Wier53

From: Andy (2012-04-24)
Subject: Re: Completely 5-limit symmetric 53-tone, as generalized Kirnberger #1

--- In [email protected], "genewardsmith" <genewardsmith@...> wrote:
>
> Amity[53] with pure 3/2 fifths is far closer to 53 equal ....
> 
> ! amity53pure.scl
> !
> Amity[53] in pure-fifths tuning
>  53
> !
>  22.73700  ! A
>  45.47400  ! B 
>  68.21100  ! C 
>  90.22500  ! D
>  112.96200 ! E 
>  135.69900 ! F
>  158.43600 ! G 
>  181.17300 ! H
>  203.91000 ! I 
>  226.64700 ! J
>  249.38400 ! K
>  271.39800 ! L
>  294.13500 ! M
>  316.87200 ! N
>  339.60900 ! O
>  362.34600 ! P 
>  385.08300 ! Q 
>  407.82000 ! R
>  429.83400 ! S 
>  452.57100 ! T
>  475.30800 ! U 
>  498.04500 ! V 
>  520.78200 ! W
>  543.51900 ! X
>  566.25600 ! Y 
>  588.99300 ! Z
>  611.00700 ! z
>  633.74400 ! Y 
>  656.48100 ! x
>  679.21800 ! w
>  701.95500 ! v
>  724.69200 ! u
>  747.42900 ! t
>  770.16600 ! s
>  792.18000 ! r
>  814.91700 ! q
>  837.65400 ! p
>  860.39100 ! o
>  883.12800 ! n
>  905.86500 ! m
>  928.60200 ! l
>  950.61600 ! k
>  973.35300 ! j
>  996.09000 ! i
>  1018.8270 ! h
>  1041.5640 ! g
>  1064.3010 ! f
>  1087.0380 ! e
>  1109.7750 ! d
>  1131.7890 ! c
>  1154.5260 ! b
>  1177.2630 ! a
>  2/1       ! @'
> !
> ![eof] 

in deed Gene,
that sounds almost undistinguishable alike 53-edo.
for comparision against 53-edo, see:
http://xenharmonic.wikispaces.com/53edo
http://en.wikipedia.org/wiki/53_equal_temperament

But how about Danny Wier's [2002] proposal, 
of modification:
Here compiled into an scala-file:


! Wier53.scl
Danny Wier's schismatically-altered 53-Pythagorgean scale (2002)
53
! http://launch.groups.yahoo.com/group/tuning/message/38888
!
64/63   ! A
36/35   ! B
28/27   ! C
135/128 ! D
16/15   ! E
243/224 ! F
35/32   ! G
10/9    ! H 
9/8     ! I
8/7     ! J
81/70   ! K
7/6     ! L
32/27   ! M
6/5     ! N
128/105 ! O
315/256 ! P
5/4     ! Q
81/64   ! R
9/7     ! S
35/27   ! T
21/16   ! U
4/3     ! V
27/20   ! W
48/35   ! X
112/81  ! Y
45/32   ! Z
64/45   ! z 
81/56   ! y 
35/24   ! x
40/27   ! w
3/2     ! v
32/21   ! u
54/35   ! t
14/9    ! s
128/81  ! r
8/5     ! q
512/315 ! p 
105/64  ! o
5/3     ! n
27/16   ! m
12/7    ! l
140/81  ! k 
7/4     ! j
16/9    ! i
9/5     ! h
64/35   ! g
448/243 ! f
15/8    ! e
256/135 ! d
27/14   ! c
35/18   ! b
63/32   ! a
2/1     ! @'
!
![eof]

Idea:
Restrict the whole 5ths-cycle completely within 5-limit ratios:

1/1=D. A. E. B. F# -schisma=|15 -8,-1>=~-1.95cents  C# G# D# A# ... 
...F+ C+ G+ D+ A+ E+=F- Monzisma=|54 -37,2>=~+0.29c B+=C- G- D- A-... 
...Gb Db Ab Eb Bb -schisma=|15 -8,-1>=~-1.95cents F. C. G. D.=1/1

That got executed by 5-limit trisection of Mercator's-comma

|-84 53> = 2*(|-15 8,1>) -|54 -37,2> =~3.62...cents

More verbose en detail:

+-0 D. 1/1 unison 
+ 1 A. 3/2
+ 2 E. 9/8 major-tone
+ 3 B. 27/16
+ 4 F# 81/64 ! end of Pythagorean 3-limit domain

-schisma=|15 -8,-1>=~-1.95cents

+ 5 C# 256/135 ! begin with syntonic 5-limit domain
+ 6 G# 64/45
+ 7 D# 16/15
+ 8 A# 8/5
+ 9 F/ 6/5 minor-3rd
+10 C/ 9/5
+11 G/ 27/20
+12 D/ 81/80 syntonic-comma
+13 A/ 243/160
+14 E/ 729/640
+15 B/ 2,187/1,280
+16 F& 6,561/5,120 = |-10 8,-1> F#/ 
+17 C& 19,683/10,240 = |-11 9,-1>
+18 G& 59,049/40,960 = |-13 10,-1>
+19 D& 177,147/163,840 = |-15 11,-1>
+20 A& 531,441/327,680 = |-16 12,-1> A#/
+21 F+ 1,594,323/1,310,720 = |-18 13,-1> F//
+22 C+ 4,782,969/2,621,440 = |-19 14,-1>
+23 G+ 14,348,907/10,485,760 = |-21 15,-1>
+24 D+ 43,046,721/41,943,040 = |-23 16,-1>
+25 A+ 129,140,163/83,886,080 = |-24 17,-1>
+26 E+=E// 387,420,489/335,544,320 = |-26 18,-1> F-=F\\

+Monzisma = |54 -37,2> =~+0.29cents at the symmetric mirror-axis

-26 C-=C\\ 671,088,640/387,420,489 = |27 -18,1> B+=B//
-25 G- 167,772,160/129,140,163 = |25 -17,1>
-24 D- 83,886,080/43,046,721 = |24 -16,1> 
-23 A- 20,971,520/14,348,907 = |22 -15,1>
-22 E- 5,242,880/4,782,969 = |20 -14,1> E\\=FB=Fb\
-21 B- 5,242,880/4,782,969 = |19 -13,1> B\\=CB=Cb\
-20 GB 655,360/531,441 = |17 -12,1> Gb\
-19 DB 327,680/177,147 = |16 -11,1>
-18 AB 81,920/59,049 = |14 -10,1>
-17 EB 20,480/19,683 = |12 -9,1>
-16 BB 10,240/6,561 = |11 -8,1> Bb\
-15 F\ 2,560/2,187 = |9 -7,1>
-14 C\ 1,280/729 = |8 -6,1>
-13 G\ 320/243
-12 D\ 160/81 inverse-SC
-11 A\ 40/27
-10 E\ 10/9 minor-tone
- 9 B\ 5/3 JI-sixth
- 8 Gb 5/4 JI-tierce
- 7 Db 15/8 JI-7th
- 6 Ab 45/32
- 5 Eb 135/128 end of syntonic 5-limit domain

-schisma=|15 -8,-1>=~-1.95cents

- 4 Bb 128/81 return back again to Pythagorean 3-limit domain 
- 3 F. 32/27 
- 2 C. 16/9 Pyth. minor 7th
- 1 G. 4/3 quarte 
+-0 D. 1/1 unison, 53-cycle closed

The same sounds in terms of an "Scala"-file:

! Sp53tone5limit.scl
!
Sparschuh's tri-section of Mercator's-comma into (schisma)*2-Monzisma 
53
!
81/80               ! A | -4   4,-1>  D/ syntonic-comma
43046721/41943040   ! B |-23  16,-1>  D+
20480/19683         ! C | 12  -9, 1>  EB 
135/128             ! D | -7   3, 1>  Eb
16/15               ! E |  4  -1,-1>  D#
177147/163840       ! F |-15  11,-1>  D&  
5242880/4782969     ! G | 20 -14, 1>  E-   
10/9                ! H |  1  -2, 1>  E\
9/8                 ! I | -3   2>     E.
729/640             ! J | -7   6,-1>  E/   
387420489/335544320 ! K |-26  18,-1>  E+=F-
2560/2187           ! L |  9  -7, 1>  F\
32/27               ! M |  5  -3>     F.    
6/5                 ! N |  1   1,-1>  F/
1594323/1310720     ! O |-18  13,-1>  F+ 
655360/531441       ! P | 17 -12, 1>  GB
5/4                 ! Q | -2   0, 1>  Gb
81/64               ! R | -6   4>     F#
6561/5120           ! S |-10   8,-1>  F&
167772160/129140163 ! T | 25 -17, 1>  G-
320/243             ! U |  6  -5, 1>  G\
4/3                 ! V |  2  -1>     G.
27/20               ! W | -2   3,-1>  G/
14348907/10485760   ! X |-21  15,-1>  G+
81920/59049         ! Y | 14 -10, 1>  AB
45/32               ! Z | -5   2, 1>  Ab
! <<<=======symmetric-mirror-axis=======>>> !
64/45               ! z |  6  -2,-1>  G#
59049/40960         ! y |-13  10,-1>  G&
20971520/14348907   ! x | 22 -15, 1>  A-
40/27               ! w |  3  -3, 1>  A\
3/2                 ! v | -1   1>     A.
243/160             ! u | -5   5,-1>  A/
129140163/83886080  ! t |-24  17,-1>  A+
10240/6561          ! s | 11  -8, 1>  BB 
128/81              ! r |  7  -4>     Bb
8/5                 ! q |  3   0,-1>  A#
531441/327680       ! p |-16  12,-1>  A&
5242880/4782969     ! o | 19 -13, 1>  B- 
5/3                 ! n |  0  -1, 1>  B\
27/16               ! m | -4   3>     B.
2187/1280           ! l | -8   7,-1>  B/
671088640/387420489 ! k | 27 -18, 1>  B+=C-
1280/729            ! j |  8  -6, 1>  C\
16/9                ! i |  4  -2>     C.
9/5                 ! h |  0   2,-1>  C/
4782969/2621440     ! g |-19  14,-1>  C+
327,680/177,147     ! f | 16 -11, 1>  DB
15/8                ! e | -3   1, 1>  Db
256/135             ! d |  8  -3,-1>  C#
19683/10240         ! c |-11   9,-1>  C&
83886080/43046721   ! b | 24 -16, 1>  D-
160/81              ! a |  5  -4,-1>  D\
2/1                 ! @'|  1>         D.

Remark:
That 53-cycle includes exactly the first attempt of
http://en.wikipedia.org/wiki/Kirnberger_temperament
as 12-out-of-53 dodecatonic subset:

A. E. B. F# -schisma C# G# D# A# F/ C/ G/ D/ 80/81=-SC A.

coincident at the same labeling of pitch-names.

Attend:
K1 contains inbetween the quinte: D/ and A. 
an ugly syntonic 'wolf'-5th of 
40/27 = (3/2)%(81/80) =~680.4...cents
that most tuners consider as unacceptable.

See deeper for more details:
http://groenewald-berlin.de/text/text_T001.html
http://groenewald-berlin.de/tabellen/TAB-001.html
http://groenewald-berlin.de/graphik-tabelle/GRA-001.html
Sorry that's only in german-language available :-(

bye
Andy

From: Steve Cullinane (2001-09-28)
Subject: 34-tone ET scale

The 53-tone equal-temperament (ET) scale is well-known as giving a 
very good approximation to "just" intervals.  The following websites 
make the case that the next-best ET scale is the 34-tone ET scale.

The Harmony Problem -- 
http://m759.freeservers.com/harmony.html

Natural Temperament --
http://m759.freeservers.com/natural.html

Comments are welcome.

-- Steve Cullinane (m759)

From: Paul Erlich (2001-09-28)
Subject: Re: 34-tone ET scale

--- In tuning-math@y..., "Steve Cullinane" <m759@p...> wrote:
> The 53-tone equal-temperament (ET) scale is well-known as giving a 
> very good approximation to "just" intervals.  The following 
websites 
> make the case that the next-best ET scale is the 34-tone ET scale.
> 
> The Harmony Problem -- 
> http://m759.freeservers.com/harmony.html
> 
> Natural Temperament --
> http://m759.freeservers.com/natural.html
> 
> Comments are welcome.
> 
> -- Steve Cullinane (m759)

34-tET was discussed by Larry Hanson and others before 1980. But it 
unfortunately suffers from the famous "comma problem" in diatonic 
common-practice music. Therefore it has not been very important 
historically. Tunings that eliminate the comma, such as 19-tET, 31-
tET, 43-tET, 55-tET, have been much more important historically, with 
19- and 31-tone keyboards dating from as early as the 16th century.

From: [email protected] (2001-09-28)
Subject: Re: 34-tone ET scale

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> 34-tET was discussed by Larry Hanson and others before 1980.

J. Murray Barbour mentions it in his classic "Tuning and 
Temperament", published in 1951, and before that in "Music and 
Ternary Continued Fractions", American Mathematical Monthly vol. 55, 
1948, p. 545. Barbour's analysis has the same two features--it 
considers only the 5-limit, and it ignores the diatonic comma. One 
can however consider not being meantone a feature, not a bug, and the 
34 division has its own individual merits; it is also worth looking 
at it from the point of view of the 17-et contained within it, or 
putting it inside of a 68-et (inside of a 612-et, a division which 
Barbour also mentions, is going a little too far.)

From: Paul Erlich (2001-09-28)
Subject: Re: 34-tone ET scale

--- In tuning-math@y..., genewardsmith@j... wrote:

> One 
> can however consider not being meantone a feature, not a bug,

Absolutely -- if one is not simply composing according to common-
practice schemata (which the use of conventional notation ABCDEFG#b 
tends to imply).

> it is also worth looking 
> at it from the point of view of the 17-et contained within it, or 
> putting it inside of a 68-et

Exactly -- things I mentioned in the "bicycle chain" discussion.

> (inside of a 612-et, a division which 
> Barbour also mentions, is going a little too far.)

Ha -- I never noticed that 612 was 36*17!

From: Paul Erlich (2001-10-01)
Subject: Re: 34-tone ET scale

34 is the next entry after 12 in this accounting of periodicity 
blocks:

http://www.kees.cc/tuning/s235.html

What that says to me is that, if one is going to use strict 5-limit 
JI, and one is seeking an "even" system with more than 12 notes but 
fewer than 53, one must go to 34. Not 34-tET, but 34-tJI.

From: Andy (2012-04-18)
Subject: Symmetric 53-tone well-temperament via 19-limit

--- In [email protected], "Paul Erlich" <paul@...> wrote:
> The 53-tone equal-temperament (ET) scale is well-known as giving a 
> very good approximation to "just" intervals.  

Cycle of 53 quintes, 
with 46 just ones and seven tempered a little bit downwards 

+-0 D. 1/1 unison
+ 1 A. 3/4 
+ 2 E. 9/8
+ 3 B. 27/32
+ 4 F# 81/64
+ 5 C# 243/256 limma upwards
+ 6 G# 729/512 tritone (9/8)^3
+ 7 D# 2187/2048 apotome
+ 8 A# 6561/4096 = |-12 8> last 3-limit interval

* 39,845,888/39,858,025 = |21 -13,-2 0 0,0 0 1> =~-0.529...cents  

+ 9 F/ 2432/2025 begin 19-limit block
+10 C/ 608/675
+11 G/ 304/225
+12 D/ 76/75 
+13 A/ 19/25 = |0 0,-2 0 0,0 0 1>
+14 E/ 57/50 end 19-lim. block

* 20,000/20,007 = |5 -4,4 0 0,-1 0 -1>  =~-0.605...cents

+15 B/ 100/117 begin 13-lim. block
+16 F& 50/39 F#/
+17 C& 25/26 = |-1 0,-2 0 0,-1>
+18 G& 75/52
+19 D& 225/208
+20 A& 675/832 A#/ 
+21 F+ 2025/1664 F// end 13-lim. block

* 4,100,096/4,100,625 = |12 -8,-4 1 1,1> =~-0.223...cents 

+22 C+ 616/675 begin 11-lim. block
+23 G+ 154/225
+24 D+ 77/75 = |0 -1,-2 1 1>
+25 A+ 77/50
+26 E+ 231/200 F- enharmonics

* 160,000/160,083 = |8 -3,4 -2 -2> =~-0.897...cents @ symm. center

-26 C- 200/231 B+ enharmonics
-25 G- 50/77
-24 D- 75/77 = |0 1,2 -1 -1>
-23 A- 225/154
-22 E- 675/616 end 11-lim. block

* 4,100,096/4,100,625 = |12 -8,-4 1 1,1> =~-0.223...cents 

-21 B- 1664/2025 B\\ begin 13-lim. block
-20 GB 832/675 Gb\
-19 DB 208/225
-18 AB 52/75
-17 EB 26/25 = |1 0,-2 0 0,1> 
-16 BB 39/50 Bb\
-15 F\ 117/100 end 13-lim. block

* 20,000/20,007 = |5 -4,4 0 0,-1 0 -1>  =~-0.605...cents

-14 C\ 50/57 begin 19-lim block
-13 G\ 25/19 = |0 0,2 0 0,0 0 -1>
-12 D\ 75/76
-11 A\ 225/304
-10 E\ 675/608
- 9 B\ 2025/2432 end 19-limit block

* 39,845,888/39,858,025 = |21 -13,-2 0 0,0 0 1> =~-0.529...cents  

- 8 Gb 4096/6561 = |12 -8> return back to Pythagorean 3-limit block
- 7 Db 2048/2187 inverse apotome
- 6 Ab 512/729
- 5 Eb 256/243 limma
- 4 Bb 64/81
- 3 F. 32/27 Pythagorean minor-3rd
- 2 C. 8/9 Pyth. diminished-7th
- 1 G. 4/3 quarte
-+0 D. 1/1 unison


Then consider that 53-cycle closer in an concise overview, 
in order to examine the 7 tempering steps 

1/1=D. 
A. E. B. F# C# D# A# @ 3-lim
|21 -13,-2 0 0,0 0 1> =~-0.529...cents  
F/ C/ G/ D/ A/ E/ @ 19-lim
|5 -4,4 0 0,-1 0 -1>  =~-0.605...cents
B/ F& C& G& D& A& F+ @ 13-lim
|12 -8,-4 1 1,1> =~-0.223...cents 
C+ G+ D+ A+ E+=F- @ 11-lim
|8 -3,4 -2 -2> =~-0.897...cents @ symm. center
B+=C- G- D- A- E- @ 11-lim
|12 -8,-4 1 1,1> =~-0.223...cents 
B- GB DB AB BB F\ @ 13-lim
|5 -4,4 0 0,-1 0 -1>  =~-0.605...cents
C\ G\ D\ A\ E\ B\ @ 19-lim
|21 -13,-2 0 0,0 0 1> =~-0.529...cents  
Gb Db Ab Bb F. C. G. @ 3-lim
D.=1/1


Control the above output, 
by summing up just that seven tempering-steps alltogether:

2*(|21 -13,-2 0 0,0 0 1> + |5 -4,4 0 0,-1 0 -1> + |12 -8,-4 1 1,1> )
+  |8 -3,4 -2 -2> 

=|84 -53> =~-3.615...cents 

collectively within Pythagorean 3-limit
all-in-all over the complete 53-cycle,
because all the 5-7-11-13-19-limit powers 
do cancel out each others by the addition of the prime-vectors.
As a matter of course the integration of the corresponding
cent-values yield the same result.


Reference:
http://xenharmonic.wikispaces.com/Mercator%27s+comma
Quote:
"... |-84 53>, known as Mercator's comma or the 53-comma,
is a small comma of 3.615 cents which is the amount by which 53 fifths exceed 31 octaves..."

But do not confuse my above ratios with the similar sounding
http://en.wikipedia.org/wiki/53_equal_temperament

bye
Andy

From: genewardsmith (2012-04-18)
Subject: Re: Symmetric 53-tone well-temperament via 19-limit

--- In [email protected], "Andy" <a_sparschuh@...> wrote:
>
> 
> 
> --- In [email protected], "Paul Erlich" <paul@> wrote:
> > The 53-tone equal-temperament (ET) scale is well-known as giving a 
> > very good approximation to "just" intervals.  
> 
> Cycle of 53 quintes, 
> with 46 just ones and seven tempered a little bit downwards 

If you take a generator of 71\99 (or 251\350, etc) you get a MOS with seven large steps and 46 small ones: 7L46s. This is amity temperament, or hitchcock if you want to push on to the 11-limit. Just enough sharpness to add flavor.

From: Andy (2012-04-18)
Subject: Symmetric 53-tone 'well'-temperament via two 19-limit transitions

> --- In [email protected], "Paul Erlich" <paul@> wrote:
> The 53-tone equal-temperament (ET) scale is well-known as giving a 
> very good approximation to "just" intervals.  
> 
Here comes an new rational cycle of 53 quintes. 
That 'well'-temperature consists in 46 just pure 5ths of exactly 3/2. 
Hence it includes barely seven remaining tempered 5ths, 
in order to compensate the famous 'Mercator's-comma" |84 53>. 
Follow the partition-pattern as excuted by the distribution-scheme
via two intermittent transitions into 19-limit intervals: 
 
+-0 D. 1/1 unison
+ 1 A. 3/4 
+ 2 E. 9/8
+ 3 B. 27/32
+ 4 F# 81/64
+ 5 C# 243/256 limma upwards
+ 6 G# 729/512 tritone (9/8)^3
+ 7 D# 2187/2048 apotome
+ 8 A# 6561/4096 = |-12 8> depart last 3-limit interval

* 39,845,888/39,858,025 = |21 -13,-2 0 0,0 0 1> =~-0.529...cents   
 
+ 9 F/ 2432/2025 begin with 19-limit block 
+10 C/ 608/675
+11 G/ 304/225
+12 D/ 76/75 
+13 A/ 19/25 = |0 0,-2 0 0,0 0 1>
+14 E/ 57/50 end 19-lim. block
 
* 20,000/20,007 = |5 -4,4 0 0,-1 0 -1>  =~-0.605...cents

+15 B/ 100/117 begin 13-lim. block
+16 F& 50/39 F#/
+17 C& 25/26 = |-1 0,-2 0 0,-1>
+18 G& 75/52
+19 D& 225/208
+20 A& 675/832 A#/ 
+21 F+ 2025/1664 F// end 13-lim. block
 
* 4,100,096/4,100,625 = |12 -8,-4 1 1,1> =~-0.223...cents 

+22 C+ 616/675 begin 11-lim. block
+23 G+ 154/225
+24 D+ 77/75 = |0 -1,-2 1 1>
+25 A+ 77/50
+26 E+ 231/200 F- enharmonics

* 160,000/160,083 = |8 -3,4 -2 -2> =~-0.897...cents @ symm. center

-26 C- 200/231 B+ enharmonics
-25 G- 50/77
-24 D- 75/77 = |0 1,2 -1 -1>
-23 A- 225/154
-22 E- 675/616 end 11-lim. block

* 4,100,096/4,100,625 = |12 -8,-4 1 1,1> =~-0.223...cents 

-21 B- 1664/2025 B\\ begin 13-lim. block
-20 GB 832/675 Gb\
-19 DB 208/225
-18 AB 52/75
-17 EB 26/25 = |1 0,-2 0 0,1> 
-16 BB 39/50 Bb\
-15 F\ 117/100 end 13-lim. block
 
* 20,000/20,007 = |5 -4,4 0 0,-1 0 -1>  =~-0.605...cents

-14 C\ 50/57 begin 19-lim block
-13 G\ 25/19 = |0 0,2 0 0,0 0 -1>
-12 D\ 75/76
-11 A\ 225/304
-10 E\ 675/608
- 9 B\ 2025/2432 end 19-limit block
 
* 39,845,888/39,858,025 = |21 -13,-2 0 0,0 0 1> =~-0.529...cents  

- 8 Gb 4096/6561 = |12 -8> return back to Pythagorean 3-limit block
- 7 Db 2048/2187 inverse apotome
- 6 Ab 512/729
- 5 Eb 256/243 limma
- 4 Bb 64/81
- 3 F. 32/27 Pythagorean minor-3rd
- 2 C. 8/9 Pyth. diminished-7th
- 1 G. 4/3 quarte
-+0 D. 1/1 unison back again, 
ready done.  
 
Now consider that verbose 53=46+7 cycle under closer look 
in more concise representation for the 46 just pure 5ths: 
Instead of that 46 just ones 
examine and expand the 7 critical tempering-steps deeper en detail:

Start from unison 1/1=D. 
A. E. B. F# C# D# A# @ 3-lim
|21 -13,-2 0 0,0 0 1> =~-0.529...cents  3==>19-lim bridge-comma
F/ C/ G/ D/ A/ E/ @ 19-lim
|5 -4,4 0 0,-1 0 -1>  =~-0.605...cents  19==>13-lim bridge-comma
B/ F& C& G& D& A& F+ @ 13-lim
|12 -8,-4 1 1,1> =~-0.223...cents  13==>11-lim bridge-comma
C+ G+ D+ A+ E+=F- @ 11-lim
|8 -3,4 -2 -2> =~-0.897...cents @ symm. center <mirror-axis>-comma
B+=C- G- D- A- E- @ 11-lim
|12 -8,-4 1 1,1> =~-0.223...cents  11==>13-lim bridge-comma again
B- GB DB AB BB F\ @ 13-lim
|5 -4,4 0 0,-1 0 -1>  =~-0.605...cents 13==>19-lim bridge-comma again
C\ G\ D\ A\ E\ B\ @ 19-lim
|21 -13,-2 0 0,0 0 1> =~-0.529...cents  19==>3-lim bridge-comma again
Gb Db Ab Bb F. C. G. @ 3-lim
D.=1/1 back home @ unison 

Revise that approach:
Control again the total amount of output,
while roaming 5th-wise forwards through the whole tempering-process.
Perform that check by summing up over the seven deviation-steps 
by stacking them alltogether at once over all the 7 "Monzo"s:
 
2*(|21 -13,-2 0 0,0 0 1> + |5 -4,4 0 0,-1 0 -1> + |12 -8,-4 1 1,1> )
+  |8 -3,4 -2 -2> 

=|84 -53> =~-3.615...cents  yields 'Mercator's-comma all-in-all.
Proof succesfully completed.

Result:
The 53 cirle compensates |84 -53> 'Mercator's-comma exactly.

For better reminding: 
Study once more the above indicated 7 positions 
where the seven-fold tempering acts: 
Listen: 
How do sound the 7 with-intend disturbed 5ths individually?
Locate that seven concerned 5ths concrete at the positions:

A#~F/ , E/~B/ , F+~C+ ,(E+=F-)~(B+=C-) E-~B- F\~C\ and B\~Gb

Against that 7 specific demanded deviations:
Insist in keeping all the other remaining 46 quintes just pure 3/2,
in order to conclude properly the complete 53=46+7 cycle,
without the least disturbance, as good as yours ears do allow.

Observation:
All the 7 bridges via the intermediate 5,7,11,13&19-limit powers,
do cancel out each others during the addition of the prime-vectors.
So we yield finally as sum |84 -53> over the whole 53 circle.
Consequently we got completely rid of all the intermediate occuring
higher limit powers 

|m n> := |m n, ? ? ?, ? ? ?, ? ? ?> 

except than the naturally remaining 3-limit |84 -53> M's-comma.

As a matter of course, 
the integration of the corresponding cent-approximations 
must yield the same result again...

2*(~0.529... +~0.605... +~0.223...) +~0.897... = ~3.611...cents

Self-evident only within the limitation of inherent rounding-errors, 
that occurs here in the last decimal, 
due to cut-off failure during the cent-conversion. 

Excursion:
Compare that 7 amounts against the traditional 53-edo.
The 53-edo contains even more tiny tempering-deviations. 
But that got distributed equally over all the 5ths uniformly:

~3.615...cents/53 = ~0.0682...cents @ each of the 53 quintes, 
That's roughly about ~~1/15~~ of a single cent
@ each so smoothened 5th.

Reference reccomendation:
http://xenharmonic.wikispaces.com/Mercator%27s+comma
Quote:
"... |-84 53>, known as Mercator's comma or the 53-comma,
is a small comma of 3.615 cents which is the amount by which 53 fifths exceed 31 octaves..."

But please do not confuse my 53 ratios 
with the more stupid or at least boring
http://en.wikipedia.org/wiki/53_equal_temperament
without any desirable 'key-characteristics'
http://biteyourownelbow.com/keychar.htm

But let's return back to the actual matter:
Now rearrange that chain of 53 pitches into up-wards ascending order:
Attend along that way as side-condition the constraint:
Locate them all mirror-inverted around the central unison=1/1 axis 
in strictly consequent symmetric placement:

z -26 G# 729/1024 tritone octaved down ((9/8)^3)/2
y -25 G& 75/104 G#/
x -24 A- 225/308 A\\
w -23 A\ 225/304
v -22 A. 3/4
u -21 A/ 19/25 = |0 0,-2 0 0,0 0 1>
t -20 A+ 77/100 A//
s -19 BB 39/50 Bb\
r -18 Bb 64/81
q -17 A# 6561/8192 = |-13 8>
p -16 A& 675/832
o -15 B- 1664/2025
n -14 B\ 2025/2432
m -13 B. 27/32
l -12 B/ 100/117
k -11 C- 200/231 B+
j -10 C\ 50/57
i - 9 C. 8/9
h - 8 C/ 608/675
g - 7 C- 616/675
f - 6 DB 208/225
e - 5 Db 2048/2187 ! apotome down-wards
d - 4 C# 243/256   ! limma down-w's
c - 3 C& 25/26
b - 2 D- 75/77
a - 1 D\ 75/76
@ +-0 D. 1/1 unison @ symmetric center <=== mirror-axis ===>
A + 1 D/ 76/75
B + 2 D+ 77/75
C + 3 EB 26/25
D + 4 Eb 256/243   ! limma up-wards
E + 5 D# 2187/2048 ! apotome up-w's
F + 6 D& 225/208
G + 7 E- 675/616
H + 8 E\ 675/608
I + 9 E. 9/8 
J +10 E/ 57/50
K +11 E+ 231/200 F-
L +12 F\ 117/100
M +13 F. 32/27 Pythagorean 3-limit minor-3rd
N +14 F/ 2432/2025
O +15 F+ 2025/1664
P +16 GB 832/675
Q +17 Gb 8192/6561 = |13 -8>
R +18 F# 81/64
S +19 F& 50/39
T +20 G- 100/77
U +21 G\ 25/19 = |0 0,2 0 0,0 0 -1>
V +22 G. 4/3 ! quarte 
W +23 G/ 304/225
X +24 G+ 308/225
Y +25 AB 104/75
Z +26 Ab 1024/729 = |10 -6> inverse tritone

Finally exhibit them also in the terminology of the
http://www.huygens-fokker.org/scala/scl_format.html


! Sp53via19lim.scl
!
Sparschuh's Symmetric 53-tone well-temperament via 19-limit (2012)
53
!
! 1/1     ! @ unison
76/75     ! A
77/75     ! B
26/25     ! C
256/243   ! D
2187/2048 ! E
225/208   ! F
675/616   ! G
675/608   ! H
9/8       ! I
57/50     ! J
231/200   ! K
117/100   ! L
32/27     ! M
2432/2025 ! N
2025/1664 ! O
832/675   ! P
8192/6561 ! Q
81/64     ! R
50/39     ! S
100/77    ! T
25/19     ! U
4/3       ! V
304/225   ! W
308/225   ! X
104/75    ! Y
1024/729  ! Z
! <======== mirror-symmetry-axis =========> !
729/512   ! z
75/52     ! y
225/154   ! x
225/152   ! w
3/2       ! v
38/25     ! u
77/50     ! t
39/25     ! s
128/81    ! r
6561/4096 ! q
675/416   ! p
3328/2025 ! o
2025/1216 ! n
27/16     ! m
200/117   ! l
400/231   ! k
100/57    ! j
16/9      ! i
1216/675  ! h
1232/675  ! g
416/225   ! f
4096/2187 ! e
243/128   ! d
50/26     ! c
150/77    ! b
75/38     ! a
2/1       ! @' octave
!
![eof]

Especially regard an 'Scala'-specific side-note: 
There in the 'scala'-reprensentation appears 
the inherent mirror-symmetry again as the phenomenon:

Aa = Bb = Cc = Dd =.... = Yy = Zz = 2/1 = octave = @'

That characteristic octave-property 
occurs exactly 26 := (53-1)/2 times,
when excluding the singular unison @=1/1
and its corresponding counterpart:
the octave @'=2/1
from the statistics of intervals, 
to be omitted from counting, becasue considered as "implicit".

That pardigm can be found in the specification of the:
http://www.huygens-fokker.org/scala/scl_format.html
Quote from:
"The rules:...

* The second line contains the number of notes....
The lower limit is 0, which is possible since 
degree 0 of 1/1 is implicit. 

* The first note of 1/1 or 0.0 cents is implicit and not in the files.
"

How about to implement that newbe for instance on?
http://www.cortex-design.com/projects_terp2.htm
http://www.h-pi.com/TPX28keyboard.html
http://www.anaphoria.com/hanson.PDF
http://en.wikipedia.org/wiki/Generalized_keyboard
http://www.wendycarlos.com/photos/bosanquet53.jpg
http://tardis.dl.ac.uk/FreeReed/organ_book/node18.html
http://www.starrlabs.com/products/keyboards/microzone-u-990
just in order to mention at least a few example
of possible instruments that may fit potentially?

bye
Andy

From: Andy (2012-04-18)
Subject: Re: Symmetric 53-tone well-temperament via 19-limit

--- In [email protected], "genewardsmith" <genewardsmith@...> wrote:

> If you take a generator of 71\99 (or 251\350, etc) you get a MOS 
> with seven large steps and 46 small ones: 7L46s. 
> This is amity temperament, 

http://xenharmonic.wikispaces.com/Ragismic+microtemperaments#Amity

> or hitchcock 

http://xenharmonic.wikispaces.com/Ragismic+microtemperaments#Hitchcock

> if you want to push on to the 11-limit.

before considering that, i do prefer to improve the lower limits,
using the higher limits barely as surrogate sub-agents in order
to close the cycle more properly at lower ratios.

> Just enough sharpness to add flavor.
Yep, i like that kind of diversified 'flavor', 
that occurs under changeing tonality, 
due to desireable variation in key-characteristic.
That's one among reasons, 
why i do prefer most of the 5ths just pure 3/2 
alike my personal guide J.S.Bach
http://launch.groups.yahoo.com/group/tuning/message/104314

bye
Andy

From: genewardsmith (2012-04-18)
Subject: Re: Symmetric 53-tone well-temperament via 19-limit

--- In [email protected], "Andy" <a_sparschuh@...> wrote:

> before considering that, i do prefer to improve the lower limits,
> using the higher limits barely as surrogate sub-agents in order
> to close the cycle more properly at lower ratios.

Amity[53] with pure 3/2 fifths is far closer to 53 equal than a tuning in 99 would be, but the septimal harmony is entirely usable--no surprise, as the same is true of 53edo.

! amity53pure.scl
!
Amity[53] in pure-fifths tuning
 53
!
 22.73700
 45.47400
 68.21100
 90.22500
 112.96200
 135.69900
 158.43600
 181.17300
 203.91000
 226.64700
 249.38400
 271.39800
 294.13500
 316.87200
 339.60900
 362.34600
 385.08300
 407.82000
 429.83400
 452.57100
 475.30800
 498.04500
 520.78200
 543.51900
 566.25600
 588.99300
 611.00700
 633.74400
 656.48100
 679.21800
 701.95500
 724.69200
 747.42900
 770.16600
 792.18000
 814.91700
 837.65400
 860.39100
 883.12800
 905.86500
 928.60200
 950.61600
 973.35300
 996.09000
 1018.82700
 1041.56400
 1064.30100
 1087.03800
 1109.77500
 1131.78900
 1154.52600
 1177.26300
 2/1

From: Andy (2012-04-24)
Subject: Re: Completely 5-limit symmetric 53-tone, as generalized Kirnberger #1

--- In [email protected], "genewardsmith" <genewardsmith@...> wrote:
>
> Amity[53] with pure 3/2 fifths is far closer to 53 equal ....
> 
> ! amity53pure.scl
> !
> Amity[53] in pure-fifths tuning
>  53
> !
>  22.73700  ! A
>  45.47400  ! B 
>  68.21100  ! C 
>  90.22500  ! D
>  112.96200 ! E 
>  135.69900 ! F
>  158.43600 ! G 
>  181.17300 ! H
>  203.91000 ! I 
>  226.64700 ! J
>  249.38400 ! K
>  271.39800 ! L
>  294.13500 ! M
>  316.87200 ! N
>  339.60900 ! O
>  362.34600 ! P 
>  385.08300 ! Q 
>  407.82000 ! R
>  429.83400 ! S 
>  452.57100 ! T
>  475.30800 ! U 
>  498.04500 ! V 
>  520.78200 ! W
>  543.51900 ! X
>  566.25600 ! Y 
>  588.99300 ! Z
>  611.00700 ! z
>  633.74400 ! Y 
>  656.48100 ! x
>  679.21800 ! w
>  701.95500 ! v
>  724.69200 ! u
>  747.42900 ! t
>  770.16600 ! s
>  792.18000 ! r
>  814.91700 ! q
>  837.65400 ! p
>  860.39100 ! o
>  883.12800 ! n
>  905.86500 ! m
>  928.60200 ! l
>  950.61600 ! k
>  973.35300 ! j
>  996.09000 ! i
>  1018.8270 ! h
>  1041.5640 ! g
>  1064.3010 ! f
>  1087.0380 ! e
>  1109.7750 ! d
>  1131.7890 ! c
>  1154.5260 ! b
>  1177.2630 ! a
>  2/1       ! @'
> !
> ![eof] 

in deed Gene,
that sounds almost undistinguishable alike 53-edo.
for comparision against 53-edo, see:
http://xenharmonic.wikispaces.com/53edo
http://en.wikipedia.org/wiki/53_equal_temperament

But how about Danny Wier's [2002] proposal, 
of modification:
Here compiled into an scala-file:


! Wier53.scl
Danny Wier's schismatically-altered 53-Pythagorgean scale (2002)
53
! http://launch.groups.yahoo.com/group/tuning/message/38888
!
64/63   ! A
36/35   ! B
28/27   ! C
135/128 ! D
16/15   ! E
243/224 ! F
35/32   ! G
10/9    ! H 
9/8     ! I
8/7     ! J
81/70   ! K
7/6     ! L
32/27   ! M
6/5     ! N
128/105 ! O
315/256 ! P
5/4     ! Q
81/64   ! R
9/7     ! S
35/27   ! T
21/16   ! U
4/3     ! V
27/20   ! W
48/35   ! X
112/81  ! Y
45/32   ! Z
64/45   ! z 
81/56   ! y 
35/24   ! x
40/27   ! w
3/2     ! v
32/21   ! u
54/35   ! t
14/9    ! s
128/81  ! r
8/5     ! q
512/315 ! p 
105/64  ! o
5/3     ! n
27/16   ! m
12/7    ! l
140/81  ! k 
7/4     ! j
16/9    ! i
9/5     ! h
64/35   ! g
448/243 ! f
15/8    ! e
256/135 ! d
27/14   ! c
35/18   ! b
63/32   ! a
2/1     ! @'
!
![eof]

Idea:
Restrict the whole 5ths-cycle completely within 5-limit ratios:

1/1=D. A. E. B. F# -schisma=|15 -8,-1>=~-1.95cents  C# G# D# A# ... 
...F+ C+ G+ D+ A+ E+=F- Monzisma=|54 -37,2>=~+0.29c B+=C- G- D- A-... 
...Gb Db Ab Eb Bb -schisma=|15 -8,-1>=~-1.95cents F. C. G. D.=1/1

That got executed by 5-limit trisection of Mercator's-comma

|-84 53> = 2*(|-15 8,1>) -|54 -37,2> =~3.62...cents

More verbose en detail:

+-0 D. 1/1 unison 
+ 1 A. 3/2
+ 2 E. 9/8 major-tone
+ 3 B. 27/16
+ 4 F# 81/64 ! end of Pythagorean 3-limit domain

-schisma=|15 -8,-1>=~-1.95cents

+ 5 C# 256/135 ! begin with syntonic 5-limit domain
+ 6 G# 64/45
+ 7 D# 16/15
+ 8 A# 8/5
+ 9 F/ 6/5 minor-3rd
+10 C/ 9/5
+11 G/ 27/20
+12 D/ 81/80 syntonic-comma
+13 A/ 243/160
+14 E/ 729/640
+15 B/ 2,187/1,280
+16 F& 6,561/5,120 = |-10 8,-1> F#/ 
+17 C& 19,683/10,240 = |-11 9,-1>
+18 G& 59,049/40,960 = |-13 10,-1>
+19 D& 177,147/163,840 = |-15 11,-1>
+20 A& 531,441/327,680 = |-16 12,-1> A#/
+21 F+ 1,594,323/1,310,720 = |-18 13,-1> F//
+22 C+ 4,782,969/2,621,440 = |-19 14,-1>
+23 G+ 14,348,907/10,485,760 = |-21 15,-1>
+24 D+ 43,046,721/41,943,040 = |-23 16,-1>
+25 A+ 129,140,163/83,886,080 = |-24 17,-1>
+26 E+=E// 387,420,489/335,544,320 = |-26 18,-1> F-=F\\

+Monzisma = |54 -37,2> =~+0.29cents at the symmetric mirror-axis

-26 C-=C\\ 671,088,640/387,420,489 = |27 -18,1> B+=B//
-25 G- 167,772,160/129,140,163 = |25 -17,1>
-24 D- 83,886,080/43,046,721 = |24 -16,1> 
-23 A- 20,971,520/14,348,907 = |22 -15,1>
-22 E- 5,242,880/4,782,969 = |20 -14,1> E\\=FB=Fb\
-21 B- 5,242,880/4,782,969 = |19 -13,1> B\\=CB=Cb\
-20 GB 655,360/531,441 = |17 -12,1> Gb\
-19 DB 327,680/177,147 = |16 -11,1>
-18 AB 81,920/59,049 = |14 -10,1>
-17 EB 20,480/19,683 = |12 -9,1>
-16 BB 10,240/6,561 = |11 -8,1> Bb\
-15 F\ 2,560/2,187 = |9 -7,1>
-14 C\ 1,280/729 = |8 -6,1>
-13 G\ 320/243
-12 D\ 160/81 inverse-SC
-11 A\ 40/27
-10 E\ 10/9 minor-tone
- 9 B\ 5/3 JI-sixth
- 8 Gb 5/4 JI-tierce
- 7 Db 15/8 JI-7th
- 6 Ab 45/32
- 5 Eb 135/128 end of syntonic 5-limit domain

-schisma=|15 -8,-1>=~-1.95cents

- 4 Bb 128/81 return back again to Pythagorean 3-limit domain 
- 3 F. 32/27 
- 2 C. 16/9 Pyth. minor 7th
- 1 G. 4/3 quarte 
+-0 D. 1/1 unison, 53-cycle closed

The same sounds in terms of an "Scala"-file:

! Sp53tone5limit.scl
!
Sparschuh's tri-section of Mercator's-comma into (schisma)*2-Monzisma 
53
!
81/80               ! A | -4   4,-1>  D/ syntonic-comma
43046721/41943040   ! B |-23  16,-1>  D+
20480/19683         ! C | 12  -9, 1>  EB 
135/128             ! D | -7   3, 1>  Eb
16/15               ! E |  4  -1,-1>  D#
177147/163840       ! F |-15  11,-1>  D&  
5242880/4782969     ! G | 20 -14, 1>  E-   
10/9                ! H |  1  -2, 1>  E\
9/8                 ! I | -3   2>     E.
729/640             ! J | -7   6,-1>  E/   
387420489/335544320 ! K |-26  18,-1>  E+=F-
2560/2187           ! L |  9  -7, 1>  F\
32/27               ! M |  5  -3>     F.    
6/5                 ! N |  1   1,-1>  F/
1594323/1310720     ! O |-18  13,-1>  F+ 
655360/531441       ! P | 17 -12, 1>  GB
5/4                 ! Q | -2   0, 1>  Gb
81/64               ! R | -6   4>     F#
6561/5120           ! S |-10   8,-1>  F&
167772160/129140163 ! T | 25 -17, 1>  G-
320/243             ! U |  6  -5, 1>  G\
4/3                 ! V |  2  -1>     G.
27/20               ! W | -2   3,-1>  G/
14348907/10485760   ! X |-21  15,-1>  G+
81920/59049         ! Y | 14 -10, 1>  AB
45/32               ! Z | -5   2, 1>  Ab
! <<<=======symmetric-mirror-axis=======>>> !
64/45               ! z |  6  -2,-1>  G#
59049/40960         ! y |-13  10,-1>  G&
20971520/14348907   ! x | 22 -15, 1>  A-
40/27               ! w |  3  -3, 1>  A\
3/2                 ! v | -1   1>     A.
243/160             ! u | -5   5,-1>  A/
129140163/83886080  ! t |-24  17,-1>  A+
10240/6561          ! s | 11  -8, 1>  BB 
128/81              ! r |  7  -4>     Bb
8/5                 ! q |  3   0,-1>  A#
531441/327680       ! p |-16  12,-1>  A&
5242880/4782969     ! o | 19 -13, 1>  B- 
5/3                 ! n |  0  -1, 1>  B\
27/16               ! m | -4   3>     B.
2187/1280           ! l | -8   7,-1>  B/
671088640/387420489 ! k | 27 -18, 1>  B+=C-
1280/729            ! j |  8  -6, 1>  C\
16/9                ! i |  4  -2>     C.
9/5                 ! h |  0   2,-1>  C/
4782969/2621440     ! g |-19  14,-1>  C+
327,680/177,147     ! f | 16 -11, 1>  DB
15/8                ! e | -3   1, 1>  Db
256/135             ! d |  8  -3,-1>  C#
19683/10240         ! c |-11   9,-1>  C&
83886080/43046721   ! b | 24 -16, 1>  D-
160/81              ! a |  5  -4,-1>  D\
2/1                 ! @'|  1>         D.

Remark:
That 53-cycle includes exactly the first attempt of
http://en.wikipedia.org/wiki/Kirnberger_temperament
as 12-out-of-53 dodecatonic subset:

A. E. B. F# -schisma C# G# D# A# F/ C/ G/ D/ 80/81=-SC A.

coincident at the same labeling of pitch-names.

Attend:
K1 contains inbetween the quinte: D/ and A. 
an ugly syntonic 'wolf'-5th of 
40/27 = (3/2)%(81/80) =~680.4...cents
that most tuners consider as unacceptable.

See deeper for more details:
http://groenewald-berlin.de/text/text_T001.html
http://groenewald-berlin.de/tabellen/TAB-001.html
http://groenewald-berlin.de/graphik-tabelle/GRA-001.html
Sorry that's only in german-language available :-(

bye
Andy

From: Andy (2012-05-14)
Subject: push 53-tone to 13-limit, was: Re: Symmetric 53-tone well-temperament....

--- In [email protected], "genewardsmith" <genewardsmith@...> wrote:
> ... amity temperament,
>  or hitchcock if you want to push on to the 11-limit. 
>  Just enough sharpness to add flavor...

Agreed Gene,

let's overrun 11-limit, 
so that even the next higher 13-limit 'flavor' 
got included in order to enrich an meliorate 53-cycle:

+-0 D. 1/1 unison-root
+ 1 A. 3/2 quinte
+ 2 E. 9/8 tone
+ 3 B. 27/16
+ 4 F# 81/64

see: http://tech.groups.yahoo.com/group/tuning-math/message/17405
* (14336/14337) =~-0.12...cents

+ 5 C# 112/59 = (243*128)(14336/14337) = (256/135)(945/944) limma

* (944/945) =~-1.83...cents

+ 6 G# 64/45
+ 7 D# 16/15
+ 8 A# 8/5
+ 9 F/ 6/5
+10 C/ 9/5
+11 G/ 27/20

* (2080/2079)

+12 D/ 78/77 = (81/80)(2080/2079) = (64/63)(351/352) as Werckmeister

* (352/351)

+13 A/ 32/21
+14 E/ 8/7
+15 B/ 12/7
+16 F& 9/7 F#/
+17 C& 27/14 = (52/27)(729/728)

* (728/729)

+18 G& 13/9 = (81/56)(728/729)
+19 D& 13/12
+20 A& 13/8
+21 F+ 39/32 F// 

* (352/351)

+22 C+ 11/6 C//
+23 G+ 11/8
+24 D+ 33/32 ( > 36/35 > 40/39 )

* (385/384)

+25 A+ 54/35 A// ( > 20/13 )

* (351/350)

+26 E+ 15/13 F-

* (676/675)

-26 C- 26/15 B+

* (351/350) 

-25 G- 35/27 G\\

* (385/384)

-24 D- 64/33 D\\
-23 A- 16/11
-22 E- 12/11

* (352/351)

-21 B- 64/39 B\\
-20 GB 16/13 Gb\
-19 DB 24/13
-18 AB 18/13

* (728/729)

-17 EB 28/27 = (27/26)(728/729)
-16 BB 14/9
-15 F\ 7/6
-14 C\ 7/4
-13 G\ 21/16

* (352/351)

-12 D\ 77/39

* (2080/2079)

-11 A\ 40/27
-10 E\ 10/9
- 9 B\ 5/3
- 8 Gb 5/4
- 7 Db 15/8
- 6 Ab 45/32

* (944/945)

- 5 Eb 59/56 Arabic-limma = (135/128)(944/945)=(256/243)(14336/14337)

* (14336/14337) 

- 4 Bb 128/81 returned back into Pythahorean 3-limit
- 3 F. 32/27 Pythag. minor-3rd
- 2 C. 16/9 Pythag. minor-7th
- 1 G. 4/3 quarte
+-0 D. 1/1 


! Sp53in13lim.scl
!
Sparschuh's overtone-series 1:3:5:7:9:11:13:15 interpolation (2012)
53
!
78/77  ! A  +1 D/
33/32  ! B  +2 D+
28/27  ! C  +3 EB
59/56  ! D  +4 Eb 19.666.../18.666...
16/15  ! E  +5 D#
13/12  ! F  +6 D&
12/11  ! G  +7 E-
10/9   ! H  +8 E\
9/8    ! I  +9 E.
8/7    ! J +10 E/
15/13  ! K +11 E+ 7.5/6.5 F-
7/6    ! L +12 F\
32/27  ! M +13 F. 6.4/5.4
6/5    ! N +14 F/
39/32  ! O +15 F+
16/13  ! P +16 GB
5/4    ! Q +17 Gb 1.25      tierce or major-3rd or 5th-partial
81/64  ! R +18 F# 1.265625  ditone (9/8)^2
9/7    ! S +19 F& 4.5/3.5
35/27  ! T +20 G- 4.375/3.375
21/16  ! U +21 G\ 4.2/3.2
4/3    ! V +22 G. quarte
27/20  ! W +23 G/
11/8   ! X +24 G+ 3.666.../2.666...
18/13  ! Y +25 AB 3.6/2.6
45/32  ! Z +26 Ab tritone
! ========  central symmetric-mirror-axis ==================
64/45  ! z -26 G# inverse-tritone
13/9   ! y -25 G& 3.25/2.25
16/11  ! x -24 G# 3.2/2.2
40/27  ! w -23 A\
3/2    ! v -22 A. quinte
32/21  ! u -21 A/ 2.9090../1.9090..
54/35  ! t -20 A+
14/9   ! s -19 BB 2.8/1.8
128/81 ! r -18 Bb
8/5    ! q -17 A# 2.666.../1.666...
13/8   ! p -16 A& 2.6/1.6
64/39  ! o -15 B- 2.56/1.56
5/3    ! n -14 B\ 2.5/1.5
27/16  ! m -13 B. 2.4545../1.4545..
12/7   ! l -12 B/ 2.4/1.4
26/15  ! k -11 B+ 2.3636../1.3636.. C-
7/4    ! j -10 C\ 2.333.../1.333...
16/9   ! i  -9 C.
9/5    ! h  -8 C/ 2.25/1.25
11/6   ! g  -7 C+ 2.2/1.2
24/13  ! f  -6 DB 2.1818../1.1818..
15/8   ! e  -5 Db
112/59 ! d  -4 C#
27/14  ! c  -3 C&
64/33  ! b  -2 D-
77/39  ! a  -1 D\
2/1    ! @'+-0 D.
!
![eof]

Attend, that it contains the odd harmonic 2n-1 'overtone-series'
at the note-names for the generalized tonic 8-fold chord:

@:v:Q:j:I:X:p:e == 1:3:5:7:9:11:13:15 == D.:A.:Gb:C\:E.:G+:A&:Db

that fits also for the corresponding dominat- and subdominat chord.

Quest:
Sounds that 'addition-of-flavor' against 53-edo 'sharp-enough'
likewise in yours ears too, when perceiving by listening to that?

bye
Andy

From: Keenan Pepper (2012-05-14)
Subject: push 53-tone to 13-limit, was: Re: Symmetric 53-tone well-temperament....

I'm interested in what actual music using this system would sound like. I'm having trouble imagining it.

Keenan

--- In [email protected], "Andy" <a_sparschuh@...> wrote:
>
> --- In [email protected], "genewardsmith" <genewardsmith@> wrote:
> > ... amity temperament,
> >  or hitchcock if you want to push on to the 11-limit. 
> >  Just enough sharpness to add flavor...
> 
> Agreed Gene,
> 
> let's overrun 11-limit, 
> so that even the next higher 13-limit 'flavor' 
> got included in order to enrich an meliorate 53-cycle:
> 
> +-0 D. 1/1 unison-root
> + 1 A. 3/2 quinte
> + 2 E. 9/8 tone
> + 3 B. 27/16
> + 4 F# 81/64
> 
> see: http://tech.groups.yahoo.com/group/tuning-math/message/17405
> * (14336/14337) =~-0.12...cents
> 
> + 5 C# 112/59 = (243*128)(14336/14337) = (256/135)(945/944) limma
> 
> * (944/945) =~-1.83...cents
> 
> + 6 G# 64/45
> + 7 D# 16/15
> + 8 A# 8/5
> + 9 F/ 6/5
> +10 C/ 9/5
> +11 G/ 27/20
> 
> * (2080/2079)
> 
> +12 D/ 78/77 = (81/80)(2080/2079) = (64/63)(351/352) as Werckmeister
> 
> * (352/351)
> 
> +13 A/ 32/21
> +14 E/ 8/7
> +15 B/ 12/7
> +16 F& 9/7 F#/
> +17 C& 27/14 = (52/27)(729/728)
> 
> * (728/729)
> 
> +18 G& 13/9 = (81/56)(728/729)
> +19 D& 13/12
> +20 A& 13/8
> +21 F+ 39/32 F// 
> 
> * (352/351)
> 
> +22 C+ 11/6 C//
> +23 G+ 11/8
> +24 D+ 33/32 ( > 36/35 > 40/39 )
> 
> * (385/384)
> 
> +25 A+ 54/35 A// ( > 20/13 )
> 
> * (351/350)
> 
> +26 E+ 15/13 F-
> 
> * (676/675)
> 
> -26 C- 26/15 B+
> 
> * (351/350) 
> 
> -25 G- 35/27 G\\
> 
> * (385/384)
> 
> -24 D- 64/33 D\\
> -23 A- 16/11
> -22 E- 12/11
> 
> * (352/351)
> 
> -21 B- 64/39 B\\
> -20 GB 16/13 Gb\
> -19 DB 24/13
> -18 AB 18/13
> 
> * (728/729)
> 
> -17 EB 28/27 = (27/26)(728/729)
> -16 BB 14/9
> -15 F\ 7/6
> -14 C\ 7/4
> -13 G\ 21/16
> 
> * (352/351)
> 
> -12 D\ 77/39
> 
> * (2080/2079)
> 
> -11 A\ 40/27
> -10 E\ 10/9
> - 9 B\ 5/3
> - 8 Gb 5/4
> - 7 Db 15/8
> - 6 Ab 45/32
> 
> * (944/945)
> 
> - 5 Eb 59/56 Arabic-limma = (135/128)(944/945)=(256/243)(14336/14337)
> 
> * (14336/14337) 
> 
> - 4 Bb 128/81 returned back into Pythahorean 3-limit
> - 3 F. 32/27 Pythag. minor-3rd
> - 2 C. 16/9 Pythag. minor-7th
> - 1 G. 4/3 quarte
> +-0 D. 1/1 
> 
> 
> ! Sp53in13lim.scl
> !
> Sparschuh's overtone-series 1:3:5:7:9:11:13:15 interpolation (2012)
> 53
> !
> 78/77  ! A  +1 D/
> 33/32  ! B  +2 D+
> 28/27  ! C  +3 EB
> 59/56  ! D  +4 Eb 19.666.../18.666...
> 16/15  ! E  +5 D#
> 13/12  ! F  +6 D&
> 12/11  ! G  +7 E-
> 10/9   ! H  +8 E\
> 9/8    ! I  +9 E.
> 8/7    ! J +10 E/
> 15/13  ! K +11 E+ 7.5/6.5 F-
> 7/6    ! L +12 F\
> 32/27  ! M +13 F. 6.4/5.4
> 6/5    ! N +14 F/
> 39/32  ! O +15 F+
> 16/13  ! P +16 GB
> 5/4    ! Q +17 Gb 1.25      tierce or major-3rd or 5th-partial
> 81/64  ! R +18 F# 1.265625  ditone (9/8)^2
> 9/7    ! S +19 F& 4.5/3.5
> 35/27  ! T +20 G- 4.375/3.375
> 21/16  ! U +21 G\ 4.2/3.2
> 4/3    ! V +22 G. quarte
> 27/20  ! W +23 G/
> 11/8   ! X +24 G+ 3.666.../2.666...
> 18/13  ! Y +25 AB 3.6/2.6
> 45/32  ! Z +26 Ab tritone
> ! ========  central symmetric-mirror-axis ==================
> 64/45  ! z -26 G# inverse-tritone
> 13/9   ! y -25 G& 3.25/2.25
> 16/11  ! x -24 G# 3.2/2.2
> 40/27  ! w -23 A\
> 3/2    ! v -22 A. quinte
> 32/21  ! u -21 A/ 2.9090../1.9090..
> 54/35  ! t -20 A+
> 14/9   ! s -19 BB 2.8/1.8
> 128/81 ! r -18 Bb
> 8/5    ! q -17 A# 2.666.../1.666...
> 13/8   ! p -16 A& 2.6/1.6
> 64/39  ! o -15 B- 2.56/1.56
> 5/3    ! n -14 B\ 2.5/1.5
> 27/16  ! m -13 B. 2.4545../1.4545..
> 12/7   ! l -12 B/ 2.4/1.4
> 26/15  ! k -11 B+ 2.3636../1.3636.. C-
> 7/4    ! j -10 C\ 2.333.../1.333...
> 16/9   ! i  -9 C.
> 9/5    ! h  -8 C/ 2.25/1.25
> 11/6   ! g  -7 C+ 2.2/1.2
> 24/13  ! f  -6 DB 2.1818../1.1818..
> 15/8   ! e  -5 Db
> 112/59 ! d  -4 C#
> 27/14  ! c  -3 C&
> 64/33  ! b  -2 D-
> 77/39  ! a  -1 D\
> 2/1    ! @'+-0 D.
> !
> ![eof]
> 
> Attend, that it contains the odd harmonic 2n-1 'overtone-series'
> at the note-names for the generalized tonic 8-fold chord:
> 
> @:v:Q:j:I:X:p:e == 1:3:5:7:9:11:13:15 == D.:A.:Gb:C\:E.:G+:A&:Db
> 
> that fits also for the corresponding dominat- and subdominat chord.
> 
> Quest:
> Sounds that 'addition-of-flavor' against 53-edo 'sharp-enough'
> likewise in yours ears too, when perceiving by listening to that?
> 
> bye
> Andy
>

From: Andy (2012-05-18)
Subject: push 53-tone to 13-limit, was: Re: Symmetric 53-tone well-temperament....

--- In [email protected], "Keenan Pepper" <keenanpepper@...> wrote:
>
> I'm interested in what actual music using this system
> would sound like. I'm having trouble imagining it.
> 
Hi Keenan

a good way to comprehned and percieve the 13th partial
deeper en detail...

> > @:v:Q:j:I:X:p:e == 1:3:5:7:9:11:13:15 == D.:A.:Gb:C\:E.:G+:A&:Db

...consists in mastering of performing that series
with your own lips on an brass-instrument, especially on an:

http://en.wikipedia.org/wiki/Natural_trumpet
quote
"...by playing in the extreme upper register and "lipping" the notes of the 11th and 13th harmonics (that is, flattening or sharpening those impure harmonics into tune with the embouchure), it was possible to play diatonic major and minor scales (and, hence, actual melodies) on a natural trumpet. The most talented players were even able to produce certain chromatic notes outside the harmonic series by this process (such as lipping a natural C down to B), although these notes were mostly used as brief passing tones. (In Germany, this technique was called Heruntertreiben, literally "driving down".) Other "impure" harmonics (such as the 7th and 14th - B♭ on an instrument pitched in C - which are very flat) were avoided by most composers, but were sometimes deliberately used, for example, where their unusual sonic qualities would complement the accompanying text in a sacred work..."

References:
http://upload.wikimedia.org/wikipedia/commons/e/e6/Harmonic_Series.png
There the postition and the nor of the 13th partial within the overtone-series is labeled in red color, 
for indicating +41cents difference against 
the detuned ordinary 12-edo concept.

See also:
http://en.wikipedia.org/wiki/Harmonic
http://de.wikipedia.org/wiki/Naturtonreihe

For comprehending that natural enriched sonority, 
try to internalize the sound-examples of that euphony 
by studying the pertinent demonstrations in:
http://en.wikipedia.org/wiki/Harmonic_series_%28music%29 
http://en.wikipedia.org/wiki/Overtone

Or simply just listen how the voices do actually sound:

http://en.wikipedia.org/wiki/List_of_overtone_musicians
records
http://www.overtone.cc/

Hope that information helps in order to 'imagine' this canorousness.
bye
Andy

From: Keenan Pepper (2012-05-19)
Subject: push 53-tone to 13-limit, was: Re: Symmetric 53-tone well-temperament....

--- In [email protected], "Andy" <a_sparschuh@...> wrote:
>
> --- In [email protected], "Keenan Pepper" <keenanpepper@> wrote:
> >
> > I'm interested in what actual music using this system
> > would sound like. I'm having trouble imagining it.
> > 
> Hi Keenan
> 
> a good way to comprehned and percieve the 13th partial
> deeper en detail...
> 
> > > @:v:Q:j:I:X:p:e == 1:3:5:7:9:11:13:15 == D.:A.:Gb:C\:E.:G+:A&:Db
> 
> ...consists in mastering of performing that series
> with your own lips on an brass-instrument, especially on an:
> 
> http://en.wikipedia.org/wiki/Natural_trumpet
> quote
> "...by playing in the extreme upper register and "lipping" the notes of the 11th and 13th harmonics (that is, flattening or sharpening those impure harmonics into tune with the embouchure), it was possible to play diatonic major and minor scales (and, hence, actual melodies) on a natural trumpet. The most talented players were even able to produce certain chromatic notes outside the harmonic series by this process (such as lipping a natural C down to B), although these notes were mostly used as brief passing tones. (In Germany, this technique was called Heruntertreiben, literally "driving down".) Other "impure" harmonics (such as the 7th and 14th - B♭ on an instrument pitched in C - which are very flat) were avoided by most composers, but were sometimes deliberately used, for example, where their unusual sonic qualities would complement the accompanying text in a sacred work..."
> 
> References:
> http://upload.wikimedia.org/wikipedia/commons/e/e6/Harmonic_Series.png
> There the postition and the nor of the 13th partial within the overtone-series is labeled in red color, 
> for indicating +41cents difference against 
> the detuned ordinary 12-edo concept.
> 
> See also:
> http://en.wikipedia.org/wiki/Harmonic
> http://de.wikipedia.org/wiki/Naturtonreihe
> 
> For comprehending that natural enriched sonority, 
> try to internalize the sound-examples of that euphony 
> by studying the pertinent demonstrations in:
> http://en.wikipedia.org/wiki/Harmonic_series_%28music%29 
> http://en.wikipedia.org/wiki/Overtone
> 
> Or simply just listen how the voices do actually sound:
> 
> http://en.wikipedia.org/wiki/List_of_overtone_musicians
> records
> http://www.overtone.cc/
> 
> Hope that information helps in order to 'imagine' this canorousness.
> bye
> Andy

It seems like you completely misunderstood me. I'm quite familiar with the sound of the 13th harmonic in the context of the harmonic series. But the scale you posted was not the harmonic series; it was something quite different.

Where is actual music that uses *that scale*?

Keenan

From: Andy (2012-05-22)
Subject: push 53-tone to 13-limit, was: Re: Symmetric 53-tone well-temperament....

--- In [email protected], "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In [email protected], "Andy" <a_sparschuh@> wrote:

> > a good way to comprehend and percieve the 13th partial
> > deeper en detail...

> >  @:v:Q:j:I:X:p:e == 1:3:5:7:9:11:13:15 == D.:A.:Gb:C\:E.:G+:A&:Db

> It seems like you completely misunderstood me. 
> I'm quite familiar with the sound of the 13th harmonic
> in the context of the harmonic series.
> But the scale you posted was not the harmonic series; 
> it was something quite different.

ok Keenan,
but i only wrote, that my 53 ratios do contain
the harmonic series up to 13-limit as subset
of the earlier similar 53-tone concept from:

http://www.microtonal-synthesis.com/scale_53tet.html
"
@ +-0 D.  1   0      1/1    +0cents   unison
A  +1 D/  2   22.642 66/65  -3.7901   first step
B  +2 D+  3   45.283 33/32  -7.9899   undecimal comma, 33rd harmonic
C  +3 EB  4   67.925 26/25  +0.0243   -
D  +4 Eb  5   90.566 21/20  +6.0988   minor semitone
E  +5 D#  6  113.208 16/15  +1.4763   minor diatonic semitone
F  +6 D&  7  135.849 13/12  -2.7236   tridecimal 2/3-tone
G  +7 E-  8  158.491 11/10  -6.5137   4/5-tone, Ptolemy's second
H  +8 E\  9  181.132 10/9   -1.2716   minor whole tone
I  +9 E. 10  203.774 9/8    -0.1364   major whole tone
J +10 E/ 11  226.415 8/7    -4.7590   septimal whole tone [E+=F-]
K +11 E+ 12  249.057 15/13  +1.3156   -
L +12 F\ 13  271.698 7/6    +4.8272   septimal minor third
M +13 F. 14  294.340 13/11  +5.1299   tridecimal minor third
N +14 F/ 15  316.981 6/5    +1.3398   minor third
O +15 F+ 16  339.623 11/9   -7.7853   undecimal neutral third
P +16 GB 17  362.264 16/13  +2.7918   tridecimal neutral third
Q +17 Gb 18  384.906 5/4    -1.4081   major third
R +18 F# 19  407.547 33/26  -5.1981   tridecimal major third
S +19 F& 20  430.189 9/7    -4.8954   septimal major third, BP third
T +20 G- 21  452.830 13/10  -1.3838   -
U +21 G\ 22  475.472 21/16  +4.6908   narrow fourth
V +22 G. 23  498.113 4/3    +0.0682   perfect fourth 
W +23 G/ 24  520.755 35/26  +6.1428   -
X +24 G+ 25  543.396 11/8   -7.9217   undecimal semi-augmented fourth
Y +25 AB 26  566.038 18/13  +2.6554   -
Z +26 Ab 27  588.679 7/5    +6.1671   septimal or Huygens' tritone 
z -26 G# 28  611.321 10/7   -6.1671   Euler's tritone
y -25 G& 29  633.962 13/9   -2.6554   -
x -24 A- 30  656.604 16/11  +7.9217   undecimal semi-diminished fifth
w -23 A\ 31  679.245 52/35  -6.1428   -
v -22 A. 32  701.887 3/2    -0.0682   perfect fifth
u -21 A/ 33  724.528 32/21  -4.6908   wide fifth
t -20 A+ 34  747.170 20/13  +1.3838   -
s -19 BB 35  769.811 14/9   +4.8954   septimal minor sixth
r -18 Bb 36  792.453 52/33  +5.1981   -
q -17 A# 37  815.094 8/5    -1.4081   minor sixth
p -16 A& 38  837.736 13/8   -2.7918   tridecimal neutral sixth
o -15 B- 39  860.377 18/11  +7.7853   undecimal neutral sixth
n -14 B\ 40  883.019 5/3    -1.3398   major sixth, BP sixth
m -13 B. 41  905.660 22/13  -5.1299   -
l -12 B/ 42  928.302 12/7   -4.8272   septimal major sixth
k -11 B+ 43  950.943 26/15  -1.3156   - [B+=C-]
j -10 C\ 44  973.585 7/4    +4.7590   harmonic seventh
i  -9 C. 45  996.226 16/9   +0.1364   Pythagorean minor seventh
h  -8 C/ 46 1018.868 9/5    +1.2716   just minor seventh, BP seventh
g  -7 C+ 47 1041.509 20/11  +6.5137   large minor seventh
f  -6 DB 48 1064.151 24/13  +2.7236   -
e  -5 Db 49 1086.792 15/8   -1.4763   classic major seventh
d  -4 C# 50 1109.434 40/21  -6.0988   acute major seventh
c  -3 C& 51 1132.075 25/13  -0.0243   -
b  -2 D- 52 1154.717 35/18  +3.4874   septimal semi-diminished octave
a  -1 D\ 53 1177.358 63/32  +4.6226   octave - septimal comma
@'+-0 D. 1' 1200     2/1              just-octave

But against that estimable forerunner, 
my own proposal roams more smooth through the 5ths.

> Where is actual music that uses *that scale*?

Sorry,
at the moment performances do exist
still only in life improvisations on my 
horn and violon-cello.

bye
Andy

From: Keenan Pepper (2012-05-23)
Subject: push 53-tone to 13-limit, was: Re: Symmetric 53-tone well-temperament....

--- In [email protected], "Andy" <a_sparschuh@...> wrote:
> > Where is actual music that uses *that scale*?
> 
> Sorry,
> at the moment performances do exist
> still only in life improvisations on my 
> horn and violon-cello.

Sounds like it could be totally sweet. You should record them.

Keenan