Re: [tuning] Re: Dantonicity

Jacky Ligon wrote,
> when I analyized the "All Prime Scale" with the Scala program, it
informed me that the scale was "strictly proper".

Hmm. I don't use Scala so I'm not sure what went wrong, but the 1/1
19/17 13/11 29/23 7/5 3/2 11/7 5/3 23/13 13/7 2/1 scale would be an
"improper" scale:

0 193 289 401 583 702 782 884 988 1072 1200
0 97 209 390 509 590 692 795 879 1007 1200
0 112 293 413 493 595 699 782 911 1103 1200
0 181 301 381 483 586 670 799 991 1088 1200
0 119 200 302 405 489 617 810 907 1019 1200
0 81 182 286 370 498 691 787 899 1081 1200
0 102 205 289 418 610 707 819 1000 1119 1200
0 103 187 316 508 605 717 898 1018 1098 1200
0 84 212 405 501 614 795 914 995 1097 1200
0 128 321 418 530 711 830 911 1013 1116 1200
0 193 289 401 583 702 782 884 988 1072 1200

While a lot of the overlaps are quite trivial (as a lot of these
various intervals are so close that they'd have the same cardinality
as far as how they'd be perceived), others definitely are not...

When it comes to scale construction, I liken it to the way people
dress and whatnot: if it works for you it works for me! Propriety,
much like consistency, operates best when it is taken in some very
specific context; what questions are you asking of this or that tuning
or scale... perhaps what it is that you want to accomplish in a given
piece before you've settled on a given tuning or scale, etc. (BTW,
your examples sound good and the scale is constructed on an
interesting and creative premise, so it all seems plenty sound enough
to me!)

ds

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> Jacky Ligon wrote,
> > when I analyzed the "All Prime Scale" with the Scala program, it
> informed me that the scale was "strictly proper".
>
> Hmm. I don't use Scala so I'm not sure what went wrong, but the 1/1
> 19/17 13/11 29/23 7/5 3/2 11/7 5/3 23/13 13/7 2/1 scale would be an
> "improper" scale:

Dan,

There seems to have been some kind of mistake about the ratios used
in the complete scale, so I'll clarify them here (from my JI
spreadsheet):

Ratio Cents Value 12TET Cents Adjust
Consecutive
1/1 0 0 0.00
31/29 115.4583783 100 15.46
115.4583783
19/17 192.5576066 200 -7.44
77.09922832
13/11 289.2097194 300 -10.79
96.65211277
29/23 401.3028469 400 1.30
112.0931275
17/13 464.4277477 500 -35.57
63.12490085
7/5 582.5121926 600 -17.49
118.0844449
3/2 701.9550009 700 1.96
119.4428083
11/7 782.4920359 800 -17.51
80.53703503
5/3 884.358713 900 -15.64
101.8666771
23/13 987.7466855 1000 -12.25
103.3879725
13/7 1071.701755 1100 -28.30
83.9550698
2/1 1200 1200 0.00
128.2982447

It is probable that since you didn't have all the ratios, it might be
why your results are different from that of Scala.

>
> 0 193 289 401 583 702 782 884 988 1072 1200
> 0 97 209 390 509 590 692 795 879 1007 1200
> 0 112 293 413 493 595 699 782 911 1103 1200
> 0 181 301 381 483 586 670 799 991 1088 1200
> 0 119 200 302 405 489 617 810 907 1019 1200
> 0 81 182 286 370 498 691 787 899 1081 1200
> 0 102 205 289 418 610 707 819 1000 1119 1200
> 0 103 187 316 508 605 717 898 1018 1098 1200
> 0 84 212 405 501 614 795 914 995 1097 1200
> 0 128 321 418 530 711 830 911 1013 1116 1200
> 0 193 289 401 583 702 782 884 988 1072 1200
>
> While a lot of the overlaps are quite trivial (as a lot of these
> various intervals are so close that they'd have the same cardinality
> as far as how they'd be perceived), others definitely are not...

I would like to kindly request that you show me how to do the above
analysis. I can see that you have modally transposed the original
scale to each successive row.
Explain the "overlaps". And most importantly - how do I interpret the
data? What am I looking for?

>
> When it comes to scale construction, I liken it to the way people
> dress and whatnot: if it works for you it works for me! Propriety,
> much like consistency, operates best when it is taken in some very
> specific context; what questions are you asking of this or that
tuning
> or scale... perhaps what it is that you want to accomplish in a
given
> piece before you've settled on a given tuning or scale, etc. (BTW,
> your examples sound good and the scale is constructed on an
> interesting and creative premise, so it all seems plenty sound
enough
> to me!)
>
Thanks for analyzing this today, it helped me to realize that the
full scale may not be posted correctly. The All Prime Scales, of
which the above is the "1st Order 12 Pitch Scale" - meaning the
ratios contained in the scale are the first set of ratios that fit
the criteria of 1. that the numerator and denominator are prime, and
2. the ratio fits the function of the given scale degree. Of course,
one could go to the next highest set of ratio fitting these
requirements - giving the 2nd Order All Prime Scale. To me the 2nd
Order scale was much more melodically interesting. Perhaps I'll post
the values for some of the others I constructed.

Below is the Scala "show data" information for the 1st Order All
Prime Scale. This does show that it is "Strictly Proper", but I think
it would be proper to say that your results were affected by not
having the full scale during your computations.

At this point, I would like to ask 3 questions:

1. What is a "Guide Tone"?
2. How is it calculated?
3. Why does Scala give the following message for the Guide Tone for
this scale: "not computable"?

Bonus question (aren't you lucky!): What does: "Rothenberg
stability : 1.000000 = 1" mean?

All Numerators and Denominators are Prime.
0: 1/1 0.000 unison, perfect prime
1: 31/29 115.458
2: 19/17 192.558
3: 13/11 289.210
4: 29/23 401.303
5: 17/13 464.428
6: 7/5 582.512 septimal tritone
7: 3/2 701.955 perfect fifth
8: 11/7 782.492 undecimal augmented fifth
9: 5/3 884.359 major sixth
10: 23/13 987.747
11: 13/7 1071.702 16/3-tone
12: 2/1 1200.000 octave
|
Number of notes : 12
Smallest interval : 391/377, 63.1249 cents
Average interval (divided octave) : 100.000 cents
Average / Smallest interval : 1.584161
Largest interval of one step : 14/13, 128.2982 cents
Largest / Average interval : 1.282982
Largest / Smallest interval : 2.032451
Least squares average interval : 98.4456 cents
Median interval of one step : 102.627 cents
Interval standard deviation : 19.2304 cents
Interval skew : 0.0019 cents
Scale is strictly proper
Scale is sum-free (all different intervals)
Number of different intervals : 128 = 11.63636 / class
Rothenberg stability : 1.000000 = 1
Lumma stability : 0.401372
Prime limit : 31
Odd number limit : 31 (O: 31 U: 29)
Fundamental : 1/340510170,-28.343 octaves,
0.000 Hz.
Guide tone : not computable
Exponens Consonantiae : 6.82929E+19, 65.8884 octaves
Euler's gradus suavitatis : 251
Wille's k value : 255382627
Vogel's harmonic complexity : 1.12583E+02
Wilson's harmonic complexity : 266
Rectangular lattice dimensions : 18
Triangular lattice dimensions : 2
Prime exponents' range and average:
2: -1 .. 1 0.00000
3: -1 .. 1 0.00000
5: -1 .. 1 0.00000
7: -1 .. 1 -0.08333
11: -1 .. 1 0.00000
13: -1 .. 1 0.00000
17: -1 .. 1 0.00000
19: 0 .. 1 0.08333
23: -1 .. 1 0.00000
29: -1 .. 1 0.00000
31: 0 .. 1 0.08333
Average exponent except of 2 : 0.08333
Average absolute exponent except of 2: 21 / 12 = 1.75000
Average distance from equal tempered : 14.8827 cents, 0.148827
steps
Standard deviation from equal tempered : 5.3703 cents, 0.053702 steps
Maximum distance from equal tempered : 35.5723 cents, 0.355722
steps
Geometric average of pitches 0..n : 590.286 cents
Arithmetic average of pitches 0..n : 629.567 cents
Harmonic average of pitches 0..n : 551.005 cents
Geometric average of pitches 1..n : 639.477 cents
Arithmetic average of pitches 1..n : 673.001 cents
Harmonic average of pitches 1..n : 605.953 cents
Geometric average of pitches 1..n-1: 588.520 cents
Arithmetic average of pitches 1..n-1: 616.074 cents
Harmonic average of pitches 1..n-1: 560.966 cents