Yahoo Tuning Groups Ultimate Backup tuning Balzano's 9-out-of-20

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Hi Kalle, and all,
>
> I just jump in and didn't follow the Balzano 9 out of 20 story,
> but your note is quite true about 2^(9/20).

Hello Jacques,

I was hoping for you to jump in because of your knowledge on
difference tone-based scales! I believe your recurrent sequence
stuff is also directly relevant to what Michael Sheiman is doing
i.e. approximating irrationally generated scales with ratios.

Regarding the Balzano story, there was no story as I don't really
know much about his construction. Another rotation is listed in
Scala as Rothenberg Generalized Diatonic. Supposedly it is derived
from Group Theory and academic Diatonic Set Theory-type stuff
which are very different from standard tuning-list ideas. Curiously
this construction has nothing to do with (sqrt(3)+1)/2!

> (V3 + 1) / 2 = 1,366 025 404... which I called "Zinith" is the generator of
> one of the first fractal waveforms I found after the Phi waveform, in the
> eighties.
> I was fascinated by its very specific timber and I used it in many
> photosonic disks and compositions using its series and also its very dynamic
> rythmn (basically 4 4 4 3 / 4 4 4 3 / 4 4 4 3 / 4 4 3 / and so on)
> Among noble numbers it's perhaps the one that has the less "inharmonic"
> sound, much less than for example Phi or of course 3rd degree or higher
> noble numbers.
> I am not sure why, of course the waveform reflects the differencial
> coherence 15 - 11 = 4, but Phi has lots of "fibonaccisations" in simple
> ratios and does not have this strange "harmonic" quality.
> I already noticed the cycle of 20, approximating 2^9, but forgot it was that
> close. That's quite rare. I used it up to ^11, which like 9 leads also to a
> balanced octave division.

Balzano and Zweifel investigated 11-out-of-20 too, of course from
a completely different perspective.

> Zinith's main serie is 4 > 11 15 > 41 56 > 153 209 > 571 780 > & so on
> (when I write ">" it means you need octaviation, to V3 + 1)
> And of course as you said it verifies 41 - 30 = 11, 56 -41 = 15 & so on.
>
> In fact, half the square of (V3 + 1), V3 + 2 = 3,732 050 808..., which I
> call Zira'at is according to certain criterias, even more convergent than
> Zinith.
> You may ask why I talk of V3 + 2 instead of (V3 + 2) / 2, since we refer to
> an octave period, but if you try the successive powers of both, you will
> understand why the real generator is V3 + 2.
> Note that Zira'at is happy with even a division of 10 of the octave and the
> series gives a very approaching rational approximation (or in the other way)
> of 1/10 of an octave in the (both spectral and differential quasi-coherent)
> ratio 209 : 224 - quite interesting since 209 = 11.19 and 224 = 7.2^5
>
> 224/209 = 1,071 770 335
> 2^(1/10) = 1,071 773 463
> Therefore 2^(1/10) is quasi-rational, if you want to see it that way.
> (note that Z^8 is also close to 3.7^2.2^8)
>
> Actually, Zinith series have more melodic possibilities than Zira'at and I
> would think of them as double-Zira'at. That's what's audible also in the
> waveform progression.
> The scale is very jazzy to my ears, don't know why, may be because it is
> quite chromatic.

Thanks Jacques, this is fascinating stuff! Perhaps you could
help Michael with his Phi-derived rational scale? He treats Phi as
the period and 2:1 as the generator but I guess we get the same
intervals either way? Michael, are you listening, here is your guru!

Kalle Aho

🔗djtrancendance@...

--Thanks Jacques, this is fascinating stuff! Perhaps you could
---help Michael with his Phi-derived rational scale?

Believe me, I would greatly appreciate it. I'm just starting to stumble across the vast possibilities of these "noble" ratios.
Jacques, I'm also wondering, have you done any work with the Silver Ratio, if so, and what have you found so far?

And, as a side-note, I noticed solving the "quadratic" form
>>>> x = x^2 + c <<< yields some interesting symmetrical ratios.
For example
A) Solving x = x^2 - 1 gives the golden ratio
B) Solving x = x^2 - 1/2 gives "Zinith" AKA 1.366025403
C) Solving x = 1/2(x^2) - 1/2 gives the Silver Ratio AKA 2.4142135623
D) Solving x = sqrt(x) + 1/2 gives x = 1.866025403 (any special purpose/advantage of formal
name you know of for this ratio, Jacques)?
************************************************************************************************
--He treats Phi as the period and 2:1 as the generator but I guess we get the same
--intervals either way?

I think we're on the same page: my basic formula is PHI^y/2^x...though I wonder if there's a better divisor to use than 2 for the sake of symmetry. In this case, I think my terminology is backward: I have in the past said PHI is the generator, my mistake.
After that I simply select tones from that tuning that are not too close together (to obey the critical band) and, lastly, round them to their nearest rational equivalents so JI-fanatic people on this list don't jump at me for being the "crazy irrational number guy". :-D So, still, it's really more a PHI scale system than a JI one.

   BTW, any ideas what sort of divisors to use instead of 2
(as in 2^y in my formula) to get the notes in the scale more closely spaced? At least 9, and preferably approaching 12, notes between 1 and 2/1 would be ideal. :-)
**********************************************************************************
---Michael, are you listening, here is your guru!
   Indeed. :-)
   For the first time in a long while I'm actually looking over this info and thinking we really are on the same page and, thank the Lord, there are people who've done solid research on difference tones (outside of the difference tones (or should I say consistent tone) between any two partials of the harmonic series.

***************************************************************************************
Seriously, this is pretty awesome: let's see if we can break some ground here...I'm pretty sure I'm just touching the surface (IE PHI may very well NOT be the best ratio) and
the possibilities with such scales seems endless.

-Michael

--- On Mon, 3/30/09, Kalle Aho <kalleaho@...> wrote:

From: Kalle Aho <kalleaho@...>
Subject: [tuning] Re:Balzano's 9-out-of-20
To: tuning@yahoogroups.com
Date: Monday, March 30, 2009, 1:35 PM

--- In tuning@yahoogroups. com, Jacques Dudon <fotosonix@. ..> wrote:

> Hi Kalle, and all,

> I just jump in and didn't follow the Balzano 9 out of 20 story,

> but your note is quite true about 2^(9/20).

Hello Jacques,

I was hoping for you to jump in because of your knowledge on

difference tone-based scales! I believe your recurrent sequence

stuff is also directly relevant to what Michael Sheiman is doing

i.e. approximating irrationally generated scales with ratios.

Regarding the Balzano story, there was no story as I don't really

know much about his construction. Another rotation is listed in

Scala as Rothenberg Generalized Diatonic. Supposedly it is derived

from Group Theory and academic Diatonic Set Theory-type stuff

which are very different from standard tuning-list ideas. Curiously

this construction has nothing to do with (sqrt(3)+1)/ 2!

> (V3 + 1) / 2 = 1,366 025 404... which I called "Zinith" is the generator of

> one of the first fractal waveforms I found after the Phi waveform, in the

> eighties.

> I was fascinated by its very specific timber and I used it in many

> photosonic disks and compositions using its series and also its very dynamic

> rythmn (basically 4 4 4 3 / 4 4 4 3 / 4 4 4 3 / 4 4 3 / and so on)

> Among noble numbers it's perhaps the one that has the less "inharmonic"

> sound, much less than for example Phi or of course 3rd degree or higher

> noble numbers.

> I am not sure why, of course the waveform reflects the differencial

> coherence 15 - 11 = 4, but Phi has lots of "fibonaccisations" in simple

> ratios and does not have this strange "harmonic" quality.

> I already noticed the cycle of 20, approximating 2^9, but forgot it was that

> close. That's quite rare. I used it up to ^11, which like 9 leads also to a

> balanced octave division.

Balzano and Zweifel investigated 11-out-of-20 too, of course from

a completely different perspective.

> Zinith's main serie is 4 > 11 15 > 41 56 > 153 209 > 571 780 > & so on

> (when I write ">" it means you need octaviation, to V3 + 1)

> And of course as you said it verifies 41 - 30 = 11, 56 -41 = 15 & so on.

> In fact, half the square of (V3 + 1), V3 + 2 = 3,732 050 808..., which I

> call Zira'at is according to certain criterias, even more convergent than

> Zinith.

> You may ask why I talk of V3 + 2 instead of (V3 + 2) / 2, since we refer to

> an octave period, but if you try the successive powers of both, you will

> understand why the real generator is V3 + 2.

> Note that Zira'at is happy with even a division of 10 of the octave and the

> series gives a very approaching rational approximation (or in the other way)

> of 1/10 of an octave in the (both spectral and differential quasi-coherent)

> ratio 209 : 224 - quite interesting since 209 = 11.19 and 224 = 7.2^5

> 224/209 = 1,071 770 335

> 2^(1/10) = 1,071 773 463

> Therefore 2^(1/10) is quasi-rational, if you want to see it that way.

> (note that Z^8 is also close to 3.7^2.2^8)

> Actually, Zinith series have more melodic possibilities than Zira'at and I

> would think of them as double-Zira' at. That's what's audible also in the

> waveform progression.

> The scale is very jazzy to my ears, don't know why, may be because it is

> quite chromatic.

Thanks Jacques, this is fascinating stuff! Perhaps you could

help Michael with his Phi-derived rational scale? He treats Phi as

the period and 2:1 as the generator but I guess we get the same

intervals either way? Michael, are you listening, here is your guru!

Kalle Aho

🔗Cameron Bobro <misterbobro@...>

> From: Kalle Aho <kalleaho@...>
> Subject: [tuning] Balzano's 9-out-of-20
> To: tuning@yahoogroups.com
> Date: Sunday, March 29, 2009, 11:01 AM

>
>
> Hello everyone,
>
>
>
> I think the following fits very well the recent topic of tunings that
>
> are not based on JI or small integer ratios.
>
>
>
> I know next to nothing about the reasoning behind Balzano's scale but I
>
> noticed a strange thing:
>
>
>
> this scale is a 9-tone MOS of 2^(9/20)-generator.
>
>
>
> Now 2^(9/20) is extremely close to (sqrt(3)+1)/ 2 which
>
> has an interesting property related to difference tones:
>
>
>
> if g = (sqrt(3)+1)/ 2
>
>
>
> g^2 - g = 1/2.
>
>
>
> That's really weird because I don't think this feature had
>
> anything to do with Balzano's construction of this scale.
>
>
>
> Kalle Aho
>

In my "shadow tunings", which rely on points of maximum "fuzziness" or "asonance" or, to my ears, such "dissonance" that we percieve a kind of far-away consonance or softness (as I've mentioned here numerous over the last couple of years), I use the square root of 3 in conjunction with phi.

Both belong to this series:

2cos(Pi/n)

where n is an integer.

When n is 4, you get the half octave (600 cents). When it is 5, you get phi, and with 6 you get the square root of 3. The series converges on 2.

Discovered this by accident some time ago, starting with a geometry approach to fretting. You can imagine the diameter as a string length, encompassing it with a circle and kind of dropping a plumb line from equal divisions of the circle to find fretting points.

Obviously you can elaborate from here 2cos(xPi/n) etc.

The square root of 3 within the octave has some interesting properties- check out the interval it makes with its inversion against the octave, for instance.

951 and 249 cents are very interesting (not to mention lovely) in a lot of ways. The "fuzzy" effect covers a region a couple of cents wide- a couple of years ago I posted a scale extracted from a tuning based on 81/70 for example, and earlier I had been poking at 196/169 (which is a fascinating interval in its own right).

By the way my math interest in the "shadow" intervals comes from having found them by ear, trying to find the "floating" points between Just intervals, and then trying to figure out how to describe them mathematically.

To my ears, shadow intervals work in "temperaments" as well as tunings. You don't get that nagging soggy feeling of knowing what an interval "should" be (due to the reference in the spectra), which often happens (to my ears) with "good" approximations. Rather, you get a harmonic-series-neutral, so to speak, interval. This happens in the 19-tone tuning Graham just posted (with its 2cos(Pi/6) interval filling out a tetrad nicely, rather than a pure or tempered 7/4).
So- it's so "out of tune" that it doesn't sound out of tune, for you cannot scan what it "should" be and settle on hearing it on the one ear for what it "is", a sound, and in the other ear in terms of scale step/interval.

🔗Kalle Aho <kalleaho@...>

🔗Jacques Dudon <fotosonix@...>

Thanks Kalle,

I appreciate your openess to such different aproaches.
Never heard of those theories. I am curious, if anybody has some info about it...

Some times very different realms meet, and these are good examples of what is harmony.
About difference tones, equal-beating models, and recurrent sequences, I observe only
little interest in the microtonal world, while I still think these are very complementary to usual approaches, considering their immense psychoacoustic and musical applications.
Everyone for example has heard of the experiments of Sethares concerning adequation of timbres and scales.
When I first heard Sethares brillant demonstrations, it striked me that I had arrived at the very same cases myself by complete different ways, only using differential coherence, and therefore just intonation.

By the way this is a point I may suggest to debate some day on this list :
(and 1st of april is a good day to joke !) -

Do noble numbers belong to just intonation ?

According to common JI theories, irrationnals are excluded of JI.
According to the differential coherence theory (if you accept it as a just intonation theory), they are.
Not because the ratios of all whole numbers recurrent series converge towards irrationnals, because
in a pure mathematical sense, they will never be attained, in theory, by whole number ratios
(besides non-coherent ratios such as Vn also can be approached by whole number ratios),
BUT because they verify perfect differential coherences, like many JI scales.

The question then is :
is differential coherence acceptable as a just intonation theory ?

exemple : Ishku :
16 36 80 178 396 881 1960...
x^2 - 2x = 1/2
and (V6 + 2) / 2 = 2,22474487139
?
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques Dudon

> Hello Jacques,

> I was hoping for you to jump in because of your knowledge on
> difference tone-based scales! I believe your recurrent sequence
> stuff is also directly relevant to what Michael Sheiman is doing
> i.e. approximating irrationally generated scales with ratios.
> Regarding the Balzano story, there was no story as I don't really
> know much about his construction. Another rotation is listed in
> Scala as Rothenberg Generalized Diatonic. Supposedly it is derived
> from Group Theory and academic Diatonic Set Theory-type stuff
> which are very different from standard tuning-list ideas. Curiously
> this construction has nothing to do with (sqrt(3)+1)/2!

🔗djtrancendance@...

--"About difference tones, equal-beating models, and recurrent --sequences, I observe only little
interest in the microtonal world, while I --still think these are very
complementary to usual approaches,
--considering their immense
psycho-acoustic and musical applications."

    I think of difference tones in that same way: complementary.
for example.....
Most people think of the harmonic series as working well because it contains whole number multiples of the root tone and its overtones reduce to small-numbered fractions (both important in the basis of just-intonation).
   But a fact I've found most people ignore is that the difference tones between any two consecutive overtones in the harmonic series are equal (IE the same # of frequencies apart) and that also helps explain why the harmonic series sounds so relaxed/consonant.
--------------------------------------------------------------------------------
   In fact, I'd even go so far as to say that difference-frequencies and whole-numbered ratios are equally important concepts and that, since only the harmonic series seems to match perfect on
both concepts, more effort should be put
into making scales that consider a fairly equal balance between these two concepts (trading one for the other until a happy compromise is reached).

---"Do noble numbers belong to just intonation ?"
Not quite...but rather I believe noble numbers can be used to enhance Just Intonation and/or form noble-number-based scales which can then be optimized via following use of just-intonation intervals as much as possible without distorting the ratios from their original values too much.

Using JI-style ratios to estimate the results of PHI and Silver Ratio scales in my own experiments, for example, has improved the sound of both a good deal without losing much symmetry in terms of difference tones. The result is a scale that's neither completely JI nor a pure noble-numbered scale...but, according to those who've listened to my experiments, preferable to both the pure JI and pure noble-number-generated
versions.
********************************************************************
---is differential coherence acceptable as a just intonation theory ?
Not so much as JI itself, but as something that lies somewhere in-between JI and the pure harmonic series
that can be used to help create scales outside JI that sound more consonant, optimize existing JI scales. And, of course...JI can be used to optimize/round values of noble-numbered scales for improvement in terms of consonance in a similar manner.

And, again, the difference-tone symmetry of the harmonic series proves how well the "coherence" concept and relatively small-numbered ratios can work together.
***********************************************
It begs the following challenge: instead of simply matching whole-numbered ratios in scales via JI, how much should we also bend the patterns of such scales to fit, for example, noble-numbered scale patterns better and further optimize their difference-tone properties?

--- On Wed, 4/1/09, Jacques Dudon <fotosonix@...> wrote:

From: Jacques Dudon <fotosonix@...>
Subject: [tuning] Re:Balzano's 9-out-of-20
To: tuning@yahoogroups.com
Date: Wednesday, April 1, 2009, 9:55 AM

Thanks Kalle,
I appreciate your openess to such different aproaches.
Never heard of those theories. I am curious, if anybody has some info about it...
Some times very different realms meet, and these are good examples of what is harmony.About difference tones, equal-beating models, and recurrent sequences, I observe only little interest in the microtonal world, while I still think these are very complementary to usual approaches, consider ing their immense psychoacoustic and musical applications.Everyone for example has heard of the experiments of Sethares concerning adequation of timbres and scales. When I first heard Sethares brillant demonstrations, it striked me that I had arrived at the very same cases myself by complete different ways, only using differential coherence, and therefore just
intonation.
By the way this is a point I may suggest to debate some day on this list :(and 1st of april is a good day to joke !) -
Do noble numbers belong to just intonation ?
According to common JI theories, irrationnals are excluded of JI.According to the differential coherence theory (if you accept it as a just intonation theory), they are.Not because the ratios of all whole numbers recurrent series converge towards irrationnals, becausein a pure mathematical sense, they will never be attained, in theory, by whole number ratios(besides non-coherent ratios such as Vn also can be approached by whole number ratios),BUT because they verify perfect differential coherences, like many JI
scales.
The question then is :is differential coherence acceptable as a just intonation theory ?
exemple : Ishku :16 36 80 178 396 881 1960...x^2 - 2x = 1/2and (V6 + 2) / 2 = 2,22474487139?
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Jacques Dudon

> Hello Jacques,
> I was hoping for you to jump in because of your knowledge
on> difference tone-based scales! I believe your recurrent sequence > stuff is also directly relevant to what Michael Sheiman is doing > i.e. approximating irrationally generated scales with
ratios. > Regarding the Balzano story, there was no story as I don't really> know much about his construction. Another rotation is listed in> Scala as Rothenberg Generalized Diatonic. Supposedly it is
derived> from Group Theory and academic Diatonic Set Theory-type stuff> which are very different from standard tuning-list ideas. Curiously> this construction has nothing to do with (sqrt(3)+1)/
2!

🔗Jacques Dudon <fotosonix@...>

Hi Michael,

Glad to see your enthusiasm !
I am quite busy with the garden these days (and I love it !) and
I have limited time to spend on the list but I 'll try to help -
And I hope others will join on this suject too...

"Silver ratio" or 2.4142135623 : I would like to know, from where did
you find this naming ?
I am calling it Aksaka, and this is also one of the first noble
ratios I resolved geometrically, in the form of a "fractal waveform",
after Phi.
I used to work on these ratios, and still now, mainly to create new
timbers for my instrument, the photosonic disk.
Then I developped scales from them, and also rhythmns. I gave it the
name "Aksaka" from its hypnotic rhythmn, perfectly identical in its
first developpements to the 5 and 7 beats "Aksak" rhythmns from the
Balkans, then to some of the fantastic 12 beats Aka & Baka (Pygmy
tribes), that also work for the irregular "ovoid" rythms (according
to jean During) from Balouchistan .
The timber is very solemn and questionning. I used it in a piece
called "Chandrakaus 235", dedicated to the moon, for my photosonic
disk solo concerts named "Objets celestes", while Zinith 1.366025403
& Zira'at were used in the piece "Pulsation Soleil" from the same
concert and some new pieces. Sorry, none of those are edited yet.
Aksaka main serie is 2 5 12 29 70 etc... (x^2 = 2x + 1) and the
second serie 3 7 17 41 99 etc.
You ask if half-Zira'at = 1.866025403 would have a special purpose.
As such it is octave-coherent with both 1 and 2 in a serie.
That's why I would call it "Zira'atco", but V3 + 2 is the real seed
of those series, that you would have otherwise to multiply endlessly
by 2^n with 1.866025403.
One thing about those names, I found them quickly necessary, as I
deal now with near to a thousand of different noble numbers. But year
after year, names are subject to change, because I extend my logic of
classification to more and more aspects and connexions, and because
of their increasing number and proximity. If anyone thinks about
other sensible naming systems, it can be added.
Also, neither these ratios nor their series belong to me. Just like
patenting living species is totally outrageous I think, I would
suggest that no one using noble numbers or their series should be
allowed to claim any property on them (even Fibonacci ! but this
exception can be done because his series is the world famous
mathematical exemple).
In my collection to this day, only three noble numbers can be
considered as "traditionnal" :
The golden ratio (out of which I resolved the fractal waveform called
"Phidiane"), present in many plants.
Narayana (known by the indians and mentionned in the famous tale
"Narayanan 's cows") = 1,4655712319
Natté (known by the greeks and also named "number of the
architects") = 1,3247179573
These are the only three I heard of related to traditionnal sources,
after I did my own experiments.
If anyone knows of any others, or has any sources on them, I would be
grateful to know.
Of course I heard of the Mt Meru scales, since almost each time I
would mention any coherent recurrent sequence on the tuning list,
their link would have been posted. I have a total admiration for Erv
Wilson, that I take for one the greatest living microtonal music
theoricians and more than that, and I hope to have the chance to meet
him some day because we surely would have much to exchange. But just
for information, before I ever heard of the Mt Meru Scales, I had
been working for years on these subjects, and had already composed
disks and music in most of the Mt Meru scales.
Anyway those are part of my collection, along with a few others from
different searchers and I will be glad to welcome any more.

I believe these frequencies to allow the maximum of differential
coherence between inside the whole scale (including lower phi-aves of
course).
You can test it by multiplying each of these numbers by any Fibonacci
number (over 55 in that division), and check that you get something
close to a whole number, ex. here :
55 58 60 63 66 68 71 73 76 79
81 84 87 89, or
144 152 157 165 173 178 186 191 199 207 212 220 228
233, etc...
This will give you an infinity of rationnal approximations, each Phi-
ave offering different versions.
Of course you could think of choosing between different Phi-aves in
order to collect some of the simplest possible ratios. Why not, but
doing like so would result in a global loss of differential coherence.
Normally if you have already some of the same ratios in your scale
these should be from the series F, 2F, 4F, 8F etc., if I understand
what you did. You'll find out that I am using not only those but 3F,
etc. and also different golden series such as the series P, L, S :
1,381966 for example comes from the plain P series. You will have no difficulty I think to find its exact ratio issued from V5.

Does this helps your purpose ?
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

Posted by: "djtrancendance@..." djtrancendance@...
djtrancendance
Mon Mar 30, 2009 2:40 pm (PDT)

--Thanks Jacques, this is fascinating stuff! Perhaps you could
---help Michael with his Phi-derived rational scale?

Believe me, I would greatly appreciate it. I'm just starting to
stumble across the vast possibilities of these "noble" ratios.
Jacques, I'm also wondering, have you done any work with the
Silver Ratio, if so, and what have you found so far?

And, as a side-note, I noticed solving the "quadratic" form
>>>> x = x^2 + c <<< yields some interesting symmetrical ratios.
For example
A) Solving x = x^2 - 1 gives the golden ratio
B) Solving x = x^2 - 1/2 gives "Zinith" AKA 1.366025403
C) Solving x = 1/2(x^2) - 1/2 gives the Silver Ratio AKA 2.4142135623
D) Solving x = sqrt(x) + 1/2 gives x = 1.866025403 (any special purpose/advantage of formal
name you know of for this ratio, Jacques)?
************************************************************************
************************
--He treats Phi as the period and 2:1 as the generator but I guess we
get the same
--intervals either way?

I think we're on the same page: my basic formula is PHI^y/
2^x...though I wonder if there's a better divisor to use than 2 for
the sake of symmetry. In this case, I think my terminology is
backward: I have in the past said PHI is the generator, my mistake.
After that I simply select tones from that tuning that are not too
close together (to obey the critical band) and, lastly, round them to
their nearest rational equivalents so JI-fanatic people on this list
don't jump at me for being the "crazy irrational number guy". :-D
So, still, it's really more a PHI scale system than a JI one.

🔗djtrancendance@...

Jacquez,

--Also, neither these ratios nor their series belong to me. Just like --patenting living species is totally outrageous I think, I would suggest --that no one using noble numbers or their series should be allowed to --claim any property on them
Agreed...in general I also believe the whole idea of patenting anything artistic, especially involving things like scales, often simply serves to hinder development of that art. It's just odd because the other
side is sometimes I find myself saying "the use of it in a context IE taking special strategic subsets of notes from the series, is original" is simply to get people to take such ideas seriously enough to want to
experiment with them and try to find new possibilities (rather than think it's a problem that's already been solved by someone else in
history).

---The timber is very solemn and questioning
I noticed that as well with the Silver Ratio tuning vs. the Golden Ratio one...it's much darker...and also found it translates less precisely into JI: the difference tones between are harder to summarize in JI-like low-numbered-fractions (more like x/29 maximum instead of about x/20 maximum for PHI for the most part...and only x/26 for about one note).

*********************
As a side question...what do you think are the noble numbers that can create scales where both the ratios in the scales themselves AND the difference tones can be summarized closely (meaning within 10 cents or so) by lowest numbered fractions?

And what are the scales they make that can be best summarized into such fractions?

For example about 9-12 notes per 2/1 "standard octave" and a denominator of 20 or less would be ideal). I managed to
estimate a 9 note per 1.618034 octave PHI-based scale using both fractions of x/26 or less and difference tones between any 2 consecutive notes in the scale of x/26 or less)...but I'm pretty sure you could help this form of art exceed that. :-)
*********************
Lastly, what are your 10 favorite noble numbers so far as music/scale-construction and why? :-)
*********************

--That's why I would call it "Zira'atco", but V3 + 2 is the real seed of those series, that you would have --otherwise to multiply endlessly by 2^n with 1.866025403.
What does v3 stand for? I can understand your notation for Aksaka (x^2 = 2x + 1...solve for x), but not this. Also, how do you form the "spiral of zira" (the equivalent of how the circle of 5ths forms meantone)...or do you create Zira-based scales in another way entirely?

--Of course I heard of the Mt Meru scales, since almost each time I would mention
any coherent recurrent --sequence on the tuning list, their link would have been posted.
By coincidence, Wilson and Sethares are probably my two favorite microtonalists. I'm interested: in your own words, how do the Mt. Meru scales related to difference tones and/or noble numbers? I always considered
Wilson on the cutting-edge of low-numbered-ratio type scales and JI...but I'd be interesting to hear how/what his patterns have in common with non-harmonic-series-type difference tones...especially since, up until now,
I always thought of his work as much closer to JI than irrational numbers.

--- On Wed, 4/1/09, Jacques Dudon <fotosonix@...> wrote:

From: Jacques Dudon <fotosonix@wanadoo.fr>
Subject: [tuning] Re: Balzano's 9-out-of-20
To: tuning@yahoogroups.com
Date: Wednesday,
April 1, 2009, 12:40 PM

Hi Michael,
Glad to see your enthusiasm !I am quite busy with the garden these days (and I love it !) and
I have limited time to spend on the list but I 'll try to help -And I hope others will join on this suject too...
"Silver ratio" or 2.4142135623 : I would like to know, from where did you find this naming ?I am calling it Aksaka, and this is also one of the first noble ratios I resolved geometrically, in the form of a "fractal waveform", after Phi.I used to work on these ratios, and still now, mainly to create new timbers for my instrument, the photosonic disk. Then I developped scales from them, and also rhythmns. I gave it the name
"Aksaka" from its hypnotic rhythmn, perfectly identical in its first developpements to the 5 and 7 beats "Aksak" rhythmns from the Balkans, then to some of the fantastic 12 beats Aka & Baka (Pygmy tribes), that also work for the irregular "ovoid" rythms (according to jean During) from Balouchistan .The timber is very solemn and questionning. I used it in a piece called "Chandrakaus 235", dedicated to the moon, for my photosonic disk solo concerts named "Objets celestes", while Zinith 1.366025403 & Zira'at were used in the piece "Pulsation Soleil" from the same concert and some new pieces. Sorry, none of those are edited yet.Aksaka main serie is 2 5 12 29 70 etc... (x^2 = 2x + 1) and the second serie 3 7 17 41 99
etc.You ask if half-Zira'at = 1.866025403 would have a special purpose. As such it is octave-coherent with both 1 and 2 in a serie.That's why I would call it "Zira'atco", but V3 + 2 is the real seed of those series, that you would have otherwise to multiply endlessly by 2^n with 1.866025403.One thing about those names, I found them quickly necessary, as I deal now with near to a
thousand of different noble numbers. But year after year, names are subject to change, because I extend my logic of classification to more and more aspects and connexions, and because of their increasing number and proximity. If anyone thinks about other sensible naming systems, it can be added.Also, neither these ratios nor their series belong to me. Just like patenting living species is totally outrageous I think, I would suggest that no one using noble numbers or their series should be allowed to claim any property on them (even Fibonacci ! but this exception can be done because his series is
the world famous mathematical exemple).In my collection to this day, only three noble numbers can be considered as "traditionnal" :The golden ratio (out of which I resolved the fractal waveform called "Phidiane"), present in many plants.Narayana (known by the indians and mentionned in the famous tale "Narayanan 's cows") = 1,4655712319Natté (known by the greeks and also named "number of the architects") = 1,3247179573These are the only three I heard of related to traditionnal sources, after I did my own experiments.If anyone knows of any others, or has any sources on them, I would be grateful to know.Of course I heard of the Mt Meru scales, since almost each time I would mention any coherent recurrent sequence on the tuning list, their link would have been posted. I have a total admiration for Erv Wilson, that I take for one the greatest living microtonal music theoricians and more than that, and I hope to have the chance to meet him some day
because we surely would have much to exchange. But just for information, before I ever heard of the
Mt Meru Scales, I had been working for years on these subjects, and had already composed disks and music in most of the Mt Meru scales. Anyway those are part of my collection, along with a few others from different searchers and I will be glad to welcome any more.

About divisions of the Phi-ave (1,618) :What I have been using, in enough small intervals as to get a glissando-wind effect between them (out of the Phi fractal timber), was divisions in increasing number, each in the next Fibonacci number each Phi-ave going up : 1 2 3 5 8
13 21 34 55 89 144 233 (I went up to 288)...This leads to some kind of "golden" harmonic series and I am giving you the ratios for the division in 13 :1,00000001,055728091,09016991,1458981,201626121,23606797751,291796071,326237921,3819661,43769411,4721359551,5278640451,583592135 I believe these frequencies to allow the maximum of differential coherence between inside the whole scale (including lower phi-aves of course).You can test it by multiplying each of these numbers by any Fibonacci number (over 55 in that division), and check that you get something close to a whole number, ex. here : 55 58 60 63 66 68 71 73 76 79 81 84 87 89, or144 152 157 165 173 178 186 191 199 207 212 220 228 233, etc...This will give you an infinity of rationnal approximations, each Phi-ave offering different versions.Of course you could think of choosing between
different Phi-aves in order to collect some of the simplest possible ratios. Why not, but doing like so would result in a global loss of differential coherence.Normally if you have already some of the same
ratios in your scale these should be from the series F, 2F, 4F, 8F etc., if I understand what you did. You'll find out that I am using not only those but 3F, etc. and also different golden series such as the series P, L, S : 1,381966 for example comes from the plain P series. You will have no difficulty I think to find its exact ratio issued from V5. Does this helps your purpose ? - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Jacques

Posted by: "djtrancendance@ yahoo.com" djtrancendance@ yahoo.com djtrancendanceMon Mar 30, 2009 2:40 pm (PDT)--Thanks Jacques, this is fascinating stuff! Perhaps you could---help Michael with his Phi-derived rational scale?
   Believe me, I would greatly appreciate it. I'm just starting to stumble across the vast possibilities of these "noble" ratios. Jacques, I'm also wondering, have you done any work with the Silver Ratio, if so, and what have you found so far?
And, as a side-note, I noticed solving the "quadratic" form>>>> x = x^2 + c <<< yields some interesting symmetrical
ratios.For exampleA) Solving x = x^2 - 1 gives the golden ratioB) Solving x = x^2 - 1/2 gives "Zinith" AKA 1.366025403C) Solving x = 1/2(x^2) - 1/2 gives the Silver Ratio AKA 2.4142135623D) Solving x = sqrt(x) + 1/2 gives x = 1.866025403 (any special purpose/advantage of formalname you know of for this ratio, Jacques)?************ ********* ********* ********* ********* ********* ********* ********* ********* ********* ***--He treats Phi as the period and 2:1 as the generator but I guess we get the same--intervals either way?
   I think we're on the same page: my basic formula is PHI^y/2^x... though I wonder if there's a better divisor to use than 2 for the sake of symmetry. In this case, I think my terminology is backward: I have in the past said PHI is the generator, my mistake. After that I simply select tones from that tuning that are not too close together (to obey the critical band) and, lastly, round them to their nearest rational equivalents so JI-fanatic people on this list don't jump at me for being the "crazy irrational number guy". :-D So, still, it's really more a PHI scale
system than a JI one.
   BTW, any ideas what sort of divisors to use instead of 2(as in 2^y in my formula) to get the notes in the scale more closely spaced? At least 9, and preferably approaching 12, notes between 1 and 2/1 would be ideal. :-)************ ********* ********* ********* ********* ********* *********

🔗Jacques Dudon <fotosonix@...>

🔗daniel_anthony_stearns <daniel_anthony_stearns@...>

🔗Aaron Krister Johnson <aaron@...>

🔗djtrancendance@...

--I have had experience with the general phenomenon of something --'being
attractive on paper' and being less interesting, or MUCH less
--interesting musically, and vice-versa-- -things which are theoretically
--considered 'useless' or 'dull' prove to be quite inspiring musically.
Such often is the case. I found an 8-note scale under 19TET (which is completely wreck-less numerically (absolutely no o-tonal or u-tonal relationships between the notes and intervals whatsoever)...translated into direct inspiration to write a song about becoming very old and knowing exactly how things work, but not be able to feel them physically anymore. So, to myself and apparently several others...it comes across as very positive and not chaotic...despite using on-the-surface chaotic-seeming scales.

I wonder how many people about here have wrote pieces about "intelligence in a state of struggle" IE songs that sound very tense yet have a sense about them which invokes a sense of a steady goal among vast chaos...kind of like a sing about landing an airplane on water or something of the like. Any of you ever heard a
really odd scale and were then complelled to make a song about that sort of thing?

--------------------------
Also, to note...the one thing I've found that seems to almost always translate from "attractive on paper" to "interesting musically" is symmetry: be it in JI, 12TET, or completely unrelated scales. But...I've found good composition skills can serve as a steady guide through even the weirdest scales...in a sense having "predictable high dissonance" is easier to follow than having dissonance that wavers from low to very low randomly without much sense of purpose. Sometime composition, I swear, can develop its own symmetry...which can make things predictable enough to feel very solid even if the scale used for it seems chaotic on the surface.
****************************
Yet another point...I've found many of the least chaotic scales are NOT complicated ones nor do they require long
mathematical proofs. The harmonic series itself is about the simplest scale imaginable and the latest version of my PHI scale turned out to be simply 2 harmonic series (one reversed, one forward)...but the 1,2,3 harmonic-type ratios came in terms of derivatives from other notes IE difference tones from the previous tone rather than difference tones from the root.

So, IMVHO...often the best forms of scales...are NOT the ones that can only be summarized in very complex equations...but rather those that start their lives as complex equations and simplify as the scale is developed until, finally, they reduce to something that seems utterly boring: until you hear it or try to compose with it.

--- On Thu, 4/2/09, Aaron Krister Johnson <aaron@...m> wrote:

From: Aaron Krister
Johnson <aaron@...>
Subject: [tuning] Re: Balzano's 9-out-of-20
To: tuning@yahoogroups.com
Date: Thursday, April 2, 2009, 10:29 AM

I don't have experience with this particular scale, but I agree with the jist of what you say--I have had experience with the general phenomenon of something 'being attractive on paper' and being less interesting, or MUCH less interesting musically, and vice-versa-- -things which are theoretically considered 'useless' or 'dull' prove to be quite inspiring musically.

-Aaron.

--- In tuning@yahoogroups. com, "daniel_anthony_ stearns" <daniel_anthony_ stearns@. ..> wrote:

> FWiW, i've had a 20-tone equal tempered guitar for almost 20 years now. and having worked hands-on with this scale (and its cousin, the 11-out-of-20) for many years, i'd have to say that i personally find it no more useful than a lot of other 20-tone scales .it sure does attract a lot attention theoretically ,myself included, but in practice i for one have yet to justify that attention... and personally i think that's an instructive point, but not one that's very well-suited to perpetuating forums like this one.

> daniel

> --- In tuning@yahoogroups. com, "Kalle Aho" <kalleaho@> wrote:

> >

> > Hello everyone,

> >

> > I think the following fits very well the recent topic of tunings that

> > are not based on JI or small integer ratios.

> >

> > I know next to nothing about the reasoning behind Balzano's scale but I

> > noticed a strange thing:

> >

> > this scale is a 9-tone MOS of 2^(9/20)-generator.

> >

> > Now 2^(9/20) is extremely close to (sqrt(3)+1)/ 2 which

> > has an interesting property related to difference tones:

> >

> > if g = (sqrt(3)+1)/ 2

> >

> > g^2 - g = 1/2.

> >

> > That's really weird because I don't think this feature had

> > anything to do with Balzano's construction of this scale.

> >

> > Kalle Aho

> >

🔗daniel_anthony_stearns <daniel_anthony_stearns@...>

well simply put,in my personal opinion, i've always been far more impressed by something like the sounds, reach, and application of the DIY EMI cult than i ever have been by the comparative machinations of the micro forum phenomena. Some people, like Dudon (a surprise contributor to this thread BtW, and a real genius IMo) seem to be able to straddle these waters pulling fascinating gold from multiple sieves. But he's really the rare type of exception that proves the norm ImO.
Anyway, I've always been interested in microtonality, even such as it is here.....but i've also been disturbed for quite a while now by the nagging feeling that it's also really _really_ lacking as regards the sounds produced relative to the theory (et al) proposed on these forums.

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> --I have had experience with the general phenomenon of something --'being
> attractive on paper' and being less interesting, or MUCH less
> --interesting musically, and vice-versa-- -things which are theoretically
> --considered 'useless' or 'dull' prove to be quite inspiring musically.
>     Such often is the case. I found an 8-note scale under 19TET (which is completely wreck-less numerically (absolutely no o-tonal or u-tonal relationships between the notes and intervals whatsoever)...translated into direct inspiration to write a song about becoming very old and knowing exactly how things work, but not be able to feel them physically anymore. So, to myself and apparently several others...it comes across as very positive and not chaotic...despite using on-the-surface chaotic-seeming scales.
>
>    I wonder how many people about here have wrote pieces about "intelligence in a state of struggle" IE songs that sound very tense yet have a sense about them which invokes a sense of a steady goal among vast chaos...kind of like a sing about landing an airplane on water or something of the like. Any of you ever heard a
> really odd scale and were then complelled to make a song about that sort of thing?
>
> --------------------------
>      Also, to note...the one thing I've found that seems to almost always translate from "attractive on paper" to "interesting musically" is symmetry: be it in JI, 12TET, or completely unrelated scales. But...I've found good composition skills can serve as a steady guide through even the weirdest scales...in a sense having "predictable high dissonance" is easier to follow than having dissonance that wavers from low to very low randomly without much sense of purpose. Sometime composition, I swear, can develop its own symmetry...which can make things predictable enough to feel very solid even if the scale used for it seems chaotic on the surface.
> ****************************
>    Yet another point...I've found many of the least chaotic scales are NOT complicated ones nor do they require long
> mathematical proofs. The harmonic series itself is about the simplest scale imaginable and the latest version of my PHI scale turned out to be simply 2 harmonic series (one reversed, one forward)...but the 1,2,3 harmonic-type ratios came in terms of derivatives from other notes IE difference tones from the previous tone rather than difference tones from the root.
>
>    So, IMVHO...often the best forms of scales...are NOT the ones that can only be summarized in very complex equations...but rather those that start their lives as complex equations and simplify as the scale is developed until, finally, they reduce to something that seems utterly boring: until you hear it or try to compose with it.
>
>
>
> --- On Thu, 4/2/09, Aaron Krister Johnson <aaron@...> wrote:
>
> From: Aaron Krister
> Johnson <aaron@...>
> Subject: [tuning] Re: Balzano's 9-out-of-20
> To: tuning@yahoogroups.com
> Date: Thursday, April 2, 2009, 10:29 AM
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> I don't have experience with this particular scale, but I agree with the jist of what you say--I have had experience with the general phenomenon of something 'being attractive on paper' and being less interesting, or MUCH less interesting musically, and vice-versa-- -things which are theoretically considered 'useless' or 'dull' prove to be quite inspiring musically.
>
>
>
> -Aaron.
>
>
>
> --- In tuning@yahoogroups. com, "daniel_anthony_ stearns" <daniel_anthony_ stearns@ ..> wrote:
>
> >
>
> > FWiW, i've had a 20-tone equal tempered guitar for almost 20 years now. and having worked hands-on with this scale (and its cousin, the 11-out-of-20) for many years, i'd have to say that i personally find it no more useful than a lot of other 20-tone scales .it sure does attract a lot attention theoretically ,myself included, but in practice i for one have yet to justify that attention... and personally i think that's an instructive point, but not one that's very well-suited to perpetuating forums like this one.
>
> > daniel
>
> >
>
> > --- In tuning@yahoogroups. com, "Kalle Aho" <kalleaho@> wrote:
>
> > >
>
> > > Hello everyone,
>
> > >
>
> > > I think the following fits very well the recent topic of tunings that
>
> > > are not based on JI or small integer ratios.
>
> > >
>
> > > I know next to nothing about the reasoning behind Balzano's scale but I
>
> > > noticed a strange thing:
>
> > >
>
> > > this scale is a 9-tone MOS of 2^(9/20)-generator.
>
> > >
>
> > > Now 2^(9/20) is extremely close to (sqrt(3)+1)/ 2 which
>
> > > has an interesting property related to difference tones:
>
> > >
>
> > > if g = (sqrt(3)+1)/ 2
>
> > >
>
> > > g^2 - g = 1/2.
>
> > >
>
> > > That's really weird because I don't think this feature had
>
> > > anything to do with Balzano's construction of this scale.
>
> > >
>
> > > Kalle Aho
>
> > >
>
> >
>

🔗djtrancendance@...

---The question then is :---is differential coherence acceptable as a just intonation theory ?
Getting off my PHI high horse (admittedly)...I challenged myself to match my PHI scale's ability to have massive chords without losing much consonance with a "plain-old" JI scale built to maximize relations between difference tones (all tones, NOT just the root and thirds/triads as diatonic JI does).

What I got was the following scale:
1
17/16    17th harmonic
19/16    19th harmonic
4/3       perfect fourth
3/2       perfect fifth
19/12   un-decimal minor sixth
11/6     un-decimal neutral seventh
2/1

   Pluses: it DOES sound more consonant
than my PHI scale, no doubt: and can create a seem-less 7-note within 2/1 octave chord very very easily (you can just barely hear the beating in such a chord with this scale!). Also, for JI nuts, the two harmonic series it is composed of (x/16 and x/12) are easily visible and all denominators are factors of 2.
------------------
   Minuses: PHI does 7-note per 2/1 standard period (octave) chords not too much worse (like comparing "sour" 12TET to 5-limit JI consonance) and allows up to about 9-notes per 2/1 with hardly any difference in consonance and up to about 11 notes per 2/1 octave worth of melodic possibilities.
******************************************************************************************
      Conclusion: difference tones make a WORLD of a difference, even if you hate the idea of using noble numbers to create scales and just want to use difference tones to optimize your
JI.

Side question: "undecimal" type intervals pop up a whole lot in both this scale and my PHI scale. Apparently, I like the sound of them a lot. How are they derived and why, perhaps, do they seem to work so well?

-Michael

--- On Wed, 4/1/09, Jacques Dudon <fotosonix@...> wrote:

From: Jacques Dudon <fotosonix@...>
Subject: [tuning] Re:Balzano's 9-out-of-20
To: tuning@yahoogroups.com
Date: Wednesday, April 1, 2009, 9:55 AM

> Hello Jacques,
> I was hoping for you to jump in because of your knowledge on> difference tone-based scales! I believe your recurrent sequence > stuff is also directly relevant to what Michael Sheiman is doing > i.e. approximating irrationally generated scales with ratios. > Regarding the Balzano story, there was no story as I don't really> know much about his construction. Another rotation is listed in> Scala as Rothenberg Generalized Diatonic. Supposedly it is derived> from Group Theory and academic Diatonic Set Theory-type stuff> which are very different from standard tuning-list ideas. Curiously> this construction has nothing to do with (sqrt(3)+1)/ 2!

🔗djtrancendance@...

Daniel,

---but i've also been disturbed for quite a while now by the nagging feeling that it's also really ---_really_ lacking as regards the sounds produced relative to the theory (et al) proposed on ---these forums.
It's a HUGE issue in this forum far as I'm concerned; an insane proportion of the time when I (or most people, for that matter) post something I get about 8 people complaining about the math. For example, if I'm very lucky, maybe one who actually listened to what said "evil" scale sounds like. :-D Isn't that what we are going for here
anyhow...making scales that sound good and not just perusing through history about scale systems?

This is one of the reasons I, for one, try to post sound examples as much as possible. The theory for some of the things on here, like Wilson's Mt. Meru scales, goes way over my head and far beyond basic o-tonal and u-tonal relationships. But, mind my criticism...such scales (yes, I have listened to them, the 6 tone ones going all the way up to the 9-tone ones) seem to fall short of what you'd think their complex theories (which almost look like the code to a commercial 3D game) would yield something that would kick the crap out of 12TET but, instead, they merely add a tad of purity to it most of the time rather than invent music in a completely different context.

Recently, I made a scale as follows (took me about an hour to make from scratch):

1
17/16    17th
harmonic
19/16    19th harmonic
4/3       perfect fourth
3/2       perfect fifth
19/12   un-decimal minor sixth
11/6     un-decimal neutral seventh
2/1

....and ALL I did to make it was take the purest parts of the x/16 harmonic series and tried to find the other fractions that, combined with them, created the most low-numbered fraction difference tones, mostly by ear and partly by just looking at the difference tones via SCALA. No huge historical document, no funky matrices... And yet, pardon my enthusiasm, this dead simple scale sounds better to me than a huge proportion of scales I've heard on this list: just about everything I compose with it sounds very resolved and free/upbeat.

Now I'm just waiting for someone to find some major flaw with the math in my new scale
and claim I am blasphemously ripping off someone from scale history who I've most likely never heard of (LOL)...but I'd be even more eager to hear how people react to actually listening to the scale or trying to compose with it. :-)

-Michael

--- On Thu, 4/2/09, daniel_anthony_stearns <daniel_anthony_stearns@...> wrote:

From: daniel_anthony_stearns <daniel_anthony_stearns@...>
Subject: [tuning] Re: Balzano's 9-out-of-20
To: tuning@yahoogroups.com
Date: Thursday, April 2, 2009, 7:56 PM

Anyway, I've always been interested in microtonality, even such as it is here.....but i've also been disturbed for quite a while now by the nagging feeling that it's also really _really_ lacking as regards the sounds produced relative to the theory (et al) proposed on these forums.

--- In tuning@yahoogroups. com, djtrancendance@ ... wrote:

> --I have had experience with the general phenomenon of something --'being

> attractive on paper' and being less interesting, or MUCH less

> --interesting musically, and vice-versa-- -things which are theoretically

> --considered 'useless' or 'dull' prove to be quite inspiring musically.

> Such often is the case. I found an 8-note scale under 19TET (which is completely wreck-less numerically (absolutely no o-tonal or u-tonal relationships between the notes and intervals whatsoever). ..translated into direct inspiration to write a song about becoming very old and knowing exactly how things work, but not be able to feel them physically anymore. So, to myself and apparently several others...it comes across as very positive and not chaotic...despite using on-the-surface chaotic-seeming scales.

> I wonder how many people about here have wrote pieces about "intelligence in a state of struggle" IE songs that sound very tense yet have a sense about them which invokes a sense of a steady goal among vast chaos...kind of like a sing about landing an airplane on water or something of the like. Any of you ever heard a

> really odd scale and were then complelled to make a song about that sort of thing?

> ------------ --------- -----

> Also, to note...the one thing I've found that seems to almost always translate from "attractive on paper" to "interesting musically" is symmetry: be it in JI, 12TET, or completely unrelated scales. But...I've found good composition skills can serve as a steady guide through even the weirdest scales...in a sense having "predictable high dissonance" is easier to follow than having dissonance that wavers from low to very low randomly without much sense of purpose. Sometime composition, I swear, can develop its own symmetry...which can make things predictable enough to feel very solid even if the scale used for it seems chaotic on the surface.

> ************ ********* *******

> Yet another point...I've found many of the least chaotic scales are NOT complicated ones nor do they require long

> mathematical proofs. The harmonic series itself is about the simplest scale imaginable and the latest version of my PHI scale turned out to be simply 2 harmonic series (one reversed, one forward)...but the 1,2,3 harmonic-type ratios came in terms of derivatives from other notes IE difference tones from the previous tone rather than difference tones from the root.

> So, IMVHO...often the best forms of scales...are NOT the ones that can only be summarized in very complex equations... but rather those that start their lives as complex equations and simplify as the scale is developed until, finally, they reduce to something that seems utterly boring: until you hear it or try to compose with it.

> --- On Thu, 4/2/09, Aaron Krister Johnson <aaron@...> wrote:

> From: Aaron Krister

> Johnson <aaron@...>

> Subject: [tuning] Re: Balzano's 9-out-of-20

> To: tuning@yahoogroups. com

> Date: Thursday, April 2, 2009, 10:29 AM

> I don't have experience with this particular scale, but I agree with the jist of what you say--I have had experience with the general phenomenon of something 'being attractive on paper' and being less interesting, or MUCH less interesting musically, and vice-versa-- -things which are theoretically considered 'useless' or 'dull' prove to be quite inspiring musically.

> -Aaron.

> --- In tuning@yahoogroups. com, "daniel_anthony_ stearns" <daniel_anthony_ stearns@ ..> wrote:

> >

> > FWiW, i've had a 20-tone equal tempered guitar for almost 20 years now. and having worked hands-on with this scale (and its cousin, the 11-out-of-20) for many years, i'd have to say that i personally find it no more useful than a lot of other 20-tone scales .it sure does attract a lot attention theoretically ,myself included, but in practice i for one have yet to justify that attention... and personally i think that's an instructive point, but not one that's very well-suited to perpetuating forums like this one.

> > daniel

> >

> > --- In tuning@yahoogroups. com, "Kalle Aho" <kalleaho@> wrote:

> > >

> > > Hello everyone,

> > >

> > > I think the following fits very well the recent topic of tunings that

> > > are not based on JI or small integer ratios.

> > >

> > > I know next to nothing about the reasoning behind Balzano's scale but I

> > > noticed a strange thing:

> > >

> > > this scale is a 9-tone MOS of 2^(9/20)-generator.

> > >

> > > Now 2^(9/20) is extremely close to (sqrt(3)+1)/ 2 which

> > > has an interesting property related to difference tones:

> > >

> > > if g = (sqrt(3)+1)/ 2

> > >

> > > g^2 - g = 1/2.

> > >

> > > That's really weird because I don't think this feature had

> > > anything to do with Balzano's construction of this scale.

> > >

> > > Kalle Aho

> > >

> >

🔗Daniel Forro <dan.for@...>

🔗Jacques Dudon <fotosonix@...>

Hi Daniel,

I am only half surprised these 9 or 11 out of 20 edo do not have a special interest on a guitar.
This is very much like Sethares examples of scales that do not sound very good in usual harmonic timbres, but would reveal their flavor with more specific inharmonic waveforms.
A guitar string has mainly L5 harmonics, that are not suited to such a scale.
Do you have a harmonizer ? I would suggest you try harmonizing your guitar with different powers of Zinith = 1,366025404, that is exactly 539,9811762 cents, or roughly 540 cents, which is exactly 9 steps of your 20 ET.
And why not feedback the transposition in chains of 540 c. as well.

Also I don't know how your guitar is tuned in open state but if it contains offscale strings they might interfere badly.
I would suggest you to try only using open strings tuned in 1/1, 15/8 (and eventually 7/4 ?), and their octaves.
Combine with harmonizing if you can, and see how it feels.

Or else, I will send you a tape with some of my photosonic sounds in that system. I would like to hear how a 20-ET guitar sounds with it.

Then, can you give an example of better chords or scales for you on this guitar ?
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques Dudon

Posted by: "daniel_anthony_stearns" daniel_anthony_stearns@... daniel_anthony_stearns
Wed Apr 1, 2009 8:17 pm (PDT)

FWiW, i've had a 20-tone equal tempered guitar for almost 20 years now. and having worked hands-on with this scale (and its cousin, the 11-out-of-20) for many years, i'd have to say that i personally find it no more useful than a lot of other 20-tone scales .it sure does attract a lot attention theoretically ,myself included, but in practice i for one have yet to justify that attention...and personally i think that's an instructive point, but not one that's very well-suited to perpetuating forums like this one.
daniel

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> Hello everyone,
>
> I think the following fits very well the recent topic of tunings that
> are not based on JI or small integer ratios.
>
> I know next to nothing about the reasoning behind Balzano's scale but I
> noticed a strange thing:
>
> this scale is a 9-tone MOS of 2^(9/20)-generator.
>
> Now 2^(9/20) is extremely close to (sqrt(3)+1)/2 which
> has an interesting property related to difference tones:
>
> if g = (sqrt(3)+1)/2
>
> g^2 - g = 1/2.
>
> That's really weird because I don't think this feature had
> anything to do with Balzano's construction of this scale.
>
> Kalle Aho

🔗Jacques Dudon <fotosonix@...>

🔗djtrancendance@...

--Sorry Michael, that was a perhaps unconventional typing shortcut I --use for "square root of n" :--V3 = 1,732...
Ah, ok. As a programmer I'm just so used to seeing people use sqrt() for square root that I would have never guessed your notation.

--Just like you use AKA to mean perhaps "as known as" ?? , while for me "Aka" refers ---to one of the main pygmy ethnies from Central Africa ...
Exactly...sorry about that; I can't live without my funny acronyms like "also known as" either. :-D And, also, IE (when I use it) means "In Example" not "Internet Explorer"... :-)

--BTW (by the way ?), you did not reply as from where you find the ---name "silver ratio" for 2^(1/2) + 1...
First I found/solved it by basic
quadratic math...then I looked it up in the wiki-pedia. Interestingly enough I could NOT find a good entry there for noble numbers...is there anywhere I could find a good list of those so I don't have to run across them by trial and error?

--and there is no contradiction with regular 2/1 octaves repetitions as --well and probably less than with Phi.
Quite convenient...that way it hopefully won't confused people so much. :-)

-Michael

--- On Fri, 4/3/09, Jacques Dudon <fotosonix@...> wrote:

From: Jacques Dudon <fotosonix@...>
Subject: [tuning] Re: Balzano's 9-out-of-20
To: tuning@yahoogroups.com
Date: Friday, April 3, 2009, 3:15 AM

Sorry Michael, that was a perhaps unconventionnal typing shortcut I use for "square root of n" :V3 = 1,732...Just like you use AKA to mean perhaps "as known as" ?? , while for me "Aka" refers to one of the main pygmy ethnies from Central Africa ...
BTW (by the way ?), you did not reply as from where you find the name "silver ratio" for 2^(1/2) + 1...Do we get all metals like that ?And what's the ratio that changes lead into gold ? ;-)
Yes, you may generate scales with Zinith and/or Zira'at the same way I indicated with the golden ratio, by Phi-aves of different sets of series. These would be transposed by
Zinith-aves and/or Zira'at-aves. That's how I arrived at 9 or 11 tones per octave, and there is no contradiction with regular 2/1 octaves repetitions as well and probably less than with Phi.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Jacques

Posted by: "djtrancendance@ yahoo.com" djtrancendance@ yahoo.com
djtrancendanceWed Apr 1, 2009 3:17 pm (PDT)--That's why I would call it "Zira'atco", but V3 + 2 is the real seed of those series, that you would have --otherwise to multiply endlessly by 2^n with 1.866025403. What does v3 stand for? I can understand your notation for Aksaka (x^2 = 2x + 1...solve for x), but not this. Also, how do you form the "spiral of zira" (the equivalent of how the circle of 5ths forms meantone)... or do you create Zira-based scales in another way entirely?

🔗djtrancendance@...

Daniel,
--No flaw, why, just another diatonic scale...
--Rather limited, don't you think?
    Yes, actually that is my partly point (I'm actually using a lot of this to prove there a hidden problem most diatonic scales don't usually solve well). :-D....but the other point was to test the application of difference tone matching in JI (THIS, for example, explains why 1.25 or anything very close to it is mysteriously absent from my diatonic scale).
---------------------------------
    After both a lot of criticism for being the guy who "blatantly ignored diatonic scales" and suspicion that my new knowledge of difference tones could be used to improve them I just had to try.
    I also wanted to be honest with myself and not so "PHI scale biased".
I already had compared my PHI scale against 12TET and diatonic JI and was quite sure the results were more consonant...but I suspected
JI could be made to do a good deal better than standard diatonic JI so far as consonance.
   Again, the one thing I swore my scale had the others did not was a system for making simple ratio difference tones work between virtually ALL ratios, and NOT just the ratio vs. the root or triads (as diatonic JI does). Jacques Dudon asked the question if difference tones are relevant as JI theory and I was strongly suspecting the answer could easily be "YES!".
----------------------------------
   So I figured "you know...I bet I can make a diatonic JI scale that actually can give my PHI scale some much more serious competition if I optimize it for difference tones.

And when I did, my girlfriend (who was sitting in the room) without being asked I think, said "wow, MUCH clearer...you should use that" when I switched from diatonic JI to the diatonic scale I just made from for a 7-note-per-octave
chord.

********"Or what's your idea about harmony?"*********************
Great question! My new theory about harmony...is that it comes JUST as much (if not more) from aligning difference tones between all other notes in the scale as harmonizing to the root: and this can be done either "by nature" via noble-number based scales or by comparative mathematics using JI.
The special case is that noble numbers seem to generate tunings that do this automatically assuming you use timbres that match and also automatically for the harmonic series timbre with special subsets of the noble number tunings.
******************************************************************
In summary: it seems blatantly obvious both the alignment of difference tones in JI and use of noble-number generated scales to align difference tones deserves much exploration AND a combination of both extensive
listening/composing to develop such scale and math to help summarize and simplify/"round" the discoveries to terms people can use to help easily create new, very consonant scales.

--- On Thu, 4/2/09, Daniel Forro <dan.for@...> wrote:

From: Daniel Forro <dan.for@...>
Subject: Re: [tuning] Re: Balzano's 9-out-of-20
To: tuning@yahoogroups.com
Date: Thursday, April 2, 2009, 10:47 PM

No flaw, why, just another diatonic scale... Hm, I would say, from

compositional point of view, what's your idea about composing with

such 7-note scale with a structure not so far from the other

traditional 7-note scales? Rather limited, don't you think? You can

get from it just some diatonic music... , to use different central

notes, tonal, modal, atonal, serial approach, write in heterophony,

polyphony, homophony... there's nothing special in it in which it

would differ from the work with let's say white keys diatonic in 12ET

(just a little bit out of tune :-) ). You can create chords from one

interval only (second clusters, triadic, quartal) or to combine

intervals, maybe on the base of some numerology or symmetry, the most

dense chord can have only 7 tones, work with consonance and

dissonance will be rather limited, too... Or what's your idea about

harmony?

So what kind of music do you want to do with it?

Daniel Forro

On 3 Apr 2009, at 12:38 PM, djtrancendance@ yahoo.com wrote:

> Recently, I made a scale as follows (took me about an hour to make

> from scratch):

> 1

> 17/16 17th harmonic

> 19/16 19th harmonic

> 4/3 perfect fourth

> 3/2 perfect fifth

> 19/12 un-decimal minor sixth

> 11/6 un-decimal neutral seventh

> 2/1

> ....and ALL I did to make it was take the purest parts of the x/16

> harmonic series and tried to find the other fractions that,

> combined with them, created the most low-numbered fraction

> difference tones, mostly by ear and partly by just looking at the

> difference tones via SCALA. No huge historical document, no funky

> matrices... And yet, pardon my enthusiasm, this dead simple scale

> sounds better to me than a huge proportion of scales I've heard on

> this list: just about everything I compose with it sounds very

> resolved and free/upbeat.

> Now I'm just waiting for someone to find some major flaw with

> the math in my new scale and claim I am blasphemously ripping off

> someone from scale history who I've most likely never heard of

> (LOL)...but I'd be even more eager to hear how people react to

> actually listening to the scale or trying to compose with it. :-)

> -Michael

🔗Kalle Aho <kalleaho@...>

🔗Daniel Forro <dan.for@...>

Thanks for answer, Michael. You have selected very pure, minimalistic, ascetic approach. I feel sympathy to all those going by narrow thorny paths :-). It will be a challenge to do some music from it, good luck! Result can be interesting.

I personally would prefer not to create just a scale as you did (in your case scale = temperament), but to use some tuning/temperament system with smaller steps (let's say between 20 - 117.3258999 Cents), not necessarily of the same size, and not necessarily closed into the octave, and thus to get a bigger arsenal of possible subsets of it (used as scales or chords) for compositional work. It would offer more freedom, contrasts, possible "inner worlds" with different sizes of steps, and different number of tones. And this all well locked together thanks to the main "roof" tuning. It would allow to use compositional techniques like continual revealing of temperament, crossfading of different subsets or chaining and alternating them, combine them, apply principle of modulation, use more consonant or more dissonant subsets etc...

I have nothing against all that numerological magic having been using always lot of numbers in my music, but all this is just a tool. For me the musical result is more important. Yes, music is mathematics, but in my understanding not more than 50%. Less means more random, more means more organized, but extremes in both directions started to sound the same boring. So last years I don't hesitate not to keep, change or cancel even my own rules or some complex order, functions, equations, matrices, series, processes - selected, planned and prefabricated before compositional process itself, when they somehow started to work against the music.

As it was said few messages back - something which looks well on the paper not necessarily sounds well, and many times what sounds well can look strange in the score.

Daniel Forro

On 4 Apr 2009, at 12:20 AM, djtrancendance@... wrote:

> Daniel,
> --No flaw, why, just another diatonic scale...
> --Rather limited, don't you think?
> Yes, actually that is my partly point (I'm actually using a lot > of this to prove there a hidden problem most diatonic scales don't > usually solve well). :-D....but the other point was to test the > application of difference tone matching in JI (THIS, for example, > explains why 1.25 or anything very close to it is mysteriously > absent from my diatonic scale).
> ---------------------------------
> After both a lot of criticism for being the guy who "blatantly > ignored diatonic scales" and suspicion that my new knowledge of > difference tones could be used to improve them I just had to try.
> I also wanted to be honest with myself and not so "PHI scale > biased". I already had compared my PHI scale against 12TET and > diatonic JI and was quite sure the results were more > consonant...but I suspected JI could be made to do a good deal > better than standard diatonic JI so far as consonance.
> Again, the one thing I swore my scale had the others did not was > a system for making simple ratio difference tones work between > virtually ALL ratios, and NOT just the ratio vs. the root or triads > (as diatonic JI does). Jacques Dudon asked the question if > difference tones are relevant as JI theory and I was strongly > suspecting the answer could easily be "YES!".
> ----------------------------------
> So I figured "you know...I bet I can make a diatonic JI scale > that actually can give my PHI scale some much more serious > competition if I optimize it for difference tones.
>
> And when I did, my girlfriend (who was sitting in the room) > without being asked I think, said "wow, MUCH clearer...you should > use that" when I switched from diatonic JI to the diatonic scale I > just made from for a 7-note-per-octave chord.
>
> ********"Or what's your idea about harmony?"*********************
> Great question! My new theory about harmony...is that it comes > JUST as much (if not more) from aligning difference tones between > all other notes in the scale as harmonizing to the root: and this > can be done either "by nature" via noble-number based scales or by > comparative mathematics using JI.
> The special case is that noble numbers seem to generate tunings > that do this automatically assuming you use timbres that match and > also automatically for the harmonic series timbre with special > subsets of the noble number tunings.
> ******************************************************************
> In summary: it seems blatantly obvious both the alignment of > difference tones in JI and use of noble-number generated scales to > align difference tones deserves much exploration AND a combination > of both extensiv e listening/composing to develop such scale and > math to help summarize and simplify/"round" the discoveries to > terms people can use to help easily create new, very consonant scales.
>

🔗djtrancendance@...

--It will be a challenge to do some music from
--it, good luck! Result can be interesting.
Thank you!...and, of course, I'm going to try (and post my results here)...I figure that, most likely, the result will sound a bit "too pure"/"without enough tonal color", if anything else. But I think, indirectly, it will stress the importance of having difference tones that can be summarized into about a x/20 or less form (just like my PHI scale and this one) for consonance.

--but to use some tuning/temperament
--system with smaller steps (let's say between 20 - 117.3258999 Cents),
Well, that's what I built my "crazy" PHI scale for (it has boat-loads of tonal-color vs. most diatonic scales). :-D Steps in that scale go down to about 67 cents and you can have about 13 notes per octave (and about a 9-note-per octave
chord that sounds consonant).

--Yes, music is mathematics,
--but in my understanding not more than 50%. Less means more --random, more means more organized, but extremes in both directions ---started to sound the same boring.
Exactly...the most organized, IMVHO, is the harmonic series, which sounds deadly boring to play in (it has virtually no sense of tonal color)...while on the other side you just things like 21TET which are so complex and take so much mental effort to listen to them you can't really relax and enjoy the piece a lot of the time.

From personal experience, in both the above diatonic scale and my PHI scale I found MANY MANY intervals that worked out mathematically, INCLUDING in terms of difference tones, but just didn't add enough randomness/"tonal color" for my tastes.

Indeed, I agree that at least 50% "feel" (and not 80%+ math) is a good way to think
about tuning. Even if you look at my diatonic scale notice how I at least managed to push in x/12 type fractions and not just x/16 type ones...a main reason for that is to add at least some tonal color to the mix. :-)

-Michael

--- On Fri, 4/3/09, Daniel Forro <dan.for@...> wrote:

From: Daniel Forro <dan.for@...>
Subject: Re: [tuning] Re: Balzano's 9-out-of-20
To: tuning@yahoogroups.com
Date: Friday, April 3, 2009, 9:37 AM

Thanks for answer, Michael. You have selected very pure,

minimalistic, ascetic approach. I feel sympathy to all those going by

narrow thorny paths :-). It will be a challenge to do some music from

it, good luck! Result can be interesting.

I personally would prefer not to create just a scale as you did (in

your case scale = temperament) , but to use some tuning/temperament

system with smaller steps (let's say between 20 - 117.3258999 Cents),

not necessarily of the same size, and not necessarily closed into the

octave, and thus to get a bigger arsenal of possible subsets of it

(used as scales or chords) for compositional work. It would offer

more freedom, contrasts, possible "inner worlds" with different sizes

of steps, and different number of tones. And this all well locked

together thanks to the main "roof" tuning. It would allow to use

compositional techniques like continual revealing of temperament,

crossfading of different subsets or chaining and alternating them,

combine them, apply principle of modulation, use more consonant or

more dissonant subsets etc...

I have nothing against all that numerological magic having been using

always lot of numbers in my music, but all this is just a tool. For

me the musical result is more important. Yes, music is mathematics,

but in my understanding not more than 50%. Less means more random,

more means more organized, but extremes in both directions started to

sound the same boring. So last years I don't hesitate not to keep,

change or cancel even my own rules or some complex order, functions,

equations, matrices, series, processes - selected, planned and

prefabricated before compositional process itself, when they somehow

started to work against the music.

As it was said few messages back - something which looks well on the

paper not necessarily sounds well, and many times what sounds well

can look strange in the score.

Daniel Forro

On 4 Apr 2009, at 12:20 AM, djtrancendance@ yahoo.com wrote:

> Daniel,

> --No flaw, why, just another diatonic scale...

> --Rather limited, don't you think?

> Yes, actually that is my partly point (I'm actually using a lot

> of this to prove there a hidden problem most diatonic scales don't

> usually solve well). :-D....but the other point was to test the

> application of difference tone matching in JI (THIS, for example,

> explains why 1.25 or anything very close to it is mysteriously

> absent from my diatonic scale).

> ------------ --------- --------- ---

> After both a lot of criticism for being the guy who "blatantly

> ignored diatonic scales" and suspicion that my new knowledge of

> difference tones could be used to improve them I just had to try.

> I also wanted to be honest with myself and not so "PHI scale

> biased". I already had compared my PHI scale against 12TET and

> diatonic JI and was quite sure the results were more

> consonant... but I suspected JI could be made to do a good deal

> better than standard diatonic JI so far as consonance.

> Again, the one thing I swore my scale had the others did not was

> a system for making simple ratio difference tones work between

> virtually ALL ratios, and NOT just the ratio vs. the root or triads

> (as diatonic JI does). Jacques Dudon asked the question if

> difference tones are relevant as JI theory and I was strongly

> suspecting the answer could easily be "YES!".

> ------------ --------- --------- ----

> So I figured "you know...I bet I can make a diatonic JI scale

> that actually can give my PHI scale some much more serious

> competition if I optimize it for difference tones.

> And when I did, my girlfriend (who was sitting in the room)

> without being asked I think, said "wow, MUCH clearer...you should

> use that" when I switched from diatonic JI to the diatonic scale I

> just made from for a 7-note-per-octave chord.

> ********"Or what's your idea about harmony?"*** ********* *********

> Great question! My new theory about harmony...is that it comes

> JUST as much (if not more) from aligning difference tones between

> all other notes in the scale as harmonizing to the root: and this

> can be done either "by nature" via noble-number based scales or by

> comparative mathematics using JI.

> The special case is that noble numbers seem to generate tunings

> that do this automatically assuming you use timbres that match and

> also automatically for the harmonic series timbre with special

> subsets of the noble number tunings.

> ************ ********* ********* ********* ********* ********* *********

> In summary: it seems blatantly obvious both the alignment of

> difference tones in JI and use of noble-number generated scales to

> align difference tones deserves much exploration AND a combination

> of both extensiv e listening/composing to develop such scale and

> math to help summarize and simplify/"round" the discoveries to

> terms people can use to help easily create new, very consonant scales.

Balzano's 9-out-of-20

🔗Kalle Aho <kalleaho@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Jacques Dudon <fotosonix@...>

🔗Kalle Aho <kalleaho@...>

🔗djtrancendance@...

🔗Cameron Bobro <misterbobro@...>

🔗Kalle Aho <kalleaho@...>

🔗Jacques Dudon <fotosonix@...>

🔗djtrancendance@...

🔗Jacques Dudon <fotosonix@...>

🔗djtrancendance@...

🔗Jacques Dudon <fotosonix@...>

🔗daniel_anthony_stearns <daniel_anthony_stearns@...>

🔗Aaron Krister Johnson <aaron@...>

🔗djtrancendance@...

🔗daniel_anthony_stearns <daniel_anthony_stearns@...>

🔗djtrancendance@...

🔗djtrancendance@...

🔗Daniel Forro <dan.for@...>

🔗Jacques Dudon <fotosonix@...>

🔗Jacques Dudon <fotosonix@...>

🔗djtrancendance@...

🔗djtrancendance@...

🔗Kalle Aho <kalleaho@...>

🔗Daniel Forro <dan.for@...>

🔗djtrancendance@...

🔗Mark Rankin <markrankin95511@...>