Yahoo Tuning Groups Ultimate Backup tuning PHI interval tuning (for Michael S)

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> Hi Michael,
>
> I don't know if this is old news but I was thinking of how to resolve PHI with octave equivalence and came up with the following:
>
> 2^x = 1.618 (close enough)
> x = log1.618/log2 = 0.6942...which is close to 25/36 (= 0.694444)
> i.e. 2^(25/36) = 1.61826115, close enough for jazz.
> Given that the ear can't determine past a few decimal places, this has all of the usual properties of PHI: its inverse is 2^(-25/36)= 0.61794..i.e. equal to itself minus 1, and squaring gives (2^(25/36))^2 = 2.61876..., itself plus 1.
>
> But the advantage of this approach might be that it implies a 36 tet system, base intervals 2^(1/36). Since 2^(3/36) = 2^(1/12),the conventional 12 tet semitone, then nothing is lost and perhaps something is gained.
>
> I'd be interested to know how this sounds. Hope it is helpful,
>
> cheers
>
> Rick
>
PS: Since this sets up a modulus mod(36), then like all mods, odd numbered intervals which do not divide into the mod can be seen as generators of the entire scale. Hence, what we are calling PHI 25/36 can be applied repeatedly to produce the MOS at 36 x 25 = 900 (i.e. 50/36 = 1.3888... and 0.3888...x 36 = 14, 75/36 = 2.08333... and 0.08333...x 36 = 3 etc until 900 gives 1,2,3,...35,0.)

Also, its 8ve inverse 36 - 25 = 11 gives 2^(11/36)= 1.235894466 which is closer to a (slightly) flat major 3 than a sharp minor 3. But I can't (as yet) find any interesting properties of this number like phi.

On the other hand, this might have promise: squaring now means multiplying 25 by 2, then 2 again etc...i.e. 2^(50/36) = 2.618..., 2^(100/36) = 6.8579... to 200, 400 and so on. But the mods of these are 14/36 = 7/18, 7/9, 5/9, 1/9, 2/9, 4/9, 8/9, etc which seems to give some type of Phi based scale, weird since they all seem to have 9 in the denominator or multiples of.

🔗Michael Sheiman <djtrancendance@...>

Rick> "But the advantage of this approach might be that it implies a 36-tet system, base intervals 2^(1/36)."

This is definitely an interesting way of looking at resetting a PHI scale to match the 2/1 octave period (if I have absorbed what you said correctly).

> "Hence, what we are calling PHI 25/36 can be applied repeatedly to
produce the MOS at 36 x 25 = 900 (i.e. 50/36 = 1.3888... and 0.3888...x
36 = 14, 75/36 = 2.08333... and 0.08333...x 36 = 3 etc until 900 gives
1,2,3,...35, 0.)"
Interesting, I am still trying to grasp how you got from 50 to 75 (since 70+14 = 74).
But, if I have it right, the way you are creating your scale is
2^(x/36)...but what confuses me is how you figure out which values to use for x (realizing that crowding too many values too close together will cause terrible critical band dissonance).

The only way to know for sure what I think of this system, of course, is to test it by ears. Although I will say, I am very impressed you managed to hit both the 1.618034 and 0.618034 "PHI-taves" very close to perfectly with this system...and could possibly help hit the 2/1 octave and "2 and a 5th" tri-tave almost dead on thus providing some extra clean sounding JI-style interval.

So what values would/should I use for x? Once I know this I can generate the scale and try it... :-)

-Michael

--- On Wed, 4/22/09, rick_ballan <rick_ballan@...> wrote:

From: rick_ballan <rick_ballan@...>
Subject: [tuning] Re: PHI interval tuning (for Michael S)
To: tuning@yahoogroups.com
Date: Wednesday, April 22, 2009, 8:55 AM

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

> Hi Michael,

> I don't know if this is old news but I was thinking of how to resolve PHI with octave equivalence and came up with the following:

> 2^x = 1.618 (close enough)

> x = log1.618/log2 = 0.6942...which is close to 25/36 (= 0.694444)

> i.e. 2^(25/36) = 1.61826115, close enough for jazz.

> Given that the ear can't determine past a few decimal places, this has all of the usual properties of PHI: its inverse is 2^(-25/36)= 0.61794..i.e. equal to itself minus 1, and squaring gives (2^(25/36))^ 2 = 2.61876..., itself plus 1.

> But the advantage of this approach might be that it implies a 36 tet system, base intervals 2^(1/36). Since 2^(3/36) = 2^(1/12),the conventional 12 tet semitone, then nothing is lost and perhaps something is gained.

> I'd be interested to know how this sounds. Hope it is helpful,

> cheers

> Rick

PS: Since this sets up a modulus mod(36), then like all mods, odd numbered intervals which do not divide into the mod can be seen as generators of the entire scale. Hence, what we are calling PHI 25/36 can be applied repeatedly to produce the MOS at 36 x 25 = 900 (i.e. 50/36 = 1.3888... and 0.3888...x 36 = 14, 75/36 = 2.08333... and 0.08333...x 36 = 3 etc until 900 gives 1,2,3,...35, 0.)

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> Rick> "But the advantage of this approach might be that it implies a 36-tet system, base intervals 2^(1/36)."
>
> This is definitely an interesting way of looking at resetting a PHI scale to match the 2/1 octave period (if I have absorbed what you said correctly).
>
> > "Hence, what we are calling PHI 25/36 can be applied repeatedly to
> produce the MOS at 36 x 25 = 900 (i.e. 50/36 = 1.3888... and 0.3888...x
> 36 = 14, 75/36 = 2.08333... and 0.08333...x 36 = 3 etc until 900 gives
> 1,2,3,...35, 0.)"
> Interesting, I am still trying to grasp how you got from 50 to 75 (since 70+14 = 74).
> But, if I have it right, the way you are creating your scale is
> 2^(x/36)...but what confuses me is how you figure out which values to use for x (realizing that crowding too many values too close together will cause terrible critical band dissonance).
>
Hi Mike,

Yeah it can get pretty confusing sometimes. Ok, what you need to look at (or probably know already but are a bit rusty), is modulus and indices maths. Under the standard 12 edo system the intervals are 2^(x/12) where x = ,0,1,2,3,...11 and the modulus maths comes from the usual math'l properties of indices found in calculus books. For eg, since multiplying or dividing intervals is the same as adding or subtracting the indices, then 2^(x/12) x 2^(y/12) = 2^((x+y)/12) etc...The mod numbers x = 0,1,2,...11 are simply an abbreviation of the indices, which we add and subtract. Other basic info is that 2^(0/12) = 2^0 = 1, our base frequency, 2^(12/12) = 2^1 = 2 the 8ve, and both are set equal to 0 in mod(12). The inverse is 1/[2^(x/12)] = 2^-(x/12) which gives you the same number of semitones down, or it can be obtained by subtracting from the number from 12 eg; 4(the maj3rd), is inverse to 8 (min6th) and vice-versa. For values x greater than 12 (our mod), we divide by that mod to get a number N+r where N is whole and r is the remainder, and x = r x 12 (mod 12). eg, x=14, x/12 = 1.16666...and 12 x 0.16666...= 2, so that 14 = 2 (mod 12). I think that's about it really.

So by reasoning backwards, what I found was that setting up mod 36 (which btw includes mod 12 since 3x12=36) then the interval 2^(25/36) gives a very good phi. Its inverse is 2^-(25/36)= 0.61797, squaring gives (2^(25/36))^2 = 2^(50/36) = 2.618. (Note, 2^(11/36)is an 8ve above the inverse i.e. dividing by 2 gives 2^(11/36).2^-1 = 2^-(25/36).) So in answer to your questions, first, x = 0,1,2,3,...a whole number and second, I got 75 because it is a multiple of 25. To bring it back into one octave we have 75/36 = 2.083333 and 0.083333 x 36 = 3. Oh and regarding what you say below, it does hit the 8ve exactly, while the fifth is the usual tempered one in 12 edo.
I suppose the most logical first test would be taking the intervals at both sides of 1, that is 2^-(25/36):1:2^(25/36). Love to hear it so don't forget to send it to me.

Cheers

Rick

> The only way to know for sure what I think of this system, of course, is to test it by ears. Although I will say, I am very impressed you managed to hit both the 1.618034 and 0.618034 "PHI-taves" very close to perfectly with this system...and could possibly help hit the 2/1 octave and "2 and a 5th" tri-tave almost dead on thus providing some extra clean sounding JI-style interval.
>
> So what values would/should I use for x? Once I know this I can generate the scale and try it... :-)
>
> -Michael
>
>
> --- On Wed, 4/22/09, rick_ballan <rick_ballan@...> wrote:
>
> From: rick_ballan <rick_ballan@...>
> Subject: [tuning] Re: PHI interval tuning (for Michael S)
> To: tuning@yahoogroups.com
> Date: Wednesday, April 22, 2009, 8:55 AM
>
>
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> --- In tuning@yahoogroups. com, "rick_ballan" <rick_ballan@ ...> wrote:
>
> >
>
> > Hi Michael,
>
> >
>
> > I don't know if this is old news but I was thinking of how to resolve PHI with octave equivalence and came up with the following:
>
> >
>
> > 2^x = 1.618 (close enough)
>
> > x = log1.618/log2 = 0.6942...which is close to 25/36 (= 0.694444)
>
> > i.e. 2^(25/36) = 1.61826115, close enough for jazz.
>
> > Given that the ear can't determine past a few decimal places, this has all of the usual properties of PHI: its inverse is 2^(-25/36)= 0.61794..i.e. equal to itself minus 1, and squaring gives (2^(25/36))^ 2 = 2.61876..., itself plus 1.
>
> >
>
> > But the advantage of this approach might be that it implies a 36 tet system, base intervals 2^(1/36). Since 2^(3/36) = 2^(1/12),the conventional 12 tet semitone, then nothing is lost and perhaps something is gained.
>
> >
>
> > I'd be interested to know how this sounds. Hope it is helpful,
>
> >
>
> > cheers
>
> >
>
> > Rick
>
> >
>
> PS: Since this sets up a modulus mod(36), then like all mods, odd numbered intervals which do not divide into the mod can be seen as generators of the entire scale. Hence, what we are calling PHI 25/36 can be applied repeatedly to produce the MOS at 36 x 25 = 900 (i.e. 50/36 = 1.3888... and 0.3888...x 36 = 14, 75/36 = 2.08333... and 0.08333...x 36 = 3 etc until 900 gives 1,2,3,...35, 0.)
>
>
>
> Also, its 8ve inverse 36 - 25 = 11 gives 2^(11/36)= 1.235894466 which is closer to a (slightly) flat major 3 than a sharp minor 3. But I can't (as yet) find any interesting properties of this number like phi.
>
>
>
> On the other hand, this might have promise: squaring now means multiplying 25 by 2, then 2 again etc...i.e. 2^(50/36) = 2.618..., 2^(100/36) = 6.8579... to 200, 400 and so on. But the mods of these are 14/36 = 7/18, 7/9, 5/9, 1/9, 2/9, 4/9, 8/9, etc which seems to give some type of Phi based scale, weird since they all seem to have 9 in the denominator or multiples of.
>

🔗djtrancendance@...

Rick> "For values x greater than 12 (our mod), we divide by that mod to get a

number N+r where N is whole and r is the remainder"
.....
"To bring it back into one octave we have 75/36 = 2.083333 and 0.083333 x 36 = 3."
****************************************************
So 75 is the "value" and, since it is greater than 12, you divide it by 36 (the mod) to get 2.083333 where 2 is the whole and 0.083333 is the remainder...as I understand it...and then multiply by the mod to get 3 (still not quite so sure what to do with the 3 once I obtain it).

Regardless...2^(-25/36) and 2^(25/36) sound fine. Your suggestion of 2^(21/36) does indeed make a very normal sounding fifth (virtually the same as in 12TET). Also 2^(11/36) AKA 1.23589 sounds very very good between 1 and PHI as a
triad. And, of course, 2^(36/36) is the standard octave.

I am still not so sure how you figure out which number to use to get the other notes (now that I've used 11/36,21/36,25/36, and 36/36)...and a 4 note per octave scale seems a bit shallow).

However, I have found 2^(20/36) AKA 1.46973 and 2^(7/36) AKA 1.144283 and 1.38724 AKA 2^(17/36) (notes derived from estimated of notes found in my own experiments) can combine with your notes to form probably the best sounding PHI scale I've heard yet:

1
1.144283 2^(7/36)
1.23589 2^(11/36)
1.38724 2^(17/36)
1.46973 2^(20/36)
1.61826 2^(25/36)

This is a very convenient coincidence because my latest PHI scale turned out to be:

1
1.1459
1.23606
1.3819
1.47214
1.618034
(so numerically close it's almost scary :-) )

Note that (as with your scale) that when you take 1.23606 times 1.618034 (the "phi-tave")...it "magically" brings you directly to the 2/1 octave!

Note
1.47214 is a bit "off" the 12TET fifth but, to my ears, it fits into the other notes in the scale (aside from the root tone) a bit better.
*******************************************************
I derived my PHI scale simply by taking the space between 0.618034 and 0...and dividing it by multiplying 0.618034 by itself...and then repeating the process until I hit 1.1459.
That gave me
0.618 (*0.618034 = 0.3819)
0.3819 (*0.618034 = 0.23606)
0.23606 (*0.618034 = 0.145898)
0.145898

I was eager to find in an extra note (the make it an 8-note per 2/1 octave scale) so I took 0.618034 - 0.1458 to "split the area in the opposite direction". This gave me the extra 0.47214. Next I figured I was taking a line between 0 and 0.618034 so, to "push it back into place", I "translated" everything back by adding 1 (making it the equivalent of splitting between 1 and 1.618034).

**************************************************************

Correct me if I am wrong, but it looks like we could be running into an
intersection here...and may indeed be able to introduce a PHI scale that both preserves the octave and fits just about perfectly into 36TET. :-)

-Michael

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> Rick> "For values x greater than 12 (our mod), we divide by that mod to get a
>
> number N+r where N is whole and r is the remainder"
> .....
> "To bring it back into one octave we have 75/36 = 2.083333 and 0.083333 x 36 = 3."
> ****************************************************
> So 75 is the "value" and, since it is greater than 12, you divide it by 36 (the mod) to get 2.083333 where 2 is the whole and 0.083333 is the remainder...as I understand it...and then multiply by the mod to get 3 (still not quite so sure what to do with the 3 once I obtain it).
>
>Ok, my fault Michael, I should have been clearer on this. In the first paragraph I'm just using the familiar 12 edo i.e. mod 12, to summarize the usual maths. But the tuning system I am now proposing is a 36 edo and therefore a 36 mod. So your sentence above should read "So 75 is the "value" and, since it is greater than 36 etc...", not "greater than 12". Everything else you did is correct. So we now have 36 notes equally distributed within the octave, where 2^(1/36) is our new smallest interval(it is now called "1", whereas in mod 12 the semitone is called 1). We name the notes 0,1,2,3,...35 and then start over again. Since 2^(3/36) = 2^(1/12) we see that we reach a conventional semitone every third note i.e. the semitone is now called "3"; the 12 edo system is now 0,3,6,9,...33. This system therefore includes the 12 edo system and ipso facto all of the usual harmony if we wish. In other words, it is not necessarily just a Phi scale. Nevertheless, we CAN arrive at this semitone "3" via Phi also. Since 25 is our new phi, then 50 is phi squared, 75 phi cubed, and so on. But because 50 and 75 exceed 36, they are replaced by 14 and 3 respectively.

> Regardless...2^(-25/36) and 2^(25/36) sound fine. Your suggestion of 2^(21/36) does indeed make a very normal sounding fifth (virtually the same as in 12TET). Also 2^(11/36) AKA 1.23589 sounds very very good between 1 and PHI as a
> triad. And, of course, 2^(36/36) is the standard octave.

In light of what I just said you can now see that (21/36) = (7/12) and is exactly the same interval as in 12 tet i.e. 21 = 3 x 7. And both 11 and 25 are the two generating phi intervals so interesting to hear they sound good (love to hear them together and successively).
>
> I am still not so sure how you figure out which number to use to get the other notes (now that I've used 11/36,21/36,25/36, and 36/36)...and a 4 note per octave scale seems a bit shallow).

So this should be clear now?
>
> However, I have found 2^(20/36) AKA 1.46973 and 2^(7/36) AKA 1.144283 and 1.38724 AKA 2^(17/36) (notes derived from estimated of notes found in my own experiments) can combine with your notes to form probably the best sounding PHI scale I've heard yet:
>
> 1
> 1.144283 2^(7/36)
> 1.23589 2^(11/36)
> 1.38724 2^(17/36)
> 1.46973 2^(20/36)
> 1.61826 2^(25/36)
>
> This is a very convenient coincidence because my latest PHI scale turned out to be:
>
> 1
> 1.1459
> 1.23606
> 1.3819
> 1.47214
> 1.618034
> (so numerically close it's almost scary :-) )

Yep, love that about maths. I love to see unexpected things working once the groundwork is in place or to arrive at the same conclusions from (seemingly) unrelated points of view. After working with this maths for years my ears pricked up the minute I saw a number close to 36 appear in the denominator because I knew that everything would then lock into place.
>
> Note that (as with your scale) that when you take 1.23606 times 1.618034 (the "phi-tave")...it "magically" brings you directly to the 2/1 octave!

Ah but this is no accident. This is the definition of a number and its inverse. From 25, its inverse is 36 - 25 = 11 so that 25 + 11 = 36 (the 8ve). (And remember that adding indices is the same as multiplying, subtracting same as dividing). By slightly detuning phi to within very negligible boundaries, we preserve the 8ve.
>
> Note
> 1.47214 is a bit "off" the 12TET fifth but, to my ears, it fits into the other notes in the scale (aside from the root tone) a bit better.

Yes 20 is only 1 away from the usual fifth 21. I can't test it (haven't got scalar) but usual fifth 21 is closer to just fifth 5/4 or 81/64 than 20. But if you're coming from phi, that should be a whole different ball game!(major chords can sound naff in serial music).
> *******************************************************
> I derived my PHI scale simply by taking the space between 0.618034 and 0...and dividing it by multiplying 0.618034 by itself...and then repeating the process until I hit 1.1459.
> That gave me
> 0.618 (*0.618034 = 0.3819)
> 0.3819 (*0.618034 = 0.23606)
> 0.23606 (*0.618034 = 0.145898)
> 0.145898
>
> I was eager to find in an extra note (the make it an 8-note per 2/1 octave scale) so I took 0.618034 - 0.1458 to "split the area in the opposite direction". This gave me the extra 0.47214. Next I figured I was taking a line between 0 and 0.618034 so, to "push it back into place", I "translated" everything back by adding 1 (making it the equivalent of splitting between 1 and 1.618034).

Good reasoning. If you translate this into the tuning it should get easier now.
>
> **************************************************************
>
> Correct me if I am wrong, but it looks like we could be running into an
> intersection here...and may indeed be able to introduce a PHI scale that both preserves the octave and fits just about perfectly into 36TET. :-)

>
> -Michael
>
You're absolutely right. Not just introduces phi and preserves the octave but the whole of 12 tone harmony as well. And who knows what other types of harmonies we can get now that we seem to have three minor thirds, three majors etc (or are they now minor or major??). And don't forget to post some phi things so I can hear them. Thanks

-Rick

🔗djtrancendance@...

Far as sound examples of noble-number generated scales, check this
out (made in a scale generated from the Silver Ratio using the same
scale method I used for PHI):

http://www.geocities.com/djtrancendance/PHI/silverrain.mp3

It sounds like we're both on to something...a sort of intersection
of a true PHI scale, 36TET, and the ability to generate a PHI scale
that actually preserves traditional harmony pretty well without the
use of traditional intervals.

Between my own experiments and your findings, I considered it worth
a shot to try and translate the silver ratio of 2.414 (where
1 / 2.414 =0.414) and multiply 0.414 by itself and add 1 to all of
the results to create the scale
1.07107
1.1715
1.24257
1.34314
1.414
(period / "silver-tave")

This results in a full 10 notes per 2/1 octave and, best yet,
virtually all combinations are very harmonically use-able. Though
I must warn
A) The vibe is a bit dark-sounding
B) You may want to round the period from 1.414 to about 1.406 to
alleviate conflicts in upper-range overtones of the harmonic series
by bringing the period closer to 7/5 IE 1.4. This will also help
greatly to solve the one major outstanding problem with the above
scale: playing notes exactly 1 period above each other beat a bit
too much to sound smooth.

BTW, if you could find a good denominator for 1.414 or 1.406....
I would be interested to see if you could fit the Silver Ratio scale
well into a TET-type construction (just as you did quite well for
my newest PHI scale with 36TET for "x/36")

The result of "what the silver ratio scale would sound like if
this scale was used in composition" are above...and I'm admittedly
not even that good a composer and I still can't get over the sound
of it.

Rick wrote:
>"Ah but this is no accident. This is the definition of a
>number and its inverse. From 25, its inverse is 36 - 25 = 11
>so that 25 + 11 = 36 (the 8ve)."

Come to think of it, exactly, the exponentials "add up". In
additional, what seems to make noble numbers that make good scale
generators unique is that they satisfy
x = 1/x + any whole number... and the resulting scales can be
described as both additively and exponentially symmetrical.

>"Since 25 is our new phi, then 50 is phi squared, 75 phi cubed,
>and so on. But because 50 and 75 exceed 36, they are replaced by
>14 and 3 respectively."

Ah, now I get it...you are reducing exponentials of PHI to fit
within a single octave. So you would get 25,14,3,(75-(36*2)),
28 (100-(36*2)), etc.

And, of course, you're right that 12TET is encapsulated perfectly
as every third note in 36TET. Although our methods of PHI scales
now differ (I take (1/PHI)^x) + 1 to get my scale where you take a
form of PHI^x (which, actually, is a lot like my old method of
making the scale). Regardless...both the "new and old" ways of
making the PHI scale, along with 12TET, fit into 36TET...and one
good question may be "what kind of combinations may be able to
combined 12TET harmonies with PHI-based tones as 'blue notes'"?

>"And who knows what other types of harmonies we can get now that
>we seem to have three minor thirds, three majors etc (or are they
>now minor or major??)."

Exactly! The thing I have noticed with noble scales in general is
instead of just trying to optimize the situation
A) the overtones/harmonics of an instrument intersect with notes
higher in the scale (which obviously, produces no beating as the
two tones are at the same frequency)...they also get to take
advantage of
B) the overtones DON'T intersect at all at some parts but, in
those parts, are at least a semi-tone or so away from where they
would intersect...just far enough away to avoid extremities in
beating altogether on both sides of the "perfect intersection".

-Michael

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> Far as sound examples of noble-number generated scales, check this
> out (made in a scale generated from the Silver Ratio using the same
> scale method I used for PHI):
>
> http://www.geocities.com/djtrancendance/PHI/silverrain.mp3
>
> It sounds like we're both on to something...a sort of intersection
> of a true PHI scale, 36TET, and the ability to generate a PHI scale
> that actually preserves traditional harmony pretty well without the
> use of traditional intervals.

Hi Michael,

Well for starters 1.414 already is in both 12 and 36 tet. It is sqrt2, 2^(6/12) = 2^(18/36) = 2^(1/2) = 1.414. While 7/4 gives a rough estimate, 45/32 = 1.40625 while 181/128 = 1.414062. The use of the "devil's interval" would probably explain why it sounds dark. 2.414 comes between 45 and 46 in the 36 system i.e. two of the minor thirds 9 and 10 taken up the octave (+36) so I don't think the silver fits into the system any where near as well as the phi. (incidentally, I noticed that 8 gives a very good approx to 7/6). Besides, I think we're getting into the arena of intervals so fine that it would be hard to hear the difference. I still would like to hear the phi intervals -25,0,25 and 0,11,25 if you've got a moment. Thanks mate

-Rick
>
> Between my own experiments and your findings, I considered it worth
> a shot to try and translate the silver ratio of 2.414 (where
> 1 / 2.414 =0.414) and multiply 0.414 by itself and add 1 to all of
> the results to create the scale
> 1.07107
> 1.1715
> 1.24257
> 1.34314
> 1.414
> (period / "silver-tave")
>
> This results in a full 10 notes per 2/1 octave and, best yet,
> virtually all combinations are very harmonically use-able. Though
> I must warn
> A) The vibe is a bit dark-sounding
> B) You may want to round the period from 1.414 to about 1.406 to
> alleviate conflicts in upper-range overtones of the harmonic series
> by bringing the period closer to 7/5 IE 1.4. This will also help
> greatly to solve the one major outstanding problem with the above
> scale: playing notes exactly 1 period above each other beat a bit
> too much to sound smooth.
>
> BTW, if you could find a good denominator for 1.414 or 1.406....
> I would be interested to see if you could fit the Silver Ratio scale
> well into a TET-type construction (just as you did quite well for
> my newest PHI scale with 36TET for "x/36")
>
> The result of "what the silver ratio scale would sound like if
> this scale was used in composition" are above...and I'm admittedly
> not even that good a composer and I still can't get over the sound
> of it.
>
> Rick wrote:
> >"Ah but this is no accident. This is the definition of a
> >number and its inverse. From 25, its inverse is 36 - 25 = 11
> >so that 25 + 11 = 36 (the 8ve)."
>
> Come to think of it, exactly, the exponentials "add up". In
> additional, what seems to make noble numbers that make good scale
> generators unique is that they satisfy
> x = 1/x + any whole number... and the resulting scales can be
> described as both additively and exponentially symmetrical.
>
> >"Since 25 is our new phi, then 50 is phi squared, 75 phi cubed,
> >and so on. But because 50 and 75 exceed 36, they are replaced by
> >14 and 3 respectively."
>
> Ah, now I get it...you are reducing exponentials of PHI to fit
> within a single octave. So you would get 25,14,3,(75-(36*2)),
> 28 (100-(36*2)), etc.
>
> And, of course, you're right that 12TET is encapsulated perfectly
> as every third note in 36TET. Although our methods of PHI scales
> now differ (I take (1/PHI)^x) + 1 to get my scale where you take a
> form of PHI^x (which, actually, is a lot like my old method of
> making the scale). Regardless...both the "new and old" ways of
> making the PHI scale, along with 12TET, fit into 36TET...and one
> good question may be "what kind of combinations may be able to
> combined 12TET harmonies with PHI-based tones as 'blue notes'"?
>
> >"And who knows what other types of harmonies we can get now that
> >we seem to have three minor thirds, three majors etc (or are they
> >now minor or major??)."
>
> Exactly! The thing I have noticed with noble scales in general is
> instead of just trying to optimize the situation
> A) the overtones/harmonics of an instrument intersect with notes
> higher in the scale (which obviously, produces no beating as the
> two tones are at the same frequency)...they also get to take
> advantage of
> B) the overtones DON'T intersect at all at some parts but, in
> those parts, are at least a semi-tone or so away from where they
> would intersect...just far enough away to avoid extremities in
> beating altogether on both sides of the "perfect intersection".
>
> -Michael
>

🔗Herman Miller <hmiller@...>

🔗djtrancendance@...

Rick
> Besides, I think we're getting into the arena of intervals so
> fine that it would be hard to hear the difference.

Indeed...I agree with you we're aiming perhaps for a degree
of accuracy so small the ear would not be able to tell anyhow.
Case closed on that one...I figure anything within about 5 cents
of "perfect" accuracy should be good enough an estimate from here
on in. :-)

>I still would like to hear the phi intervals -25,0,25 and
>0,11,25 if you've got a moment. Thanks mate."

This is actually a great "back to basics" test for PHI fractal
intervals (using Jacques' terminology now), thank you for thinking
of it.

Anyhow, here you go:
http://www.geocities.com/djtrancendance/PHI/neg25_0_25_and_0_11_25.mp3

This is an example of the chords
2^(-25/36),2^(0/36),2^(25/36)
..followed by....
2^(0/36),2^(11/36),2^(25/36)

You know what's especially freaky? You really have to listen
for the difference between these two chords to tell they have even
changed at all...they seem to point VERY strongly to the same
"virtual pitch"!

So, in short, this is very good news: maybe your past idea
of a PHI-equivalent of the harmonic series (and perhaps other
equivalents) isn't so far fetched after all. :-)

-Michael

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Michael Sheiman <djtrancendance@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> Rick
> > Besides, I think we're getting into the arena of intervals so
> > fine that it would be hard to hear the difference.
>
> Indeed...I agree with you we're aiming perhaps for a degree
> of accuracy so small the ear would not be able to tell anyhow.
> Case closed on that one...I figure anything within about 5 cents
> of "perfect" accuracy should be good enough an estimate from here
> on in. :-)
>
> >I still would like to hear the phi intervals -25,0,25 and
> >0,11,25 if you've got a moment. Thanks mate."
>
> This is actually a great "back to basics" test for PHI fractal
> intervals (using Jacques' terminology now), thank you for thinking
> of it.
>
> Anyhow, here you go:
> http://www.geocities.com/djtrancendance/PHI/neg25_0_25_and_0_11_25.mp3
>
> This is an example of the chords
> 2^(-25/36),2^(0/36),2^(25/36)
> ..followed by....
> 2^(0/36),2^(11/36),2^(25/36)
>
> You know what's especially freaky? You really have to listen
> for the difference between these two chords to tell they have even
> changed at all...they seem to point VERY strongly to the same
> "virtual pitch"!
>
> So, in short, this is very good news: maybe your past idea
> of a PHI-equivalent of the harmonic series (and perhaps other
> equivalents) isn't so far fetched after all. :-)
>
> -Michael
>
Wow, you can really hear two distinct beat frequencies, first a high one and then, if you listen a little closer, a low one. I suppose this is because of the nature of phi constantly reinforcing the original frequencies. And yes, both eg's seem to give the same beats and sound very similar. They also seem to sound better after the initial attack of the notes (probably due to extraneous upper harmonics?). As for the virtual pitch, if you mean the GCD, the closest ratio I could find to 25 was 1657/1024 which is so high it doesn't seem very helpful, but it's early days yet. (And these posts are coming to me at 2am these days so I'm usually tired).

You know I've been thinking that there's stacks we can do with a 36 mod, anything from standard jazz harmonies (or new variations of) to completely phi based compositions to 36 tone rows (even from a probability perspective there are 36! i.e. 36 factorial, possible combinations which reaches into the billions). And of course since both 11 and 25 are odd then they can be seen as generators of all 36 notes. Looking at the intervals successively we have 11 - 0 = 11 and 25 - 11 = 14, suggests the symmetrical chord 0:11:14:25 (which you might want to test). And naturally we can then treat 11 and 25 as our new 0 giving 25:0:3:14 (i.e. Transpose 11 or T11 meaning subtract 11, where 0 = 36) and 11:22:25:0 (T25) i.e. we take phi from phi, and play it altogether 0:3:11:14:22:25. But then 8 appears which we can then add, and so on. (But the problem is that being generators we end up with all 36 notes, so where to stop?? Are they ordered notes?).

At any rate you might want to check out a great book on 12 tone serial method (mod 12 maths) by the composer Charles Wuorenin called 'simple composition' (though there's nothing simple about it) and extend its techniques to mod 36. You'll start to see parallels b/w this 36 maths and phi such as inverses etc...

Thanks for sending the examples. I'll keep on listening to them.

Rick

🔗djtrancendance@...

Rick wrote:
>"And yes, both eg's seem to give the same beats and sound very similar.
"
Indeed...

>They also seem to sound better after the initial attack of the notes
(probably due to >extraneous upper harmonics?)."
I think the most likely reason for the problem with the initial attack is that the overtones are spaced very close to the point of dissonance and the added overtones in the attack make them flow over the limit for a second and collide with each other. But, yes, I definitely agree it has to do with the extra harmonics/overtones added in during the attack.

>"As for the virtual pitch, if you mean the GCD, the closest ratio I
could find to 25 was" >"1657/1024 which is so high it doesn't seem very
helpful, but it's early days yet."
Technically, you're right, it is, of course, the actual GCD between the two chords is far from in common. But, as you said before the beating between the two sounds very similar. My point is that...even though the a simple GCD does not occur...the mind/ear in many ways can't tell the difference between the "aligned beating" of the PHI scale and a true GCD; both seem to have the same effect of pointing the a similar root tone.

>"But then 8 appears which we can then add, and so on. (But the problem
is that being" >"generators we end up with all 36 notes, so where to
stop?? Are they ordered notes?)."

This is a fundamental issue I have with using PHI^x and particularly many PHI scales other than my own...they use PHI as an infinite generator that goes 'in circles' until it hits just about everything. Which is a bad idea for consonance if you go to far...of course.
The only way I've figured out to make every note within a period truly a power of PHI is to use 1 + (1/PHI^x) as the generator. This gives your (2^) 0,11,25 (out of 36) chord plus the additional notes 16 (about 1.38), 20 ,and 7...and makes the gaps between any two successive notes equal to 1/PHI.

>0:3:11:14:22:25.
This is almost precisely what my old PHI^x/2^y scale gave...in fact 14 is almost exactly phi^2/2. I still am pretty convinced the new theory above sounds much more consonant, though...of course I am going to give sound examples
later IE in my next message and when I have the time to make them of your chord and the nearest equivalent in my new "1/PHI" system.
Note that...in general, I have found any two note closer than about the 1.055 interval begin to beat wildly...but the good news is between 0 and 3...and 11 and 14 in your chord are just wide enough to avoid this beating problem, at least for the root tone (though not so sure about the overtone which is why I'll have to try a sound example). :-)

>"suggests the symmetrical chord 0:11:14:25 (which you might want to
test). And naturally" >"we can then treat 11 and 25 as our new 0 giving
25:0:3:14"
Ah, ok...I see how you are using inversions of the chord 0:11:14:25 to build your '36TET PHI scale'...you are changing the 'root' the the second note and taking intervals from there...very clear explanation the time around. :-)

>"great book on 12 tone serial method (mod 12 maths) by the composer Charles Wuorenin" >"called 'simple composition'"
Yes I will definitely try that book...I realize that the use of inverses to create scales is a method I have not tried and, agreed, if he does in with "mod 12/12TET"...the techniques should be adaptable to fit mod 36.

-Michael

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> --- In tuning@yahoogroups.com, djtrancendance@ wrote:
> >
Actually Michael while I think of it, you said <Although our methods of PHI scales now differ (I take (1/PHI)^x) + 1 to get my scale where you take a form of PHI^x >, I'm not exactly sure what you mean.

Firstly, if you meant that Phi^x referred to taking 25 and squaring, cubing, etc...then I was only using this as one demo that it agrees perfectly with all of the usual properties of phi i.e. squaring now is equal to adding 1 and so on.

Secondly,(1/PHI)^x = (2^-(25/36))^x = 2^[-(25/36)+x] so that your method is this + 1. eg x = 1 = (36/36) and index becomes -25 + 36 = 11 giving [2^(11/36)]. However, I don't see the need of then adding 1 to this. Have I understood you correctly that x = 1,2,3,...?

Rick

Far as sound examples of noble-number generated scales, check this
> > out (made in a scale generated from the Silver Ratio using the same
> > scale method I used for PHI):
> >
> > http://www.geocities.com/djtrancendance/PHI/silverrain.mp3
> >
> > It sounds like we're both on to something...a sort of intersection
> > of a true PHI scale, 36TET, and the ability to generate a PHI scale
> > that actually preserves traditional harmony pretty well without the
> > use of traditional intervals.
>
> Hi Michael,
>
> Well for starters 1.414 already is in both 12 and 36 tet. It is sqrt2, 2^(6/12) = 2^(18/36) = 2^(1/2) = 1.414. While 7/4 gives a rough estimate, 45/32 = 1.40625 while 181/128 = 1.414062. The use of the "devil's interval" would probably explain why it sounds dark. 2.414 comes between 45 and 46 in the 36 system i.e. two of the minor thirds 9 and 10 taken up the octave (+36) so I don't think the silver fits into the system any where near as well as the phi. (incidentally, I noticed that 8 gives a very good approx to 7/6). Besides, I think we're getting into the arena of intervals so fine that it would be hard to hear the difference. I still would like to hear the phi intervals -25,0,25 and 0,11,25 if you've got a moment. Thanks mate
>
> -Rick
> >
> > Between my own experiments and your findings, I considered it worth
> > a shot to try and translate the silver ratio of 2.414 (where
> > 1 / 2.414 =0.414) and multiply 0.414 by itself and add 1 to all of
> > the results to create the scale
> > 1.07107
> > 1.1715
> > 1.24257
> > 1.34314
> > 1.414
> > (period / "silver-tave")
> >
> > This results in a full 10 notes per 2/1 octave and, best yet,
> > virtually all combinations are very harmonically use-able. Though
> > I must warn
> > A) The vibe is a bit dark-sounding
> > B) You may want to round the period from 1.414 to about 1.406 to
> > alleviate conflicts in upper-range overtones of the harmonic series
> > by bringing the period closer to 7/5 IE 1.4. This will also help
> > greatly to solve the one major outstanding problem with the above
> > scale: playing notes exactly 1 period above each other beat a bit
> > too much to sound smooth.
> >
> > BTW, if you could find a good denominator for 1.414 or 1.406....
> > I would be interested to see if you could fit the Silver Ratio scale
> > well into a TET-type construction (just as you did quite well for
> > my newest PHI scale with 36TET for "x/36")
> >
> > The result of "what the silver ratio scale would sound like if
> > this scale was used in composition" are above...and I'm admittedly
> > not even that good a composer and I still can't get over the sound
> > of it.
> >
> > Rick wrote:
> > >"Ah but this is no accident. This is the definition of a
> > >number and its inverse. From 25, its inverse is 36 - 25 = 11
> > >so that 25 + 11 = 36 (the 8ve)."
> >
> > Come to think of it, exactly, the exponentials "add up". In
> > additional, what seems to make noble numbers that make good scale
> > generators unique is that they satisfy
> > x = 1/x + any whole number... and the resulting scales can be
> > described as both additively and exponentially symmetrical.
> >
> > >"Since 25 is our new phi, then 50 is phi squared, 75 phi cubed,
> > >and so on. But because 50 and 75 exceed 36, they are replaced by
> > >14 and 3 respectively."
> >
> > Ah, now I get it...you are reducing exponentials of PHI to fit
> > within a single octave. So you would get 25,14,3,(75-(36*2)),
> > 28 (100-(36*2)), etc.
> >
> > And, of course, you're right that 12TET is encapsulated perfectly
> > as every third note in 36TET. Although our methods of PHI scales
> > now differ (I take (1/PHI)^x) + 1 to get my scale where you take a
> > form of PHI^x (which, actually, is a lot like my old method of
> > making the scale). Regardless...both the "new and old" ways of
> > making the PHI scale, along with 12TET, fit into 36TET...and one
> > good question may be "what kind of combinations may be able to
> > combined 12TET harmonies with PHI-based tones as 'blue notes'"?
> >
> > >"And who knows what other types of harmonies we can get now that
> > >we seem to have three minor thirds, three majors etc (or are they
> > >now minor or major??)."
> >
> > Exactly! The thing I have noticed with noble scales in general is
> > instead of just trying to optimize the situation
> > A) the overtones/harmonics of an instrument intersect with notes
> > higher in the scale (which obviously, produces no beating as the
> > two tones are at the same frequency)...they also get to take
> > advantage of
> > B) the overtones DON'T intersect at all at some parts but, in
> > those parts, are at least a semi-tone or so away from where they
> > would intersect...just far enough away to avoid extremities in
> > beating altogether on both sides of the "perfect intersection".
> >
> > -Michael
> >
>

🔗djtrancendance@...

Rick>"Firstly, if you meant that Phi^x referred to taking 25 and squaring"
>",cubing, etc...(but in Rick's new version) squaring now is"
>"equal to adding 1 and so on. "
Right...but I am saying the methods, though your one is expressed as 2^(x/36) and mine as PHI^x end up intersecting most of the same value IE (est. 1.309). So you end up with very similar tunings from which to build scales. I know I am a bit deviant for this...but if my ear hears the same thing and the values are generally very close, I basically consider it the same thing as it has essentially the same musical functionality.

Rick> "Secondly,(1/ PHI)^x = (2^-(25/36)) ^x = 2^[-(25/36)+ x] so that"
>" your method is this + 1. eg x = 1 = (36/36) and index becomes -25"
>" + 36 = 11 giving [2^(11/36)]. "
> However, I don't see the need of then"
>"adding 1 to this. Have I understood you correctly that x = 1,2,3,...?"

Yes, x is 1,2,3 in all of our methods, including both my new and old one. I understand that in your way of math for these methods does everything exponentially IE 1/x = x^-1 and everything is estimated to the closest number that can be generated by 2^(x/36).

And, in that case (using your method), sure you don't need to add "1 +". But mine new method simply uses 0.614^x, hence the need to add 1 to, for example 0.614 to get 1.614 and add 1 to 0.23 (known in your method as 2^(11/36)) to get the approximate same 1.23 value as your method obtains from 2^(11/36).

My point again is, to some extent, we are apparently using two different ways to end up at the same place IE with virtually the same values. And, to clarify, my old method, my new method, and your method ALL include 0.614, 1, 1.23, and 1.614 as a valid "PHI chord" and share all of those tones in common. So is it true that we can agree these are all a good "bare minimum" for PHI tones, regardless of the mathematical method used to obtain them?

But, beyond those tones, tones like 1.30 simply do not occur in my new phi^(1/x) AKA phi^(-x) method, but do in both my old phi^x method and your 2^(x/36) method.

I am going to get down to making sound examples of your chords later today...but, I am just saying in advance just looking at the decimal values generated from you scale, all evidence seems to point to the result's being almost indistinguishable close to my phi^x method when perceived by human hearing.
One thing to note though...your method is more efficient than my old one in finding such values: phi^x displays a tuning with many extra values that don't work (where some have to be eliminated purely by ear and process of elimination to to form a good scale) while your method seems to generate the most consonant notes first from the sequence.
So I have a hunch if someone wanted to "use my old method"...I'd simply tell them they would be better off using yours to generate virtually the same values much more efficiently.

-Michael

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> Rick wrote:
> >"And yes, both eg's seem to give the same beats and sound very similar.
> "
> Indeed...
>
> >They also seem to sound better after the initial attack of the notes
> (probably due to >extraneous upper harmonics?)."
> I think the most likely reason for the problem with the initial attack is that the overtones are spaced very close to the point of dissonance and the added overtones in the attack make them flow over the limit for a second and collide with each other. But, yes, I definitely agree it has to do with the extra harmonics/overtones added in during the attack.

Hi Mike,

I got so carried away I forgot to mention silverain (and it is a good title btw. You should write a whole composition). What is really interesting is that a melody in beats seems to come out all on its own, going from 5th to 7th to maj 6 back to 7th. Very interesting indeed. Really like this one. I've been thinking that the clash in the beginning of notes might be caused by the piano sample you're using which tend to have a lot of treble harmonics. Why not try more neutral instruments like strings or flutes (I often find that clarinets are good for checking harmony)? Or perhaps ask Bill his opinion as to what tone you should go for. I'd love to hear the result.
>
> >"As for the virtual pitch, if you mean the GCD, the closest ratio I
> could find to 25 was" >"1657/1024 which is so high it doesn't seem very
> helpful, but it's early days yet."
> Technically, you're right, it is, of course, the actual GCD between the two chords is far from in common. But, as you said before the beating between the two sounds very similar. My point is that...even though the a simple GCD does not occur...the mind/ear in many ways can't tell the difference between the "aligned beating" of the PHI scale and a true GCD; both seem to have the same effect of pointing the a similar root tone.
>
> >"But then 8 appears which we can then add, and so on. (But the problem
> is that being" >"generators we end up with all 36 notes, so where to
> stop?? Are they ordered notes?)."
>
> This is a fundamental issue I have with using PHI^x and particularly many PHI scales other than my own...they use PHI as an infinite generator that goes 'in circles' until it hits just about everything. Which is a bad idea for consonance if you go to far...of course.
> The only way I've figured out to make every note within a period truly a power of PHI is to use 1 + (1/PHI^x) as the generator. This gives your (2^) 0,11,25 (out of 36) chord plus the additional notes 16 (about 1.38), 20 ,and 7...and makes the gaps between any two successive notes equal to 1/PHI.

I can't remember what "period" is in this context (I'm conditioned to think of it as inverse of frequency). And do you mean 1 + 1/(phi^x) = 1 + (phi^-x) where x = 0,1,2,...? But this would give the first number when x = 0 as 2. Also 1/phi = 2^(11/36) which would give 0,11,22,33,44=8,19,etc...until all 36 notes are included (i.e. 11 doesn't go into 36 and so can generate the whole scale). I'm sure you don't mean this so you'll have to be more precise.
>
> >0:3:11:14:22:25.
> This is almost precisely what my old PHI^x/2^y scale gave...in fact 14 is almost exactly phi^2/2. I still am pretty convinced the new theory above sounds much more consonant, though...of course I am going to give sound examples
> later IE in my next message and when I have the time to make them of your chord and the nearest equivalent in my new "1/PHI" system.
> Note that...in general, I have found any two note closer than about the 1.055 interval begin to beat wildly...but the good news is between 0 and 3...and 11 and 14 in your chord are just wide enough to avoid this beating problem, at least for the root tone (though not so sure about the overtone which is why I'll have to try a sound example). :-)

Yeah this would exclude 1 and 2. Incidentally, 3 is the cubed root of 9 (the 12tet min 3rd) and 7 the cubed root of 21, the tempered fifth. These seem to suggest new symmetric chords similar to the diminished and augmented etc...
>
> >"suggests the symmetrical chord 0:11:14:25 (which you might want to
> test). And naturally" >"we can then treat 11 and 25 as our new 0 giving
> 25:0:3:14"
> Ah, ok...I see how you are using inversions of the chord 0:11:14:25 to build your '36TET PHI scale'...you are changing the 'root' the the second note and taking intervals from there...very clear explanation the time around. :-)

But that's the beauty of this mod maths, that it all begins to meet up whatever you do. You can invert, transpose by odd numbers non divisible by the mod...can't remember all of them, but there's a whole lot of techniques for applying the one basic idea over and over to arrive at a whole composition. Really like the book (except for Milton Babbit's time-point system which I don't like). Till tmw then,

-Rick
>
> >"great book on 12 tone serial method (mod 12 maths) by the composer Charles Wuorenin" >"called 'simple composition'"
> Yes I will definitely try that book...I realize that the use of inverses to create scales is a method I have not tried and, agreed, if he does in with "mod 12/12TET"...the techniques should be adaptable to fit mod 36.
>
> -Michael
>
>
PS:I've just tried to install scala - I had no idea it was freeware - but wasn't successful (it said click on scalaX, not scala, but its not there?? I'm on mac OSX so if you know of anyone else who's had this problem.

🔗djtrancendance@...

Rick> "I got so carried away I forgot to mention silverain (and it is a
>good title btw. You should write a whole composition)."
Thanks and, for the record, I plan to. :-)

>the clash in the beginning of notes might be
>caused by the piano
>sample you're using which tend to have a lot of
>treble harmonics.

Agreed...that or the fact there's a loop within the actual wave sample that gives a slight audible click: I don't exactly own the best sample sets. Either way, it's a weakness that may occur with high-attack instruments based on the harmonic series (as opposed the 'Sethares-ian' instruments with suitably bent-to-match timbres.

I am definitely looking for an instrument with smoother overtones for the final version. I indeed may ask Bill Sethares is odd-harmonic-intensive instruments like flutes would work better or something else entirely (I wonder if he could concoct some instruments with custom-made/"bent" timbres that would do the trick).
As I have found...strings (those in the background in "Silver Rain" work great due to their low attack noise...but I am afraid using low attack instruments for everything will kill some of the momentum and brightness of the song and am still looking for suitable replacements.

>"I can't remember what "period" is in this context (I'm conditioned"
>" to think of it as inverse of frequency). "
Thanks for bringing this up...we are in fact not talking about the signal-processing/physics definition you mentioned above, but instead the period of symmetry IE the equivalent of an "octave" (though I refuse to use the term 'phi-tave') due to some people getting very angry at me for calling something other than 2/1 an "octave". So, in short, 1.618034 is the period in both my old and new version PHI scales. And 1.414 (1/2.414 + 1) is the period in my Silver Ratio scale.

Mike>"11 and 14 in your chord are just wide enough to avoid this
>>"problem, at least for the root tone (though not so sure about the"
>>"overtone which is why I'll have to try a sound example). :-)"
Rick>"Yeah this would exclude 1 and 2."
Indeed...noble numbered scales have the inherent danger of
leading toward numbers with interval gaps smaller than 1.05 (but larger than about 1.01) that simply produce too much roughness to be good for consonance. It is, IMVHO, a great way to single out frequencies in a scale you know will cause dissonance due to critical band roughness...and the best scale-creating formulas, IMVHO, avoid creating/leading to frequencies which break this "critical band rule" altogether.

Mike> "Ah, ok...I see how you are using inversions of the chord"
>"0:11:14:25 to build your '36TET PHI scale'...you are changing the"
>" 'root' the the second note and taking intervals from there...very clear"
>"explanation the time around. :-)"

Rick>"But that's the beauty of this mod maths, that it all begins to meet"
>"up whatever you do. You can invert, transpose by odd numbers non" >"divisible by the mod..."
Exactly...it simplifies things immensely if you can make a scale fit beautifully into a MOD and make inversions that, at the same time, don't violate the critical band. Not to mention the ease of transposing/"moving every note in the scale up by one step" within equal temperament tunings. The question from myself to you then becomes...can you think of a good formula that fits any fractal number into an ET/mod?
For example consider the "fractal numbers":
2.4142135 where x^2 = 2x + 1 (the "Silver Ratio")
3.3027756 where x^2 = 3x + 1
4.2360679 where x^2 = 4x + 1
(these are actually many of the same ones from Jacques Dudon's post).

The thing with my 1+(1/fractal)^x is that it can produce consonant scales from all of the above...it would be great if you could devise a formula to closely estimate any scale generated by the above fractal numbers into an ET tuning.
If you did so, I would be happy to write a program that would automatically find the best MOD given any generator and create scales using that MOD.

-Michael

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> Rick> "I got so carried away I forgot to mention silverain (and it is a
> >good title btw. You should write a whole composition)."
> Thanks and, for the record, I plan to. :-)
>
> >the clash in the beginning of notes might be
> >caused by the piano
> >sample you're using which tend to have a lot of
> >treble harmonics.
>
> Agreed...that or the fact there's a loop within the actual wave sample that gives a slight audible click: I don't exactly own the best sample sets. Either way, it's a weakness that may occur with high-attack instruments based on the harmonic series (as opposed the 'Sethares-ian' instruments with suitably bent-to-match timbres.
>
> I am definitely looking for an instrument with smoother overtones for the final version. I indeed may ask Bill Sethares is odd-harmonic-intensive instruments like flutes would work better or something else entirely (I wonder if he could concoct some instruments with custom-made/"bent" timbres that would do the trick).
> As I have found...strings (those in the background in "Silver Rain" work great due to their low attack noise...but I am afraid using low attack instruments for everything will kill some of the momentum and brightness of the song and am still looking for suitable replacements.

Hi Mike,
But here is the difference between composition and music theory, a piece of music and an experiment. Instruments come with their own special character and I often find that, given a "neutral" score, assigning instruments according to their range usually works out just fine. If you want long extended notes, go the strings or woodwind, short stabbing notes, accentuate with the brass. If the strings sound wishy-washy, then this can be easily rectified by doubling up the beginning of notes with trumpets or by making the strings staccato or marcato etc...The problem with alternate tunings on the piano (samples) is that they tend to sound like one of those honky tonk saloon pianos found in the old cowboy movies.

> >"I can't remember what "period" is in this context (I'm conditioned"
> >" to think of it as inverse of frequency). "
> Thanks for bringing this up...we are in fact not talking about the signal-processing/physics definition you mentioned above, but instead the period of symmetry IE the equivalent of an "octave" (though I refuse to use the term 'phi-tave') due to some people getting very angry at me for calling something other than 2/1 an "octave". So, in short, 1.618034 is the period in both my old and new version PHI scales. And 1.414 (1/2.414 + 1) is the period in my Silver Ratio scale.

Yeah I remember. I don't know why people adopt already well-defined terms in a completely new context. Why don't they use "generating interval" or something because that's exactly what it seems to be? But I'm still not clear as to what your generating interval is. Here you say it's phi but in the last post you gave 1 + phi^-x, while above you write for the silver ratio 1.414 (1/2.414 + 1). I suspect you mean that 1.414 = (1/2.414 + 1), which it does, instead of 1.414 x (1/2.414 + 1) which is what it looks like. There are certain math'l conventions which help to make things clearer. Writing ax for eg means their product a times x. And I see below that you're beginning to watch your brackets. Incidentally, 1.414...is the square root of 2 which corresponds to the flat-fifth interval, 2^(1/2). This divides the 8ve in half.
>
> Mike>"11 and 14 in your chord are just wide enough to avoid this
> >>"problem, at least for the root tone (though not so sure about the"
> >>"overtone which is why I'll have to try a sound example). :-)"
> Rick>"Yeah this would exclude 1 and 2."
> Indeed...noble numbered scales have the inherent danger of
> leading toward numbers with interval gaps smaller than 1.05 (but larger than about 1.01) that simply produce too much roughness to be good for consonance. It is, IMVHO, a great way to single out frequencies in a scale you know will cause dissonance due to critical band roughness...and the best scale-creating formulas, IMVHO, avoid creating/leading to frequencies which break this "critical band rule" altogether.

Sorry Mike, everything else I agree with but I don't know what IMVHO stands for.
>
> Mike> "Ah, ok...I see how you are using inversions of the chord"
> >"0:11:14:25 to build your '36TET PHI scale'...you are changing the"
> >" 'root' the the second note and taking intervals from there...very clear"
> >"explanation the time around. :-)"
>
>
> Rick>"But that's the beauty of this mod maths, that it all begins to meet"
> >"up whatever you do. You can invert, transpose by odd numbers non" >"divisible by the mod..."
> Exactly...it simplifies things immensely if you can make a scale fit beautifully into a MOD and make inversions that, at the same time, don't violate the critical band. Not to mention the ease of transposing/"moving every note in the scale up by one step" within equal temperament tunings. The question from myself to you then becomes...can you think of a good formula that fits any fractal number into an ET/mod?
> For example consider the "fractal numbers":
> 2.4142135 where x^2 = 2x + 1 (the "Silver Ratio")
> 3.3027756 where x^2 = 3x + 1
> 4.2360679 where x^2 = 4x + 1
> (these are actually many of the same ones from Jacques Dudon's post).
>
> The thing with my 1+(1/fractal)^x is that it can produce consonant scales from all of the above...it would be great if you could devise a formula to closely estimate any scale generated by the above fractal numbers into an ET tuning.
> If you did so, I would be happy to write a program that would automatically find the best MOD given any generator and create scales using that MOD.
>
> -Michael
>
It is certainly a nice idea but making it a reality is a different matter. For 2^(1/2) (i.e. silver minus 1), it is already in 12 and 36 MOD as 6 and 18. Being the only interval (dyad) that divides the 8ve into two equal parts, its recursive properties are well known (i.e. Sqrt2/1 = 2/sqrt2, so which is the tonic?). And the discovery (at least for me) that phi can be represented by 25 and 11 in 36 MOD, which includes 12 MOD and all of the traditional harmony, came as a complete surprise. This makes it extremely practical. But as for the silver ratio, sqrt2 + 1, the closest interval I could deduce was 2^(2543/2000)= 2.414124, implying a 2000 MOD (bringing into one 8ve, 2543 - 2000 = 543 and 2^(543/2000)). Although electronics could reach it, I'm sure that we are far beyond the range of human recognition here. And besides, the technique you are already adopting seems much simpler. So unless I'm mistaken, the phi tuning seems to be a one-off, shame but that's math's for you.

-Rick

🔗Cameron Bobro <misterbobro@...>

🔗Chris Vaisvil <chrisvaisvil@...>

I tried the silver ratio tuning last night. It didn't fit right for me. I'm
not sure why. I will try it again.
Then I looked for PHI tunings in my scala folder and didn't find anything
that sounded like what you had proposed. The PHI tuning files I found almost
all just add a single PHI based note.

That's how I ended up with the Phillips 22 (see [MMM]) improvisation.

I'm very interested in a PHI scale that has steps based on PHI which is
what I understood your last PHI proposals to be.

Chris

On Sat, Apr 25, 2009 at 9:31 PM, <djtrancendance@...> wrote:

>
>
> Chris>"Can I get a scala file of the silver tuning?"
> Of course... :-)
> Here is a link:
> http://www.geocities.com/djtrancendance/SILVERRATIO.scl
>
> WORD OF CAUTION...
> As a realistic limitation for composing please note that, in any one
> chord, you can not use any more than three notes a 'silver half step' apart
> and expect 12TET chord-like consonance: go figure the scale has a rather
> dense 10 notes per 2/1 octave. For example my silver ratio scale is (in
> cents)
> 0 (the period)
> 118
> 274
> 375
> 510
> 589 (the period)
>
> So, for example
> A) the chord 118,274,375,space,589 is fine because of the gap between 375
> and 589
> B) the chord 274 375 510, space,118 after the next period is fine: gap
> between 510 and 118
> BUT
> C) the chord 274 375 510,589 will produce some annoying dissonance due to
> having more than 3 notes that are close together.
>
> On the other hand, if you want a quick and easy way to add tension, go
> ahead and shove in 4 tones in a row with no interval gap.
>
> Michael
>
>

🔗djtrancendance@...

Rick> "Here you say it's phi but in the last post you gave 1 + phi^-x, while above you write for the silver ratio 1.414 (1/2.414 + 1). I suspect you mean that 1.414 = (1/2.414 + 1),"

That is exactly what I mean IE 1/2.414 + 1 = 1.414 or 1/1.618034 + 1 = 1.618034. It's a way to get from the inverse of PHI back to PHI itself...sorry if I was unclear before.

>"The problem with alternate tunings on the piano (samples) is that they
tend to sound like one of those honky tonk saloon pianos found in the
old cowboy movies."
Right...I'll take a shot at your staccato string and brass "attack" tricks to add that staccato/brightness to the lead parts instead of using the "old west de-tuned piano" (LOL).

>"Sorry Mike, everything else I agree with but I don't know what IMVHO stands for."
Means "In my Very Humble Opinion" IE I realize well others could argue it is not fact (and 'even' be right)...but I strongly believe all scales aiming for consonance should avoid any notes within the intervals of between 1.015 and 1.05 of each other since those intervals, to my own ear, are the ones where dissonance becomes unbearable.

> "But as for the silver ratio, sqrt2 + 1, the closest interval I could
deduce was 2^(2543/2000) = 2.414124, implying a 2000 MOD (bringing into
one 8ve, 2543 - 2000 = 543 and 2^(543/2000) ). Although electronics
could reach it, I'm sure that we are far beyond the range of human
recognition here. And besides, the technique you are already adopting
seems much simpler. So unless I'm mistaken, the phi tuning seems to be a one-off, shame but that's math's for you."

I agree...it seems to be out of reach to find a near-perfect match for the Silver Ratio using the mod AKA "ET" technique...and no one with practical instruments (or even most working with electronic ones) is likely going to want to mess with 2000-TET. :-D

While the technique is using, I believe, is very simple (mostly just taking 1/fractal_number squared + 1 to get all the numbers in the scale, which works for any fractal number)...I am watching out for the following:
A) To help counter my own bias: it's easier to truly test my theory if I have something that can approximate it that can create notes in the same "ballpark" and compete with it directly...in the same way 36TET does a great job at approximating both your PHI scales and my new ones.

For example...the concept of MOS, IMVHO, is the corollary to JI, which prevents it from having a monopoly...and the same goes for Sethares derivation of JI assuming a simple-ratio timbre and applying it to consonance curves to derive JI: they are many paths to get to a similar and competitive result.

B) Universalism: I am highly convinced that an ideal formula will work for any fractal number, and not just PHI and/or the Silver Ratio...in the same way MOS can work for turning any number, fractal numbers included, into scales which match the 2/1 octave.

C) Documentation: I hope to be able to sum up how to generate scales with noble numbers in a way simple enough so an average musician can pick it up. I hope to eventually create something as easily to sum up as, say, the concept of MOS scales is to creating just about any consonant scale within the 2/1 octave (since it can easily derive JI, pentatonic, diatonic...IE it's quite universal for 2/1 octave scales).

And I realize fitting fractal numbered scales into ET may be an incredibly tricky, or, as you seem to say, virtually impossible task. I'm just eager to find a corollary to my own methods for the reasons above...and, of course, to avoid it "having a monopoly" as the only scale that hits frequencies in the same area, which is not exactly useful for finding ways to challenge and ultimately improve it.

-Michael

🔗djtrancendance@...

Chris> "I tried the silver ratio tuning last night. It didn't fit right for me. I'm not sure why. I will try it again."
Hmm...well could you show me what you made with it? If I could hear it, I could likely point you to what is going wrong. One of the most likely possibilities is you need to adjust the "silver-tave"/period to 1.406 IE 589.915 cents instead of 1.414 if you plan to use a lots of octaves (IE "C5 and C6") at once. Using 1.414 in that scale can cause too much beating between overtones if you plan to use a lot of such "unison octaves" (which many musicians like to do)...so I strongly suspect that's your problem.

Chris> "Then I looked for PHI tunings in my scala folder and didn't find
anything that sounded like what you had proposed. The PHI tuning files
I found almost all just add a single PHI based note."

I'm glad someone realizes the problem I also have with about 98% of other people's PHI tunings: they indeed shove things in and, in general, sound nothing like my own nor preserve the actual artistic principle of the golden ratio. :-)
How to explain...when you look at the wikipedia example:

http://upload.wikimedia.org/wikipedia/commons/thumb/4/44/Golden_ratio_line.svg/225px-Golden_ratio_line.svg.png

...you will notice that the proportions of PHI are based on essentially splitting a line between 0 and 1 with a point at 0.618 (this is the exact same proportion as splitting a line between 1 and 2 at 1.618)!

Then I asked myself the question...how would I get the same split I did for the line between 0 and 1 for the smaller line between 0 and 0.618? Aha...I would take 0.618*0.618 = .3819. And, if I keep doing
that and "work recursively", I keep finding "divisions of PHI" that actually resemble how PHI splits a line in a graphical sense and truly does have steps based on PHI (and does not involve dividing by 2^x, taking an exponential of 2, or anything to distort the use of PHI). This is how I came up with my "true to PHI" (1/PHI)^x + 1 formula.

That's why I truly believe my scale represents a rare method to represent PHI in music in a way that's 100% true to the way PHI proportions work in art...unlike things like PHI-mean-tone which just shove it in, split it, force it into the octave via 2^(x/phi)...but don't preserve it's original artistic "line-splitting" use.
******************************************

-Michael

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
Hi Chris,

Why don't you try the phi intervals from the 36 EDO I have been discussing with Michael? The notes are 2^(n/36), n = 0,1,2,3,...35 and the PHI intervals are n = 25 and its inverse n = 11. Furthermore, since 25 and 11 do not divide into 36 then all notes can be generated from these.

-Rick

I tried the silver ratio tuning last night. It didn't fit right for me. I'm
> not sure why. I will try it again.
> Then I looked for PHI tunings in my scala folder and didn't find anything
> that sounded like what you had proposed. The PHI tuning files I found almost
> all just add a single PHI based note.
>
> That's how I ended up with the Phillips 22 (see [MMM]) improvisation.
>
> I'm very interested in a PHI scale that has steps based on PHI which is
> what I understood your last PHI proposals to be.
>
> Chris
>
> On Sat, Apr 25, 2009 at 9:31 PM, <djtrancendance@...> wrote:
>
> >
> >
> > Chris>"Can I get a scala file of the silver tuning?"
> > Of course... :-)
> > Here is a link:
> > http://www.geocities.com/djtrancendance/SILVERRATIO.scl
> >
> > WORD OF CAUTION...
> > As a realistic limitation for composing please note that, in any one
> > chord, you can not use any more than three notes a 'silver half step' apart
> > and expect 12TET chord-like consonance: go figure the scale has a rather
> > dense 10 notes per 2/1 octave. For example my silver ratio scale is (in
> > cents)
> > 0 (the period)
> > 118
> > 274
> > 375
> > 510
> > 589 (the period)
> >
> > So, for example
> > A) the chord 118,274,375,space,589 is fine because of the gap between 375
> > and 589
> > B) the chord 274 375 510, space,118 after the next period is fine: gap
> > between 510 and 118
> > BUT
> > C) the chord 274 375 510,589 will produce some annoying dissonance due to
> > having more than 3 notes that are close together.
> >
> > On the other hand, if you want a quick and easy way to add tension, go
> > ahead and shove in 4 tones in a row with no interval gap.
> >
> > Michael
> >
> >
>

🔗djtrancendance@...

Rick> "and the PHI intervals are n = 25 and its inverse n = 11. "
I will agree the notes formed by n= 25, n = 11 n = -25 and n = 0 in Rick's formula 2^(n/36) are "created straight from PHI intervals"

...and note that the result of n = 11 in his scale is apx. 1.23 which is about 2 / PHI. And...all of those above notes are in common between my own 1+(1/PHI)^x scale and his (no coincidence).

Beyond that though, the rest of the notes in Rick's scale, as I understand it, are generated by taking inversions of that chord so the construction method becomes different from mine at that point.

But, depending in how "true/legal" you view the concept of a chord inversion to creating a scale, Rick's scale is in deed loyal to the properties of PHI...and still, in my opinion, already more loyal than most scales as it bases itself directly off the 1/PHI 1 2/PHI PHI chord mentioned above.

So, to clear things up, I recommend his scale more than a huge proportion of other so called "PHI scales" I have heard...and realize my own view that my own scale is "more closely based on how PHI works visually in art" may or may not be relevant to your cause.

-Michael

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Michael Sheiman <djtrancendance@...>

>"Have you tried something like this construct

1.618 to the power of (1/12) = 1.040914 = twelfth root of PHI as the generator?"

Are you talking to myself or Rick?

In the case of myself, in my formula (1 + (1/PHI)^x) if you use x = 6 you get 1.0570 and using x = 7 gives 1.03444...and since 1.03444 is too close to the point of maximum entropy/harmonic roughness I don't use it.

In short, my scale has nothing to do with splitting the phi-tave into the 12th root of PHI and none of the values in my scale approach 1.040914 (which is also too close to the maximum point of critical band roughness, in my opinion, unlike ratios of 1.05+ which come out as fine to my ears).

As for Rick's scale, I can't speak for him...but I get a strong impression he's deriving the scale from the fact 2^(25/36) is uncannily close to 1.618034 (in fact, it is 1.618026) and the fact 2^(-25/36) is uncannily close to the inverse of PHI IE 1/PHI. I don't think it has anything to do with taking PHI^(x/12)...the 36 seems to be divisible by 12 just by coincidence.

But, if you want to take the PHI^(x/12) route I would recommend PHI^(2/12) = 1.0835 as your generator, which would avoid the critical band roughness issue with the interval PHI^(1/12) give you the scale
1
1.0835 (2/12)
1.1739 (4/12)
1.2720 (6/12)
1.3782 (8/12)
1.4933 (10/12)
1.618034 (12/12)

Note...the above scale is not mine does not generate PHI proportions as my 1+(1/PHI)^x scale does (IE it does not split note sections the way PHI splits a line)...but it is based directly on powers of PHI and, since it obeys root-tone critical band roughness, should have at least half-decent consonance.

-Michael

🔗chrisvaisvil@...

🔗djtrancendance@...

>"Might I suggest your consonance quest might be better served with the 12edo solution of skipping notes?"

That obviously helps...but the side-effect, of course, is that the number of notes and degrees of expression (my other goal) is hurt a lot by doing that
For example, here is my 1 + (1/PHI^x) scale
1
1.0557
1.14589
1.23606
1.38196
1.52505 (1.618034 * 1/1.0557)
1.61 (rounded from 1.618 to clarify the PHI-tave to sound more like the 5-limit fraction 8/5)

The next step up, so far as "spreading out the root tones" would involved making the scale 1,1.23606, 1.38196, 1.61 (a bit more than a standard whole-tone apart) to minimize dissonance of the root tones...but then you would have a terribly inflexible 4 tone per octave scale and, in my opinion, fairly negligible gain in increased consonance level.

I also don't see...why all of a sudden you have this issue with using tones about a semi-tone apart in my scales. For
example, most of the tones used in my "silver rain" example are between the ratios 1.05 and 1.06 apart IE around a semi-tone apart. I don't see why my compositional examples would somehow manage to side-step problems you seem to be identifying as being huge issues.

Are you sure you are using my PHI scale correctly (and/or have you tried composing using the scale and simple instruments such as guitars, strings, and pianos as I did with Silver Rain)?

-Michael

🔗chrisvaisvil@...

🔗Michael Sheiman <djtrancendance@...>

Chris>"When I tried the silver version I just didn't find appealing intervals."

Hmm...for starters...try listening to this.
http://www.geocities.com/djtrancendance/PHI/silvertest.mp3

This shows all of the tones in the silver ratio scale played at once (and all possible intervals at once), droning as a sustained chord with no notes left un-played (IE "worst case scenario"). You should be able to duplicate the result using the SCALA file I gave you...if not, perhaps I gave you the wrong intervals.
BTW, which note(s) sound sour to you in the above example?

>"As for common usage I think it would be a reasonable expectation that composers will try every permutation of your tuning. "

And that is, of course, a fair assumption...although I am wondering how much of this is a side-effect of stacking too many tones too close together (IE, just as with 12TET...you get the purest chords by skipping a lot of 'semi-tones'...yet the semi-tones still, of course, exist in the scale and sound fine if you play, for example a and a# but stacking a,a#,b is not).

But again, perhaps it's much a matter of personal opinion, but the above example does not skip any tones and, at least to my ears, it sounds fine and certainly there is not any heavy beating.

-Michael
__,_._,

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> >
> > .
> >
> >
> >
> Have you tried something like this construct
>
> 1.618 to the power of (1/12) = 1.040914 = twelfth root of PHI as the
> generator?
>
> which is analogous to
>
> 2 to the power of (1/12) = twelfth root of 2 = 1.059463 = generator for
> 12EDO
>
>
> That is what I thought you were driving at.
>
> Chris
>
Hi Chris,

There are a few problems with this approach. First is that applying PHI successively spirals off into infinity (so that all pitch classes would eventually end up having the same note name). Second, the 12th root of PHI gives a very small interval of the order of 1.04, which is below the threshold of critical band roughness. Besides, there is really no reason to believe that dividing PHI by 12 would be the same as 12 EDO which evolved by approximating basic JI intervals.

On the other hand, 36 EDO not only includes 12 EDO as the series 0,3,6,9,...i.e. 2^(3/36) = 2^(1/12), the semitone, so that all traditional 12 tet harmonies are preserved, but it also allows access to PHI intervals 25 and its inverse 11 (i.e. [2^(11/36)]/2 = 2^-(25/36)). Note also that the difference between 2^(25/36) and PHI is well below 5 cents. Since 25 and 11 don't divide into 36 then they can be seen as scale generators i.e. 25 + 25 + 25...and bringing them within the 8ve will give all 35 notes, so that it can in fact be seen as a PHI tuning. One new symmetrical chord you might want to look at is 0:5:10:15:20:25. I'm having difficulties downloading Scala so I wouldn't mind hearing it if you've got the time (PS: Is there a Scala forum for troubleshooting? The only thing I get on the net is some programming language called Scala).

Thanks

-Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
Hi Mike,

Well the way I see it is that you already have the formula for fractal numbers x^2 = Nx + 1, N = 1,2,3,...and solving for x. The question now is whether the generating interval (period) for each tuning is simply x. If so, then the series would be simply x, x^2, x^3, and so on. Since x = [Nx + 1]^(1/2), then x^3 = [Nx + 1]^(3/2), x^4 = [Nx + 1]^2, etc...Better not to over complicate things.

(PS: while I like the title 'silverrain' I would strongly recommend against writing a phi piece called "goldenshowers". LOL) Oh and thanks for your support below.

-Rick

>
> Rick> "Here you say it's phi but in the last post you gave 1 + phi^-x, while above you write for the silver ratio 1.414 (1/2.414 + 1). I suspect you mean that 1.414 = (1/2.414 + 1),"
>
> That is exactly what I mean IE 1/2.414 + 1 = 1.414 or 1/1.618034 + 1 = 1.618034. It's a way to get from the inverse of PHI back to PHI itself...sorry if I was unclear before.
>
> >"The problem with alternate tunings on the piano (samples) is that they
> tend to sound like one of those honky tonk saloon pianos found in the
> old cowboy movies."
> Right...I'll take a shot at your staccato string and brass "attack" tricks to add that staccato/brightness to the lead parts instead of using the "old west de-tuned piano" (LOL).
>
>
> >"Sorry Mike, everything else I agree with but I don't know what IMVHO stands for."
> Means "In my Very Humble Opinion" IE I realize well others could argue it is not fact (and 'even' be right)...but I strongly believe all scales aiming for consonance should avoid any notes within the intervals of between 1.015 and 1.05 of each other since those intervals, to my own ear, are the ones where dissonance becomes unbearable.
>
> > "But as for the silver ratio, sqrt2 + 1, the closest interval I could
> deduce was 2^(2543/2000) = 2.414124, implying a 2000 MOD (bringing into
> one 8ve, 2543 - 2000 = 543 and 2^(543/2000) ). Although electronics
> could reach it, I'm sure that we are far beyond the range of human
> recognition here. And besides, the technique you are already adopting
> seems much simpler. So unless I'm mistaken, the phi tuning seems to be a one-off, shame but that's math's for you."
>
> I agree...it seems to be out of reach to find a near-perfect match for the Silver Ratio using the mod AKA "ET" technique...and no one with practical instruments (or even most working with electronic ones) is likely going to want to mess with 2000-TET. :-D
>
> While the technique is using, I believe, is very simple (mostly just taking 1/fractal_number squared + 1 to get all the numbers in the scale, which works for any fractal number)...I am watching out for the following:
> A) To help counter my own bias: it's easier to truly test my theory if I have something that can approximate it that can create notes in the same "ballpark" and compete with it directly...in the same way 36TET does a great job at approximating both your PHI scales and my new ones.
>
> For example...the concept of MOS, IMVHO, is the corollary to JI, which prevents it from having a monopoly...and the same goes for Sethares derivation of JI assuming a simple-ratio timbre and applying it to consonance curves to derive JI: they are many paths to get to a similar and competitive result.
>
> B) Universalism: I am highly convinced that an ideal formula will work for any fractal number, and not just PHI and/or the Silver Ratio...in the same way MOS can work for turning any number, fractal numbers included, into scales which match the 2/1 octave.
>
> C) Documentation: I hope to be able to sum up how to generate scales with noble numbers in a way simple enough so an average musician can pick it up. I hope to eventually create something as easily to sum up as, say, the concept of MOS scales is to creating just about any consonant scale within the 2/1 octave (since it can easily derive JI, pentatonic, diatonic...IE it's quite universal for 2/1 octave scales).
>
>
> And I realize fitting fractal numbered scales into ET may be an incredibly tricky, or, as you seem to say, virtually impossible task. I'm just eager to find a corollary to my own methods for the reasons above...and, of course, to avoid it "having a monopoly" as the only scale that hits frequencies in the same area, which is not exactly useful for finding ways to challenge and ultimately improve it.
>
> -Michael
>

🔗vaisvil <chrisvaisvil@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗djtrancendance@...

Chris> "I'm not sure I see the point in just having the PHI and 1/PHI interval.
I thought the idea was to build a tuning system like those Wilson?
Horograms? (Which I still have to get my hands on)"

Far from it...although I generally admire hora-grams and MOS scales as a superb way to get great sounding scales so long as they are built deliberately to fit the 2/1 octave.

But at least in my case and, I presume, Rick's as well...that appears not to be the point. Both Rick and my scales start with 1/PHI, 1, 2/PHI, and PHI...and then build up from there in different ways: there's nothing deliberately MOS/horagram-like about the construction processes.

Far as I understand them, the hora-gram scales are really just MOS scales; thus based on the idea of finding intervals which, when taken to the ^x power and then divided by a power of 2 (to fit the exponential series within the octave), can generate scales which two distinct interval sizes that intersect perfectly with the 2/1 octave.

Although Rick's scale has apx. 2/PHI in it, which can be multiplied to get 2/1, it certainly is not simply the result of taking PHI^x/2^y (which would produce some 20 or so notes per 2/1 octave, not anything like the 6-9 tone hora-gram scales).
In both Rick and my scales, I think it's fair to say the apx. 2/1 interval is a side-effect of having the ratio 2/PHI and the PHI-tave in the same scale...rather than a deliberate fitting to MOS, for example.

-Michael

🔗djtrancendance@...

🔗Chris Vaisvil <chrisvaisvil@...>

please listen to the mp3s again - look at the midi files.

both are unique songs.

the similarities are only because of my lack of fresh ideas improvising
today.

On Wed, Apr 29, 2009 at 11:00 PM, <djtrancendance@...> wrote:

>
>
>
> I'm trying to understand. These sound almost like the same song, and it
> does sound like you are indeed using my SCALA files, but the way the octaves
> are separated are very very extreme.
>
> Of course, if you play notes about a 1.05 interval apart under 200hz you're
> going to start having miserable dissonance problems due to the critical band
> getting wider as you hit those lower frequencies...which I think is what
> happend to both scales...and, of course, the PHI one more so as it has less
> notes per octave and thus more extreme pitches.
>
> One idea...you might want to try making a song from scratch rather than
> throwing the tuning at an existing song (which is kinda like throwing an
> octave-based song at a tritave-based tuning IE Bohlen & Pierce's)....and
> try staying within about the C4-C7 frequency range...like I warned you
> about before, as well, the scales I gave you used normalized
> phi-tave/silver-tave estimated which distort as you get further from C5
> (particularly for lower notes where critical band becomes more of an issue
> and the scales are so closely spaced). So if you are using "C2"...of course
> you're going to have some problems.
>
> -Michael
>
>
>

🔗Michael Sheiman <djtrancendance@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "vaisvil" <chrisvaisvil@...> wrote:
>
>
> >
> > On the other hand, 36 EDO not only includes 12 EDO as the series 0,3,6,9,...i.e. 2^(3/36) = 2^(1/12), the semitone, so that all traditional 12 tet harmonies are preserved, but it also allows access
> >
> > Thanks
> >
> > -Rick
> >
>
> Hi Rick,
>
> For what it is worth I tried 36 EDO via bend commands in the late 90's in a tracker.
>
> http://clones.soonlabel.com/mp3/goose.mp3
>
> Be warned the piano sample sucks. We are spoiled now :-)
>
> I'm not sure I see the point in just having the PHI and 1/PHI interval. I thought the idea was to build a tuning system like those Wilson? horograms? (Which I still have to get my hands on)
>
> Thanks,
>
> Chris
>
> I have had luck with the 1/silver - will post later tonight.
> Off to 1/PHI
>
Thanks Chris,

A few interesting sections, kind of hyper jazzy, but unfortunately because of the old technology I suppose you couldn't avoid making it sound like an exercise in note bending rather than specific intervals i.e. I'd like to hear specific things outside usual 12 edo. But as you said, we are spoiled now.

Well the point is that its not just the PHI and 1/PHI because its inverse IS also PHI from another perspective. It's a bit like saying "I don't see the point of a major triad" as if we could only ever play it from the tonic. For starters, watch how it captures all the usual recursive properties of PHI. Inverting should be equal to itself - 1: 2^-(25/36) = 0.618...(Bringing it into the same 8ve gives 2^(11/36)). Squaring should be equal to itself + 1: [2^(25/36)]^2 = 2^(50/36)= 2.618...The list goes on. Of course we can then apply all of this to our newly gained notes and repeat the process i.e. taking Phi of Phi etc...And as I said, since 11 and 25 don't divide into 36 then all notes will eventually be reached so it IS a complete phi tuning system already. Therefore there is nothing in the audible range of PHI that should not be able to be captured by this system (including Wilson's holograms). PHI does seem to bring together the two worlds of ratios and sums/differences. Beats are created which seem to reinforce the original frequencies.

-Rick

🔗Carl Lumma <carl@...>

🔗djtrancendance@...

🔗Carl Lumma <carl@...>

🔗djtrancendance@...

🔗Carl Lumma <carl@...>

🔗djtrancendance@...

Carl and others interested in testing decimal-ratio-based entry in SCALA,

To reproduce the problem

Click on
File -> New -> Scale

Now look under options and click the radio button "linear factors/frequencies"

Now type in the decimal values on each line and click yes when it asks "change the root tone to 1hz".

BTW...I DID notice if you click NO to the above box it will AUTOMATICALLY set the root tone to 261.6255 HZ IE middle C (exactly what I wanted it to do: assume 1.0000 = 261.6255hz)...but I still don't consider the question that intuitive.

One suggestion would be click ok to set the hz value to 1hz or cancel to set it to 261.6255hz IE the middle C.

Yet another (and perhaps more easy to use) option would simply be to add a ratio button which says "input values as decimal ratios where 1 = middle C". The SCALA interface as of now seems very round-about for entering decimal valued ratios.

-Michael

--- On
Thu, 4/30/09, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: PHI interval tuning (for Michael S)
To: tuning@yahoogroups.com
Date: Thursday, April 30, 2009, 1:47 PM

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

> Ok, here goes:

> Say you type in decimal value IE

> 1.147

> 1.23

> 1.38

> 1.618

> .....

> into SCALA as a new scale and SCALA accepts the values (as HZ).

Typed in where?

>but will change some of them if you change 262hz back to 1hz

>and say "no" to the change.

Can you tell us what dialogs your clicking, so we can reproduce

this?

-Carl

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> Carl> "Care to offer a bug report?"
> I'm not going to take up too much space here.
> There's not really a problem with SCALA, it just seems to natively >support
> A) Hz values
> B) Fractions
> C) Cents
> ...and my DAW works with decimals. So what I'm doing is fairly >non-standard (and not built natively into SCALA's capacity) and not >exactly worth a "bug fix".

...but SCALA works fine with decimals, I do it all the time. ??????
When you open the "input" dialog, just click on the "linear factors (frequencies)" radio button. You need to enter a 1.0 as the base (otherwise it will make the first value the "1" and the others following proportionate). Then it will ask you if you want to set "1", or whatever your first value is, as your base frequency, just say "no" here and you're set, with your default base frequency as whatever it is (C 261 in your case) and the rest of the tuning proportionate to that.

>
> As for what was shifted...for example, the PHI-tave/period become >about 820 cents instead of 833 cents.

Now that is truly strange.

>One thing I have noticed is the PHI-tave (833) is, ironically, the >most sour interval in the PHI >scale...despite the fact it is >essentially for keeping all the other >interval proportionate.
>
> And changing the PHI-tave to 13/8 actually improved the >balance, so far as aligning odd harmonics in the
> timbre of instruments to the scale, a lot. So long as you work >within a couple octaves above or below of C5 (any further than that >and the accumulating error of the estimated PHI-tave begins to throw >everything off)...I'd recommend swapping the PHI-tave to 13/8 AKA >1.625.
>
> -Michael
>

If you study your tunings of the last couple of weeks you'll find that you've already gone this way unconciously, by ear- for example in one tuning you had a second exactly (one cent error) 9/13 below Phi, then a 13/10 off of that, and so on.

🔗Michael Sheiman <djtrancendance@...>

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> Carmeron> "...but SCALA works fine with decimals, I do it all the time. ??????"
>
>
> As I said in my last e-mail on the thread...I recently (after the bug) figured out the CLICK NO trick when it asks you "set the base to 1hz?" as well.
> But still I believe "guessing" correctly that the program will automatically set the base to middle C when you click no seems a lot less intuitive than, say, adding a radio button to the option list which says "frequency ratios from middle C". Such an interface/display format would make it much more obvious what the program will do vs. the trick you (and very recently, I) now use for inputting decimal ratios into SCALA.
>
>
> BTW...for the record, I now know how to input decimal ratios straight into SCALA...but I continued this issue-thread in the hopes people won't have to find the same "loop-hole" you and I found to get decimal frequencies to work.
>
> -Michael
>

I see what you mean, but I think it's really a matter of implementing tons of features. What's not intuitive is that inputting ratios and cents will automatically insert a 1/1, but inputting decimals doesn't- however, the way it works is actually more flexible, because you might be working directly with specific absolute frequencies, and with the pop-up you can choose to convert the whole thing to your base frequency, or input a new base frequency. A good solution overall.