Re:Melkis, Re:new 19-limit epimoric well-tuning, was Re: "In tune" eq

Hi Andy,
Thanks for your very documented and inspiring analysis, as always !
My "C243" temperament lacks one epimoricity on only one 3rd deviation, well, nothing's perfect...
At least both variations have all the 5ths deviations epimoric, your 19-limit with
C 728/729 G 272/273 D 288/289 A 323/324 E-B-F# 512/513 C#-G#-Eb-Bb-F-C
My "29-limit" with
C 728/729 G 272/273 D 204/203 A 609/608 E-B-F# 512/513 C#-G#-Eb-Bb-F-C

I would be interested to know what this epimoric quality of the deviations means for you, in terms of acoustics or others ?

In my sense, my 1024/1015 in here is not so bad knowing that "C# - A" = 1024 - 1015 = 9 has the particularity to show a quadruple equal-beating with "Bb - F#" = "Eb - F#" (=864 - 855) = "C - Eb" (=729 - 720) = 9 (= Eb...).

On the other hand I would say your A = 256/153 has another quality in its relation to Bb, to be 17/16 apart, a highly auto-coherent powerful and colorful interval.
It's the same A and Bb we find in my 19-limit but rather-17 "Flamenca.scl", conceived for the guitar, that we can hear in both "Saudade Mi KoraSound" of Francois Breton and "H17" of Denis Grandclement in the Ethno demos :
https://www.dropbox.com/s/9grkajrdg7uo6ds

Flamenco chromatic scale around the 17th harmonic, in A (= guitar)
12
!
160/153
512/459
32/27
64/51
4/3
1216/867
76/51
80/51
256/153
16/9
4096/2187
2/1
! Dudon 2005

Anyway it's pleasant to see we arrived to very close variations (one note different) so hopefully there should be something harmonious in their global features.
My experience, confirmed by other users, is that Melkis-qualifying temperaments (as both are and many other 19-limit temperaments or -c extensions of 19-limit) are highly satisfying for Eastern-European and Tzigan/Rom harmonies, rich in minor chords, minor thirds and chromatisms, while factor 17- intervals and chromatisms are definitively more Spanish-Gypsy, and compared to average temperaments both should offer a large range of useful tonics.

Harmonically yours,
- - - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
> At least both variations have all the 5ths deviations epimoric,
> your 19-limit with
> C 728/729 G 272/273 D 288/289 A 323/324 E-B-F# 512/513 C#
> C#-G#-Eb-Bb-F-C
> My "29-limit" with
> C 728/729 G 272/273 D 204/203 A 609/608 E-B-F# 512/513 C#
> C#-G#-Eb-Bb-F-C

Salut Jacques,
meanwhile i even reduced such epimoric decompositions of the PC
even down to 11-limit superparticular ratios:

F 9800/9801 C 539/540 G 384/385 D 242/243 A 384/385 E 440/441 B ... F

see the consecutive entries in the corresponding integer-sequence:
http://en.wikipedia.org/wiki/Smooth_number
http://oeis.org/classic/A051038
http://oeis.org/classic/b051038.txt

or when expressed in 'Monzo' prime-number vectors:
using the labeling of interval-names from
http://www.huygens-fokker.org/docs/intervals.html

F
|3,-4,2,2,-2> = 9800/9801 'Gauss'-comma: kalisma
C
|-2,-3,-1,2,1> = 539/540 Swet's comma
G
|7,1,-1,-1,1> = 384/385 undecimal kleisma
D
|1,-5,0,0,2> = 242/243 neutral third comma
A
|7,1,-1,-1,1> = 384/385 undecimal kleisma (again)
E
|3,-2,1,-2,1> = 440/441 Werckmeister's "spetenarian" schisma
B
that yields as total sum over all that six 11-limit commata the PC:
|19,-12,0,0,0>

That results in the scala-file format:

!SpUndecanarian.scl
Sparschuh's [2010] epimoric 11-limit decomposition of the PC
12
!F 9800/9801 C 539/540 G 384/385 D 242/243 A 384/385 E 440/441 B ... F
!
3872/3675 ! C# (256/243)(9801/9800) = |5,-1,-2,-2,2> ~limma
28/25 ! D(10/9)(126/125)=(9/8)(224/225)=|2,0,-2,1> "middle-second"
1452/1225 ! Eb (32/27)(9801/9800) = |2,1,-2,-2,2> ~Pyth.minor-3rd
1408/1125 ! E (5/4)(5632/5625)=[804+4/7]/[803+4/7]) = |7,-2,-3,0,1>
3267/2450 ! F (4/3)(9801/9800) = |-1,3,-2,-2,1> ~JI-4th
15488/11025 ! F# (1024/729)(9801/9800) = |7,-2,-2,-2,2> ~tritone
539/360 ! G (3/2)(539/540) = |-2,-2,-1,2,1> ~JI-5th
1936/1225 ! G# (128/81)(9801/9800) = |4,0,-2,-2,2> ~dim-6th
3388/2025 ! A (5/3)(3388/3375=[260+8/13]/[259+8/13]) = |2,-4,-2,1,2>
2178/1225 ! Bb (16/9)(9801/9800) = |1,2,-2,-2,2> ~dim-7th
61952/33075 ! B (15/8)([974+359/509]/[973+359/509]) = |9,-3,-2,-2,2>
2/1
!
![eof]

by that procedure i got rid of 17 and 19 powers, not to mention 29.

au bientot
Andy

--- In tuning@yahoogroups.com, "Andy" <a_sparschuh@...> wrote:

> see the consecutive entries in the corresponding integer-sequence:
> http://en.wikipedia.org/wiki/Smooth_number
> http://oeis.org/classic/A051038
> http://oeis.org/classic/b051038.txt

Also relevant:

http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem
http://oeis.org/classic/A002071
http://oeis.org/classic/A117581
http://oeis.org/classic/A117582
http://oeis.org/classic/A117583

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> > F 9800/9801 C 539/540 G 384/385 D 242/243 A 384/385 E 440/441 B...

Hi Gene & Jacques,

http://oeis.org/classic/A117581
"2, 9, 81, 4375, 9801, 123201,...."

appearently those number-sequence refers to the cases

2-limit: 2/1

3-limit: 9/8 := (3/2)(3/4) the Pyth. whole-tone

5-limit: 81/80 := (9/8)(9/10) the SC

7-limit: 4375/4374 := 7 * 5^4 / (3^7 *2) the 'Ragisma'

11-limit 9801/9800 := (99/98)(99/100) the 'Gauss'-comma kalisma, see:
/tuning-math/message/16961

13-limit 123201/123200 := (351/350)(351/352)
Who knows the name of that very tiny one, of about ~0.014...Cents?
or circa ~1/71 Cents?
...&.c.t....

But let us return back to the above 11-limit distribution of the PC. That can be simplified an refined again,
when abandoning the epimoric property inbetween the 5ths:

Ansatz in tree steps:
At first sub-divide the Pyth.-Comma into SC*schisma

1. C ... 80/81 ... B 32768/32805 F# C# G# Eb Bb F C

then split that SC=80/81 by trisection into three 7-limit parts
80/81 = (224/225)*(3125/3136)*(225/225) over the 5ths C~G~D~A~E~B

2. C ...224/225... D 3125/3136 A ...224/225... B 32768/32805 F#...C

and finally half within 11-limit both 224/225=(539/540)*(384/385)
that results in an symmetric distribution of the SC over five 5ths:

3.
C 539/540 G 384/385 D 3125/3126 A 384/385 E 539/540 B 1/schisma F#...C

that procedure results as "SCALA"-file:

! Sparschuh's [2010] symmetric 11-lim. 5-fold distribution of the SC
Sp_symm_11lim_SC.scl
12
!
!C 539/540 G 384/385 D 3125/3136 A 384/385 E 539/540 B 1/schism F#...C
!=>> synchronous C-major triad C:E:G = 4 : 5*(540/539) : 6*(539/540)
!
256/243 ! C# |8,-5> the Pythagorean limma
28/25 ! D |2,0,-2,1> = (9/8)(224/225) = (10/9)(126/125) mid-tone
32/27 ! Eb |5,-3>
675/539 ! E |0,3,2,-2,-1> = (5/4)(540/539 ~+2.3... Cents sharp off)
4/3 ! F |2,-1>
1024/729 ! F# |10,-6>
539/360 ! G |-3,-2,-1,2,1> = (3/2)(539/540 ~-2.3... Cents flat off)
128/81 ! G# |-7,4>
375/224 ! A |-5,1,3,-1> = (5/3)(225/224) = (27/16)(125/126) mid-6th
16/9 ! Bb |-4,2>
15/8 ! B |-3,1,1> the solely 5-limit one pitch
2/1
!
![eof]

this gains an remarkable simplifikation against the 'undecanarius'
at the expense of the 'epimoric' property.

Some statistics on the limitness of the individual pitch-classes:
(attend the 'Monzo' prime-nuber-vectors)

2-limit: C ; unison 1/1 root and its octaves
3-limit: F-Bb-Eb-G#-C#-F# ; includes all the remote accidentials
5-limit: B ; the 'leading-tone' 16/15 to root 1/1 unison
7-limit: D-A ; that eases the tension of the false 5th: 40/27 in JI
11-limit: E-G ; makes the C-major triad C:E:G 'equal-beating'

Conclusion
As far as I do know at the moment, the above:

C 539/540 G 384/385 D 3125/3126 A 384/385 E 539/540 B 1/schisma F#...C

is i.m.h.o. the most easiest tunable 11-limit
doedcatonic well-temperament,
that I've ever applied on my old piano.

bye
Andy

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:

> > C 539/540 G 384/385 D 3125/3126 A 384/385 E 539/540 B 1/schisma F#...C
>

Salut Jacques,
even that one can be reduced again down from 11-limit
to the simpler 7-limit by:
F 32768/32805 C 32768/32805 G 5103/5210 D 3125 A 5103 E 32768/32805 B

especially for all those who do'nt like to involve the
harmonic 11th partial 11/8 out of the over-tone series:
Ok, let's get rid of 11-limit by the following tiny modifications:

! Sparschuh's [2010] 3-fold schismaitc 7-limit well-temerament
Sp3schismatic7lim.scl
12
!
! define the 'schisma' as usual:
! s := 32805/32768 = 3^8*5/2^15 and apply it 3-times at the positions:
!F 1/s C 1/s G 5103/5210 D 3125/3136 A 5103/5210 E 1/s B F# C#...Bb F
!
!=> synchronous C-major triad beats schismatic equal C:E:G = 4:5*s:6/s
!
135/128 ! C# |7,3,1> = s*256/243 = s*|8,-5>
28/25 ! D |2,0,-2,1> = (9/8)(224/225) = (10/9)(126/125) tone 1215/1024 ! Eb |-10,5,1> = s*32/27 = s*|5,-3>
164025/131072 ! E |-17,8,2> = (5/4)*s by an schisma sharpend up 3rd
10935/8192 ! F |-13,7,1> = (4/3)*s by an schisma sharpend up 4th
45/32 ! F# |-5,2,1> = s*1024/729 = s*|10,-6>
16384/10935 ! G |14,-7,1> = (3/2)/s by an s flattend down 5th
405/256 ! G# |-8,4,1> = s*128/81 = s*|7,-4>
375/224 ! A |-5,1,3,-1>= (5/3)(225/224) = (27/16)(125/126) 6th
3645/2048 ! Bb |-11,6,1> = s*16/9 = s*|4,-2>
15/8 ! B |-3,1,1> = s*4096/2187 = s*|12,-7>
2/1
!
![eof]

That results in schismatically biased intervals from the unison 1/1

Conclusion
1. 5th: C-G/s
2. 3rd: C-E*s
3. 4th: C-F*s

Also attend the schismatic enharmonics within the remote accidential notes. But here the nice 'epimoric' ratio property went lost,
that once had appeared in the 11-limit case within some of the 5ths.
By that procedure the seize of numbers arise considerable for the ratios in nominators versus denominators significantly in decimals,
when departing the former 11-limit propotions.

Quest:
Which variant do you prefer personally as acoustically supererior:
The older 11-limit or the new actual 7-limit changes?

bye-bye, au bientot
Andy

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> "Flamenca.scl", conceived for the guitar, that we can hear in both
> "Saudade Mi KoraSound" of Francois Breton and "H17" of Denis
> Grandclement in the Ethno demos :
> https://www.dropbox.com/s/9grkajrdg7uo6ds
>
> Flamenco chromatic scale around the 17th harmonic, in A (= guitar)
> 12
> !
> 160/153 ! C#
> 512/459 ! D
> 32/27 ! Eb
> 64/51 ! E
> 4/3 ! F
> 1216/867 ! F#
> 76/51 ! G
> 80/51 ! G#
> 256/153 ! A
> 16/9 ! Bb
> 4096/2187 ! B or may-be better "1624/289" or even "152/82"
> 2/1
> ! Dudon 2005

Salut Jacques,

that fine well-sounding ratios do yield an
almost epimoric chain of 5ths,
with the only exception on the note "B" inbetween the 5ths E~B~F#

C 152/153 G 512/513 D-A-E 2176/2187 B 4617/4624 F#...
...F# 170/171 C#-G# 136/135 Eb-Bb-F-C

But already Andreas Werckmeister prefered to kept the property of
all 5ths should stay within the epimoric restriction, see:
/tuning/topicId_90126.html#90228
W's definition of his 'spetenarian' monochord-string-lenghts:

C196 C#186 D176 D#165 E156 F147 F#139 G131 G#124 A117 Bb110 B104

also available from
http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_the_Septenarius_tunings
http://sites.google.com/site/240edo/equaldivisionsoflength%28edl%29

which he had initially obtained completely by epimoric 5ths construction:

C 392/393 G 132/131 D 352/351 A-E-B 416/417 F#...
...F# 278/279 C#-G# 496/495 Eb-B 440/441 F-C

for an earlier [2006] interpretation see also my:
/tuning/topicId_68023.html#68047

Hence here in yours 'flamenca' case arises now the question:
How to change in W's sense yours original ratio 4096/2187
of the concerning note "B" so, that both neighbouring 5ths also become epimoric integral too, in order to gain a more smooth
transition inbetween, as already satisfied in all the other 5ths:

.E 2176/2187=[197+9/11]/[198+9/11 B 4617/4624=[659+4/7]/[660+4/7] F#.

Here I see at least two apt cases as possible solutions
in order get rid of the above broken epimorics:

Suggested little change:
1.) Yours own 'melkis' way in 23-limit, with the revision
1624/867 ! B, that yiedls in the 5ths: ...E 203/204 B 608/609 F#...?

Somehow bigger change:
2.) or my 19-limit splitting of 152/153=(288/289)(323/324) alterates
152/82 ! B, with ...E 288/289 B 323/324 F#...?

Now, compare the Cent-values for the possible 3 variants of note B:

Analysis:
0.) 4096/2187 : ~1086.315...Cents the original 3-limit initial ratio
1.) 1624/867 : ~1086.537...Cents in 'melkis' 29-limit ?
2.) 152/81 : ~1089.693...Cents in the lower 19-limit ?

the last 2.) one 152/81 appears -by-the-way- already in [2008]:
/tuning/topicId_78659.html#78677
c243 #256 d272 #288 e304 f312 #342 g364 #392 a406 #432 b456 c'486
that meets EXACTLY yours own specification of 'melkis'-tuning.

or express the relative deviations among that three ones as ratios:

1.) vs 0.) : 147987/147968 := (1624/867)/(4096/2187) ~+0.222...Cents
2.) vs 0.) : 513/512 := (152/81)/(4096/2187) ~+3.378...Cents
3.) vs 1.) : 5491/5481 := (152/81)/(1624/867) ~+3.156...Cents

Well personally,
I try to keep the ratios as simple as needed, but not to much simple.

Quest:
Which one among that three alternatives for "B"
do you prefer now after that 'analysis'?

Hope that discussion by that example answers yours question:
> I would be interested to know what this epimoric quality
> of the deviations means for you, in terms of acoustics or others ?

It simply improves the acuostics as well as the arthmetics behind,
makes calculations easier, and helps to gain more smooth transitions in modulations, when changeing the key,
at least when listening with my ears,
the epimoric ratios do fit better in their proportion of seize:
That results in less work for the brain to compensate the bias,
with the benefit of easier accessable enjoyment
while listening oder even performing music with that smoothening
property, against the rough disturbences without that refinement.

harmonically-greetings
yours
Andy

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> My experience, confirmed by other users,
> is that Melkis-qualifying temperaments
> (as both are and many other 19-limit temperaments or -c
> extensions of 19-limit) are highly satisfying for Eastern-European
> and Tzigan/Rom harmonies, rich in minor chords, minor thirds and
> chromatisms, while factor 17- intervals and chromatisms are
> definitively more Spanish-Gypsy, and compared to average
> temperaments both should offer a large range of useful tonics.
>

Salut Jacques,

fully agreed,
but how in that context would you classify 23-limit enrichment,
alike used in the following bi-epimoric progression-scheme:

All negative Cent-values indicate the 5ths-deviations downwards
below pure 3/2, respective the according positive C.-values
do indicate the departments of the 3rds upwards form 5/4 JI.

[Recommendaton: 'Use-Fixed-Wide-Font' option]

Gb 81/80 Bb ~+21.51...Cents (the SC)
Db 82/81 F ~+21.24 attend tiny improvement by 41 refinement
Ab 91/90 C ~+19.13
Eb 96/95 G ~+18.13
Bb 136/135 D ~+12.78
|
6560/6561 ~-0.26
v
F 1025/1024 A ~+1.69
|
819/820 ~-2.11
v
C 576/575 E ~+3.01 here attend the factor 23 in the nominator
|
1728/1729 ~-1.001
v
G 456/455 B ~+3.80
|
323/324 ~-5.35
v
D 256/255 F# ~+6.78
|
152/153 ~-11.35 !
v
A 96/95 C# ~+18.13
|
2184/2185 ~-0.79
v
E 92/91 G# ~+18.92 in order to get again rid of the '23'
|
8280/8281 ~-0.21
v
B 91/90 D# ~+19.13 in order to get again rid of the 23
|
728/729 ~-2.38
v
F# 81/80 A# ~+21.51 the only 3rd within the remote black keys

Try it out yourself in scala:

! Sp_41_23_bi_epi.scl
Sparschuh's 41- and 23-limit bi-epimoric well-temperament [2010]
12
!
96/91
102/91
108/91
144/115
3280/2457
128/91
2592/1729
144/91
152/91
162/91
15552/8281
2/1
!
![eof]

Quest:
How do you judge about that additional factor '23' aside 17 and 19 ?

au revoir
bye-bye
Andy

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> My "29-limit" with
> C 728/729 G 272/273 D 204/203 A 609/608 E-B-F#
> F# 512/513 C#-G#-Eb-Bb-F-C
> I would be interested to know what this epimoric quality of the
> deviations means for you, in terms of acoustics or others ?

Salut Jacques,
the double-epimoric property result in more smooth
transitions when modulating:

Here my recent refinements in 43-limit
with the meanwhile improved progression of 5ths:

Ab-Eb 1700/1701 Bb F 7224/7225 C 7568/7569 G 493/495=246.5/247.5 D...
...D 288/289 A 493/495=246.5/247.5 E 7568/7569 B 1160/1161 F#-C#-G#

or when considered in the cent-units approximation

Ab - Eb ~-1.02 Bb ~-0.24 C ~-0.23 G ~-7.01 D...
...D ~-6.0008 A ~-7.01 E ~-0.23 B ~-1.49 F# - C# - G#

that chain results into an progression of 3rds, that is also epimoric

Gb 85/84 Bb ~20.48... Cents sharper arised than pure 5/4 JI
Db 86/85 F ~20.24...
Ab 87/86 C ~20.01...
Eb 88/87 G ~19.78...
Bb 126/125 D ~13.79...all 12-EDO 3rds deviate same ~13.69...Cents
F 216/215 A ~8.033...
C 1376/1375 E ~1.258... within the range of the pulling-effect
G 1376/1375 B ~1.258... also fusing downwards to 5/4 JI
D 256/255 F# ~6.677...
A 136/135 C# ~12.77...
E 88/87 G# ~19.78...
B 87/86 D# ~20.01...
F# 85/84 A# ~20.48... enharmonic same again as first Gb ~ Bb

!Sp43lim_high_contr.scl
Sparschuh's 43-limit 'high-key-contrast' bi-epimoric well-temp. [2010]
!
12
!
1376/1305 ! C# := 2^5 x 3^-2 x 5^-1 x 29^-1 x 43
1462/1305 ! D := 2 x 3^-2 x 17 x 29^-1 x 43
172/145 ! Eb := 2^2 x 5^-1 x 29^-1 x 43
344/275 ! E := 2^3 x 5^-2 x 11^-1
29584/22185 ! F := 2^4 x 3^-2 x 5^-1 x 17^-1 x 29^-1 x 43^2
5504/3915 ! F# := 2^7 x 3^-3 x 5^-1 x 29^-1 x 43
3784/2523 ! G := 2^3 x 3^-1 x 11 29^-2 x 43
688/435 ! G# := 2^4 x 3^-1 x 5^-1 x 29^-1 x 43
4128/2465 ! A := 2^5 x 3 x 5^-1 x 29^-1 x 43
29240/16443 ! Bb := 2^3 x 3^4 x 5 x 7^-1 x 17 x 29^-1 x 43
118366/63075 ! B := 2^6 x 3^-1 x 5^-2 x 29^-2 x 43^2
2/1
!
![eof]

for the intensional 'pulling-effect' of the 3rds C-E and G-B
refer for instance to:

http://www.arpschnitger.nl/sgro02.html
there especially to the organ of
http://www.arpschnitger.nl/snoordb.html
quote:
"
The Aa-Kerk organ departs only slightly from equal temperament, thus lending the F-sharp-minor and E-major prel­udes a mild character. The sound produced by individual pipes, ranks in different combinations and the full ensemble is so harmonious that the “sour thirds” of the neariy equal temperament do not disturb. The “pulling effect” in the tuning can be heard very clearly in the long chords and is produced by the mutual influence of the pipes which are arranged in thirds on the windchest.

or more technically
http://www.aes.org/e-lib/browse.cfm?elib=6621
"
Coupling between Simultaneously Sounded Organ Pipes

The interaction of two, simultaneously sounded flue organ pipes of the same nominal tone were investigated in laboratory experiments. Three different phenomena were observed: simple superposition of the two sounds; coupling of the two pipes through the surrounding air; and -melting- the two sounds into a new, common sound with a bit higher frequency. The last two cases appear only for neighboring pipes. The results of the experiments and the possible physical explanations are discussed.
"
http://adsabs.harvard.edu/abs/2000ASAJ..108.2592S
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000119000004002467000001&idtype=cvips&gifs=yes

http://www.worldscinet.com/ijbc/17/1710/S021812740701924X.html
SYNCHRONIZATION OF HIGHER HARMONICS IN COUPLED ORGAN PIPES
Abstract:
We report results on the synchronization of two organ pipes positioned side by side. Special attention is put on the synchronization of the higher harmonics. As possible explanation, classical theory provides the amplitude death as explanation for the reduction to almost silence of two coupled organ pipes. With our measurements we exclude this scenario. The higher harmonics show a behavior in perfect coincidence with synchronization theory. In addition we investigate the dependence on the coupling of two pipes by varying their distance. In the context of synchronization in networks, a new synchronization effect is observed for extended systems with two distributed, slightly different delays.
"

And finally here an reference paper for free down-load:
http/:arxiv.org/pdf/physics/0506094
sound examples of the research group
http://www.stat.physik.uni-potsdam.de/~organ/index.php?page=Sound

bye
Andy

Hi Andy,
Leaving in a few hours for Portugal where I give a concert in Boom Festival...
Will try to have a better look later, but seems to me 23 is not implicated in intervals where it would have a direct meaning, like 17 and 19 have here, but may be I'm wrong. So 23 and 41 could well have a relation with the tuning, but in extensions of the scale.
My excuses for not replying to your previous messages either, not that I was not interested -
but I had to work on my music, and other technical aspects.
Keep on tuning :)
Will be back in a few weeks,
- - - - - - - - - - -
Jacques

Andy wrote :

> Salut Jacques,
>
> fully agreed,
> but how in that context would you classify 23-limit enrichment,
> alike used in the following bi-epimoric progression-scheme:
>
> ! Sp_41_23_bi_epi.scl
> Sparschuh's 41- and 23-limit bi-epimoric well-temperament [2010]
> 12
> !
> 96/91
> 102/91
> 108/91
> 144/115
> 3280/2457
> 128/91
> 2592/1729
> 144/91
> 152/91
> 162/91
> 15552/8281
> 2/1
> !
> ![eof]
>
> Quest:
> How do you judge about that additional factor '23' aside 17 and 19 ?
>
> au revoir
> bye-bye
> Andy