Topic: dyadic-rational 53 microtonal AP scale starting from dodecatonic 440Hz kernel

1 scales

Thread (1 messages)

From: Andreas Sparschuh (2007-07-31)
Subject: dyadic-rational 53 microtonal AP scale starting from dodecatonic 440Hz kernel

First of all:
Get rid of yahoo's garbeled spaceing by:
Click first on button: (Show Meassage Option)
than also on: (Use Fixed Widht Font)
for proper display of the charts.

Additional accidentials:
'/' elevation of about an comma upwards sharpened
'\' depression of about an comma downwards flattened

Here comes a chain of 53 partially tempered 5ts
starting from 256Hz expressed in terms of
http://en.wikipedia.org/wiki/Dyadic_fraction
s as exact frequencies in the middle octave:

Freq: :Pitch: :Octave-equivalences: :Further comments
-----------------------------------------------------
256Hz C\\     1  unison @ middle-C\\ equivalent to 2^8
384   G\\     3 * 128 '3rd partial of C\\
288   D\\     9 * 32  '9th partial of C\\
432   A\\     27 * 8  'Guiseppe Verdi's a'=432Hz
324   E\\=Fb\ 81 * 4
486   B\\=Cb\ 243 * 2 = 3^5*2
364.5 Gb\     729 / 2 = 3^6/2 '3-limit tritone (9/8)^3
273.5 Db\     (3^7=2187)2188 1094 547/2 '~apotome 3^7/2^11
410.5 Ab\     (1641)1642 821/2 
308   Eb\     (2463)2464 1232 308. 154 77=11*7 '7th partial of F=11
462   Bb\     231 * 2   
693   F\      693.
520   C\      (2079)2080 1040 520. 260 130 65=13*5 '3rd of Ab=13
390.5 G\      (195 390 780)781 / 2
293   D\      (2343)2344 1172 586 293.
----begin Schlick's "eleven-eyes" dodecatonic subset from 440Hz---
440   A\  (879)880 440. 220 110 55 http://en.wikipedia.org/wiki/A440
660   E\=Fb   165 * 4 ' 3rd of C=33 'in C-E\-G == C-Fb-G == 44:55:66 
495   B\=Cb   495.
371   Gb      (1485)1484 742 371.
556   Db      (1113)1112 556. 278 139
416   Ab      (417)416. 208 104 52 26 13 '13th partial of C\\ unison
312.5 Eb      (39 78 156 312 624)625 = 5^4 = 5*5*5*5
469   Bb      (1875)1876 938 469.
352   F       (1407)1408 704 352. 176 88 44 22 11 '11th partial of C\\
264   C       33 * 16 ' middle C for a'=440Hz dodecatonics
396   G       99 * 4 ' Pseudo-Odo's & Newton's GAMMA-ut GG ~100 Hz
297   D       297.
----end dodecatonic 440Hz 12-tone subset-----------------------------
445   A       (891)890 445.Hz 'Karajan's high pitch 89/88 above normal
333.5 E       (1335)1334 667 / 2
500   B       (2001)2000 1000 500. 250 125 = 5^3 = 5*5*5
375   F#      375.
281.5 C#      (1125)1126 563 / 2
422   G#      (1689)1688 844 422. 211
316.5 D#      633 / 2
475   A#      (1899)1900 950 475.
356   F/=E#   (1425)1424 712 356. 178 89
267   C/=B#   267.
400   G/      (801)800 400.Hz 200 100 50 25 = 5*5 'french opera-tone 
300   D/      75 * 4
448   A/      (225)224 112 56 28 14 7 ' harmonic 7th partial of C\\
336   E/      21 * 16
524   B/      63 * 8
378   F#/     189 * 2
284   C#/     (567)568 284 142 71
426   G#/     213 * 2
320   D#/     (639)640 320. 160 80 40 20 10 5 '3rd of C\\ unison
480   A#/     15 * 64
360   F//=E#/ 45 * 8
270   C//=B#/ 135 * 2     
405   G//     405.
303.5 D//     (1215)1214 607 / 2
455   A//     (1821)1820 910 455.
316.5 E//=F\\ (1365)1366 683/2 
256   B//=C\\ (2049)2048 1024 512 256. 128 64 32 16 8 4 2 1 = unison

AP possessors easily select among that 53 pitch-classes
in an secure way out the 12-tone kernel 
consisting in the usually known 12 pitches of:

A\ E\ B\ Gb Db Ab Eb Bb F C G D

!eleven_eyes_dodecatonics.scl
!
12 subset 5ths out of 53 starting from a'=440Hz
 12
!
! 1/1   ! C  *264*Hz = 33Hz*8 unison "middle-C"
139/132 ! Db *278* C#\
  9/8   ! D  *297*
625/528 ! Eb *312.5* = 5^4/2
  5/4   ! E\ *330* or Fb
  4/3   ! F  *352* = 11 * 32 '11th partial to C\\
371/264 ! Gb *371* F#\
  3/2   ! G  *396*
 52/33  ! Ab *416* 208 104 52 26 13th partial to C\\
  5/3   ! A\ *440*Hz standard normal-tone
469/264 ! Bb *469*
 15/8   ! B\ *495*
  2/1   ! C' *528* octave

...without hesitating in labeling the note names correctly
corresponding to the absolute frequencies as reference.

In some AP cases it's also possible 
-by some further exercise-
to develop from that initial coarse *12-tone*
upgrading to an 53-tone refinement as goal,
gained stepwise in commatatically
ascending increments,
while the initial *12-tone* base 
remains kept as embedded subset: 

AP53.scl
!
dyadic-rational 53-tone not only for AP possessors
 53
!
! 1/1   ! C\\ 256Hz unsion "middle-C\\" for A\=440Hz
 65/64  ! C\  260 
 33/32  ! C  *264*
267/256 ! C/  267
135/128 ! C// 270
547/512 ! Db\ 273.5
139/128 ! Db *278*
563/512 ! C#  281.5
 71/64  ! C#/ 284
  9/8   ! D\\ 288
293/256 ! D\  293
297/256 ! D  *297*
 75/64  ! D/  300
607/512 ! D// 303.5
 77/64  ! Eb\ 308
625/712 ! Eb *312.5*
633/512 ! D#  316.5
  5/4   ! D#/ 320
 81/64  ! E\\ 324
165/128 ! E\ *330*
667/512 ! E   333.5
 21/16  ! E/  336
683/512 ! E// 341.5
693/512 ! F\  346.5
 11/8   ! F  *352*
 89/64  ! F/  356
 45/32  ! F// 360
729/512 ! Gb\ 364.5
371/256 ! Gb *371*
375/256 ! F#  375
189/128 ! F#/ 378
  3/2   ! G\\ 384
781/512 ! G\  390.5
 99/64  ! G  *396*
 25/16  ! G/  400
405/256 ! G// 405
821/512 ! Ab\ 410.5
 13/8   ! Ab *416*
211/128 ! G#  422
213/128 ! G#/ 426
 27/16  ! A\\ 432
 55/32  ! A\ *440*Hz
445/256 ! A   445
  7/4   ! A/  448
455/256 ! A// 455
231/128 ! Bb\ 462
469/256 ! Bb *469*  
475/256 ! A#  475
 15/8   ! A#/ 480
243/128 ! B\\ 486
495/256 ! B\ *495*
125/64  ! B   500
 63/32  ! B/  504
2/1 ! B//=C\\'512 octave to initial unison
!
!

Real AP posessors should be able to sing by voice 
-without any further training- at least
the embedded *12 pitch-classes*  within
an accuracy of ~1/3--~1/2 comma 
or max. ~0.5--~0.7 % deviation from
the intended exact pitch-frequency,
as indicated in the above list.

Remark:
Sorry, but without that AP aptitude: 
There's hardly any chance!
Without that demand it makes imho. no sense to proceed.

The goal:
After he/she has become more (hopefully not less)
secure within that well approved und tested limits,
it's likely that he/she is able to develop the same
skill -by and by- for the other 41=53-12 pitch-classes
inbetween too. After a while AP identification works
for 53 detection as well as for 12 cognotion.

Advice for training:
Listen to old Karajan's a'=445Hz recordings,
but intead of tolerating them in black despair as 5Hz to much
sharp detuned versus the really 440Hz meant, 
just consider fully aware that his performances
turn out to be sounding as one comma-class higher 
shifted than usually normal:
Learning outcome:
Understand K's ~+5Hz detuning as transposition
of about an whole comma sharper elevated,
as an change in pitchclass about a dozen of 5ths:
Than yours tortured ears acclimate even to bear the skirl Karajan.

Conclusion:
Whenever examine different types of AP possessors, 
the usual 12tone-AP should be clearly discerned from 
more demanding 53tone-AP refinement. 

Quest:
Are here in that group any others 53tone-AP possessors too?
And if so:
Has anybody developed an alternative set of 
53 absolute frequencies
as individual preference of personal taste?

Never mind if you are lacking the AP aptitude,
even without that competence it's possible
to improve the cabability of discerning pitches
in the finer microtonal 53-tone graduation,
howsoever independent if you own AP or RP
or even none of them both?

A.S.