Re:new 19-limit epimoric well-tuning, was Re: "In tune" equal temperam

Andreas wrote :
(Digest number 6720, 13 th of July 2010)

> ! Sp19limWell.scl
> !
> Sparschuh's 19-limit well-temperament [2010] with epimoric 5ths & 3rds
> 12
> !
> 256/243 ! C# |8,-5>
> 272/243 ! D |4,-5,0,0,0,0,1>
> 32/27 ! Eb |4,-3>
> 304/243 ! E |4,-5,0,0,0,0,0,1>
> 4/3 ! F |2,-1>
> 38/27 ! F# |1,-3,0,0,0,0,0,1>
> 364/243 ! G |2,-5,0,1,0,1>
> 128/81 ! G# |7,-4>
> 256/153 ! A |8,-2,0,0,0,0,-1>
> 16/9 ! Bb |4,-2>
> 2/1 ! C' |1>
> !
> ![eof]
>
> Quest.
> Are there any thoughts about that particular 19-limit well-tuning?

Dear Andy,
Sorry for this late reply, have been 4 weeks without internet and I review some the past mails...
I think it's a beautiful rational temperament ! And I love the way the temperament starts softly with a 729/728 in C:G = 364/243
In fact it differs of only one note from Jak Dudon's Melkis-WT
presented with other harmonic temperaments in 2005, but that I probably found years before, since it uses the simplest possible Melkis series, that I discovered in 1992.
Instead of 256/153, it uses 406/243 which belongs to the harmonic series :
243 91 17 203 19 57 171 1 3 9 27 81
where it applies the Melkis -c of the minor third :
203 - 171 = 32 (A - F# = C#)
Thus displaying a Melkis series of 8 terms :
(54) 72 96 128 171 228 304 406 that verify the same -c algorithm :
x^4 = 2x + 1/2
(the same series is used in full 12 notes in Ethno2's Tango_hwt.scl)
What I think is that 256/153 is elegant and probably proposes an correct temperament, but with the 17^-1 factor misses the simplicity of the harmonic series issued from C#.
So even if 203 raises it to a theorical 29-limit, I still prefer my version.

Another one quite close, this time in you might be more interested in the Ethno collection (download it from the Tuning list files) is my 19L-Rocky-hwt.scl (North-America folder), that combines Eair and Melkis recurrent series.
I am completing it here partially with your notation system for better reading :

! 19-l_rocky_hwt.scl
!
19-limit well-temperament, C to B achieving eq-b of bluesy DEG-type chords
12
!
256/243 ! C#
272/243 ! D
32/27 ! Eb
304/243 ! E
4/3 ! F
1024/729 ! F#
364/243 ! G
128/81 ! G#
1216/729 ! A
16/9 ! Bb
4096/2187 ! B
2/1
! three occurences of Eair eq-b recurrent sequence 2x^3 - 3 = x^2
! 3C - 2G = 3A - 5C, etc. (and complements) all in synchronous-beating
! Dudon 2005

- - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
> Dear Andy,
>... uses the simplest possible
> Melkis series, that I discovered in 1992.
> Instead of 256/153, it uses 406/243 which belongs to the harmonic
> series :
> 243 91 17 203 19 57 171 1 3 9 27 81

Salut Jacques,
but yours choice of A=203 delivers an none-epimoric sharpness
in the deviation of the 3rd: A-C#

A 1024/1015=([113+7/9] / [112/+7/9]) C#

or more gerneally the within the complete second stack of 3rds
within: F-A-C#-F

F
406/405 |1,-4,-1,1>*29 ~+4.269...Cents
A
1024/1015 |10,-3,-1,-1>/29 ~+15.283...Cents
C#
81/80 |-4,4,-1> ~+21.506...Cents the SC
F

when compared against my own refinement of: A=203+5/17
please see again:
/tuning/topicId_90656.html#90896
"
2.) Stack of 3rds: F-A-C#-F
---------------------------
F
256/255 |8,-1,-1,0,0,0,-1> ~+6.775...Cents
A
136/135 |3,-3,-1,0,0,0,1> ~+12.776...Cents
C#
81/80 |-4,4,-1> ~+21.506...Cents the SC
F

or more concise:
yours: F 406/405 A 1024/1015=([113+7/9]/[112/+7/9]) C# 81/80 F
my own:F 256/255 A 136/135 C# 81/80 F
that's epimoric in all the dozen 3rds without any exceptions
as in yours case inbetween A-C#.
Confirm that all the other there mentioned stacks of 3rds

1.) Bb 136/135 D 171/170 F# 96/95 Bb
3.) C 1216/1215 E 96/95 G# 81/81 C
4.) G 456/455 B 96/95 Eb 91/90 G

do agree in both tunings.

But also my change cuts yours 29-limit
down to the lower 19-limit.

> What I think is that 256/153 is elegant and probably proposes an
> correct temperament,
> but with the 17^-1 factor misses the simplicity
> of the harmonic series issued from C#.

But I do like to have the deviation of all 3rds
also in epimoric bias as the already all the 5ths:

C 728/729 G 272/273 D 288/289 A 323/324 E-B-F# 512/513 C#-G#-Eb-Bb-F-C

Attend that i overtook "323/324" from Kirnberger_III
/tuning/topicId_90656.html#91245

> So even if 203 raises it to a theorical 29-limit,
> I still prefer my version.

So am I in favour too for my double-superparticular property.

au bientot
Andy