Yahoo Tuning Groups Ultimate Backup tuning Pure Chords

Here's my latest development. I used my Interval Calculator v.7.0 (in the JohnOSullivan folder in the "Files" section) to work out all of the "good" intervals that occur in my NPT scale which are not more than two octaves wide. I then used these intervals to build chords where every possible interval/dyad was "good" (according to the calculator which so far seems consistent).

If you're is interested in testing these chords you'll need to tune your keyboard to NPT...
1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.
If the tonic is 'E' then lower or raise the frequencies of the notes by the following amounts (in cents).

E......0.0c
F....+19.4c
F#....+3.9c
G....+15.6c
G#...-13.7c
A.....-2.0c
A#...-17.5c
B.....+2.0c
C....+13.7c
C#...-15.6c
D....+17.6c
D#...-11.7c

Set the tonic to E. Here are the chords (all consecutive notes are not more than 6 semitones apart)...

E, G#, B, E, G#, B, E.

F, G#, B, D#, F, G#, B, D#.

F#, B, D, F#, B, D, F#.

G, A#, C, D, G.

G, B, D, G, B, D, G.

G#, B, D#, F, G#, B, D#.

A, B, E, G, B, E, G.

A#, C, D, G.

A#, C, E, G, A#, C.

B, D, F#, B, D, F#, B.

C, D, G, D, G, B.

C#, D#, F, G#, B, D#, G#, B.

D, F#, B, D, F#, B, D.

D#, F, G#, B, D#, F, G#, B, D#.

I have tried to squeeze as many notes as possible into most of the chords listed above. I have found that many times in music less is more and by omitting one or more notes from each chord above often makes the chord stronger. Try comparing the chords using 12TET instead of NPT, particularly with the A# chords.

If you work out all possible intervals (including the unisons) that occur in my 12 key NPT scale that are not more than two octaves wide there are 300 intervals (12 keys by 25 intervals = 300). If my calculator is accurate then of these 300, 154 of the intervals are "good". That means that just over half of the intervals are good which is a pretty good result. Because of this I don't think I need to temper my scale.

John.

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

>
> If you work out all possible intervals (including the unisons) that occur
> in my 12 key NPT scale that are not more than two octaves wide there are 300
> intervals (12 keys by 25 intervals = 300). If my calculator is accurate then
> of these 300, 154 of the intervals are "good". That means that just over
> half of the intervals are good which is a pretty good result. Because of
> this I don't think I need to temper my scale.
>

Hi John,

I think you will still find the need to temper your scale.
Try playing some common practice music in it, for instance Beethoven's Drei
Equale no1, MIDI available at www.develde.net
A "tonic" or tonal music has a much broader palette of pitches.

Also, I think you will find that by changing your 15/14 to 21/20 you'll have
a more harmonious scale.
1/1 21/20 9/8 6/5 5/4 4/3 7/5 3/2 8/5 5/3 9/5 15/8 2/1
I'm not sure how your calculator works, but it should like the above scale
better.

In my opinion the above scale works from a single "harmonic root", and is a
mix of "harmonic-6-limit" and 2 'harmonic-7-limit" intervals.
However there are far too few 7-limit intervals for this scale to function
for 7-limit music.
(and again, for common practice 6-limit music you'd need a model for
changing the root of this scale in the music)

Marcel

🔗john777music <jfos777@...>

Hi Marcel,

some common practice music clearly won't work with my scale and I'm not worried if it doesn't.

I chose 15/14 as the second note because when you do this my scale has two tonics of equal strength: 1/1 and 3/2. One tonic is better for ascending music, and the other for descending music.

As regards my two 7 limit notes, according to my calculator the 15/14 goes with 5/4, 3/2, 15/8, 15/7, 5/2, 3/1, 15/4 and 30/7. The 7/5 goes with 8/5, 9/5, 2/1, 12/5, 14/5, 16/5 and 28/5.

I tried a 5 limit version of my symmetric NPT scale, I used 16/15 instead of 15/14 and 45/32 instead of 7/5 but this yielded fewer "good" intervals.

Check out the calculator (in the JohnOSullivan folder in the "Files" section) and let me know what you think of it.

John.

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> >
> > If you work out all possible intervals (including the unisons) that occur
> > in my 12 key NPT scale that are not more than two octaves wide there are 300
> > intervals (12 keys by 25 intervals = 300). If my calculator is accurate then
> > of these 300, 154 of the intervals are "good". That means that just over
> > half of the intervals are good which is a pretty good result. Because of
> > this I don't think I need to temper my scale.
> >
>
> Hi John,
>
> I think you will still find the need to temper your scale.
> Try playing some common practice music in it, for instance Beethoven's Drei
> Equale no1, MIDI available at www.develde.net
> A "tonic" or tonal music has a much broader palette of pitches.
>
> Also, I think you will find that by changing your 15/14 to 21/20 you'll have
> a more harmonious scale.
> 1/1 21/20 9/8 6/5 5/4 4/3 7/5 3/2 8/5 5/3 9/5 15/8 2/1
> I'm not sure how your calculator works, but it should like the above scale
> better.
>
> In my opinion the above scale works from a single "harmonic root", and is a
> mix of "harmonic-6-limit" and 2 'harmonic-7-limit" intervals.
> However there are far too few 7-limit intervals for this scale to function
> for 7-limit music.
> (and again, for common practice 6-limit music you'd need a model for
> changing the root of this scale in the music)
>
> Marcel
>

🔗a_sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.

Hi John,
simply mutiply all of that ratios by the common factor 420Hz :=7*5*3*4,
That procedure yields:

a' 420 Hz Mozart's tuning-fork in absolute-pitch
#' 450
b' 472.5
c" 504 tenor_C5
#" 525
d" 560
#" 588
e" 630
f" 672
#" 700
g" 756
#" 787.5
a" 840

or as analyis by an cycle of biased 5ths deviations

A: A105 a210 a'420 Hz
E: e'315 := (A*3) e"630:=(a*3 pure)
B: b236.25 b'472.5 b"945 := (e'*3 pure)
F#: f#175 f#'350 f#"700 (< 708.75 := e'*3, an SC=81/80 ~21.506C down)
C#: c#"525 := (f#*3 pure)
G#: g'393.75 g"787.5 g'''1575 := (f"*3 pure)
D#: 147 294 d588 1176 (< 1181.25 := g'*3 or 225/224 ~7.71C upwards)
Bb: (7..224)<a225 a'450=9*50(>(49*9=441:=D#*3 wolf-5th 50/49 ~+35C)
F: FFF21:=7*3 FF42 F84 f168 f'336 f"672
C: C63 c126 c'252=middleC4 c"504 pure
G: g189 g'378 g"756 pure
D: DD35 D70 d140 d'280 d"560 pure
A: A105 pure

In order to get rid of the two coarse wolf-5ths
1: B-F#: 40/27 = ~680.4...Cents, an SC to much flattend downwards
2: D#-Bb: 150/98 = 736.93...Cents, all to much sharpend upwards
here i do suggest some refinement changes,
for obtaining an more subtle well-temperament without any wide 5ths:

Bb: 7
F: 21
C: 63
G: (D/3=47 94 188 <) 189 := C*3
D: (E/9=Bb*5=35 70 140 <) 141 := 47*3
A: E/3=105 210 420Hz
E: 315
B: (59 118 236 472 944<) 945 := E*3)
F# 177 := 59*3
C# 531
G# 1593
D# 4779
Bb 7 14 28 56 112 224 448 896 1792 3584 7168 14336 (< 14337 := 59*3^5)

or in chromatic ascending order

c' 252 middle_C4
#' 265.5
d' 282
#' 298.6875 = 298+11/16
e' 315
f' 336
#' 354
g' 378
#' 398.25 = 398+1/4
a' 420 Hz
#' 448
b' 472.5
c" 504 tenor_C5

! SpWell420Hz.scl
Sparschuh's almost-JI synchrone-beating well-tuning @ abs.-pitch 420Hz
12
!
531/448 ! C# (135/128)*(944/945) = (256/243)*(14337/14336)
47/42 ! D (10/9)*(141/140 ~+13.3C) = (9/8)*(188/189 ~-9.2C)
4479/3584 ! D# (32/27)*(14337/14336 +~0.120757092...Cents)
5/4 ! E
4/3 ! F
177/112 ! F# (1024/729)*(14337/14336) = (45/32)*(944/945)
3/2 ! G
1539/896 ! G# (128/81)*(14337/14336) = (135/128)*(944/945)
5/3 ! A
16/9 ! Bb (405/256)*(32805/32768 = the schisma)
15/8 ! B
2/1
!
![eof]

Here attend in the 5 remote accidentials F#-C#-G#-D#-Bb
the both approximations of 3-limit versus 5-limit ratios,
due to the superparticular-ratio bisection of the schisma:

/tuning-math/message/17405
"
2; 59,7-limit
(945/944)*(14337/14336) := (7*5*3^3/59/2^4)*(59*3^5/7/2^11)
~1.8329637...Cents + ~0.120757092...Cents
"
and also be aware of the bisection of the SC=81/80
inbetween the central 5ths G-D-A into
(81/80 ~+21.5) = (189/188 ~+9.2C)*(141/140 ~+13.3C)
too.

or concise for division of the PC 3^12/2^19 ~23.46...Cents
into 4 subparts, that satisfy the superparticular ratio property:

In the exact fractions:
C-G 188/189 D 140/141 A-E-B 944/945 F#-C#-G#-D# 14336/14337 Bb-F-C

or for those that do prefer approximations in Cent-units:
C-G-~-9.2...~D-~-13.3...~A-E-B-~1.8...~F#-C#-G#-D#-~-0.12...~Bb-F-C

I hope that advanced variant without the rough wolfs
sounds also more pleasant in yours ears too.

bye
A.S.

🔗a_sparschuh <a_sparschuh@...>

> --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@> wrote:

> In my opinion the above scale works from a single "harmonic root",
> and is a mix of "harmonic-6-limit" and 2 'harmonic-7-limit"
> intervals.
> However there are far too few 7-limit intervals for this scale to
> function for 7-limit music.
> (and again, for common practice 6-limit music you'd need a model for
> > changing the root of this scale in the music)
>
Hi Marcel & John,
how about to try this in an cycle of 53 conscutive 5hts modulo octaves?

[In order to read here the intended spaceings more properly]
[plaese click under the option 'Show-Meassage-Info' on the button ']
[for the option "Use-Fixed-Width-Font"]

! Sp53Ragismatic.scl
Sparschuh's [2010] almost-JI Ragismatic 4375/4374 based 53-tone
53
!
! as cycle of 53 consecutive 5ths by an generalized 3n-1 sequence
!
!Nr: Name Ratio "low-decimal-representation" in absolute-pitch
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! 0: C- [1/1] 1.0 unison
! 1: G- [3/2] 3.0
! 2: D- [9/8] 9.0 18 36 72 (= 71+1,see #41:C&)
! 3: A- [27/16] 27.0
! 4: E- [81/64] 81.0
! 5: B- [243/128] 243.0
! 6: GB [729/512] 729.0
! 7: DB [2187/2048] 2187.0
! 8: AB [205/128] (0.1...102.4 <) 102.5 205 410 ...6560(<6561:=3^8)
! 9: EB [6/5] 0.3
!10: BB [9/5] 0.9
!11: F\ [27/20] 2.7
!12: C\ [81/80] Syntonic-Comma 8.1
!13: G\ [243/160] 24.3
!14: D\ [729/640] 72.9
!15: A\ [2187/1280] 218.7
!16: E\ [41/32] (0.01 ... 10.24 <) 10.25 20.5...656(<656.1=3^8/10)
!17: B\ [48/25] 0.03
!18: Gb [36/25] 0.09
!19: Db [27/25] 0.27
!20: Ab [81/50] 0.81
!21: Eb [243/200] 2.43
!22: Bb [729/400] 7.29
!23: F. [2187/1600] 21.87
!24: C. [41/40] (0.001 ... 1.024 <) 1.025 2.05 4.1 ... 65.6(<65.61)
!25: G. [192/125] 0.003
!26: D. [144/125] 0.009
!27: A. [216/125] 0.027
!28: E. [162/125] 0.081
!29: B. [243/125] 0.243
!30: F# [729/500] 0.729
!31: C# [2187/2000] 2.187
!32: G# [41/25] (0.0001...0.1024<) 0.1025 0.205 ... 6.56(<6.561)
!33: D# [768/625] 0.0003
!34: A# [1152/625] 0.0009
!35: F/ [864/625] 0.0027
!36: C/ [648/625] 0.0081
!37: G/ [972/625] 0.0243
!38: D/ [729/625] 0.0729
!39: A/ [7/4] (0.2187 0.4374 >) 0.4375 ... 0.175 3.5 7.0 Ragisma
!40: E/ [21/16] 21.0
!41: B/ [63/32] 63.0
!42: F& [189/128] 189.0
!43: C& [567/512] (71 142 284 568>)567.0
!44: G& [213/128] 213.0 := 71*3 = 639/3 wide 5hts
!45: D& [5/4] 5.0 10 20 40 ... 320 640 (>639)
!46: A& [15/8] 15.0
!47: F+ [45/32] 45.0
!48: C+ [135/128] 135.0
!49: G+ [405/256] 405.0
!50: D+ [1215/1024] 1215.0 (final schisma trisection)
!51: A+ [911/512] 911.0 1822 3644 (<3645 := 1215*3)
!52:E+=F-[683/512] 683.0 1366 2732 (<2733 := 911*3)
!53:C-=B+ [1/1] unison 1.0 2 .. 2048 (<2049 := 683*3)
!
!
!Ratios,Note Nr: Name=decimal-value, name of Absolute-Pitch frequency
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! 1/1 ! 0: C- = 1.0 unison c-' 256.0 Hz middle_C-
81/80 ! 1: C\ = 1.012,5 c\' 259.2
41/40 ! 2: C. = 1.025 c.' 262.4
648/625 ! 3: C/ = 1.036,8 c/' 265.420,8
135/128 ! 4: C+ = 1.054.687,5 c+' 270.0
2187/2048 ! 5: DB = 1.067,871,093,75 dB' 273.375
27/25 ! 6: Db = 1.08 db' 276.48
2187/2000 ! 7: C# = 1.093,5 c#' 279.936
567/512 ! 8: C& = 1.107,421,875 c&' 283.5
9/8 ! 9: D- = 1.125 d-' 288.0
729/640 ! 10: D\ = 1.139,062,5 d\' 291.6
144/125 ! 11: D. = 1.152 d.' 294.912
729/625 ! 12: D/ = 1.1664 d/' 298.598,4
1215/1024 ! 13: D+ = 1.186.523,437,5 d+' 303.75
6/5 ! 14: EB = 1.2 eB' 307.2
243/200 ! 15: Eb = 1.215 eb' 311.04
768/625 ! 16: D# = 1.228,8 d#' 314.572,8
5/4 ! 17: D& = 1.25 d&' 320.0
81/64 ! 18: E- = 1.265,625 e-' 324.0
41/32 ! 19: E\ = 1.281,25 e\' 328.0
162/125 ! 20: E. = 1.296 e.' 331.776
21/16 ! 21: E/ = 1.312,5 e/' 336.0
683/512 ! 22:E+F-= 1.333,984,375 e+=f-' 341.5 or (4/3)*(2049/2048)
27/20 ! 23: F\ = 1.35 f\' 345.6
2187/1600 ! 24: F. = 1.366,875 f.' 349.92
864/625 ! 25: F/ = 1.382,4 f/' 353.894,4
45/32 ! 26: F+ = 1.406,25 f+' 360.0
729/512 ! 27: GB = 1.423,828,125 gB' 364.5
36/25 ! 28: Gb = 1.44 gb' 368.64
729/500 ! 29: F# = 1.458 f#' 373.248
189/128 ! 30: F& = 1.476,562,5 f&' 378.0
3/2 ! 31: G- = 1.5 g-' 384.0
243/160 ! 32: G\ = 1.518,75 g\' 388.8
192/125 ! 33: G. = 1.536 g.' 393.216
972/625 ! 34: G/ = 1.555,2 g/' 398.131,2
405/256 ! 35: G+ = 1.582,031,25 g+' 405.0 ~J.S.Bach in Köthen~
205/128 ! 36: AB = 1.601,562,5 aB' 410.0 A.Werckmeister[?1681?]
81/50 ! 37: Ab = 1.62 ab' 414.72 ~neo-Baroque~ a'
41/25 ! 38: G# = 1.64 g#' 419.84 ~W.A.Mozart,Vienna~
213/128 ! 39: G& = 1.664,062,5 g&' 426.0 or (5/3)*(639/640)
27/16 ! 40: A- = 1.687,5 a-' 432.0 Mersenne/Sauveur pitch
2187/1280 ! 41: A\ = 1.708,593,75 a\' 437.4 ~Luciano-Pavarotti~
216/125 ! 42: A. = 1.728 a.' 442.368 Hz my~acustic~piano
7/4 ! 43: A/ = 1.75 a/' 448.0 = 7th-partial of 64Hz
911/512 ! 44: A+ = 1.779,296,875 a+' 455.5 ~Bach's~organ~Halle~
9/5 ! 45: BB = 1.8 bB' 460.8
729/400 ! 46: Bb = 1.822,5 bb' 466.56
1152/625 ! 47: A# = 1.843,2 a#' 471.859,2
15/8 ! 48: A& = 1.875 a&' 480.0
243/128 ! 49: B- = 1.898,437,5 b-' 486.0
48/25 ! 50: B\ = 1.92 b\' 491.52
243/125 ! 51: B. = 1.944 b.' 497.664
63/32 ! 52: B/ = 1.968,75 b\' 504 Archytas-Comma below 2/1
2/1 ! 53:B+C-= 2.0 b+'=c-" 512 tenor_C-
!
![eof]

Statistics of interval occurence:
1. Number of exact 5ths (3/2): 39
2. Number of exact 3rds (5/4): 36
3. Number of exact 7ths (7/4): 04

Attend:
Septimal chords do appear on:
C- : D& : G- : A/ : D- == 4 : 5 : 6 : 7 : 9
and also over the fundamentals G- , D- and A- respectively.

Here some addional links about
the historically origin and the technically properties of the:
http://en.wikipedia.org/wiki/Ragisma
or the so called:

http://hexadecimal.florencetime.net/from_Nippur_cubit_to_Viennese_ell.htm
"poppy(-seed komma)
4375 / 4374 := (5^4 × 7^1) : (2^1 × 3^7) = ~1.000,229...."

See for further information:
http://de.wikipedia.org/wiki/Karlspfund
(Sorry, available only in German or Esperanto)
http://it.wikipedia.org/wiki/Utente:Carlomorino/libbra_carolingia
"
Bei historischen Längenmaßen liegt der Variationskoeffizient im allgemeinen unter 1/500, was eine Genauigkeit von ± 0,2 % bedeutet. So gelten bei den Längenmaßen z.B. 1/2400 oder 1/4374, also die 7-glatten Ratios 2401 : 2400 und 4375 : 4374, sowie ihre Reziprokwerte nicht als eigentliche Ratios, sondern nur als Kommata.
"
Google translates that into:
'
For historical length measurements is the coefficient of variation generally less than 1 / 500, which means an accuracy of ± 0.2%. Shall apply to the length dimensions e.g. 1/2400 or 1/4374, so the 7-smooth ratios 2401:2400 and 4375:4374, and their reciprocals not in actual ratios, but only as a comma.'
'
http://www.top-handwerk.de/article/Karlspfund
http://www.freelists.org/post/tuning-math/Genes-transformation

The 'Ragisma' is all about the same precision in accuracy
of well trained human ears, when discerning deviating pitches.

MIDI listening example under:
http://en.wikipedia.org/wiki/List_of_musical_intervals
"0.40Cent 4375:4374 ragisma" there the smallest given interval:
http://upload.wikimedia.org/wikipedia/en/c/ca/Ragisma_on_C.mid

bye
A.S.

🔗a_sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@> wrote:

> However there are far too few 7-limit intervals for this scale to
> function for 7-limit music.

Hi Marcel,
then try out 7-limit in the following cycle of 53 conscutive 5hts...

[In order to read here the intended spaceings more properly]
[plaese click under the option 'Show-Meassage-Info' on the button]
["Use-Fixed-Width-Font"]

! Sp53Ragismatic.scl
Sparschuh's [2010] almost-JI Ragismatic 4375/4374 based 53-tone
53
!
! as cycle of 53 consecutive 5ths by an generalized 3n-1 sequence
!
!Nr:Name Ratio 'Monzo' decimal-representation in absolute-pitch
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! 0:C- [1/1] | 0,0,0> 1.0 unison ... 256Hz
! 1:G- [3/2] |-1,1,0> 3.0
! 2:D- [9/8] |-3,2,0> 9.0
! 3:A- [27/16] |-4,3,0> 27.0
! 4:E- [81/64] |-5,4,0> 81.0
! 5:B- [243/128] |-7,5,0> 243.0
! 6:GB [729/512] |-9,6,0> 729.0
! 7:DB[2187/2048]|-11,7,0> Apotome 2187.0
! 8:AB [205/128] |-7,0,1>*41 (0.1...102.4<)102.5 205..6560(<6561:=3^8)
! 9:EB [6/5] |1, 1,-1> 0.3
!10:BB [9/5] |0, 2,-1> 0.9
!11:F\ [27/20] |-2,3,-1> 2.7
!12:C\ [81/80] |-4,4,-1> Syntonic-Comma 8.1
!13:G\ [243/160] |-5,5,-1> 24.3
!14:D\ [729/640] |-6,6,-1> 72.9
!15:A\[2187/1280]|-8,7,-1> 218.7
!16:E\ [41/32] |-5>*41 (0.01 ... 10.24 <) 10.25 20.5 ... 656(<656.1)
!17:B\ [48/25] |+3,1,-2> 0.03
!18:Gb [36/25] |+2,2,-2> 0.09
!19:Db [27/25] |+0,3,-2> 0.27
!20:Ab [81/50] |-1,4,-2> 0.81
!21:Eb [243/200] |-3,5,-2> 2.43
!22:Bb [729/400] |-4,6,-2> 7.29
!23:F.[2187/1600]|-6,7,-2> 21.87
!24:C. [41/40] |-3,0,-1>*41 (0.001..1.024<)1.025 ... 65.6(<65.61)
!25:G. [192/125] |+6,1,-3> 0.003
!26:D. [144/125] |+4,2,-3> 0.009
!27:A. [216/125] |+2,3,-3> 0.027
!28:E. [162/125] |+1,4,-3> 0.081
!29:B. [243/125] |+0,5,-3> 0.243
!30:F# [729/500] |-2,6,-3> 0.729
!31:C#[2187/2000]|-4,7,-3> 2.187
!32:G# [41/25] |0,0,-2>*41(0.0001..0.1024<)0.1025 .. 6.56(<6.561)
!33:D# [768/625] |8,1,-4> 0.0003
!34:A#[1152/625] |7,2,-4> 0.0009
!35:F/ [864/625] |5,3,-4> 0.0027
!36:C/ [648/625] |3,4,-4> 0.0081
!37:G/ [972/625] |2,5,-4> 0.0243
!38:D/ [729/625] |0,6,-4> 0.0729
!39:A/ [7/4] |-2,0,0,1> (0.2187 0.4374 >)0.4375...3.5 7.0 Ragisma
!40:E/ [21/16] |-4,1,0,1> some 7-limit 21.0
!41:B/ [63/32] |-5,2,0,1> intervals 63.0
!42:F& [189/128] |-7,3,0,1> 189.0
!43:C& [567/512] |-9,4,0,1> (71 142..568>) 567.0
!44:G& [213/128] |-7,1>*71 213.0 :=71*3 = 639/3
!45:D& [5/4] |-2,0,1> 5.0 10 ... 320 640 (>639)
!46:A& [15/8] |-3,1,1> 15.0
!47:F+ [45/32] |-5,2,1> 45.0
!48:C+ [135/128] |-7,3,1> 135.0 final
!49:G+ [405/256] |-8,4,1> 405.0 schisma
!50:D+ 1215/1024]|-10,5,1> 1215.0 trisection
!51:A+ [911/512] |-9>*911 911.0 1822 3644.0 (<3645 := 1215*3)
!52:E+ [683/512] |-9>*683 =F- 683.0 1366 2732.0 (<2733 := 911*3)
!53:C-=B+[1/1] |0> unison 1 2 4 8 ... 2048.0 (<2049 := 683*3)
!
!
!Ratios,Note Nr: Name=decimal-value, name of Absolute-Pitch frequency
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! 1/1 ! 0: C- = 1.0 unison c-' 256.0 Hz middle_C-
81/80 ! 1: C\ = 1.012,5 c\' 259.2
41/40 ! 2: C. = 1.025 c.' 262.4
648/625 ! 3: C/ = 1.036,8 c/' 265.420,8
135/128 ! 4: C+ = 1.054.687,5 c+' 270.0
2187/2048 ! 5: DB = 1.067,871,093,75 dB' 273.375
27/25 ! 6: Db = 1.08 db' 276.48
2187/2000 ! 7: C# = 1.093,5 c#' 279.936
567/512 ! 8: C& = 1.107,421,875 c&' 283.5
9/8 ! 9: D- = 1.125 d-' 288.0
729/640 ! 10: D\ = 1.139,062,5 d\' 291.6
144/125 ! 11: D. = 1.152 d.' 294.912
729/625 ! 12: D/ = 1.1664 d/' 298.598,4
1215/1024 ! 13: D+ = 1.186.523,437,5 d+' 303.75
6/5 ! 14: EB = 1.2 eB' 307.2
243/200 ! 15: Eb = 1.215 eb' 311.04
768/625 ! 16: D# = 1.228,8 d#' 314.572,8
5/4 ! 17: D& = 1.25 d&' 320.0
81/64 ! 18: E- = 1.265,625 e-' 324.0
41/32 ! 19: E\ = 1.281,25 e\' 328.0
162/125 ! 20: E. = 1.296 e.' 331.776
21/16 ! 21: E/ = 1.312,5 e/' 336.0
683/512 ! 22:E+F-= 1.333,984,375 e+=f-' 341.5 =:(4/3)*(2049/2048)
27/20 ! 23: F\ = 1.35 f\' 345.6
2187/1600 ! 24: F. = 1.366,875 f.' 349.92
864/625 ! 25: F/ = 1.382,4 f/' 353.894,4
45/32 ! 26: F+ = 1.406,25 f+' 360.0
729/512 ! 27: GB = 1.423,828,125 gB' 364.5
36/25 ! 28: Gb = 1.44 gb' 368.64
729/500 ! 29: F# = 1.458 f#' 373.248
189/128 ! 30: F& = 1.476,562,5 f&' 378.0
3/2 ! 31: G- = 1.5 g-' 384.0
243/160 ! 32: G\ = 1.518,75 g\' 388.8
192/125 ! 33: G. = 1.536 g.' 393.216
972/625 ! 34: G/ = 1.555,2 g/' 398.131,2
405/256 ! 35: G+ = 1.582,031,25 g+' 405.0 ~J.S.Bach in Koethen~
205/128 ! 36: AB = 1.601,562,5 aB' 410.0 .Werckmeister[?1681?]
81/50 ! 37: Ab = 1.62 ab' 414.72 ~neo-Baroque~ a'
41/25 ! 38: G# = 1.64 g#' 419.84 ~W.A.Mozart,Vienna~
213/128 ! 39: G& = 1.664,062,5 g&' 426.0 or (5/3)*(639/640)
27/16 ! 40: A- = 1.687,5 a-' 432. Mersenne/Sauveur-pitch
2187/1280 ! 41: A\ = 1.708,593,75 a\' 437.4 ~Luciano-Pavarotti~
216/125 ! 42: A. = 1.728 a.' 442.368 Hz my~acustic~piano
7/4 ! 43: A/ = 1.75 a/' 448.0 = 7th-partial of 64Hz
911/512 ! 44: A+ = 1.779,296,875 a+' 455.5 ~Bach's~organ~Halle~
9/5 ! 45: BB = 1.8 bB' 460.8
729/400 ! 46: Bb = 1.822,5 bb' 466.56
1152/625 ! 47: A# = 1.843,2 a#' 471.859,2
15/8 ! 48: A& = 1.875 a&' 480.0
243/128 ! 49: B- = 1.898,437,5 b-' 486.0
48/25 ! 50: B\ = 1.92 b\' 491.52
243/125 ! 51: B. = 1.944 b.' 497.664
63/32 ! 52: B/ = 1.968,75 b\' 504 Archytas-Comma below /1
2/1 ! 53:B+C-= 2.0 b+'=c-" 512 middle_B+ = tenor_C-
!
![eof]

Statistics in occurence of JI-intervals:

1. Number of exact 5ths (3/2): 39
2. Number of exact 3rds (5/4): 36
3. Number of exact 7ths (7/4): 04

Attend:
Yours 'few' requested 'septimal' chords
do appear here in distances over the first
four initial 5ths at the note-positions:

1.) 00.C- : 17.D& : 31.G- : 43.A/ : 02.D- == 4: 5: 6: 7: 9
2.) 31.G- : 48.A& : 02.D- : 21.E/ : 40.A- == 12: 15: 18: 21: 27
3.) 09.D- : 26.F+ : 40.A- : 52.B/ : 18.E- == 36: 45: 54: 63: 81
4.) 40.A- : 04:C+ : 18.E- : 30.F& : 49.B- == 108:135:162:189:243

That can by played for instance on 53-tone keyboards alike:

http://www.cortex-design.com/body-project-terpstra-1.htm
http://www.alaindanielou.org/Instrument.html
or many other
http://en.wikipedia.org/wiki/Generalized_keyboard
s

bye
A.S.

🔗Petr Parízek <p.parizek@...>

🔗a_sparschuh <a_sparschuh@...>

🔗Marcel de Velde <m.develde@...>

Hello Andreas,

Statistics in occurence of JI-intervals:
>
> 1. Number of exact 5ths (3/2): 39
> 2. Number of exact 3rds (5/4): 36
> 3. Number of exact 7ths (7/4): 04
>
> Attend:
> Yours 'few' requested 'septimal' chords
> do appear here in distances over the first
> four initial 5ths at the note-positions:
>
> 1.) 00.C- : 17.D& : 31.G- : 43.A/ : 02.D- == 4: 5: 6: 7: 9
> 2.) 31.G- : 48.A& : 02.D- : 21.E/ : 40.A- == 12: 15: 18: 21: 27
> 3.) 09.D- : 26.F+ : 40.A- : 52.B/ : 18.E- == 36: 45: 54: 63: 81
> 4.) 40.A- : 04:C+ : 18.E- : 30.F& : 49.B- == 108:135:162:189:243
>
> That can by played for instance on 53-tone keyboards alike:
>
> http://www.cortex-design.com/body-project-terpstra-1.htm
> http://www.alaindanielou.org/Instrument.html
> or many other
> http://en.wikipedia.org/wiki/Generalized_keyboard
> s
>
> bye
> A.S.

I don't really understand the logic behind this scale.
How are these pitches attained?

My own system I have 3 ways to obtain pitches.

First of all through permutations of the harmonic series.
This gives the following scales:

Harmonic-6-limit scale:
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

Harmonic-7-limit scale:
1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9 8/5 5/3 7/4 9/5
28/15 15/8 35/18 2/1

Harmonic-8-limit:

0: 1/1 0.000 unison, perfect prime
1: 36/35 48.770 septimal diesis, 1/4-tone
2: 21/20 84.467 minor semitone
3: 16/15 111.731 minor diatonic semitone
4: 15/14 119.443 major diatonic semitone
5: 35/32 155.140 septimal neutral second
6: 10/9 182.404 minor whole tone
7: 9/8 203.910 major whole tone
8: 8/7 231.174 septimal whole tone
9: 7/6 266.871 septimal minor third
10: 6/5 315.641 minor third
11: 5/4 386.314 major third
12: 9/7 435.084 septimal major third, BP third
13: 21/16 470.781 narrow fourth
14: 4/3 498.045 perfect fourth
15: 48/35 546.815 septimal semi-augmented fourth
16: 7/5 582.512 septimal or Huygens' tritone, BP
fourth
17: 10/7 617.488 Euler's tritone
18: 35/24 653.185 septimal semi-diminished fifth
19: 3/2 701.955 perfect fifth
20: 32/21 729.219 wide fifth
21: 14/9 764.916 septimal minor sixth
22: 8/5 813.686 minor sixth
23: 5/3 884.359 major sixth, BP sixth
24: 12/7 933.129 septimal major sixth
25: 7/4 968.826 harmonic seventh
26: 16/9 996.090 Pythagorean minor seventh
27: 9/5 1017.596 just minor seventh, BP seventh
28: 64/35 1044.860 septimal neutral seventh
29: 28/15 1080.557 grave major seventh
30: 15/8 1088.269 classic major seventh
31: 40/21 1115.533 acute major seventh
32: 35/18 1151.230 septimal semi-diminished octave
33: 2/1 1200.000 octave
|
1/1 36/35 21/20 16/15 15/14 35/32 10/9 9/8 8/7 7/6 6/5 5/4 9/7 21/16 4/3
48/35 7/5 10/7 35/24 3/2 32/21 14/9 8/5 5/3 12/7 7/4 16/9 9/5 64/35 28/15
15/8 40/21 35/18 2/1

etc.
These scales are made by all permutations of the harmonic series up till a
chosen limit.

What one can also do is keep for instance the 6th till 8th harmonic
undivided to make these prime-5-limit.
Which gives 1/1 16/15 10/9 9/8 6/5 5/4 4/3 3/2 8/5 5/3 16/9 9/5 15/8 2/1 for
harmonic 8-limit, prime-5-limit.
And 1/1 81/80 135/128 16/15 10/9 9/8 6/5 5/4 81/64 4/3 27/20 45/32 3/2 8/5
5/3 27/16 16/9 9/5 15/8 2/1 for harmonic-9-limit, prime-5-limit.
etc.
If one continues along the prime-5-limit line one gets a JI version of
53edo.

The other way I make scales is by taking a chosen harmonic-limit, see this
as my "harmonic model" which means I'll allow only chords within this scale,
and chord to chord transitions within this scale.
And then I'll take a fixed drone with harmonic overtones and see where these
chords occur in the original harmonic model scale, and then transpose the
harmonic model scale to all point that will still give the fixed frequency
harmonic overtone drone (you're a smart man if you can follow what I mean
lol)
I used to call this my "tonality model", it made sense to me in some ways,
but I've since stepped away from it.

The third way to make a scale is to take method number one, and allow chord
progressions to continue into eternity with some basic rules for transposing
the 1/1 point.
This will give a "chord model" based on permutations, but the scale itself
depends on the music played (which it allways does btw, also in method 2
only then it's called a modulation).
There's not really a point in using a fixed scale for this method other than
the basic harmonic permutation scale and allowing infinite transpositions of
it.

If one can't get away with using pitchbends or midi tuning standard for
tuning, and have to use a fixed scale because of the technical limitations
of the synth/sampler or something like that. I think I'll use 53edo.
53edo does prime-5-limit more than well enough for me. And it does
prime-7-limit half decent aswell.
Some would probably also suggest 31edo.

But anyhow, with all of these scales. They're not there for the sake of the
scale, they're there for the sake of the theory that goes behind it and the
chords and chord progressions that they play etc.
So actually, prime-5-limit music is pretty well channeled by 12edo even :)

Marcel