Re: [tuning] question on Scala scale

Joseph Pehrson wrote:
>
> Question on Scala scale...

> This scale, to me, seems quite resonant and nice in harmonic usage...
> but
> how is it constructed? It is seven-limit just intonation, but what
> does
> XH7+8 mean? Is this some kind of notation by Erv Wilson?

My guess:

Xenharmonicon, issues 7 & 8?

I think that's the name of the publication.

--
* D a v i d B e a r d s l e y
* xouoxno@virtulink.com
*
* 49/32 R a d i o "all microtonal, all the time"
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* http://www.virtulink.com/immp/lookhere.htm

Joseph Pehrson wrote:

> Question on Scala scale...

XH7+8 mean? Xenharmonikon 7 & 8. They were structured as an MOS and reflect the possible
appox. to just. You might try latticing it out which is the way Wilson was working allot at
this time (my educated guess)

>
>
> I have a question concerning the Scala scale 19-31ji.scl:
>
> A septimal interpretation of 19 out of 31 tones, after Wilson, XH7+8
> 0: 1/1 0.000 unison, perfect prime
> 1: 25/24 70.672 classic chromatic semitone
> 2: 16/15 111.731 minor diatonic semitone
> 3: 9/8 203.910 major whole tone
> 4: 7/6 266.871 septimal minor third
> 5: 6/5 315.641 minor third
> 6: 5/4 386.314 major third
> 7: 9/7 435.084 septimal major third
> 8: 4/3 498.045 perfect fourth
> 9: 7/5 582.512 septimal tritone
> 10: 10/7 617.488 Euler's tritone
> 11: 3/2 701.955 perfect fifth
> 12: 14/9 764.916 septimal minor sixth
> 13: 8/5 813.686 minor sixth
> 14: 5/3 884.359 major sixth
> 15: 7/4 968.826 harmonic seventh
> 16: 16/9 996.090 Pythagorean minor seventh
> 17: 15/8 1088.269 classic major seventh
> 18: 27/14 1137.039 septimal major seventh
> 19: 2/1 1200.000 octave
>
> Number of notes : 19
> Smallest interval : 64/63, 27.2641 cents
> Average interval (divided octave) : 63.158 cents
> Average / Smallest interval : 2.316523
> Largest interval of one step : 135/128, 92.1787 cents
> Largest / Average interval : 1.459496
> Largest / Smallest interval : 3.380957
> Least squares average interval : 63.3253 cents
> Median interval of one step : 28/27, 62.9609 cents
> Interval standard deviation : 18.6593 cents
> Interval skew : 0.0004 cents
> Scale is not proper
> Number of different intervals : 112 = 6.22222 / class
> Lumma stability : 0.258112
> Impropriety factor : 0.014055
> Prime limit : 7
> Odd number limit : 27 (O: 27 U: 15)
> Average absolute harmonicity : 0.147808
> Specific harmonicity : 0.117219
> Fundamental : 1/2520,-11.299 octaves, 0.104 Hz.
> Guide tone : 75600, 16.206 octaves,
> 19778892.74
> Hz.
> Exponens Consonantiae : 1.90512E+08, 27.5053 octaves
> Euler's gradus suavitatis : 42
> Wille's k value : 2362
> Vogel's harmonic complexity : 2.21053E+01
> Wilson's harmonic complexity : 44
> Rectangular lattice dimensions : 10
> Triangular lattice dimensions : 5
> Prime exponents' range and average:
> 2: -3 .. 4 0.05263
> 3: -2 .. 3 0.05263
> 5: -1 .. 2 0.10526
> 7: -1 .. 1 0.05263
> Average exponent except of 2 : 0.21053
> Average absolute exponent except of 2: 36 / 19 = 1.89474
> Average distance from equal tempered : 9.6189 cents, 0.152299 steps
> Standard deviation from equal tempered : 2.6472 cents, 0.041913 steps
> Maximum distance from equal tempered : 21.4575 cents, 0.339743 steps
> Geometric average of pitches 0..n : 602.170 cents
> Arithmetic average of pitches 0..n : 24319/16800, 640.3472 cents
> Harmonic average of pitches 0..n : 563.994 cents
> Geometric average of pitches 1..n : 633.864 cents
> Arithmetic average of pitches 1..n : 23479/15960, 668.2924 cents
> Harmonic average of pitches 1..n : 599.435 cents
> Geometric average of pitches 1..n-1: 602.412 cents
> Arithmetic average of pitches 1..n-1: 21799/15120, 633.3646 cents
> Harmonic average of pitches 1..n-1: 571.459 cents
>
> This scale, to me, seems quite resonant and nice in harmonic usage...
> but
> how is it constructed? It is seven-limit just intonation, but what
> does
> XH7+8 mean? Is this some kind of notation by Erv Wilson?
>
> Any help would be appreciated...
>
> ___________ _____ ____ __
> Joseph Pehrson
>
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-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Manuel, can you explain the following assertions on this scale:

Odd number limit : 27
Rectangular lattice dimensions : 10
Triangular lattice dimensions : 5

These don't seem right to me. Perhaps instead of "dimensions" you really
mean "diameter" or some such thing.

Joseph!
The interesting thing is that i cannot find this scale in Xen 7&8 so maybe this is not
wilson scale after all! It seems to have more tones in common with the 9-limit diamond than
any of the other scales listed but makes some rather very strange choices. I say its a wilson
forgery and not a good one. If it is after Wilson, the author should take the blame
themselves.
It lacks his symmetry as well as not having any repeating tetrachords. two of his favorite
things! look at the 22 tones scales and you will see what I mean. Lattice it out an nothing
appears as far as logic.

BUT
as we all know, the ear hears things that sometimes is not apparent on paper. If this sounds
more "resonant" than the others listed, I cannot tell you why w/o tuning it up and trying to
figure this out, don't have the time at the moment. I think you might explain it to us and
then the rest of us will learn something! But what do you mean by resonant?

Joseph Pehrson wrote:

> Question on Scala scale...
>
> I have a question concerning the Scala scale 19-31ji.scl:
>
> A septimal interpretation of 19 out of 31 tones, after Wilson, XH7+8
> 0: 1/1 0.000 unison, perfect prime
> 1: 25/24 70.672 classic chromatic semitone
> 2: 16/15 111.731 minor diatonic semitone
> 3: 9/8 203.910 major whole tone
> 4: 7/6 266.871 septimal minor third
> 5: 6/5 315.641 minor third
> 6: 5/4 386.314 major third
> 7: 9/7 435.084 septimal major third
> 8: 4/3 498.045 perfect fourth
> 9: 7/5 582.512 septimal tritone
> 10: 10/7 617.488 Euler's tritone
> 11: 3/2 701.955 perfect fifth
> 12: 14/9 764.916 septimal minor sixth
> 13: 8/5 813.686 minor sixth
> 14: 5/3 884.359 major sixth
> 15: 7/4 968.826 harmonic seventh
> 16: 16/9 996.090 Pythagorean minor seventh
> 17: 15/8 1088.269 classic major seventh
> 18: 27/14 1137.039 septimal major seventh
> 19: 2/1 1200.000 octave
>
> Number of notes : 19
> Smallest interval : 64/63, 27.2641 cents
> Average interval (divided octave) : 63.158 cents
> Average / Smallest interval : 2.316523
> Largest interval of one step : 135/128, 92.1787 cents
> Largest / Average interval : 1.459496
> Largest / Smallest interval : 3.380957
> Least squares average interval : 63.3253 cents
> Median interval of one step : 28/27, 62.9609 cents
> Interval standard deviation : 18.6593 cents
> Interval skew : 0.0004 cents
> Scale is not proper
> Number of different intervals : 112 = 6.22222 / class
> Lumma stability : 0.258112
> Impropriety factor : 0.014055
> Prime limit : 7
> Odd number limit : 27 (O: 27 U: 15)
> Average absolute harmonicity : 0.147808
> Specific harmonicity : 0.117219
> Fundamental : 1/2520,-11.299 octaves, 0.104 Hz.
> Guide tone : 75600, 16.206 octaves,
> 19778892.74
> Hz.
> Exponens Consonantiae : 1.90512E+08, 27.5053 octaves
> Euler's gradus suavitatis : 42
> Wille's k value : 2362
> Vogel's harmonic complexity : 2.21053E+01
> Wilson's harmonic complexity : 44
> Rectangular lattice dimensions : 10
> Triangular lattice dimensions : 5
> Prime exponents' range and average:
> 2: -3 .. 4 0.05263
> 3: -2 .. 3 0.05263
> 5: -1 .. 2 0.10526
> 7: -1 .. 1 0.05263
> Average exponent except of 2 : 0.21053
> Average absolute exponent except of 2: 36 / 19 = 1.89474
> Average distance from equal tempered : 9.6189 cents, 0.152299 steps
> Standard deviation from equal tempered : 2.6472 cents, 0.041913 steps
> Maximum distance from equal tempered : 21.4575 cents, 0.339743 steps
> Geometric average of pitches 0..n : 602.170 cents
> Arithmetic average of pitches 0..n : 24319/16800, 640.3472 cents
> Harmonic average of pitches 0..n : 563.994 cents
> Geometric average of pitches 1..n : 633.864 cents
> Arithmetic average of pitches 1..n : 23479/15960, 668.2924 cents
> Harmonic average of pitches 1..n : 599.435 cents
> Geometric average of pitches 1..n-1: 602.412 cents
> Arithmetic average of pitches 1..n-1: 21799/15120, 633.3646 cents
> Harmonic average of pitches 1..n-1: 571.459 cents
>
> This scale, to me, seems quite resonant and nice in harmonic usage...
> but
> how is it constructed? It is seven-limit just intonation, but what
> does
> XH7+8 mean? Is this some kind of notation by Erv Wilson?
>
> Any help would be appreciated...
>
> ___________ _____ ____ __
> Joseph Pehrson
>
> ------------------------------------------------------------------------
> Missing old school friends? Find them here:
> http://click.egroups.com/1/4055/1/_/239029/_/959141544/
> ------------------------------------------------------------------------
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@egroups.com - join the tuning group.
> tuning-unsubscribe@egroups.com - unsubscribe from the tuning group.
> tuning-nomail@egroups.com - put your email message delivery on hold for the tuning group.
> tuning-digest@egroups.com - change your subscription to daily digest mode.
> tuning-normal@egroups.com - change your subscription to individual emails.

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Paul,

> These don't seem right to me. Perhaps instead of "dimensions" you really
> mean "diameter" or some such thing.

"Diameter" is a much better word, thanks for correcting my English.

Kraig,

It's not a Wilson scale, the text says "after Wilson" and I don't remember
where it came from. The names of Wilson scales in the archive begin with
"wilson". I agree that it's not a very well structured scale and I'm
tempted to remove it from the archive, unless Joseph protests, or I find
the source. If I were to rationalize the 19-31.scl scale (also don't know
its source) then I would select this:

135/128 15/14 9/8 7/6 135/112 5/4 21/16 4/3 45/32 10/7 3/2 14/9 45/28 5/3
7/4 16/9 15/8 63/32 2/1

Number of notes : 19
Smallest interval : 64/63, 27.2641 cents
Average interval (divided octave) : 63.158 cents
Average / Smallest interval : 2.316523
Largest interval of one step : 135/128, 92.1787 cents
Largest / Average interval : 1.459496
Largest / Smallest interval : 3.380957
Least squares average interval : 63.7858 cents
Median interval of one step : 28/27, 62.9609 cents
Interval standard deviation : 24.4176 cents
Interval skew : 0.0040 cents
Scale is not proper
Most common triad is 0.0 498.045 701.955 cents, amount: 11
Scale is a Constant Structure
Number of different intervals : 78 = 4.33333 / class
Smallest interval difference : 2401/2400, 0.7212 cents
Most common intervals : 4/3, 498.0450 cents & inv., amount: 15
Number of recognisable fifths : 15, average 701.955 cents
Lumma stability : 0.185202
Impropriety factor : 0.107511
Prime limit : 7
Odd number limit : 135 (O: 135 U: 9)
Average absolute harmonicity : 0.138724
Specific harmonicity : 0.112963
Fundamental : 1/8064,-12.977 octaves, 0.032 Hz.
Guide tone : 15120, 13.884 octaves, 3955778.55 Hz.
Exponens Consonantiae : 1.219277E+08, 26.8615 octaves
Euler's gradus suavitatis : 38
Wille's k value : 4252
Vogel's harmonic complexity : 22.84211
Wilson's harmonic complexity : 34
Rectangular lattice diameter : 8
Triangular lattice diameter : 6
Prime exponents' range and average:
2: -7 .. 4 -1.63158
3: -2 .. 3 0.57895
5: 0 .. 1 0.47368
7: -1 .. 1 0.05263
Average exponent except of 2 : 1.10526
Average absolute exponent except of 2: 43 / 19 = 2.26316
Average distance from equal tempered : 14.0218 cents, 0.222011 steps
Standard deviation from equal tempered : 4.0184 cents, 0.063624 steps
Maximum distance from equal tempered : 35.8938 cents, 0.568318 steps
Geometric average of pitches 0..n : 608.358 cents
Arithmetic average of pitches 0..n : 3719/2560, 646.5251 cents
Harmonic average of pitches 0..n : 570.190 cents
Geometric average of pitches 1..n : 640.377 cents
Arithmetic average of pitches 1..n : 189/128, 674.6909 cents
Harmonic average of pitches 1..n : 606.062 cents
Geometric average of pitches 1..n-1: 609.286 cents
Arithmetic average of pitches 1..n-1: 3335/2304, 640.2553 cents
Harmonic average of pitches 1..n-1: 578.317 cents

Manuel Op de Coul coul@ezh.nl

I wrote,

>> These don't seem right to me. Perhaps instead of "dimensions" you really
>> mean "diameter" or some such thing.

Manuel wrote,

>"Diameter" is a much better word, thanks for correcting my English.

You're welcome. I still have a problem with your odd-limit. Please read the
definition of "limit" on Monz's site.

Paul wrote
>I still have a problem with your odd-limit. Please read the
>definition of "limit" on Monz's site.

Ah, you mean it should take the odd limit of the scale as a chord,
i.e. integer frequency multiples. Which for the 19-31ji scale is 4725.

Manuel Op de Coul coul@ezh.nl

Manuel wrote,

>Ah, you mean it should take the odd limit of the scale as a chord,

yes, that's probably the only unambiguous solution.

>i.e. integer frequency multiples. Which for the 19-31ji scale is 4725.

Really? Which interval is 4725-limit?