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)

--- 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.

--- 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.)

--- 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!

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.

--- In tuning-math@yahoogroups.com, "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

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

>

>

>

> --- In tuning-math@yahoogroups.com, "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.

> --- In tuning-math@yahoogroups.com, "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

--- In tuning-math@yahoogroups.com, "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

/tuning/topicId_103187.html#104314

bye

Andy

--- In tuning-math@yahoogroups.com, "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

--- In tuning-math@yahoogroups.com, "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

! /tuning/topicId_38888.html#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

--- In tuning-math@yahoogroups.com, "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: /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

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

Keenan

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

>

> --- In tuning-math@yahoogroups.com, "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: /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

>

--- In tuning-math@yahoogroups.com, "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 - BâÂ™ 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

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

>

> --- In tuning-math@yahoogroups.com, "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 - BâÂ™ 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

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

>

> --- In tuning-math@yahoogroups.com, "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

--- In tuning-math@yahoogroups.com, "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