Topic: more complicated 19 17 2 WT

1 scales

Thread (31 messages)

From: Tom Dent (2008-10-02)
Subject: more complicated 19 17 2 WT

1
1083/1024
323/288
19/16
171/136
171/128
361/256
6137/4096
3249/2048
29241/17408
57/32
61731/32768

looks unpleasant doesn't it... because C is 'out on a limb' with
respect to the factors of 19.

If I rewrite setting F=1 I get
128/171
19/24
68/81
8/9
16/17 
1
19/18
323/288
19/16
171/136
4/3
361/256

The scale has 7 19-limit semitones, 5 19/16 thirds, 2 24/19's, 2
17/12's ... and the narrowest fifth is 256/171.
~~~T~~~

From: Carl Lumma (2008-10-02)
Subject: Re: more complicated 19 17 2 WT

--- In [email protected], "Tom Dent" <stringph@...> wrote:
>
> 1
> 1083/1024
> 323/288
> 19/16
> 171/136
> 171/128
> 361/256
> 6137/4096
> 3249/2048
> 29241/17408
> 57/32
> 61731/32768
//
> The scale has 7 19-limit semitones, 5 19/16 thirds, 2 24/19's,
> 2 17/12's ... and the narrowest fifth is 256/171.

Here's a tally of the 19-limit consonances.

19/18 1
18/17 4
17/16 2
19/17 0
  9/8 2
19/16 5
24/19 2
  4/3 4
17/12 2
---------
     22

Compare to 31 and 26 in the previous RWTs we discussed.

In logland, the 5ths are much better, the flattest being
699 cents.  And 24/19 is still the sharpest 3rd.

So that's a very high number of 19-limit consonances
considering how close to 12-ET it is.

-Carl

From: Tom Dent (2008-10-02)
Subject: Re: more complicated 19 17 2 WT

--- In [email protected], "Carl Lumma" <carl@...> wrote:
>
> --- In [email protected], "Tom Dent" <stringph@> wrote:
> >
> > 1
> > 1083/1024
> > 323/288
> > 19/16
> > 171/136
> > 171/128
> > 361/256
> > 6137/4096
> > 3249/2048
> > 29241/17408
> > 57/32
> > 61731/32768
> //
> 
> Here's a tally of the 19-limit consonances.
> 
> 19/18 1
> 18/17 4
> 17/16 2
> 19/17 0
>   9/8 2
> 19/16 5
> 24/19 2
>   4/3 4
> 17/12 2
> ---------
>      22
> 
> Compare to 31 and 26 in the previous RWTs we discussed.
> 
> In logland, the 5ths are much better, the flattest being
> 699 cents.  And 24/19 is still the sharpest 3rd.
> 
> So that's a very high number of 19-limit consonances
> considering how close to 12-ET it is.
> 
> -Carl

...Not only that, the scale is symmetrical under inversion. The circle
of fifths has tempering
(a,b,c,b,a,b,1,1,b,1,1,b) where a=6137/6144, b=512/513, c=4617/4624.
~~~T~~~

From: Andreas Sparschuh (2008-10-06)
Subject: Proposal  for Mietke's  1719-1784 lost "Bach" harpsichord    Re:compl.19 17 2 WT

--- In [email protected], "Tom Dent" <stringph@...> wrote:
> > Here's a tally of the 19-limit consonances.
> > 
> > 19/18 1
> > 18/17 4
> > 17/16 2
> > 19/17 0
> >   9/8 2
> > 19/16 5
> > 24/19 2
> >   4/3 4
> > 17/12 2
> > ---------
> >      22
> (a,b,c,b,a,b,1,1,b,1,1,b) where a=6137/6144, b=512/513, c=4617/4624.

Hi Carl, Tom & all others,

yours ... 19,17,3 approachs work similar as my older attempts in:
http://launch.groups.yahoo.com/group/clavichord/message/8737
.
That proposal intends to offer an arguable
http://en.wikipedia.org/wiki/Historically_informed_performance
tuning on modern replicas of
http://en.wikipedia.org/wiki/Michael_Mietke
"Clavessin" 
"He delivered a harpsichord to the court at Cöthen in 1719 on the 
recommendation of Johann Sebastian Bach, which was probably the 
instrument for which Bach composed Brandenburg concerto no.5 as a 
show-piece."
The scholary literature offers some further details
about Bachs's journey to Berlin for buying there the instrument. 
Even the bill is still preserved.  
Dated:  Cöthen, March 1st 1719, 
It lists $130 expenses altogehter.  
Reprint in: 
Bach-Dokumente, Kassel 1969, Vol.II p.73-74, #95
Translation into english by A.Mendel, Bach reader p.431,490

But when Bach moved in 1723 to Leipzig  
that piano remained back in Coethen at least until 1784,
recorded among the court's inventory:  
"Specifickation derer Fürstlichen Instrumenten in der 
Musikalienkammer" 
lists it on  March, 8th, 1784 as "defect".

The problem:
Today, nobody knows any more how Bach had tuned that instrument.

But HIPerformers need for Mietke-replicas 
adequate modern so called "Bach"-tunings,
preferably in the coeval "Cammerthone" 
pitch of a4=(~405+-5)cps
as known from Mietke's others instruments too.

Hence a modern "Bach"-tuning should contain that specification,
in order to represent the state of the art in the research of the 
last ~10 years: 

New proposal for absolute pitches located at the frequencies:

243 middle_c'
256
272 d'
304
304 e'
312 f'
342
364 g'
392
406 a' cps or Hz
432
456 b'
486 c"

as again obtained from Bach's 1722(1723?) WTC autograph,
when reading the 'squiggle' pattern layout as an instruction
how to temper 5ths the corresponding to the "squiggles"
http://en.wikipedia.org/wiki/Well_Tempered_Clavier

Here an older photo of JSB's "decorative-ornaments" that i do prefer: 
http://www.strukturbildung.de/Andreas.Sparschuh/Bach_Handschrift.jpg

That doodle at the top can be interpreted symbolic as:  

Concise: start~2~2~2~1-1-1~3-3-3-3-3-3=end

when understood as an cylce 12 times 5ths

Or when labeld with note-names:
start=C~2G~2D~2A~2E-1B-1F#-1C#~3G#-3D#-3A#-3F-3C=end

Or even when expanded into full detailed pitches:
start=C 243 := 3^5 obtained from the last 5 ternary 'squiggles' 
~2~ G 91 182 364 728 (<729 :=3^6)
~2~ D 68 136 272 (<273 := 3*G)
~2~ A 203 (<204 := 3*68)
~1- E 19 38 76 152 304 608 (<609 := 3*A) ; 3 single 'squiggles'
-1- B 57
-1~ F# 171
~3- C# 1...512 (<513 := 3*F#) ; the concluding 5 triply 'squiggles'
-3- G# 3
-3- D# 9 := 3^2
-3- A# 27 := 3^3
-3- F 81 := 3^4
-3. C=end 243 := 3^5 

That's in today's modern terms of the:
http://www.xs4all.nl/~huygensf/scala/scl_format.html

!Sparschuh_proposal_Mietke.scl
!absolute pitches from the middle_c' on:
c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
!
12
!
256/243 !   ~90.22...(cents)
272/243 !  ~195.81... D
32/27   !  ~294.13...
304/243 !  ~387.74... E (5/4)*(1216/1215)
4/3     !  ~498.04... F
38/27   !  ~591.65...
364/243 !  ~699.58... G (3/2)*(728/729)
128/81  !  ~792.18...
406/243 !  ~888.63... A4=406cps ~ Coeval Berlin, Coethen Cammer-thone 
16/9    !  ~996.09...
152/81  ! ~1089.69... B
2/1

Attend, 
that tuning contains some consecutive partials 
out of the  overtone-series inbetween the 4 semitones:

16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1

in reference to:
http://en.wikipedia.org/wiki/Johann_Gottfried_Walther
specification in his:
"He wrote a handbook for the young Duke with the title Praecepta der 
musicalischen Composition, 1708. It remained handwritten until Peter 
Benary's edition (Leipzig, 1955)."

in german:
http://de.wikipedia.org/wiki/Johann_Gottfried_Walther
's
"Praecepta der musicalischen Composition, Weimar 1708;
Neu hrsg. von Peter Benary in: 
Jenaer Beiträge zur Musikforschung. Band 2, 
Breitkopf & Härtel, Leipzig 1955 "

bye
A.S.

From: Carl Lumma (2008-10-13)
Subject: more time-wasting on the VRWT problem

Trying to make some headway on this...

Generally, WTs are too close to ET to be worth it if
they have fewer than five pure 5ths, and too uneven to
be "Victorian" if they have more than seven pure 5ths.

I had previously defined the VRWT bounds to be:
 M3 range = 386.3 - 404.4 cents (5/4 - 24/19)
 P5 range = 696.0 - 702.0 cents (432/289 - 3/2)

I'm assuming we have to shoot for {2 3 17 19} JI here,
since all the primary intervals with 5, 7, 11, or 13 in
them are too far away from 12-ET to meet these bounds.
Or at least, that's my guess.

So I did a search of the {2 3 17 19} lattice for 5ths
in the VRWT range, and found:

 432/289     696  (  4  3 -2  0 )
 323/216     697  ( -3 -3  1  1 )
 256/171     699  (  8  2  0 -1 )
 13851/9248  699  ( -5  6 -2  1 )
 6137/4096   700  (-12  0  1  2 )

Actually the last two were outside of the convex ball I
was searching, but they occur in Tom's most recent VRWT.
And I didn't notice this until after I'd messed around
with the first three...

I didn't actually compute it, but it doesn't seem to be
possible to get a WT with five or seven pure 5ths, where
all smaller 5ths are either 432/289, 323/216, or 256/171.
But you can get close:
-----
(3/2 5) (256/171 7) (323/216 0) (432/289 0) -> 8399.8
------
(3/2 6) (256/171 4) (323/216 1) (432/289 1) -> 8398.6
(3/2 6) (256/171 4) (323/216 2) (432/289 0) -> 8399.2
And some of these might be interesting for a flat-octave
WT.  But for now, let's focus on scales with pure octaves.

With seven pure 5ths, there seems to be only one solution:
-------
(3/2 7) (256/171 2) (323/216 2) (432/289 1) -> 8400.0

And lo and behold, this is the combination of 5ths that
my own VRWT uses.  It remains to be seen whether it uses
the smoothest permutation of the 5ths.

Because I'm not allowing 81/64s (too sharp), there can
be at most three pure 5ths in a row, and indeed, we want
them in as few chunks as possible, to get all the 9/8s
we can.  That leaves two patterns to test:
 P P P x P P P x P x x x
 P P P x P P P x x P x x

And I should probably go back and check out Tom's 19-limit
5ths too.  13851/9248 might be better for total consonances
than 256/171, since it contains factors of 17.  And 432/289
is as flat of a 5th as I'd ever want, and maybe we're better
off without any of them.

'til next time...

-Carl

From: Carl Lumma (2008-10-13)
Subject: Re: more time-wasting on the VRWT problem

I wrote:
> And I should probably go back and check out Tom's 19-limit
> 5ths too.  13851/9248 might be better for total consonances
> than 256/171, since it contains factors of 17.  And 432/289
> is as flat of a 5th as I'd ever want, and maybe we're better
> off without any of them.

Of the 19-limit 5ths I identified, three have factors of
both 17 and 19 in them: Tom's two 5ths, and my 323/216.
I can't get the chain to come out exactly to the octave,
but I can get ridiculously close:

(3/2 5) (323/216 2) (13851/9248 4) (6137/4096 1) -> 8400.3
(3/2 6) (323/216 3) (13851/9248 2) (6137/4096 1) -> 8400.2
(3/2 7) (323/216 4) (13851/9248 0) (6137/4096 1) -> 8400.1

-Carl

From: Andreas Sparschuh (2008-10-14)
Subject: Re: more time-wasting on the VRWT problem

--- In [email protected], "Carl Lumma" <carl@...> wrote:
>


> I'm assuming we have to shoot for {2 3 17 19} JI here,
> since all the primary intervals with 5, 7, 11, or 13 in
> them are too far away from 12-ET to meet these bounds.
> Or at least, that's my guess.
> 
> So I did a search of the {2 3 17 19} lattice for 5ths...

Bach's pattern 

http://launch.groups.yahoo.com/group/tuning/message/78677

start~2~2~2~1-1-1~3-3-3-3-3-3=end

can be interpreted as {2 3 7 13 17 19 29} as cycle of 5ths:

start=C~2G~2D~2A~2E-1B-1F#-1C#~3G#-3D#-3A#-3F-3C=end

start=C 243 := 3^5 ; 5 times triple squiggles
~2~ G 91 182 364 728 (<729 :=3^6) with 91=13*7
~2~ D 68 136 272 (<273 := 3*G)
~2~ A 203 (<204 := 3*68) := 29*7 with J+S+B=18+9+2=29
~1- E 19 38 76 152 304 608 (<609 := 3*A) ; 3 single 'squiggles'
-1- B 57 := 19*3
-1~ F# 171 := 19*9
~3- C# 1...512 (<513 := 3*F#) ; the concluding 5 triply 'squiggles'
-3- G# 3
-3- D# 9 := 3^2
-3- A# 27 := 3^3
-3- F 81 := 3^4
-3. C=end 243 := 3^5

!Sparschuh_proposal_Mietke.scl
!absolute pitches from the middle_c' on:
c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
!
12
!
256/243 !_! ~90.22 ... C#(cents)
272/243 !_! ~195.81... D
32/27 !___! ~294.13... Eb
304/243 !_! ~387.74... E (5/4)*(1216/1215) Erasthostenes's comma
4/3 !_____! ~498.04... F
38/27 !___! ~591.65... F#
364/243 !_! ~699.58... G (3/2)*(728/729)
128/81 !__! ~792.18... G#
406/243 !_! ~888.63... A4=406cps ~ Coeval Berlin, Coethen Cammer-thone
16/9 !____! ~996.09... Bb
152/81 !__! ~1089.69.. B
2/1

sharpness of the 3rds as rational Neidhardt-Sorge Matrix

Bb 136/135 D 171/170 F# 96/95 A#

F 406/405 A 1024/1015 C# 96/95 F

C 1216/1215 E 96/95 G# 81/80(=sc) C

G 456/455 B 96/95 Eb 91/90 G

Approximation in modern Cent units: 

Bb ~12.8 D ~10.2 F# ~18.1 A#
F ~ 4.27 A ~15.3 C# ~18.1 F
C ~ 1.42 E ~18.1 G# ~21.5 C
G ~ 3.80 B ~18.1 Eb ~19.1 G

Ranking of the 3rds deviation, sharper than 5/4

1. C-E ~1.42
2. G-B ~3.80
3. F-A ~4.27
4. D-F# ~10.2
5. Bb-D ~12.8
------12-EDO has all 3rds the same ~13.7 Cents of, out of tune-----
6. A-C# ~15.3
7. F#-Bb = C#-F = E-G# = B-Eb ~18.1
8. Eb-G ~19.1
9. G#-C ~21.5 the Syntonic-Comma 

bye
A.S.

From: Andreas Sparschuh (2008-10-14)
Subject: A well-temperament with "at most 3 pure 5ths in a row" Re:...on the VRWT problem

--- In [email protected], "Carl Lumma" <carl@...> wrote:
> 
> Generally, WTs are too close to ET to be worth it if
> they have fewer than five pure 5ths, and too uneven to
> be "Victorian" if they have more than seven pure 5ths.
> 
> I had previously defined the VRWT bounds to be:
>  M3 range = 386.3 - 404.4 cents (5/4 - 24/19)
>  P5 range = 696.0 - 702.0 cents (432/289 - 3/2)
> ..... 
> Because I'm not allowing 81/64s (too sharp), there can
> be at most three pure 5ths in a row.... 

Hi Carl, Tom & all others,
How about my recent cycle of 5ths?

Verbose version:
A 110 220 440(cps or Hz)
E 329 (< 330 := 3*A)
B 987 := 3*E
F# 185 370 740 1480 2960 (< 2961 := 3*B)
C# 555 := 3*F#
G# 13 26 52 104 208 416 832 1664 (< 1665 := 3*C#)
Eb 39 := 3*G#
Bb 117 := 3*Eb
F 351 := 3*Bb
C 263 526 1052 (< 1053 := 3*F)
G 197 394 788 (< 789 := 3*C)
D (147 294 588 <) 589 (< 591 := 3*G)
A 440 (< 441 := 3*147)

In concise ratios:
A 329/330 E-B 2960/2961 F#-C# 1664/1665 G#-Eb-Bb-F 1052/1053 C...
...C 788/789 G 589/591 D 1760/1767 A

Approximation in Cents, for the amounts of flattening down the 5ths: 
A ~-5.25 E-B ~-0.58 F#-C# ~-1.04 G#-Eb-Bb-F ~-1.64 C...
...C ~-2.20 G ~-5.87 D ~-6.87 A

Comment:
Here the last 5th D-A exceeds Carl's bound:
> P5 range = 696.0 - 702.0 cents (432/289 - 3/2)
of 702-696=~6Cents only by little ~0.87 Cents lower.

!Sparschuh440well.scl
!
c526#555 d589#624 e658 f702#740 g788#832 a880#936 b987
!
555/526 ! 92.92 C#
589/526 ! 195.8 D
624/526 ! 195.8 Eb 
658/526 ! 387.6 E (5/4)*(1316/1315 ~+1.3C)
702/526 ! 499.7 F (4/3)*(1053/1052 ~+1.64C)
740/526 ! 591.0 F#
788/526 ! 699.8 G (3/2)*(788/789 ~-2.2C)
832/526 ! 793.3 G#
880/526 ! 890.9 A 
936/526 ! 997.7 Bb
987/526 ! 1089.6 B  
2/1
!
That results in:
Sharpnesses of the 3rds as Neidhardt-Sorge matrix:
Rational values vs. ~~approximation-in-Cents~~

Bb 589/ 585 ~11.8~ D 592 / 589 ~8.79~~  F# 936/925 ~20.4~~ Bb
F 352 / 351 ~4.93~ A 111 / 110 ~15.7~~  C# 936/925 ~20.4~~ F
C 1316/1315 ~1.32~ E 1664/1645 ~19.88~  G# 263/260 ~19.86~ C 
G 987 / 985 ~3.51~ B 1664/1645 ~19.88~  Eb 197/195 ~17.7~~ G

That yields in ascending order an
ranking of the 3rds, determined by the 
arising deviation away from pure 5/4 :

start
1. C-E : 1316/1315 ~1.32~ begin with the best fusing 3rd
2. G-B : 987 / 985 ~3.51~ continue with cummulative detuning 
3. F-A : 352 / 351 ~4.93~ the maximum inbetween the lower-keys
4. D-F#: 592 / 589 ~8.79~ almost twice soaring temperd
5. Bb-D: 589 / 585 ~11.8~ so far still better than ET
-------average---~13.7~cents---alike the common 12-EDO 3rds---
6. Eb-G: 197/195 ~17.7~  henceforward increasing worser than ET
7. G#-C: 263/260 ~19.86~ the remote keys should sound Pythagorean
8. E-G# = B-Eb : 1664/1645 ~19.88~ that both are the same amount sharp
9. F#-Bb = C#-F :  936/925 ~20.4~ finally, far the two worst ones
end

The last four 3rds in 8. and 9. do exceed Carl's personal bound:
>  M3 range = 386.3 - 404.4 cents (5/4 - 24/19)
of 386.3 - 404.4 = ~-18.1Cents
by
8. ~1.8 cent := 19.9-18.1 
9. ~2.3 cent := 20.4-18.1  
But Carl's limitations do sound at least
in my ears all to much near to most other ET approximations,
due to all to much weak "key-characteristics"

But meanwhile I do agree with Carl's condition 
of excluding the 81/64 ditone:
> Because I'm not allowing 81/64s (too sharp), there can
> be at most three pure 5ths in a row.... 
because that would result in an Pythagorean-3rd of 81/64 ~408~

Historically comment:
Kirnberger's 81/80 Pythagorean "ditone-3rds"  would even satisfy
http://en.wikipedia.org/wiki/Christian_Friedrich_Daniel_Schubart
's requirements
http://www.koelnklavier.de/quellen/schubart/_index.html
of early 19.th-century so called "key-characteristics",
as specified by examples in:
http://www.wmich.edu/mus-theo/courses/keys.html
Hence I do allow less restictive tolerances than Carl.

My new tempering follows now:
http://www.answers.com/topic/johann-heinrich-scheibler
's a4=440Hz normal pitch.
"Apart from his experiments in tuning and equal temperament, 
he is remembered for proposing the pitch a\u2032 = 440 as a standard 
at Stuttgart (1834)."

Observation:
Discern the subtle differences 
when modulating through the keys,
listen carefully to the intended refinements
against the stupid uniformity of ET;

How can you do that adequate?
Simply compare them on two different acoustic pianos, 
that are made preferable almost identical in construction: 

1. Tune the first piano in my abvoe tempering, 
leave against that the other one staying in: 
2. http://en.wikipedia.org/wiki/Equal_temperament

So that each one becomes tempered in the corresponding
http://en.wikipedia.org/wiki/Piano_key_frequencies
as given for the key-numbers in absolute pitches:

"Victorian" vs. ET 
1052> 64. 	c\u2032\u2032;\u2032 (3-line 8ve) 	C6 (Soprano C) 	1046.50
987 > 63. 	b\u2032\u2032 	B5 	987.767
936 < 62. 	a\u266f\u2032\u2032/b\u266d\u2032\u2032 	Ab39;5/B\u266d5 	932.328
880 = 61. 	a\u2032\u2032 	A5 	880.000
832 > 60. 	g\u266f\u2032\u2032/a\u266d\u2032\u2032 	G\u266f5/A\u266d5 	830.609
788 > 59. 	g\u2032\u2032 	G5 	783.991
740 > 58. 	f\u266f\u2032\u2032;/g\u266d\u2032\u2032 	F\u266f5/G\u266d5 	739.989
702 > 57. 	f\u2032&#8242; 	F5 	698.456
658 < 56. 	e\u2032\u2032 	E5 	659.255
624 > 55. 	d	839;\u2032\u2032/e\u266d\u2032\u2032 	D\u266f5/E\u266d5 	622.254
589 > 54. 	d\u2032\u2032 	D5 	587.330
555 > 53. 	c\u266f\u2032\u2032/d	837;\u2032\u2032 	C\u266f5/D\u266d5 	554.365
526 > 52. 	c\u2032\u2032 (2-line 8ve) 	C5 (Tenor C) 	523.251

Attend:
For proper stretching of the octaves in both acoustic pianos,
consider for each one an individual aligned:
http://en.wikipedia.org/wiki/Railsback_curve#The_Railsback_curve
according to the inharmonicity as given by the strings diameters.

Who in that group here dares also to try out 
my current tempering on his/hers own piano too?

Are there any suggestions for improvement?

bye
A.S.

From: Andreas Sparschuh (2008-10-16)
Subject: Staying within Carl's bounds,  was: Re: more time-wasting on the VRWT problem

--- In [email protected], "Carl Lumma" <carl@...> wrote:
> I had previously defined the VRWT bounds to be:
>  M3 range = 386.3 - 404.4 cents (5/4 - 24/19)
>  P5 range = 696.0 - 702.0 cents (432/289 - 3/2)
> ...
> ...in the VRWT range, and found:
> 
>  432/289     696  (  4  3 -2  0 )
more precisely: (1 200 * ln(432 / 289)) / ln(2) = ~695.954184....
>  323/216     697  ( -3 -3  1  1 )
>  256/171     699  (  8  2  0 -1 )
>  13851/9248  699  ( -5  6 -2  1 )
>  6137/4096   700  (-12  0  1  2 )
> 
Hi Carl & all others,

In order to stay within Carl's personal ~696 & ~404.4 conditions:

How about temper down the three 5ths inbetween G~D~A~E
on the empty stings of violn
http://upload.wikimedia.org/wikipedia/commons/a/a1/Violin_-_open_strings_notes.PNG

each down about ~5Cents flat?

G ~697 D ~697 A ~697 E

and then all the other 9 remaining 5ths down about ~1Cent flat:

E ~701 B ~701 F# ~701 C# ~701 G# ~701 Eb ~701 Bb ~701 F ~701 C ~701 G

PC =~24Cents := ~15Cents + ~9Cents = ~5Cents*3 + ~1Cent*9

so that (modulo 1200)

C 0
G ~701 := ~702 -~1
D ~198 := ~701G +~697 -1200
A ~895 := ~198D +~697
E ~392 := ~895A +697 -1200 
B ~1093 := ~392E +~701
F# ~594 := ~1093B +~701 -1200
C# ~ 95 := ~594F# +~701 -1200
G# ~796 := ~95C# +~701
Eb ~297 := ~796G# +~701 -1200
Bb ~998 := ~297Eb +~701
F ~499 := ~998Bb +~701 -1200
C 0 = ~499F +~701 -1200

!in_Carl_s_bounds.scl
!
12
!
C ~1 G ~5 D ~5 A ~5 E ~1 B ~F# ~1 C# ~1 G# ~1 Eb ~1 Bb ~1 F ~1 C
!
95 ! C#
198 ! D
297 ! Eb
392 ! E
499 ! F
594 ! F#
701 ! G
796 ! G#
895 ! A
998 ! Bb
1093 ! B
2/1
!
!

that yields 4 classes of 3rds

1. C-E & G-B : 392 = ~386 +~4
2. F-A & D-F# : 396 = ~386 +~10
3. Bb-D & A-C# : same 400 = ~386 +~14 as in ET
4. all 6 others: E-G#, B-Eb, F#-Bb, C#-F, G#-C, Eb-G : 404 = ~386 +~18

Even the last 4. class undercuts Carl's above
>  M3 range = 386.3 - 404.4 cents (5/4 - 24/19)  
by tiny ~0.4 Cents 

bye
A.S.

From: Carl Lumma (2008-10-16)
Subject: Staying within Carl's bounds,  was: Re: more time-wasting on the VRWT problem

Hi Andreas,

> Hi Carl & all others,
> 
> In order to stay within Carl's personal ~696 & ~404.4 conditions:
> 
> How about temper down the three 5ths inbetween G~D~A~E
> on the empty stings of violn

The 696 and 404 are my conditions for a VWT (Victorian
Well Temperament), but please note that in this conversation,
we are discussing VRWT (Victorian RATIONAL Well Temperaments).

We are trying to find a temperament in which all intervals
are rational numbers, and which has the highest number of
19-limit rational numbers, while still meeting ~696 and ~404.

-Carl

From: Andreas Sparschuh (2008-10-17)
Subject: Epimoric neo-Victorian, was Re: Staying within Carl's bounds 696 and 404

--- In [email protected], "Carl Lumma" <carl@...> wrote:
>
> The 696 and 404 are my conditions for a VWT (Victorian)...
>...we are discussing VRWT (Victorian RATIONAL Well Temperaments).
> 
>...19-limit rational numbers, while still meeting ~696 and ~404.
> 
Hi Carl & all others,

ok, let's try an modification 
of my recent neo-Baroque proposal:
http://launch.groups.yahoo.com/group/tuning/message/78740
which can easily got adapted more smooth, 
so that it keeps now even Carl's personal bounds:

C 3^5=243 486 (>485 = G/3)
G 1455 := 485*3
D (A/3 = 17 34 68 136 272 544<) 545 1090 2180 4360 (<4365 := G*3)
A 51 := 17*3
E (B/3 = 19 38 76 152 304<) 305 (<306 153 := A*3)
B 57 := E*3
F# 171 := B*3
C# (1...512<) 513 := F#*3
G# (Eb/3 = 3...1536<) 1539 := C#*3
Eb 9
Bb 27
F 81 
C 243=3^5

!neoVictorian_well.scl
!
12
!
C485/486 G872/873 D544/545 A305/306 E304/305 B F# C# G#512/513 EbBbF C
!
!C ~-3.57 G ~-1.98 D ~-3.18 A ~-5.67 E ~-5.69 B F# C# G#~-3.38 EbBbF C
!
19/18 ! C# (256/243)(513/512) Erasthostenes's 19-limit limma approx.
545/486 ! D (*)
32/27 !   Eb
305/243 ! E (*)
4/3 !     F
38/27 !   F#
485/324 ! G (*)
19/12 !   A
136/81 !  Bb
16/9 !    B
2/1
!
!
Attend: Some ratios (*) here do exceed 19-limit,
because i do consider synchronous beatings more important,
than maintainig any arbitrary limits.

3rds sharpnesses: Epimoric-ratios vs. ~approx. in 'c'ents

G-B : 2432/2425 = (347+3/7)/(346+3/7) ~+4.99c
D-F# : 2736/2725 = (248+8/11)/(247+8/11) ~+6.97c
C-E : 244/243 ~+7.11c
A-C# : 171/170 ~+10.2c
F-A : 136/135 ~+12.8c
------average ~+13.7c--as all 5ths in 12-EDO----------
E-G# : 1539/1525 = (109+13/14)/(108+13/14) ~+15.8c
Bb-D : 109/108 ~+16.0c
Eb-G : 97/96 ~+17.9c
B-Eb, F#-Bb, C#-F, G#-C all: 96/95 ~+18.1c := ~404.4c -~386.3c

For those, that do prefer: C-E lower tempered than G-B;
i do reccomend to shift the cyle by one 5th downwards to the left.

bye
A.S.

From: Carl Lumma (2008-10-17)
Subject: Epimoric neo-Victorian, was Re: Staying within Carl's bounds 696 and 404

Hi Andreas,

> ok, let's try an modification 
> of my recent neo-Baroque proposal:
> http://launch.groups.yahoo.com/group/tuning/message/78740
> which can easily got adapted more smooth, 
> so that it keeps now even Carl's personal bounds:
//
> !neoVictorian_well.scl
> !
> 12
> !
//
> !
> 19/18 ! C# (256/243)(513/512) Erasthostenes's 19-limit limma approx.
> 545/486 ! D (*)
> 32/27 !   Eb
> 305/243 ! E (*)
> 4/3 !     F
> 38/27 !   F#
> 485/324 ! G (*)
> 19/12 !   A
> 136/81 !  Bb
> 16/9 !    B
> 2/1
> !

I think we're missing G# here.

-Carl

From: Andreas Sparschuh (2008-10-17)
Subject: Correction: Epimoric neo-Victorian, Re:within Carl's bounds 696 and 404

--- In [email protected], "Carl Lumma" <carl@...> wrote:
 
> > so that it keeps now even Carl's personal bounds:

Sorry Carl, 
appearently I had forgotten to include
the line with notename G# and its ratio,
when transferring from the prefiguration:

Corrected version:

!neoVictorian_well.scl
!
12
!
C 485/486 G 872/873 D 544/545 A 305/306 E 304/305 B...
! ...B F# C# G#512/513 Eb Bb F C
!
!C~-3.57 G~-1.98 D~-3.18 A~-5.67 E~-5.69 B F# C# G#~-3.38 Eb Bb F C
!
19/18 ! C# (256/243)(513/512) Erasthostenes's 19-limit limma approx.
545/486 ! D (*)
32/27 !   Eb
305/243 ! E (*)
4/3 !     F
38/27 !   F#
485/324 ! G (*)
19/12 !   G#
136/81 !  A
16/9 !    Bb
152/81 !  B
2/1
!
!
But now all the note-names got finally completed to an full dozen :-)
> 
> I think we're missing G# here.
>
Correct observation,
I simply forgot in a hurry 
to overtake the lacking G# from the 5ths cycle.

Many thanks for the advice 
of makeing me aware 
to fix that transmission - bug.

Quest:
Is that epimoric version now acceptable so for you,
even if it contains three spots of bother;
in excceding 19-limit at the pitches D,E and G?

bye
A.S.

From: Carl Lumma (2008-10-18)
Subject: Correction: Epimoric neo-Victorian, Re:within Carl's bounds 696 and 404

Hi Andreas,

> Corrected version:
> 
> !neoVictorian_well.scl
> !
> Andreas Sparschuh's neovictorian WT.
> 12
> !
> 19/18 ! C# (256/243)(513/512) Erasthostenes's 19-limit limma approx.
> 545/486 ! D (*)
> 32/27 !   Eb
> 305/243 ! E (*)
> 4/3 !     F
> 38/27 !   F#
> 485/324 ! G (*)
> 19/12 !   G#
> 136/81 !  A
> 16/9 !    Bb
> 152/81 !  B
> 2/1
> !

It meets the P5 and M3 bounds I suggested.  Here's a tally
of 19-limit intervals (below 600 cents):

Target consonances:
 19/18 = 3
 18/17 = 1
 17/16 = 0
 19/17 = 1
 9/8   = 4
 19/16 = 3
 24/19 = 4
 4/3   = 6
 24/17 = 1

That makes the total 23, which is one more than Tom's
scale but 8 less than mine.

It has 5 404-cent M3s, compared to 3 in my scale and 2 in
Tom's.  It has 2 696-cent P5s, compared to 3 in mine and
zero in Tom's.

-Carl

From: Carl Lumma (2008-10-20)
Subject: Re: more time-wasting on the VRWT problem

I'm now able to say that there is only one assortment of
2.3.17.19 rational 5ths, containing 6 or 7 pure 5ths and 2
other sizes of 5th, that closes exactly to 7 octaves,
where all 5ths are within my size bounds and are found
within a ball of 'radius' 7 from the origin of the 3.7.19
lattice.  And here it is:

7 * 3/2
4 * (-3 1 1) (323/216 696.6 cents)
1 * (5 -4 -4)) (16307453952/10884540241 699.9 cents)

Next time: I'll look for assortments where there are
3 sizes of nonpure 5ths, and then test chain patterns to
meet my bounds on the 3rds and optimize the key contrast
cycles.

-Carl

From: Carl Lumma (2008-10-21)
Subject: Re: more time-wasting on the VRWT problem

I just tried all 8 unique permutations of this chain of
fifths (that give the maximum number of 9/8s without
any 81/64s), and the worst 3rd can be kept to a mild
402.5 cents, but the number of 19-limit intervals is
only 22, which is miserable compared to either Tom's or
my own previous VRWTs.

On to 3 types of nonpure 5th...

-Carl

I wrote:

> I'm now able to say that there is only one assortment of
> 2.3.17.19 rational 5ths, containing 6 or 7 pure 5ths and 2
> other sizes of 5th, that closes exactly to 7 octaves,
> where all 5ths are within my size bounds and are found
> within a ball of 'radius' 7 from the origin of the 3.7.19
> lattice.  And here it is:
> 
> 7 * 3/2
> 4 * (-3 1 1) (323/216 696.6 cents)
> 1 * (5 -4 -4)) (16307453952/10884540241 699.9 cents)
> 
> Next time: I'll look for assortments where there are
> 3 sizes of nonpure 5ths, and then test chain patterns to
> meet my bounds on the 3rds and optimize the key contrast
> cycles.
> 
> -Carl

From: Tom Dent (2008-10-21)
Subject: Re: more time-wasting on the VRWT problem

I think the lack of pure intervals is due to 'one-dimensionality' in
the assortment of (non-3/2) fifths - four go along the same diagonal
in 3.17.19 space and the last has to get all the way back in the 17-
and 19-directions ... the chance of combining some group of these to
get a simple 19-limit interval is small.

As a 'gut feeling' I would think the arithmetical simplicity of each
fifth was the first thing to aim for, and after that to have several
*different* fifths (pointing in different 17.19 directions) such that
some of the 17- and 19-factors can cancel each other in useful ways.
To restrict the variety of 'tempered' fifths could even be the worst
strategy, because identical rational fifths can't add up to simple
rational intervals.
~~~T~~~

--- In [email protected], "Carl Lumma" <carl@...> wrote:
>
> I just tried all 8 unique permutations of this chain of
> fifths (that give the maximum number of 9/8s without
> any 81/64s), and the worst 3rd can be kept to a mild
> 402.5 cents, but the number of 19-limit intervals is
> only 22, which is miserable compared to either Tom's or
> my own previous VRWTs.
> 
> On to 3 types of nonpure 5th...
> 
> -Carl
> 
> I wrote:
> 
> > 
> > 7 * 3/2
> > 4 * (-3 1 1) (323/216 696.6 cents)
> > 1 * (5 -4 -4)) (16307453952/10884540241 699.9 cents)
> >

From: Carl Lumma (2008-10-21)
Subject: Re: more time-wasting on the VRWT problem

Hi Tom,

> I think the lack of pure intervals is due to 'one-dimensionality' in
> the assortment of (non-3/2) fifths - four go along the same diagonal
> in 3.17.19 space and the last has to get all the way back in the 17-
> and 19-directions ... the chance of combining some group of these to
> get a simple 19-limit interval is small.

Yup.  When I saw the 4 1, I knew it wouldn't be a winner, but
it's the only such arrangement that closes the circle of fifths
so I thought I'd report it.

> As a 'gut feeling' I would think the arithmetical simplicity of each
> fifth was the first thing to aim for,

Which is why I find 5ths by radius in 3.17.19 space.

> and after that to have several *different* fifths (pointing in
> different 17.19 directions) such that some of the 17- and
> 19-factors can cancel each other in useful ways.

Just wanted to try the simplest case first.  Fewer permutations
to wrangle with.  Of course there can't be just one tempered
5th, because something has to put in, and then take out, the
17s and 19s.

> To restrict the variety of 'tempered' fifths could even be
> the worst strategy, because identical rational fifths can't
> add up to simple rational intervals.

That's why I'm now working on 3 kinds of tempered 5ths.
I just wrote the code to test all the permutations, but I
have to go to work so I won't be able to work with it
until later.

If I don't find anything good I suppose I'll go up to 4
sizes of tempered 5ths, and so on.

-Carl

From: Charles Lucy (2008-10-22)
Subject: LucyTuned Radiohead mix

This might amuse the Radiohead fans.

http://www.radioheadremix.com/remix/?id=1289



Give it a vote if you like it.



Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

From: Carl Lumma (2008-10-22)
Subject: Re: LucyTuned Radiohead mix

I do like that, quite a bit.  Voted it up.

-Carl

--- In [email protected], Charles Lucy <lucy@...> wrote:
>
> This might amuse the Radiohead fans.
> 
> http://www.radioheadremix.com/remix/?id=1289
> 
> Give it a vote if you like it.
> 
> Charles Lucy
> lucy@...
>

From: Carl Lumma (2008-10-22)
Subject: Re: LucyTuned Radiohead mix

In fact, it's much better than the
current #1 remix (NASTY FISH RMX).

--- In [email protected], "Carl Lumma" <carl@...> wrote:
>
> I do like that, quite a bit.  Voted it up.
> 
> -Carl
> 
> --- In [email protected], Charles Lucy <lucy@> wrote:
> >
> > This might amuse the Radiohead fans.
> > 
> > http://www.radioheadremix.com/remix/?id=1289
> > 
> > Give it a vote if you like it.
> > 
> > Charles Lucy
> > lucy@
> >
>

From: Carl Lumma (2008-10-22)
Subject: Re: more time-wasting on the VRWT problem

I wrote:
> On to 3 types of nonpure 5th...

==WITH SIX PURE 5THS==

In radius 4, there is only one assortment that closes to
the octave:
   (((-2 0 -1) (-1 2 4) (3 -2 0)) (6 4 1 1))

How to read this... it's the assortment with:
   6 * 3/2
   4 * (-2 0 -1) = 3^-2 * 17^0 * 19^-1 = 256/171
   1 * (-1 2 4)
   1 * (3 -2 0)

In radius 5, there are five additional assortments:
   (((-2 0 -1) (-1 2 4) (2 -4 -5)) (6 3 2 1))
   (((-3 1 1) (-1 -1 -3) (5 -1 3)) (6 3 2 1))
   (((-4 -1 -4) (-1 2 4) (5 -4 -4)) (6 2 3 1))
   (((-3 1 1) (1 0 0) (5 -4 -4)) (6 4 1 1))
   (((-3 1 1) (-2 0 -1) (5 -1 3)) (6 1 4 1))

I wasn't absolutely rigorous (too lazy to do the symmetries
on these permutations right), but I'm fairly certain this
last assortment wins among those I've listed so far, with
52 19-limit dyads if you use one of these chain orderings:
   (p p p b p p p a b b c b)
   (p p p b p p p b b c b a)
   (p p p b p p p a b c b b)
   (p p p b p p p b c b b a)
Where p = 3/2 and a, b, and c are mapped to the above 19-limit
fifths in order from left to right.

I went all the way up to radius 10, and no (6 2 2 2) assortment
exists.

==WITH SEVEN PURE 5THS==

In radius 3, there are two assortments that close:
   (((-3 1 1) (-1 -1 -3) (3 -2 0)) (7 3 1 1))
   (((-3 1 1) (-2 0 -1) (3 -2 0)) (7 2 2 1))

In radius 4, there are two more:
   (((-3 1 1) (-2 0 -1) (4 -3 -2)) (7 3 1 1))
   (((-4 -1 -4) (-1 2 4) (3 -2 0)) (7 2 2 1)

I checked the results by eye so hopefully I didn't miss
something, but of these four assortments one has the most
19-limit dyads:
   (((-3 1 1) (-2 0 -1) (3 -2 0)) (7 2 2 1))
And it hits those 66 intervals with only one chain ordering:
   (p p p b p p p b p a c a)
Whoops, that's not true, here are two more:
   (p p p b c a p p p b p a)
   (p p p a c b p p p a p b)

I believe these are the same three 5ths I use in lumma_vrwt,
but the chain ordering there gives only 62 19-limit dyads.

So it's not totally settled, but I'd be very surprised if
66 isn't the max for these.  I'll buy you the microtonal CD
or mp3 download of your choice if you can beat it.

-Carl

From: Carl Lumma (2008-10-22)
Subject: Re: more time-wasting on the VRWT problem

I wrote:

> I checked the results by eye so hopefully I didn't miss
> something, but of these four assortments one has the most
> 19-limit dyads:
>    (((-3 1 1) (-2 0 -1) (3 -2 0)) (7 2 2 1))
> And it hits those 66 intervals with only one chain ordering:
>    (p p p b p p p b p a c a)
> Whoops, that's not true, here are two more:
>    (p p p b c a p p p b p a)
>    (p p p a c b p p p a p b)

The last two chain orderings are some kind of reflection
of eachother.  Either of them give 3 largest Maj 3rds,
whereas the first chain ordering gives 6 largest 3rds.

So here's what I think is the best key of the 2nd ordering:

!
 New VRWT (P P P b c a P P P b P a)
 12
!
 18/17
 323/288
 19/16
 64/51
 4/3
 24/17
 3/2
 19/12
 57/34
 16/9
 32/17
 2/1
!
! 66 19-limit dyads

Next time: checking scales with 5 pure 5ths and 3 other sizes
of fifth.

-Carl

From: Danny Wier (2008-10-22)
Subject: Re: [tuning] LucyTuned Radiohead mix

Hey, I like it! I wish I could do remixes. Anyway, I voted for it. ~D.

Charles Lucy wrote:
> This might amuse the Radiohead fans. 
> <http://www.radioheadremix.com/remix/?id=1289>
>
> http://www.radioheadremix.com/remix/?id=1289
>
>
>
> Give it a vote if you like it.
>

From: Charles Lucy (2008-10-22)
Subject: Re: [tuning] LucyTuned Radiohead mix - Melodyne  DNA capabilities

Thanks for the support.

I am looking forward to finding out how well the Melodyne DNA will be  
able to "retune" previously 12edo tuned tracks.

http://www.celemony.com/cms/index.php?id=dna



On 22 Oct 2008, at 18:18, Danny Wier wrote:

> Hey, I like it! I wish I could do remixes. Anyway, I voted for it. ~D.
>
> Charles Lucy wrote:
> > This might amuse the Radiohead fans.
> > <http://www.radioheadremix.com/remix/?id=1289>
> >
> > http://www.radioheadremix.com/remix/?id=1289
> >
> >
> >
> > Give it a vote if you like it.
> >
>
> 

Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

From: Tom Dent (2008-10-26)
Subject: Re: more time-wasting on the VRWT problem

I replied to this earlier but that message seemed to have got eaten.
(Just as well in one respect) Anyway... see below

--- In [email protected], "Carl Lumma" <carl@...> wrote:
>
> here's what I think is the best key of the 2nd ordering:
> 
> !
>  New VRWT (P P P b c a P P P b P a)
>  12
> !
>  18/17
>  323/288
>  19/16
>  64/51
>  4/3
>  24/17
>  3/2
>  19/12
>  57/34
>  16/9
>  32/17
>  2/1
> !
> ! 66 19-limit dyads
> 
> Next time: checking scales with 5 pure 5ths and 3 other sizes
> of fifth.
> 
> -Carl

Yes, this seems arithmetically unbeatable, particularly considering
the 'full sweep' of 17/12's.

Concerning the 6-pure-fifths possibilities, I might as well give one
of the preliminary stages before my temp with 4 pure fifths. The
ratios are

18/17 
64/57 
384/323 
6912/5491 
432/323 
24/17 
3/2 
27/17
486/289 
576/323 
10368/5491
2/1

For the tally of just intervals I get one 19/18, six 18/17, two 17/16,
two 9/8, three 19/16, two 24/19, six 4/3 and four 17/12: total 26 or
52 including inversions. 
The 'tempered' fifths are 
a) 256/171 (3x), 
b) 323/216 (2x) and 
d) 4617/4624 (1x). 
(If one rearranges to get a sequence P b d b P a P P a P P a round the
circle then there are somewhat more 19/16's and 24/19's, but the
others suffer.)

So by splitting up the 6 pure fifths further (compared with two sets
of P P P) the total consonances don't necessarily suffer, while
32/27's can be eliminated.
~~~T~~~

From: Carl Lumma (2008-10-26)
Subject: Re: more time-wasting on the VRWT problem

--- In [email protected], "Tom Dent" <stringph@...> wrote:
>
> I replied to this earlier but that message seemed to have got eaten.

That does happen sometimes with yahoo.

> > ! 66 19-limit dyads
//
> 
> Yes, this seems arithmetically unbeatable, particularly considering
> the 'full sweep' of 17/12's.
//
> So by splitting up the 6 pure fifths further (compared with two sets
> of P P P) the total consonances don't necessarily suffer, while
> 32/27's can be eliminated.

You're right.  And not only don't they suffer, they can even beat
arithmetic.  I had wrongly assumed that groups of 3 Ps were
necessary because of the double-duty of 3 & 9.  This has 68
19-limit dyads (including inversions):

!
 (P P b P P a P P b P c a) a=(-3 1 1) b=(-2 0 -1) c=(3 -2 0)
 12
!
 18/17
 9/8
 19/16
 171/136
 171/128
 24/17
 3/2
 27/17
 323/192
 57/32
 513/272
 2/1
!

It still has the full sweep of 17-limit tritones.  One mode
of it has only one fraction with three digits in the numerator.

-Carl

From: Carl Lumma (2008-10-26)
Subject: Re: more time-wasting on the VRWT problem

--- In [email protected], "Carl Lumma" <carl@...> wrote:
>
> --- In [email protected], "Tom Dent" <stringph@> wrote:
> >
> > I replied to this earlier but that message seemed to have
> > got eaten.
> 
> That does happen sometimes with yahoo.

And it seems to be happening to me at the moment.  -Carl

From: Tom Dent (2008-10-24)
Subject: Re: more time-wasting on the VRWT problem

--- In [email protected], "Carl Lumma" <carl@...> wrote:
>
> here's what I think is the best key of the 2nd ordering:
> 
> !
>  New VRWT (P P P b c a P P P b P a)
>  12
> !
>  18/17
>  323/288
>  19/16
>  64/51
>  4/3
>  24/17
>  3/2
>  19/12
>  57/34
>  16/9
>  32/17
>  2/1
> !
> ! 66 19-limit dyads

Yes, I have no idea how to improve on this arithmetically.
Particularly impressed by the 'full house' (well, six-of-a-kind) of
17/12's.

> Next time: checking scales with 5 pure 5ths and 3 other sizes
> of fifth.

Was there any reason not to use 6 pure fifths?

I thought I may as well display an intermediate stage of my working
towards the 'more complicated' RWT that had 4 pure fifths and a worst
fifth of about 698.6. The present tuning has 6 pure fifths and the
worst fifth is 323/216, almost exactly quarter-comma.

18/17
64/57
384/323
6912/5491
432/323
24/17
3/2
27/17
486/289
576/323
10368/5491
2/1

I make the tally one 19/18, six 18/17, two 17/16, two 9/8, three
19/16, two 24/19, six 4/3 and four 17/12 for a total of 26
'consonances'. I don't know if there is any better for this collection
of fifths.

If we call this one 
(P a d a P b P P a P P b), 
then ringing some changes we get
(P b d b P a P P a P P a)
with the sequence P a P P a P P a P giving you seven 19/16's and six
24/19's, but many of the 17-consonances would be lost.
~~~T~~~

From: Carl Lumma (2008-10-26)
Subject: Tying up VRWTs with 3 sizes of nonpure 5ths

So I reworked my code to try more permutations.  It's still
not completely exhaustive, but at this point I'll eat a bug
if these aren't right:

For nonpure 5ths within radius 10 of the 3.17.19 origin...

With 5 pure 5ths the greatest number of 19-limit dyads
possible is 52:
(p p p a p p a b c a b a) a=(-2 0 -1) b=(0 1 2) c=(3 -2 0)

With 6 pure 5ths, the number is 56:
(p p p b p p p a b c b a) a=(-3 1 1) b=(-2 0 -1) c=(6 -2 1)

With 7 pure 5ths, the number is 68:
(p p b p p a p p b p c a) a=(-3 1 1) b=(-2 0 -1) c=(3 -2 0)

Each of these scales uniquely hits the maximum number of
19-limit consonances in its category.

-Carl

From: Carl Lumma (2008-10-27)
Subject: Re: Tying up VRWTs with 3 sizes of nonpure 5ths

I wrote:

> With 7 pure 5ths, the number is 68:
> (p p b p p a p p b p c a) a=(-3 1 1) b=(-2 0 -1) c=(3 -2 0)
> 
> Each of these scales uniquely hits the maximum number of
> 19-limit consonances in its category.

Heh, not quite.  (P P P b P P a P P b c a) also has 68
intervals.  And despite having one 32/27 and one non-17-limit
tritone, its pattern of M3s is more equal, and its key-color
pattern is more sinusoidal on the circle of fifths.

!
 (P P P b P P a P P b c a) a=(-3 1 1) b=(-2 0 -1) c=(3 -2 0)
 12
!
 19/18
 323/288
 19/16
 64/51
 4/3
 24/17
 3/2
 19/12
 57/34
 16/9
 32/17
 2/1
!

As for 4 different nonpure 5ths... with 5 pure 5ths and
searching to radius 5, we get 19-limit intervals = 54:
(p p b c p p a p b c d a)
a=(-3 1 1) b=(-1 -1 -3) c=(0 1 2) d=(3 -2 0)

With 6 pure 5ths and radius 5, we get 60:
(p p p b p p p b c b d a)
a=(-3 1 1) b=(-2 0 -1) c=(0 1 2) d=(3 -2 0)

With 7 pure 5ths and radius 9, we get 64:
(p p b p p a p p b p c d)
a=(-3 1 1) b=(-2 0 -1) c=(-2 3 6) d=(2 -4 -5)

With this last, I think I broke Scala.  309237645312/206806264579
shows up in cents even though it's rational in the .scl file.

-Carl