

then
A/D = ~1.500000000004484770...
that confirms yours obsevation
within the numerically limit in precision
of 10 valid decimals in accuracy.

> Would that change your proposal?
not at all, because i tryed to yield epimoric bisections of the SC
for the both affected 5ths G~D and D~A, which rsulted in
the undesired effect of an other fom JI deviating 5th.

> I think Mario's proposal is exactly a rotation/transposition of
> http://groenewald-berlin.de/tabellen/TAB-004.html
Again agreed with that clever observation!
More info's here:
http://groenewald-berlin.de/text/text_T118.html
http://groenewald-berlin.de/tabellen/TAB-118.html

Also entry about the inventor of that ancient scheme:
http://en.wikipedia.org/wiki/Henricus_Grammateus

Grammateus (Schreiber's) scheme is already available as "Scala"-resource under:
http://www.h-pi.com/additionals/ScalaSortedByTones.txt
or
http://www.huygens-fokker.org/docs/scalesdir.txt
there in both links listed as:
"12 grammateus.scl
H. Grammateus (Heinrich Schreiber) (1518).
B-F# and Bb-F 1/2 P. Also Marpurg temp.nr.6"
"
hence there is no need for generating an extra new file.

Mark Lindley commented just about that way of tuning in:
http://www.law-guy.com/dummygod/Entries/S27643.htm
"
... In the wake of the Renaissance revival of Euclidian geometry, Henricus Grammateus published in his arithmetic book of 1518 a simple diagram (fig.1)
http://www.law-guy.com/dummygod/Images/006567.gif
showing how the geometrical mean between two pipe lengths in the ratio 9:8 could be determined by drawing a semicircle on the sum of the lengths 9 and 8 taken as diameter, and then measuring the perpendicular from the juncture of the two lengths. By this means the Pythagorean whole tone would be, in theory, divided into two musically equal semitones. Grammateus referred to these as `minor semitones', and other theorists later used that term to designate equal semitones on the lute (see §9 below); but the context of the prescription leaves no doubt that he intended his `amusing reckoning' (`kurtzweyllig rechnung') to be applied to organ pipes. Such a temperament would make both BF and BF sour, however, and even Barbour (1951), although ideologically attracted to it, described it as an organ tuning which `may have been used in practice but hardly by anyone who was accustomed, like Schlick, to tune by ear'. Nor does any extant Renaissance organ music avoid BF as well as BF....
"

Probably Mark would judge similar about Mario's rotated variant.

Margo remarks about that:
http://www.medieval.org/emfaq/harmony/pyth5.html
"
...While forerunners of well-temperament might be traced back as far as an English organ tuning scheme of 1373 and the proposal of Henricus Grammateus (Heinrich Schreiber) in 1518, these systems seem more akin to equal temperament, and so will be discussed under that heading; also, such systems partially tame rather than fully domesticate the Wolf, leaving at least one fifth more than 10 cents from just..."

That practical instruction is more tolerant:
http://www.hpschd.nu/index.html?nav/nav-4.html&t/welcome.html&http://www.hpschd.nu/tech/tmp/grammateus.html
"..The thirds are usable, especially when you get into the sharp keys, and the two baby wolves not impossible. Many players use this temperament for the Fitzwilliam Virginal Book."

In order to get rid of the "baby-wolf" problem
here come some needed changes of my 'new-scale'.
Please attend the fine subtle modifications,
in order to gain more smoothness and refined elegance:

Ab 7/9
Eb 7/3
Bb 7
F 21
C 63
G (D/3 = 282/3 <) 283/3 566/3 (<567/3 = 189:=C*3)
D (A/3 = 281 <) 282
A (E/3 = 841.5 <) 843
E (B/3 = 315 630 1260 2520 <) 2524.5
B (F#/3 = 59 118 236 472 944 <) 945 := 315*3
F# 177 := 59*3
C# 537
G# 7/9 14/9 ... 398+2/9=398.222222... (< 398.25 796.5 1593 := 537*3)

1.) The 5ths got reduced by the following amounts in exact ratios:
Ab - Eb - Bb - F - C 566/567 G 282/283 D ...
... D 281/282 A 561/562 E 560/561 B 944/945 F#-C# 14336/14337 G#

2. When rounded up to fractions of the Synt.-Comma: s:=81/80
Ab - Eb - Bb - F - C ~s^(-1/7) G ~s^(-2/7) D ...
... D ~s^(-2/7) A ~s^(-1/7) E ~s^(1/15) B - F# - C# ~s^(1/178) G#

3.) View all the epimorics in logarithmic 'cent'-units approximation:
Ab - Bb - F - C ~-3.056 G ~-6.128 D ...
... D ~-6.150 A ~-3.083 E ~-3.089 B ~-1.833 F# - C# ~-0.121 G#

4.)For all those that are familiar with the more modern concept of TUs
Ab - Bb - F - C ~-93.9 G ~-188.1 D ...
... D ~-188.7 A ~-94.6 E ~94.8 B ~-56.3 F# - C# ~-3.7 G#

Info about TUs
http://www.huygens-fokker.org/docs/measures.html
# Temperament Unit: 1/720 part of a Pythagorean comma
"
This measure was developed by organ builder John Brombaugh to describe very small intervals as integer values. In this measure, the syntonic comma is almost exactly 660 Temperament Units and the schisma 60. Because 720 is divisible by all numbers from 2 to 6 and more, most temperaments can be described by only integer values. In a well-temperament, -720 TU must be distributed over the cycle of fifths. One Grad is 60 TU.
"
Attend the bisection of the schisma into: 60 = 56.3 + 3.7 TUs.

!Sp7th_part_SC.scl
Sparschuh's epimoric two- and one-7th part of Snytonic-Comma [2010]
12
59/56 ! C#=(256/243)(14337/14336 ~+0.12c)=(135/128)(944/945 +~0.12c)
47/42 ! D = (9/8)(188/189 {~-9.18c}) = (10/9)(141/140 {~+12.23c})
32/27 ! Eb
561/448 ! E = (5/4)(561/560 {~+3.08c})
4/3 ! F
59/42 ! F#=(1024/729)(14337/14336 ~+0.12c)=(45/32)(944/945 +~0.12c)
283/189 ! G =(3/2)(566/567 {~-3.06c})
128/81 ! G#
281/168 ! A = (5/3)(281/280 {~+6.17c})= (27/16)(112.4/113.4 {-15.33c})
16/9 ! Bb
15/8 ! B
2/1
!
! [eof]

Welcome to the modern revival of 1/7 and 2/7 parts of the SC.

Already Vincenzo Galilei and his son Galilei liked just that
arithmetically division of into 7 consecutive parts:

81:80 = 560:561:562:563:564:565:567

see:
http://mto.societymusictheory.org/issues/mto.06.12.3/mto.06.12.3.duffin_frames.html
"
Quote:
"...Listeners can hear in the 1/3 comma version how the odd-numbered measures sound sour since they contain only open fifths, whereas the even-numbered measures are somewhat better. The 2/7 comma version sounds similar but slightly better with the improved fifths and major thirds, and the version in 1/4 comma meantone sounds better still in the open fifths (though not actually pleasant) and very euphonious in the triads. In 1/5 comma meantone, the fifths and major thirds are about the same distance from pure (though the third is wide and the fifth narrow), which means that they beat at a similar rate and create a kind of "vibrato" effect that some listeners find attractive. The 1/6 comma version is slightly better in the fifths but slightly worse in the major thirds, though they are still quite acceptable--a good compromise, perhaps. (The virtues of 1/6 comma meantone become evident with more complex harmonies.)(26) Finally, the equal tempered version exhibits excellent odd-numbered measures because its fifths are almost pure, but its major thirds are excruciatingly wide, causing the even-numbered measures to sound very sour and "jangly."...
"
Hence Galilei's quote confirms my own old view again:
12-EDO := 'Tears in my ears!'

Quest:
Are there any newer thoughts about mixing such arithmetically
1/7 and 2/7 parts SC reduced 5ths?

How does C:E:G = 4 : 5*SC^(+1/7) : 6*SC^(-1/7) sound in yours ears?

Or has anybody further hints for
improvements and refinements
of the above version an other time again?

bye
Andy