Intended uses and research for (1/PHI)^x + 1, (1/octave)^x + 1 type scales

There have been so many posts in which these have been mis-interpreted that I feel this is absolutely necessary to clarify what is and is not in these scales:

There are two main formulas for these scales

A) (1/PHI)^x + 1 where x = 1 to 6 and PHI = the period and 1/1 (and >not< 1.23606 or anything else) is the root tone

B) (1/octave)^x + 1 where x = 1 to 4 and octave=2/1 and 3/2 = the period and 1/1 is the root tone

Scale B is equivalent to a scale Daniel Forro devised in 12TET and it rotates around the spiral of fifths.

Scale A is a bit more complex, but, as you can see plugging in different values of x between 1 and 6, there is no creation of 1.309 as a tone in the mode of this scale based from the root tone.
>>
Now for something a bit more complex.

You'll notice 1.618034 /
1.0557 (1.0557 is where x = 6 in my PHI scale) = 1.5326
...and...
Also that 1.618034 - 1.09017 (1.0557 is where x = 5 in my PHI scale) = 1.527864
So these "reverse" additive and multiplicative symmetries produce almost exactly the same value and either can be used as an extra note to the scale created from the (1/PHI)^x + 1 formula.

For (1/octave)^x + 1 no such convenient "double-symmetry" exists, so the extra "reverse symmetry" tone that can be added to the results from the (1/octave)^x + 1 formula is derived from 1.0625 (the value where x = 4). Here 1.5 / 1.0625 = 1.41176.

>>
Now some questions:

1) A lot of people have debated whether or not scale A) is or is not a complete series of golden sections.
I simply consider it a way of taking consecutive PHI-ths between 2/1 and 1/1 in (and only in) the
direction toward 1/1, but several people said I should be calling this splitting method (which I admittedly borrowed directly from architecture) "taking golden sections".

How should I phrase this (or not?) without pissing people off? I've heard so many different (often conflicting) suggestions on what the proper terminology is for this it's not even funny.

2) What other scales beside A, B, and the silver-section scale (using 2.414 as the generator...) what other scales can you think of using a (1/number)^x+1 type generation system and/or what papers do you know of related to research of such a construction?

Hopefully I can finally reach some sort of consensus about all of this (without pissing people off)...

-Michael

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> There have been so many posts in which these have been mis->interpreted that I feel this is absolutely necessary to clarify what >is and is not in these scales:
>
> There are two main formulas for these scales
>
> A) (1/PHI)^x + 1 where x = 1 to 6 and PHI = the period and 1/1 >(and >not< 1.23606 or anything else) is the root tone

Now you've introduced modality to your original concept of everything being consonant with everything else. If I recall correctly, it was Hermann Miller who called this kind of scale "wind chime", which I think is a great name and gets the point across immediately.

This modality isn't necessary for this scale, though, as it works according to the standards you originally set regardless of where you do or do not put a "tonic".
>
> B) (1/octave)^x + 1 where x = 1 to 4 and octave=2/1 and 3/2 = >the period and 1/1 is the root tone

>
> Scale B is equivalent to a scale Daniel Forro devised in 12TET >and it rotates around the spiral of fifths.
>
> Scale A is a bit more complex, but, as you can see plugging in >different values of x between 1 and 6, there is no creation of 1.309 >as a tone in the mode of this scale based from the root tone.

When I posted this file:

!First_Five_ Golden_Cuts_ of_Phi.scl
!
6
!
93.88597
149.46366
235.77441
366.90970
560.06656
833.09030

You said:

" Aha! You finally have it, meaning about 90% of the scale I discovered when tooling with PHI. And the only note missing vs. the scale I found (not necessarily "made") is 733 cents, which is the equivalent of 833 cents minus the interval 1.09 (IE the one note that represents the octave inverse). Congrats...finally this is a correct derivation of my scale."

That makes this:

93.88597
149.46366
235.77441
366.90970
560.06656
733.81900
833.09030

your scale, right? Unless you've changed it since then.

Now, anyone can take take this scale for themselves and play it (remember that 833.09030 is the "octave" here), and they will discover that two steps down from 1/1 the tone forms an interval of 466.x cents (1.309...) with 1/1.

So this statement:

>there is no creation of 1.309 >as a tone in the mode of this scale >based from the root tone.

is simply, demonstratably, incorrect.

Also, all that stuff about avoiding critical band conflicts and intervals smaller than so and so simply goes out the window as soon as you extend the scale beyond one "phiave". Which doesn't matter, as it's a bunch of boloney anyway in this case, for what gives the tuning its "wind chime" quality is also the fuzzy overlapping of harmonic partials.

>
> >>
> Now some questions:
>
> 1) A lot of people have debated whether or not scale A) is or is >not a complete series of golden sections.

Not a single person here has debated such a thing, for obviously a series of golden sections can be carried on indefinitely. The only statement made about this, if I recall correctly, was to the completely opposite effect, when I pointed out that you had simply taken a few sections then stopped and called it a scale.

> I simply consider it a way of taking consecutive PHI-ths between >2/1 and 1/1 in (and only in) the
> direction toward 1/1, but several people said I should be calling >this splitting method (which I admittedly borrowed directly from >architecture) "taking golden sections".

One person called it taking successive golden sections.
>
> How should I phrase this (or not?) without pissing people >off? >I've heard so many different (often conflicting) suggestions >on what the proper terminology is for this it's not even funny.

There is no set "proper terminology", you could call it "Walloon ensquadultion" if you wanted to. The only real requirement for being "proper" is that others understand what you are talking about.

>
> 2) What other scales beside A, B, and the silver-section scale >(using 2.414 as the generator...) what other scales can you think >of using a (1/number)^x+1 type generation system and/or what papers >do you know of related to research of such a construction?

Sorry but this statement reveals that either you simply do not know what you are talking about, or seriously misrepresent what you're thinking when you write it down.

Start with the Means of ancient Greece, and go from there.

> Hopefully I can finally reach some sort of consensus about all of >this (without pissing people off)...
>

Noone is pissed off about anything. But I've had enough of trying to do damage control on your posts (I say this without rancour).

>"This modality isn't necessary for this scale, though, as it works
according to the standards you originally set regardless of where you
do or do not put a "tonic"."

   I'm a bit confused here.  The way the scale is constructed has nothing to do with where the tonic is.  I was simply enforcing the idea of a single mode starting at 1/1 to make it obvious, for example, that the way the scale was generated is not related a different mode like 1.23606 1.38 1.5326 1.618 1.708....(starting where root 1.23606)....

Cameron>
"You said:
 
'Aha! You finally have it, meaning about 90% of the scale I discovered
when tooling with PHI. And the only note missing vs. the scale I found
(not necessarily "made")'
"
     And I meant it...in that case that really was the intended scale without anything unintended thrown on top. :-)
  Note also: that posting of your scale, like my own, correctly did not contain 466 cents (AKA 1.309).  Which was correct...and made me even more surprised when both you a Jacques bashes me over the head about how the scale really did have 466 cents and I was somehow denying it.  Maybe he thinks it should but, if he adds it, that would no longer be my intended scale.  See the difference?

>"Also, all that stuff about avoiding critical band conflicts and
intervals smaller than so and so simply goes out the window as soon as
you extend the scale beyond one "phiave"."
   In can believe that a good few of the higher harmonics and tones don't align: after all this certainly is not JI.  However, as you said, the result seems more fuzzy/"assonant" than it does blatantly clashing...and when I said critical band I was referring to the first phi-tave or so where the partials are the loudest.

>"The only statement made about this, if I recall correctly, was to the
completely opposite effect, when I pointed out that you had simply
taken a few sections then stopped and called it a scale."

  Ah ok, I think I get it.  I recall you said something about there being a few consecutive sections going in one direction (toward 1 and away from 2).  Taking a large number of sections where any two points in any direction are legal would result in something like the scale 15+ tone scale Jacques mentioned before...which is a completely different beast than the one I intended despite also including my scale within it.

>"The only real requirement for being "proper" is that others understand what you are talking about."

  Well does "golden sections of PHI between 2 and 1 going in one direction from 2 to 1 stopping at 1.0557" make a fair explanation...or is something still missing?

Me>"what other scales can you think >of using a (1/number)^x+ 1 type generation system "
  I still don't get what you think is so alien or badly explained there.

The basic form is result = (1/var)^x+ 1
Some examples
result =(1/PHI)^x+ 1
result =(1/2)^x + 1
result =(1/2.414)^x + 1
......
   You talked about the Ten Means of Ancient Greece IE
http://www.mathpages.com/home/kmath462.htm

  And I know the arithmetic mean of 2 and 1 is 1.5 and the geometric means of 2 and 1 is sqrt(2*1) = 1.414.  So this does give me two of the values for the result where x = 1 in my equations above (for example (1/2.414)^x + 1 = 1.414).

   Meanwhile the harmonic mean for 2 and 1 gives (2(1)(2)/(1+2)) = 4/3).  So where x = 4/3  (1/q)^1 + 1 = 4/3 so 1/q = 1/3 and q = 3 thus giving the potential equation result =(1/3)^x + 1 giving the rather tiny scale 1.3333, 1.1111.  Which, if using 1.3333 as the period, can be extended to 1.1111,1.3333, (1.33*1.11= 1.4814), 1.777777, 1.9753

  Is this the kind of thing you were aiming toward?

-Michael