Yahoo Tuning Groups Ultimate Backup tuning [tuning] Re: PHI interval tuning (for Michael S): with "Phicicles" song example :-)

Rick>"There are those interesting beat frequencies again"
..which seems to be a fairly unique property of scales based on fractal numbers...considering this pops up in both the PHI and Silver Ratio based scales... :-)

> "Those sounds are much better than the piano."
Agreed...and it seems to be coincidence that choice of instrument matters more with the PHI scale than it does the Silver Ratio one. This time around, I used a very clean sounding guitar...which still has the brightness of the piano but without the dis-harmonies evident in the piano's timbre.
Thus I wonder...do you think it's fair the say that instruments which adhere strongly to consistent harmonic ratios, such as guitars, should be highly recommended timbres for my PHI scale?

>"I'd like to hear a
version parred back even further, perhaps using only the most consonant
sounding intervals (I want to see if its possible to build a very
consonant system to form a strong basis and go outward from there)."

Indeed...in that case I would have to space out the notes more to avoid beating to a high extent. The funny thing is the beating between any two notes, at least to my ear, sound about equally pleasant in their speed/modulation...and the only way to go further is to reduce the beating entirely by spreading the intervals out.

One thing is almost certainly for sure (and in common in both of our scale)...the purest interval set (kind of the equivalent of a major triad) within a "PHI-tave"/period appears to be 1 2/PHI PHI. Next in line is probably 1 2/PHI 1/PHI^2+1 PHI...which is kinda like major 7th chord in tensity. Anything much denser than that...and you are going to have some "semi-tone" style amounts of beating (ah...the horrors of realistic psycho-acoustics). :-D

>"Nice one."
Thanks!...I'm glad you like it (the last sound example "PHIcicles"). To be realistic, though, I think Chris hit the nail on the head
before when he said the Silver Ratio is a more consonant scale and I will admit there are some sour spots I found when I approached the challenge of using the PHI scale in a piece involving more than one 2/1 octave worth of frequency range.

In dealing with the above issue/problem, I have also noticed a good amount of consonance can be gained (if you are only playing an instrument within a 3-4 Phi-tave range) by moving the PHI-tave to 1.625 (13/8) instead of 1.618: this preserved the odd harmonics/overtones slightly better. Same goes for using around 1.406 (to estimate 1.4 IE 7/5) for the Silver Ratio scale.

-Michael

🔗rick_ballan <rick_ballan@...>

🔗chrisvaisvil@...

Man o man I don't know if I understand your request. I'm a dummy in tunings. My keyboard tonight only has 49 keys. Korg ms2000. But ill try 36 edo as soon as I up load what I just did.

Chris.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "rick_ballan" <rick_ballan@yahoo.com.au>

Date: Sat, 02 May 2009 02:22:26
To: <tuning@yahoogroups.com>
Subject: [tuning] Re: PHI interval tuning (for Michael S): with "Phicicles" song example :-)

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I'll try to get to it today or tomorrow. I owe some vocals and then a poetry
> reading to some collaborators which I want to get to tonight. Your examples
> sound as if they would be great on a hammer dulcimer.
>
> Rick: - you want me to try 36 EDO right? Can you identify by rank the pitch
> that is the Phi from the start of the tuning?
>
> (IE - the 7th or 11th note in the tuning is PHI away from the starting
> frequency.)
>
> Chris

Hi Chris,

Yeah that would be great mate. Michael already sent an example of the phi chords a few days ago but there's so many more possibilities in 36 edo. The PHI pitch is 25, that is 2^(25/36) = 1.618261 from 1, but it needs to appear with its inverse 2^-(25/36) = 0.617947 (or in one 8ve, 2^(11/36) = 1.235894).

Two symmetrical chords of interest, 1).repeated (i.e.successive) applications of 2^(4/36) = 1.08006 gives a 9 note chord which includes a type of diminished tonality very close to the minor 3rd as 7/6 AND an augmented triad together, 2). repeated applications of 2^(5/36) = 1.101057 gives all 36 notes over 5 8ves, but it does divide into PHI i.e. 25, so applying it 5 times to get you to PHI should do. If possible, I'd like to hear them as scales and then chords.

Muchly appreciated

Rick
>
> On Thu, Apr 30, 2009 at 12:09 AM, <djtrancendance@...> wrote:
>
> >
> >
> >
> > Chris.
> > I made a mistake with my PHI scale...once of the notes (at about 443.17
> > cents) was a mis-copy and sound dissonant vs. virtually any other note in
> > the scale. :-(
> >
> > I
> >
>

🔗rick_ballan <rick_ballan@...>

Hi Mike, Do you mean instruments that have fairly uniform tone? I would suggest trying many instruments, even in combo, and seeing which works best for each and every passage in a composition. But I find that using fairly uniform tone (like string or woodwind quartet for classical music) is good for checking original harmonies.
>
> >"I'd like to hear a
> version parred back even further, perhaps using only the most consonant
> sounding intervals (I want to see if its possible to build a very
> consonant system to form a strong basis and go outward from there)."
>
>
> Indeed...in that case I would have to space out the notes more to avoid beating to a high extent. The funny thing is the beating between any two notes, at least to my ear, sound about equally pleasant in their speed/modulation...and the only way to go further is to reduce the beating entirely by spreading the intervals out.

Not necessarily Mike. The original phi chord you sent had beats. Playing long extended notes would also produce them.
>
> One thing is almost certainly for sure (and in common in both of our scale)...the purest interval set (kind of the equivalent of a major triad) within a "PHI-tave"/period appears to be 1 2/PHI PHI. Next in line is probably 1 2/PHI 1/PHI^2+1 PHI...which is kinda like major 7th chord in tensity. Anything much denser than that...and you are going to have some "semi-tone" style amounts of beating (ah...the horrors of realistic psycho-acoustics). :-D

You're being unclear again. What's 1 2/PHI PHI and 1 2/PHI 1/PHI^2+1 PHI...? This could mean 12/(phi^2) or 1 x 2/phi x phi etc... Ah, got visitors, have to get back to you.
>
> >"Nice one."
> Thanks!...I'm glad you like it (the last sound example "PHIcicles"). To be realistic, though, I think Chris hit the nail on the head
> before when he said the Silver Ratio is a more consonant scale and I will admit there are some sour spots I found when I approached the challenge of using the PHI scale in a piece involving more than one 2/1 octave worth of frequency range.
>
> In dealing with the above issue/problem, I have also noticed a good amount of consonance can be gained (if you are only playing an instrument within a 3-4 Phi-tave range) by moving the PHI-tave to 1.625 (13/8) instead of 1.618: this preserved the odd harmonics/overtones slightly better. Same goes for using around 1.406 (to estimate 1.4 IE 7/5) for the Silver Ratio scale.
>
> -Michael
>

🔗Michael Sheiman <djtrancendance@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> Rick>"There are those interesting beat frequencies again"
> ..which seems to be a fairly unique property of scales based on fractal numbers...considering this pops up in both the PHI and Silver Ratio based scales... :-)
>
> > "Those sounds are much better than the piano."
> Agreed...and it seems to be coincidence that choice of instrument matters more with the PHI scale than it does the Silver Ratio one. This time around, I used a very clean sounding guitar...which still has the brightness of the piano but without the dis-harmonies evident in the piano's timbre.
> Thus I wonder...do you think it's fair the say that instruments which adhere strongly to consistent harmonic ratios, such as guitars, should be highly recommended timbres for my PHI scale?
>
> >"I'd like to hear a
> version parred back even further, perhaps using only the most consonant
> sounding intervals (I want to see if its possible to build a very
> consonant system to form a strong basis and go outward from there)."
>
>
> Indeed...in that case I would have to space out the notes more to avoid beating to a high extent. The funny thing is the beating between any two notes, at least to my ear, sound about equally pleasant in their speed/modulation...and the only way to go further is to reduce the beating entirely by spreading the intervals out.
>
Hi Mike (back again). Well beating and dissonance are not necessarily the same thing so avoiding beating is not the issue. Besides, spacing notes should (theoretically) still produce beats with these numbers. And playing long notes should produce more beats than short ones so that speed or spacing don't come into it. I think the real issue is compositional. For example, starting a piece with notes long enough for the ear to get a grip on and then building chords slowly in layers, and things like that. In other words, dissonance is usually caused when things seem out of place. The fact that beating sometimes produces undesirable notes in tonal compositions is only one case in point.

> One thing is almost certainly for sure (and in common in both of our scale)...the purest interval set (kind of the equivalent of a major triad) within a "PHI-tave"/period appears to be 1 2/PHI PHI. Next in line is probably 1 2/PHI 1/PHI^2+1 PHI...which is kinda like major 7th chord in tensity. Anything much denser than that...and you are going to have some "semi-tone" style amounts of beating (ah...the horrors of realistic psycho-acoustics). :-D
>
> >"Nice one."
> Thanks!...I'm glad you like it (the last sound example "PHIcicles"). To be realistic, though, I think Chris hit the nail on the head
> before when he said the Silver Ratio is a more consonant scale and I will admit there are some sour spots I found when I approached the challenge of using the PHI scale in a piece involving more than one 2/1 octave worth of frequency range.

In light of what I just said about dissonance, I don't buy that for a second. It might just happen to be true in your two pieces, but just to refer back to my approach as an example, a 2000 tet (for the silver)approximates far more rarified upper harmonics than a 36 (phi) tet. And there are loads of ideas which have not been explored yet. The semi-symmetrical phi chord 0:11:14:25 (including two 14's from the outsides and one 3 in the middle, equating to phi^2 and phi^3), the ordered scale based on phi squared sequence 0:25:50=14:75=3, etc...Then there is the fully symmetrical six-note scale 0:5:10:15:20:25 (and even this strange scale 0:4:8:16:20:24:28:32 which includes a diminished type chord based on the interval '8', very close to the flattened minor third 7/6, AND something close to a major third 21 giving a type of augmented triad 0:20:40=4). And while 13/8 is one tonal interval with odd numerator and 2^N=1,2,3 in the denom (i.e.8ve's), others are 207/128 = 1.617..., 829/512 = 1.619.. and 1657/1024 = 1.618...These are very small in comparison to the silver ratio I gave the other day. So my feeling is that we haven't even scratched the surface yet, way too early to talk about consonance/dissonance.

Rick
>
> In dealing with the above issue/problem, I have also noticed a good amount of consonance can be gained (if you are only playing an instrument within a 3-4 Phi-tave range) by moving the PHI-tave to 1.625 (13/8) instead of 1.618: this preserved the odd harmonics/overtones slightly better. Same goes for using around 1.406 (to estimate 1.4 IE 7/5) for the Silver Ratio scale.
>
> -Michael
>

🔗chrisvaisvil@...

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Man o man I don't know if I understand your request. I'm a dummy in tunings. My keyboard tonight only has 49 keys. Korg ms2000. But ill try 36 edo as soon as I up load what I just did.
>
> Chris.
> Sent via BlackBerry from T-Mobile

Hi Chris,

I don't have scala and haven't got a clue whether you reprogram keyboard notes or go straight to midi (I assumed the latter). Obviously you wouldn't fit 36 into the standard keyboard 8ve, but it would fit into 3 8ve's. The interval that replaces the semitone is 2^(1/36) = 1.019441 and every successive tone is reached by multiplying this with itself. eg the interval that replaces the whole tone is (1.019441)^2 = 1.039259, the note assigned to the minor third is (1.019441)^3 = 1.059463 and so on till you do it 36 times to reach the 8ve. Once the keys are retuned, we can call them 0,1,2,...36. The Phi intervals are 11 and 25 i.e. B and Db from C in usual tuning. Does that help?

Rick

>
> -----Original Message-----
> From: "rick_ballan" <rick_ballan@...>
>
> Date: Sat, 02 May 2009 02:22:26
> To: <tuning@yahoogroups.com>
> Subject: [tuning] Re: PHI interval tuning (for Michael S): with "Phicicles" song example :-)
>
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
> >
> > I'll try to get to it today or tomorrow. I owe some vocals and then a poetry
> > reading to some collaborators which I want to get to tonight. Your examples
> > sound as if they would be great on a hammer dulcimer.
> >
> > Rick: - you want me to try 36 EDO right? Can you identify by rank the pitch
> > that is the Phi from the start of the tuning?
> >
> > (IE - the 7th or 11th note in the tuning is PHI away from the starting
> > frequency.)
> >
> > Chris
>
> Hi Chris,
>
> Yeah that would be great mate. Michael already sent an example of the phi chords a few days ago but there's so many more possibilities in 36 edo. The PHI pitch is 25, that is 2^(25/36) = 1.618261 from 1, but it needs to appear with its inverse 2^-(25/36) = 0.617947 (or in one 8ve, 2^(11/36) = 1.235894).
>
> Two symmetrical chords of interest, 1).repeated (i.e.successive) applications of 2^(4/36) = 1.08006 gives a 9 note chord which includes a type of diminished tonality very close to the minor 3rd as 7/6 AND an augmented triad together, 2). repeated applications of 2^(5/36) = 1.101057 gives all 36 notes over 5 8ves, but it does divide into PHI i.e. 25, so applying it 5 times to get you to PHI should do. If possible, I'd like to hear them as scales and then chords.
>
> Muchly appreciated
>
> Rick
> >
> > On Thu, Apr 30, 2009 at 12:09 AM, <djtrancendance@> wrote:
> >
> > >
> > >
> > >
> > > Chris.
> > > I made a mistake with my PHI scale...once of the notes (at about 443.17
> > > cents) was a mis-copy and sound dissonant vs. virtually any other note in
> > > the scale. :-(
> > >
> > > I
> > >
> >
>

🔗rick_ballan <rick_ballan@...>

🔗Michael Sheiman <djtrancendance@...>

🔗rick_ballan <rick_ballan@...>

>Hi Mike,

Now that you wrote a clear definition of your approach I can analyse it. You said
<2/PHI means, literally, 2/1.618034 = 1.236067, for example. (1/PHI)^2 + 1 = (0.618034^2) + 1 = (0.3819) + 1 = 1.3819. It's really that simple...>.

Ok, your first value is simply 2^(11/36) = 1.235894. Or to just follow your logic through in indices, 2/1.618034 = 2 x 2^-(25/36) = 2^(36/36)x 2^-(25/36) = 2^(-25+36/36) = 2^(11/36). But this is easier by thinking of 2 as 36, phi as 25, and division becomes subtraction so that 2/phi becomes 36 - 25 = 11.

Now your second number here is more problematic. The value (1/PHI)^2 is ok and can be obtained as (2^-(25/36))^2 = 2^-(50/36) = 0.381859. Remember the first phi example you sent where you said we could hardly tell the difference between the two chords -25:0:25 and 0:11:25? This is because 8ve equivalence almost never seems to alter the harmonic content (This is a uniquely musical thing. Why it is I'm not sure but it's so self-evident we can't explain it). Therefore we can bring this interval within a single 8ve (up two 8ve's) to obtain : -50 + 36 + 36 = 22 and 2^(22/36) = 1.527435. Just to check that this is indeed 8ve equivalent to your original value, dividing by 4 gives 0.381859. Notice also that 22 is 14 from 36 and that 14 is one of the original phi numbers i.e. the interval between phi and its inverse, 11 and 25.

However, the problem begins when you add 1. This is because it changes the harmonic content and is no longer a PHI number. For example, given a perfect fifth as 3/2 = 1.5, adding 1 gives 5/2 and the interval is now a major tenth (maj 3rd separated by an 8ve). Adding 1 again gives 7/2 etc...As you see, the fifth is not preserved by adding 1's. The closest number I could find to your value was 2^(17/36) = 1.387245, which isn't close enough. (I can understand how you got confused. Adding or subtracting 1 to PHI or its inverse is the only exception). So just to repeat, taking 0.381859 x 2 = 0.763718 and x 4 = 1.527435, that is -14 and 22, are the correct intervals from -50.

When Chris said that the silver was more consonant than the phi I knew there was something amiss but couldn't put my finger on it.

Hope this helps

-Rick

> Mike>"Indeed...in that case I would have to space out the notes more to
> avoid beating to a high extent. The funny thing is the beating between
> any two notes, at least to my ear, sound about equally pleasant in
> their speed/modulation. ..and the only way to go further is to reduce
> the beating entirely by spreading the intervals out."
>
>
> Chris> "Not necessarily Mike. The original phi chord you sent had beats. Playing long extended notes would also produce them."
>
> True and I realize that...but I recommended (as I have several times) moving the 1.618034 PHI-tave to the almost indistinguishably different ratio of 1.625 to neutralize the odd-harmonic beats...yet you don't seem to want to do that.
>
> Chris> "You're being unclear again. What's 1 2/PHI PHI and 1 2/PHI 1/PHI^2+1
> PHI...? This could mean 12/(phi^2) or 1 x 2/phi x phi etc... Ah, got
> visitors, have to get back to you."
>
> 2/PHI means, literally, 2/1.618034 = 1.236067, for example. (1/PHI)^2 + 1 = (0.618034^2) + 1 = (0.3819) + 1 = 1.3819. It's really that simple...I don't understand why you seem to keep wanting to shove 12 (from 12TET?) into here for no apparent reason: soon as you start trying to fit this into 12TET logic...you tear apart the very basis of this scale (the PHI-tave and the inverse of it, 1/PHI, which is 0.618 instead of 1.618 which is exactly 1 under it while my formula uses + 1 in many place intentionally (see the connection?)).
>
> -Michael
>

🔗Michael Sheiman <djtrancendance@...>

Rick>"Just to check that this is indeed 8ve equivalent to your original value, dividing by 4 gives 0.381859. The closest number I could find to your value was 2^(17/36) = 1.387245, which isn't close enough."

But, you see, I'm not going by "mod 12" or "mod 36"...which makes my scale fundamentally different than yours.
Take a line going for 0 to 1. Now take 1 and keep multiplying it by 1.618034 (IE splitting it into PHI-ths). That's how you get my 1.3819, 1.23...values. Now take that same line, but move it between the values 1 and 2. Artistically/in architecture, of course, the way PHI splits the line in the exact same way my scale does. Hence the above effect is, in fact, intended.

Rick> "When Chris said that the silver was more consonant than the phi I knew
there was something amiss but couldn't put my finger on it."

Right but, if you read later on...he restated he thought the main problem had nothing to do with different versions of 1.38x or any of the other ratios, but PHI itself IE that 1 and the PHI-tave beat against each other too much. And when I told him to estimate PHI as 1.625 (which has a VERY similar tonal color to 1.618)...he noted "dramatic improvements".

Feel free to show me an example where you swap my 1.3819 with your 1.3872 and prove your version sounds much better...but, as for now, it looks to me just the be a side-issue of aligning 36TET with 12TET.

And, yes, 36TET gives slightly more accurate octave, and I've experimented with that myself by ear and do hear a bit of a difference (PHI vs. octave)...
So what does help though, I've found is
A) using 1.625 IE 13/8 instead of 1.618034
B) making my PHI-scale using the PHI tave...but then cutting the part of it between 1 and 2 (not just between 1 and 1.618) and then using those 8 or so notes as a basis for an octave-based scale (this way you center on both PHI and the octave...without using inversions). Actually, in some ways, this does show my scale somewhat adapting to 36TET...and we might want to try a version of it rounded into 36TET for grins.

-Michael

--- On Sat, 5/2/09, rick_ballan <rick_ballan@...> wrote:

From: rick_ballan <rick_ballan@....au>
Subject: [tuning] Re: PHI interval tuning (for Michael S): with "Phicicles" song example :-)
To: tuning@yahoogroups.com
Date: Saturday, May 2, 2009, 11:20 PM

>Hi Mike,

Now that you wrote a clear definition of your approach I can analyse it. You said

<2/PHI means, literally, 2/1.618034 = 1.236067, for example. (1/PHI)^2 + 1 = (0.618034^2) + 1 = (0.3819) + 1 = 1.3819. It's really that simple...>.

Now your second number here is more problematic. The value (1/PHI)^2 is ok and can be obtained as (2^-(25/36)) ^2 = 2^-(50/36) = 0.381859. Remember the first phi example you sent where you said we could hardly tell the difference between the two chords -25:0:25 and 0:11:25? This is because 8ve equivalence almost never seems to alter the harmonic content (This is a uniquely musical thing. Why it is I'm not sure but it's so self-evident we can't explain it). Therefore we can bring this interval within a single 8ve (up two 8ve's) to obtain : -50 + 36 + 36 = 22 and 2^(22/36) = 1.527435. Just to check that this is indeed 8ve equivalent to your original value, dividing by 4 gives 0.381859. Notice also that 22 is 14 from 36 and that 14 is one of the original phi numbers i.e. the interval between phi and its inverse, 11 and 25.

When Chris said that the silver was more consonant than the phi I knew there was something amiss but couldn't put my finger on it.

Hope this helps

-Rick

> Mike>"Indeed. ..in that case I would have to space out the notes more to

> avoid beating to a high extent. The funny thing is the beating between

> any two notes, at least to my ear, sound about equally pleasant in

> their speed/modulation. ..and the only way to go further is to reduce

> the beating entirely by spreading the intervals out."

> Chris> "Not necessarily Mike. The original phi chord you sent had beats. Playing long extended notes would also produce them."

> True and I realize that...but I recommended (as I have several times) moving the 1.618034 PHI-tave to the almost indistinguishably different ratio of 1.625 to neutralize the odd-harmonic beats...yet you don't seem to want to do that.

> Chris> "You're being unclear again. What's 1 2/PHI PHI and 1 2/PHI 1/PHI^2+1

> PHI...? This could mean 12/(phi^2) or 1 x 2/phi x phi etc... Ah, got

> visitors, have to get back to you."

> 2/PHI means, literally, 2/1.618034 = 1.236067, for example. (1/PHI)^2 + 1 = (0.618034^2) + 1 = (0.3819) + 1 = 1.3819. It's really that simple...I don't understand why you seem to keep wanting to shove 12 (from 12TET?) into here for no apparent reason: soon as you start trying to fit this into 12TET logic...you tear apart the very basis of this scale (the PHI-tave and the inverse of it, 1/PHI, which is 0.618 instead of 1.618 which is exactly 1 under it while my formula uses + 1 in many place intentionally (see the connection?) ).

> -Michael

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> Rick>"Just to check that this is indeed 8ve equivalent to your original value, dividing by 4 gives 0.381859. The closest number I could find to your value was 2^(17/36) = 1.387245, which isn't close enough."
>
> But, you see, I'm not going by "mod 12" or "mod 36"...which makes my scale fundamentally different than yours.
> Take a line going for 0 to 1. Now take 1 and keep multiplying it by 1.618034 (IE splitting it into PHI-ths). That's how you get my 1.3819, 1.23...values. Now take that same line, but move it between the values 1 and 2. Artistically/in architecture, of course, the way PHI splits the line in the exact same way my scale does. Hence the above effect is, in fact, intended.

Ok Mike, you asked me if I could find you a PHI tuning system and I did. It is elegant and simple, something which you look for in both mathematical proofs and musical composition. The discovery that we can reach PHI with a 36 note equal tempered system means that any phi chord or composition etc...can be transposed 35 times, which means 35 modulations. And because every PHI number can generate all 35 notes, then it can indeed be seen as a PHI based system. So statements like "And, yes, 36TET gives slightly more accurate octave", which shows such a basic error, convinces me that you don't yet understand or fully appreciate it. Because it hits the octave EXACTLY which is WHY we adopt powers to the base 2. Taking the product of the 25th and 11th intervals gives 2, the perfect 8ve. Multiplying either PHI 25 or inverse PHI 11 by themselves 36 times will arrive at perfect 8ve equivalents. And if you read my past posts correctly you'll also see that I'm multiplying/dividing by 2 all the time (i.e. or adding/subtracting 36).

Again when you say "Feel free to show me an example where you swap my 1.3819 with your 1.3872 and prove your version sounds much better...but, as for now, it looks to me just the be a side-issue of aligning 36TET with 12TET", since this has absolutely nothing to do with 12 tet, then this just shows me more proof that you are using it as an excuse to avoid studying the mathematics I send. And where did you get that I said 1.3872 was the correct interval? Cause I said that 0.3819 was the correct PHI derivative, equal to 2^-(50/36), that is, inverse Phi squared. I also said that adding 1 to this interval, which is your number, was no longer in the PHI system. And below I see you are now replacing PHI by 13/8! We could have done from the word go, so it seems like a big step backwards to me (and besides, I thought you didn't like JI?). Come on Mike, I put allot of work into these posts and you could at least have the courtesy of reading them properly. The problem is solved, just apply it.

-Rick
>
>

Rick> "When Chris said that the silver was more consonant than the phi I knew
> there was something amiss but couldn't put my finger on it."
>
> Right but, if you read later on...he restated he thought the main problem had nothing to do with different versions of 1.38x or any of the other ratios, but PHI itself IE that 1 and the PHI-tave beat against each other too much. And when I told him to estimate PHI as 1.625 (which has a VERY similar tonal color to 1.618)...he noted "dramatic improvements".
>
> Feel free to show me an example where you swap my 1.3819 with your 1.3872 and prove your version sounds much better...but, as for now, it looks to me just the be a side-issue of aligning 36TET with 12TET.
>
> And, yes, 36TET gives slightly more accurate octave, and I've experimented with that myself by ear and do hear a bit of a difference (PHI vs. octave)...
> So what does help though, I've found is
> A) using 1.625 IE 13/8 instead of 1.618034
> B) making my PHI-scale using the PHI tave...but then cutting the part of it between 1 and 2 (not just between 1 and 1.618) and then using those 8 or so notes as a basis for an octave-based scale (this way you center on both PHI and the octave...without using inversions). Actually, in some ways, this does show my scale somewhat adapting to 36TET...and we might want to try a version of it rounded into 36TET for grins.
>
> -Michael
>
>
>
>
> --- On Sat, 5/2/09, rick_ballan <rick_ballan@...> wrote:
>
> From: rick_ballan <rick_ballan@...>
> Subject: [tuning] Re: PHI interval tuning (for Michael S): with "Phicicles" song example :-)
> To: tuning@yahoogroups.com
> Date: Saturday, May 2, 2009, 11:20 PM
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> >Hi Mike,
>
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>
> Now that you wrote a clear definition of your approach I can analyse it. You said
>
> <2/PHI means, literally, 2/1.618034 = 1.236067, for example. (1/PHI)^2 + 1 = (0.618034^2) + 1 = (0.3819) + 1 = 1.3819. It's really that simple...>.
>
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> Ok, your first value is simply 2^(11/36) = 1.235894. Or to just follow your logic through in indices, 2/1.618034 = 2 x 2^-(25/36) = 2^(36/36)x 2^-(25/36) = 2^(-25+36/36) = 2^(11/36). But this is easier by thinking of 2 as 36, phi as 25, and division becomes subtraction so that 2/phi becomes 36 - 25 = 11.
>
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>
> Now your second number here is more problematic. The value (1/PHI)^2 is ok and can be obtained as (2^-(25/36)) ^2 = 2^-(50/36) = 0.381859. Remember the first phi example you sent where you said we could hardly tell the difference between the two chords -25:0:25 and 0:11:25? This is because 8ve equivalence almost never seems to alter the harmonic content (This is a uniquely musical thing. Why it is I'm not sure but it's so self-evident we can't explain it). Therefore we can bring this interval within a single 8ve (up two 8ve's) to obtain : -50 + 36 + 36 = 22 and 2^(22/36) = 1.527435. Just to check that this is indeed 8ve equivalent to your original value, dividing by 4 gives 0.381859. Notice also that 22 is 14 from 36 and that 14 is one of the original phi numbers i.e. the interval between phi and its inverse, 11 and 25.
>
>
>
> However, the problem begins when you add 1. This is because it changes the harmonic content and is no longer a PHI number. For example, given a perfect fifth as 3/2 = 1.5, adding 1 gives 5/2 and the interval is now a major tenth (maj 3rd separated by an 8ve). Adding 1 again gives 7/2 etc...As you see, the fifth is not preserved by adding 1's. The closest number I could find to your value was 2^(17/36) = 1.387245, which isn't close enough. (I can understand how you got confused. Adding or subtracting 1 to PHI or its inverse is the only exception). So just to repeat, taking 0.381859 x 2 = 0.763718 and x 4 = 1.527435, that is -14 and 22, are the correct intervals from -50.
>
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> When Chris said that the silver was more consonant than the phi I knew there was something amiss but couldn't put my finger on it.
>
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>
> Hope this helps
>
>
>
> -Rick
>
>
>
> > Mike>"Indeed. ..in that case I would have to space out the notes more to
>
> > avoid beating to a high extent. The funny thing is the beating between
>
> > any two notes, at least to my ear, sound about equally pleasant in
>
> > their speed/modulation. ..and the only way to go further is to reduce
>
> > the beating entirely by spreading the intervals out."
>
> >
>
> >
>
> > Chris> "Not necessarily Mike. The original phi chord you sent had beats. Playing long extended notes would also produce them."
>
> >
>
> > True and I realize that...but I recommended (as I have several times) moving the 1.618034 PHI-tave to the almost indistinguishably different ratio of 1.625 to neutralize the odd-harmonic beats...yet you don't seem to want to do that.
>
> >
>
> > Chris> "You're being unclear again. What's 1 2/PHI PHI and 1 2/PHI 1/PHI^2+1
>
> > PHI...? This could mean 12/(phi^2) or 1 x 2/phi x phi etc... Ah, got
>
> > visitors, have to get back to you."
>
> >
>
> > 2/PHI means, literally, 2/1.618034 = 1.236067, for example. (1/PHI)^2 + 1 = (0.618034^2) + 1 = (0.3819) + 1 = 1.3819. It's really that simple...I don't understand why you seem to keep wanting to shove 12 (from 12TET?) into here for no apparent reason: soon as you start trying to fit this into 12TET logic...you tear apart the very basis of this scale (the PHI-tave and the inverse of it, 1/PHI, which is 0.618 instead of 1.618 which is exactly 1 under it while my formula uses + 1 in many place intentionally (see the connection?) ).
>
> >
>
> > -Michael
>
> >
>

🔗djtrancendance@...

Rick> "So statements like 'And, yes, 36TET gives slightly more accurate
octave', which shows such a basic error, convinces me that you don't
yet understand or fully appreciate it."

And then you stated (a statement which seems to conflict heavily with what you said above)
>"......Because it hits the octave EXACTLY which is WHY we adopt powers to the base 2."

I do appreciate your scale and its parent tuning (36TET)'s adherence to the standard octave (which seems to be a point you're making, even if you keep denying it, then proving it, then denying it)...I simply think it and my own scales are symmetrical in different ways. I never said "one uses PHI more accurately than the other" and am getting kind of sick of what looks like your pitting your scale against mine rather than combining our discoveries to make more progress.

Before, you criticized my scale for not matching the 8ve (octave), as I understand it, and I simply told you flat out that's not the symmetry I was going for in the first place. Above (again) you say "which is why we adopt powers of base 2"...again re-enforcing the idea you thing the ideal system should, in some way, revolve primarily around the octave.
So, in short, I refuse to go so far
as randomly abandoning my system because it does not match the octave perfectly. I'm not saying either system is better but, rather, I'm not so convinced by the beauty of your system that I am willing to randomly drop my own.

>"I also said that adding 1 to this interval, which is your number, was
no longer in the PHI system."
Which can be wrong depending on how you use PHI, as I explained. If you divide a line between 0 and 1 using PHI you get 0.618. Then dividing the line between 0 and 0.618 in the same fashion gives...0.3819 (the exact number I use).

>"And below I see you are now replacing PHI
by 13/8!"
This is simply to help match the overtones of most acoustic instruments. 1.618 works 100% great with sine wave...but runs into some conflicts with higher overtones (it does in your own scale >as well as< mine, BTW). So what would and ideal instrument be for either of our scales? A Sethares-ian one with harmonics "bent" to match the scale. Which was my point there.

Rick>"Come on Mike, I put allot of work into these posts and you could at least have the courtesy of reading them properly."

Argh...here we go again. How about reading my posts properly or, if you don't understand, asking me rather than assigning random intentions to my words?

Again, I understand the basic inversion principles with your scale and am doing things like round 1.618034 to 1.625 (13/8) simply to deal with problems of overlapping overtones with PHI itself (a problem with using PHI in
any scale with instruments containing harmonic-series-type overtone, including either of our scales).

And...I agree 2/1 and 1.618 are both good period measures and it would be nice to match both perfectly. But, if you consider your scale (not the whole 36TET tuning) examples so far don't intersect both 2/1 and PHI...I hope you'll realize it's quite likely no consonant scale will intersect both the 2/1 octave and the PHI-tave perfectly between any possible set of notes.
So your scale and mine both have their own benefits and problems, both are agreeable >quite< loyal to direct use of PHI (I don't see why you keep insisting mine isn't)...and imvho the problem, in many ways, becomes "how can we make a scale that best balances symmetry between the phi-tave and the octave...since we can't hit both dead on for every combination of intervals?"

Rick, hopefully you will read this and we can both get on the
same page trying to solve problems like the above rather than doing this "my scale is right therefore your scale must be wrong" sort of b.s. I don't think your scale is wrong why are you bringing this up that mine suddenly is...it doesn't make any sense.

-Michael

🔗rick_ballan <rick_ballan@...>

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

>Hi Mike,

Yeah I agree about getting on the same page. And it might be a communication thing. Eg: It was you who said <36TET gives slightly more accurate octave> to which I answered in frustration <it hits the octave EXACTLY which is WHY we adopt powers to the base 2>. In other words, I took you to mean that 36TET doesn't hit the 8ve, whereas now I understand you meant something like "Sure, 36TET does reach the 8ve but there are other problems to consider". And for the record, I would never say that 36TET doesn't reach the 8ve so you've obviously misunderstood me somewhere along the line too. No worries.

Getting back to business, for the record I DO believe in the sanctity of the 8ve for musical tunings and that its not just "one period/interval generator among so many" like PHI. This is because it is impossible to think of an 8ve as a scale generator, which would just give the same note over and over in different...octaves. You see it's so self-evident that I can't even find words to replace it! So the challenge was to find a system which included PHI while maintaining the 8ve, which 36 does satisfy. Of course, we can find freq's that approximate 2/1 just like other intervals, 1.99879...2.00314...etc and round them off according to (say) Carl's/Erlich's harmonic entropy, so I'm not denying that you might find 8ve's in different ways.

-Rick
>
> Rick> "So statements like 'And, yes, 36TET gives slightly more accurate
> octave', which shows such a basic error, convinces me that you don't
> yet understand or fully appreciate it."
>
>
> And then you stated (a statement which seems to conflict heavily with what you said above)
> >"......Because it hits the octave EXACTLY which is WHY we adopt powers to the base 2."
>
> I do appreciate your scale and its parent tuning (36TET)'s adherence to the standard octave (which seems to be a point you're making, even if you keep denying it, then proving it, then denying it)...I simply think it and my own scales are symmetrical in different ways. I never said "one uses PHI more accurately than the other" and am getting kind of sick of what looks like your pitting your scale against mine rather than combining our discoveries to make more progress.
>
> Before, you criticized my scale for not matching the 8ve (octave), as I understand it, and I simply told you flat out that's not the symmetry I was going for in the first place. Above (again) you say "which is why we adopt powers of base 2"...again re-enforcing the idea you thing the ideal system should, in some way, revolve primarily around the octave.
> So, in short, I refuse to go so far
> as randomly abandoning my system because it does not match the octave perfectly. I'm not saying either system is better but, rather, I'm not so convinced by the beauty of your system that I am willing to randomly drop my own.
>
> >"I also said that adding 1 to this interval, which is your number, was
> no longer in the PHI system."
> Which can be wrong depending on how you use PHI, as I explained. If you divide a line between 0 and 1 using PHI you get 0.618. Then dividing the line between 0 and 0.618 in the same fashion gives...0.3819 (the exact number I use).
>
> >"And below I see you are now replacing PHI
> by 13/8!"
> This is simply to help match the overtones of most acoustic instruments. 1.618 works 100% great with sine wave...but runs into some conflicts with higher overtones (it does in your own scale >as well as< mine, BTW). So what would and ideal instrument be for either of our scales? A Sethares-ian one with harmonics "bent" to match the scale. Which was my point there.
>
> Rick>"Come on Mike, I put allot of work into these posts and you could at least have the courtesy of reading them properly."
>
> Argh...here we go again. How about reading my posts properly or, if you don't understand, asking me rather than assigning random intentions to my words?
>
> Again, I understand the basic inversion principles with your scale and am doing things like round 1.618034 to 1.625 (13/8) simply to deal with problems of overlapping overtones with PHI itself (a problem with using PHI in
> any scale with instruments containing harmonic-series-type overtone, including either of our scales).
>
> And...I agree 2/1 and 1.618 are both good period measures and it would be nice to match both perfectly. But, if you consider your scale (not the whole 36TET tuning) examples so far don't intersect both 2/1 and PHI...I hope you'll realize it's quite likely no consonant scale will intersect both the 2/1 octave and the PHI-tave perfectly between any possible set of notes.
> So your scale and mine both have their own benefits and problems, both are agreeable >quite< loyal to direct use of PHI (I don't see why you keep insisting mine isn't)...and imvho the problem, in many ways, becomes "how can we make a scale that best balances symmetry between the phi-tave and the octave...since we can't hit both dead on for every combination of intervals?"
>
> Rick, hopefully you will read this and we can both get on the
> same page trying to solve problems like the above rather than doing this "my scale is right therefore your scale must be wrong" sort of b.s. I don't think your scale is wrong why are you bringing this up that mine suddenly is...it doesn't make any sense.
>
> -Michael
>

🔗djtrancendance@...

Rick>"Eg: It was you who said <36TET gives slightly more accurate
octave> to which I answered in frustration <it hits the octave"
Ok enough drama. :-D
Consensus (I hope)...is that we both agree 36TET gives a perfect/better 2/1 octave...but there are indeed other things to consider.

BTW, I have been working on a modified version of my PHI scale which tries to take the best properties of both our scales and preserves the 2/1 octave perfectly and the PHI octave near-perfectly.

And, for the record, I think hitting both intervals dead-on is something not possible...but a good ideal to aim toward nonetheless (kind of like approaching an infinite limit).

>"So the challenge was to find a system which included PHI while maintaining the 8ve, which 36 does satisfy."
Agreed...if you use the tones (more or less) of 0,2/PHI,PHI,and 2/1...intervals we both agree are good. These are indeed, it appears, the purest notes in either of our systems...which explains why both of our systems have these tones in
common.

The problem is if you want more than those tones (and, come on, you do want more than a 3-tone scale don't you?...then you end up getting something either not perfectly symmetrical to PHI between all possible dyads or not perfectly symmetrical to the octave. So, IMVHO, we have to compromise to get those extra tones...and just duplicating everything over the 2/1 octave is going to create some distortions in PHI-tave symmetry and vice-versa.

Rick> "Of course, we can find freq's that approximate 2/1 just like other
intervals, 1.99879...2. 00314...etc and round them off"

But, you see, that's how my PHI scale works...it either hits very very close intervals (like those you mention above) or simply hits intervals like 1.9 and 2.1 that are just far enough away from the octave so they don't "fight" with it or form violent beating against it.

Again, I swear compromise is the word of the day as basing a scale both perfectly on the PHI-tave and the octave is not mathematically possible (though I agree it would be nice if it was possible).

-Michael

P.S.-
For the record my new "octave-complaint" PHI scale (which keeps the octave while going through great links to also preserve the PHI-tave as much as possible) is
1
1.05572
1.1459
1.23607
1.3819
1.52
1.625
1.77152
2/1 (octave)

I will provide a sound example of this scale in practice soon.

🔗Cameron Bobro <misterbobro@...>

I don't understand why you want to use Phi, but at the same time avoid its acoustic properties, and the proportional properties of the golden cut.

Of course it doesn't really matter, eventualy you will find what you're looking for by picking through a big field of what are ultimately more or less random possibilities.

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> Rick>"Eg: It was you who said <36TET gives slightly more accurate
> octave> to which I answered in frustration <it hits the octave"
> Ok enough drama. :-D
> Consensus (I hope)...is that we both agree 36TET gives a perfect/better 2/1 octave...but there are indeed other things to consider.
>
> BTW, I have been working on a modified version of my PHI scale which tries to take the best properties of both our scales and preserves the 2/1 octave perfectly and the PHI octave near-perfectly.
>
> And, for the record, I think hitting both intervals dead-on is something not possible...but a good ideal to aim toward nonetheless (kind of like approaching an infinite limit).
>
> >"So the challenge was to find a system which included PHI while maintaining the 8ve, which 36 does satisfy."
> Agreed...if you use the tones (more or less) of 0,2/PHI,PHI,and 2/1...intervals we both agree are good. These are indeed, it appears, the purest notes in either of our systems...which explains why both of our systems have these tones in
> common.
>
> The problem is if you want more than those tones (and, come on, you do want more than a 3-tone scale don't you?...then you end up getting something either not perfectly symmetrical to PHI between all possible dyads or not perfectly symmetrical to the octave. So, IMVHO, we have to compromise to get those extra tones...and just duplicating everything over the 2/1 octave is going to create some distortions in PHI-tave symmetry and vice-versa.
>
> Rick> "Of course, we can find freq's that approximate 2/1 just like other
> intervals, 1.99879...2. 00314...etc and round them off"
>
> But, you see, that's how my PHI scale works...it either hits very very close intervals (like those you mention above) or simply hits intervals like 1.9 and 2.1 that are just far enough away from the octave so they don't "fight" with it or form violent beating against it.
>
> Again, I swear compromise is the word of the day as basing a scale both perfectly on the PHI-tave and the octave is not mathematically possible (though I agree it would be nice if it was possible).
>
> -Michael
>
> P.S.-
> For the record my new "octave-complaint" PHI scale (which keeps the octave while going through great links to also preserve the PHI-tave as much as possible) is
> 1
> 1.05572
> 1.1459
> 1.23607
> 1.3819
> 1.52
> 1.625
> 1.77152
> 2/1 (octave)
>
> I will provide a sound example of this scale in practice soon.
>

🔗Michael Sheiman <djtrancendance@...>

>"Of course it doesn't really matter, eventually you will find what you're
looking for by picking through a big field of what are ultimately more
or less random possibilities."
Actually, my PHI theories have become progressively less random. My first set of scales was generated from the tuning PHI^x/2^y...where x = 0 to about 20...which has a massive number of tones per phi-tave (about 16 or so).
My latest method res = (0.618^x) + 1 (where res must be > 1.05 for sake of obeying the critical band) creates less than 5 results/notes, all of which can be mirrored (meaning take 2/1 - the results created above) to create up to about 9 notes that split the "line between 1 and 2" in a fashion similar to how PHI is used in real-world architecture.

Now compare those 9 tones possible above to diatonic mean-tone which generates 12 notes. If anything, my method is less random than the standard.

For the record, the acoustic property I am looking to enhance is proportionate beating...rather than simple having any two consecutive tones beat at the same rate (something the harmonic series does).

Put that together with the fact PHI itself has a similar feel to the octave (IE you can play 'random' notes in the PHI scale and when you hit PHI your mind says "octave!") and the fact 0 2/PHI PHI and -1/PHI 0 2/PHI have virtually the same sense of root tone (just like consecutive harmonics in the harmonic series)...and I think it's clear PHI does have some special and useful acoustic properties.

Not to say they are superior meant to replace anything...but rather they are relatively un-explored and offer a new alternative to musicians that are fairly on-par with more established musical systems.
Given that I've written two pieces using the above "split line fractal scale generation system" that have received 4-5 stars among everyday listeners (better than most of my 12TET songs...and I'm not that good a composer)...it seems to say, for whatever reason, that there is something acoustically special about PHI.

-Michael

--- On Tue, 5/5/09, Cameron Bobro <misterbobro@...> wrote:

From: Cameron Bobro <misterbobro@...>
Subject: [tuning] Re: PHI interval tuning (for Michael S): with "Phicicles" song example :-)
To: tuning@yahoogroups.com
Date: Tuesday, May 5, 2009, 2:15 AM

I don't understand why you want to use Phi, but at the same time avoid its acoustic properties, and the proportional properties of the golden cut.

Of course it doesn't really matter, eventualy you will find what you're looking for by picking through a big field of what are ultimately more or less random possibilities.

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

> Rick>"Eg: It was you who said <36TET gives slightly more accurate

> octave> to which I answered in frustration <it hits the octave"

> Ok enough drama. :-D

> Consensus (I hope)...is that we both agree 36TET gives a perfect/better 2/1 octave...but there are indeed other things to consider.

> BTW, I have been working on a modified version of my PHI scale which tries to take the best properties of both our scales and preserves the 2/1 octave perfectly and the PHI octave near-perfectly.

> And, for the record, I think hitting both intervals dead-on is something not possible...but a good ideal to aim toward nonetheless (kind of like approaching an infinite limit).

> >"So the challenge was to find a system which included PHI while maintaining the 8ve, which 36 does satisfy."

> Agreed...if you use the tones (more or less) of 0,2/PHI,PHI, and 2/1...intervals we both agree are good. These are indeed, it appears, the purest notes in either of our systems...which explains why both of our systems have these tones in

> common.

> The problem is if you want more than those tones (and, come on, you do want more than a 3-tone scale don't you?...then you end up getting something either not perfectly symmetrical to PHI between all possible dyads or not perfectly symmetrical to the octave. So, IMVHO, we have to compromise to get those extra tones...and just duplicating everything over the 2/1 octave is going to create some distortions in PHI-tave symmetry and vice-versa.

> Rick> "Of course, we can find freq's that approximate 2/1 just like other

> intervals, 1.99879...2. 00314...etc and round them off"

> But, you see, that's how my PHI scale works...it either hits very very close intervals (like those you mention above) or simply hits intervals like 1.9 and 2.1 that are just far enough away from the octave so they don't "fight" with it or form violent beating against it.

> Again, I swear compromise is the word of the day as basing a scale both perfectly on the PHI-tave and the octave is not mathematically possible (though I agree it would be nice if it was possible).

> -Michael

> P.S.-

> For the record my new "octave-complaint" PHI scale (which keeps the octave while going through great links to also preserve the PHI-tave as much as possible) is

> 1

> 1.05572

> 1.1459

> 1.23607

> 1.3819

> 1.52

> 1.625

> 1.77152

> 2/1 (octave)

> I will provide a sound example of this scale in practice soon.

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:

> Now compare those 9 tones possible above to diatonic
> mean-tone which generates 12 notes. If anything, my method
> is less random than the standard.

I still don't know what you're after -- you've mentioned
successive cuts of the golden ratio with itself, you've mentioned
consonant 7-tone chords that can also serve as scales, and
you've mentioned beating (as in this message).

But anyway, meantone generates series of scales like 5, 7, 12,
19, 31... And there's a very natural, strong, and fundamental
justification and logic to the meantone system.

> For the record, the acoustic property I am looking to enhance
> is proportionate beating...

Can you define "proportionate beating"?

> rather than simple having any two consecutive tones beat at the
> same rate (something the harmonic series does).

Does not. :P

>Put that together with the fact PHI itself has a similar feel
>to the octave (IE you can play 'random' notes in the PHI scale
>and when you hit PHI your mind says "octave!")

To the extent that this is true, it can be true for any
interval used as the period of a scale.

>and I think it's clear PHI does have some special and useful
>acoustic properties.

Why on earth do you think that? It isn't clear. You haven't
made a single clear example of an acoustic property phi is
supposed to have.

> Given that I've written two pieces using the above "split
> line fractal scale generation system" that have received 4-5
> stars among everyday listeners

Almost any scale or tuning system can be used to write music
enjoyed by everyday listeners, or experts, or any other
group of people you care to name.

>(better than most of my 12TET songs...and I'm not that good
>a composer)...

If it inspires you, that's great. If you want to talk about
objective things it does, that's great too, but you haven't yet.

-Carl

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
> Take a line going for 0 to 1. Now take 1 and keep multiplying it by 1.618034 (IE splitting it into PHI-ths). That's how you get my 1.3819, 1.23...values. Now take that same line, but move it between the values 1 and 2. Artistically/in architecture, of course, the way PHI splits the line in the exact same way my scale does.

I do not think that this analogy works for pitches. Yes, the line intervals 0-1, 1-2 and 2-3 are equivalent on a piece of graph paper, or indeed on an artist's canvas, but not in the context of pitches. The human ear perceives pitches according to a logarithmic scale, so that 1-2 is equivalent to 2-4 (not 2-3) and to 0.5-1 (not 0-1). That is why multiplying and dividing are much more usual than adding and subtracting, when dealing with pitches. That's not to say your way cannot work, but it is unusual.

Consider the notes 1+1/p^n where p=3/2, for n=1,2,3,4; they are 5/3, 13/9, 35/27, 97/81. None of them has an obvious similarity to the original 3/2. I think the same applies to your scale where p=1.618.

AKJ's method, using 2^p (where p=0.618, or a Fibonacci-number approximation to it) is in fact splitting the octave geometrically (rather than arithmetically) and it seems to me to be the closest aural analogy to the visual splitting of a line segment.

If you want to use 1.618 itself as the generator, that is fine, and is different. The standard theories for creating scales with useful properties make use of generators - either (a) a single generator of a small size (e.g. 2^(1/12)) or (b) two or more generators which may be of a larger size (e.g. 2^(7/12)) and 2). And to repeat the point, the generators are usually combined multiplicatively, not additively. I understand your main point to be to use 1.618 as the main generator. It is too large to be the only generator, so a promising route seems to be to look for a second one.

Again, that is not to say your way cannot work. But at the moment I do not understand how to extend from your original five notes to a larger set for use in compositions. And will there be any symmetries and self-similarities there?

Regards,
Steve M.

🔗djtrancendance@...

Carl> "I still don't know what you're after -- you've mentioned successive cuts of the golden ratio with itself, you've mentioned consonant 7-tone chords that can also serve as scales, and you've mentioned beating (as in this message)."

Basically, I'm looking for ways (not just one, but several) to push the closer to the edge of the critical band and maximize the # of available tonal colors in as many ways as possible while making consonance more-or-less nothing worse than what you would hear in a jazz chord such as a 9th or 13th chord. I've been messing with 2 main ideas: using PHI^x/2^y to generate tones (rather similar to Rick's method) and taking (1/PHI)^x + 1 where x = 0 to 6 (my new method)...but both aim to get the best possible symmetry using PHI (or any other fractal/"noble" number put into the above formula).

As you've said yourself, it is hard to get anything more than a 4-note chord (per 2/1 period) to sound fairly consonant with JI...and, in other words, I am looking to push past that limitation.

Proportionate beating and symmetry in general (although I know you disagree with me)...is still the major thrust of my argument and reason for using PHI and other highly symmetrical fractal ratios (though people here seem more eager to discuss PHI than, say, the Silver Ratio). Other parts of the argument include fine-tuning the individual notes for brighter color and slightly tempering notes off PHI (IE 1.625 (13/8) instead of 1.618034) to get virtually the same tonal color and beating symmetry as 1.618034 and hit both symmetry with the PHI-tave and the harmonic series that way. Side note: 1.625 is only about 7 cents off from 1.618034.

In fact...you could even go so far as to say I am simply trying to produce as many symmetries as possible (to the PHi-tave, octave, tritave, symmetrical beating...) to make the mind/ear feel at home to the point notes can be closer spaced and applied in more combinations to make more chords...without getting the usual sense of buzzing and/or roughness you get when you are at the edge of where two notes push near the edge of the range of critical band roughness.

BTW...the scale I am happiest with so far is the Silver Ratio scale, which hits the Silver-tave, octave, tritave, "4-tave"...almost perfectly. And it uses a full 10 tones per 2/1 period.

Want to know how it sounds? Try this (and see if you still wonder "what's the musical point of all
this"...
http://www.geocities.com/djtrancendance/PHI/silverrain.mp3
And even that's with a fairly un-pure instrument (piano)...something that matches the harmonic series more closely, like a guitar, would further improve the clarity/consonance. BTW, the above preview is actually doing better on Trax In Space than most of my 12TET songs and no one has even figured out it's micro-tonal yet according to the comments...which is kind of the greater goal: to make scales with very complex new structures that somehow fool the mind into thinking they are nearly as relaxed as simple harmony-based structure.

-Michael

🔗Carl Lumma <carl@...>

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

>Basically, I'm looking for ways ... to ... maximize the # of
>available tonal colors...

I'm not sure what this means.

>...in as many ways as possible while making consonance
>more-or-less nothing worse than what you would hear
>in a jazz chord such as a 9th or 13th chord.

OK, as I said, you're looking for a chord with around
7 notes that can be used as a scale. We've discussed
several of these already.

>I've been messing with 2 main ideas: using PHI^x/2^y to generate
>tones (rather similar to Rick's method) and taking (1/PHI)^x + 1
>where x = 0 to 6 (my new method)...but both aim to get the best
>possible symmetry using PHI (or any other fractal/"noble" number
>put into the above formula).

There are lot of ways to get symmetry... what makes you
think any of these methods will help you solve the goal
you just stated?

>As you've said yourself, it is hard to get anything more than
>a 4-note chord (per 2/1 period) to sound fairly consonant with
>JI...and, in other words, I am looking to push past that
>limitation.

I never said that and it certainly isn't true.

>Proportionate beating and symmetry in general (although I know
>you disagree with me)...

I don't disagree because you haven't said what it is. I asked
you but you seem to have ignored my question.

>In fact...you could even go so far as to say I am simply trying
>to produce as many symmetries as possible (to the PHi-tave,
>octave, tritave, symmetrical beating...)

How do you know your methods do anything like this?

-Carl

🔗djtrancendance@...

Martin>"Yes, the line intervals 0-1, 1-2 and 2-3 are equivalent on a piece of
graph paper, or indeed on an artist's canvas, but not in the context of
pitches...the human ear perceives pitches according to a logarithmic
scale..."

Right, the human ear is exponential. However, if you take a took at the harmonic series, you'll notice it is based on addition. Meaning the 3rd harmonic is the same distance from the second harmonic as the second is from the first...and not on an exponential scale, but an additive one. Yes, it's quite weird, but what my PHI-based (and silver-ratio-based and other new scales) do is attempt to balance between additive and exponential symmetry.
The most promising result so far, in my book, is that most people seem to be liking this new method better than my old exponential PHI^x/2^y generation system.

>"AKJ's method, using 2^p (where p=0.618, or a Fibonacci-number approximation to it)"
What exactly are the notes in the above method? 2^0.618 just happens to equal 1.5347, which is very close to the 1.528 value my scale...but that's just one note (what about the other 6+
notes)? Do you keep multiplying p by itself (IE 2^(0.618*0.618) = 1.30305, 2^(0.618*0.618*0.618) = 1.17777) or what exactly?

>"But at the moment I do not understand how to extend from your original five notes to a larger set for use in compositions."
Well my formula (1/PHI)^x + 1 for x = 1 to 5 gives:
1) 1.618034
2) 1.5278 (1.618034 - 1.0902: mirrored)
3) 1.3819
4) 1.23606
5) 1.14589
6) 1.0902
7) 1.05572
8) 1
Also note that 1.23606 * PHI is = 2....so it does indeed end up splitting the 2/1 octave as well.
Is that (7 notes per phi-tave) enough possible notes for you?

-Michael

--- On Tue, 5/5/09, martinsj013 <martinsj@lycos.com> wrote:

From: martinsj013 <martinsj@...>
Subject: [tuning] Re: PHI interval tuning (for Michael S):
with "Phicicles" song example :-)
To: tuning@yahoogroups.com
Date: Tuesday, May 5, 2009, 1:45 PM

--- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ ...> wrote:

> Take a line going for 0 to 1. Now take 1 and keep multiplying it by 1.618034 (IE splitting it into PHI-ths). That's how you get my 1.3819, 1.23...values. Now take that same line, but move it between the values 1 and 2. Artistically/ in architecture, of course, the way PHI splits the line in the exact same way my scale does.

Regards,

Steve M.

🔗Michael Sheiman <djtrancendance@...>

>"There are lot of ways to get symmetry... what makes you

think any of these methods will help you solve the goal

you just stated?"
Easy example (that Rick and I worked on)...try playing the chord -1/PHI,1/1,PHI. Now play 1/1,2/PHI,PHI. See how they seem to point to the same virtual root tone and beat as if they were overtones of the same root tone? That's rather like a PHI corollary to the harmonic series.

Mike>As you've said yourself, it is hard to get anything more than

>a 4-note chord (per 2/1 period) to sound fairly consonant with

>JI...and, in other words, I am looking to push past that

>limitation.

Carl>I never said that and it certainly isn't true.
Well you seemed to point to it very strongly when you gave all those examples about tetra-chords. As I recall you also gave me some 13-limit examples of 7-note chords...but those you gave me sounded far more dissonant than what I was aiming for.

Mike>"Proportionate beating and symmetry in general (although I know

>you disagree with me)..."

Carl>"I don't disagree because you haven't said what it is. I asked

>you but you seem to have ignored my question."
Look at my example of the PHI corollary to the harmonic series for an obvious example of the type of symmetry I have in mind.

Mike>In fact...you could even go so far as to say I am simply trying

>to produce as many symmetries as possible (to the PHi-tave,

>octave, tritave, symmetrical beating...)

Carl>How do you know your methods do anything like this?
Rick and I have both been discussing this. Easy example: take 1.23606 from my scale and multiply it by PHI and you get 2/1: it literally nails the octave despite not being generated by it or tempered by the octave any way.

-Michael

🔗Carl Lumma <carl@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
>>There are lot of ways to get symmetry... what makes you
>>think any of these methods will help you solve the goal
>>you just stated?"
>
> Easy example (that Rick and I worked on)...try playing the
> chord -1/PHI,1/1,PHI.

-1/phi ??

> Now play 1/1,2/PHI,PHI. See how they
> seem to point to the same virtual root tone and beat as if they
> were overtones of the same root tone? That's rather like a PHI
> corollary to the harmonic series.

I have no idea what you're talking about. The harmonic
series doesn't do what you said earlier.

>>> As you've said yourself, it is hard to get anything more
>>> than a 4-note chord (per 2/1 period) to sound fairly consonant
>>> with JI...and, in other words, I am looking to push past that
>>> limitation.
>>
>> I never said that and it certainly isn't true.
>
> Well you seemed to point to it very strongly when you gave all
> those examples about tetra-chords.

You mean tetrads? I've never given you any tetrachords.
I think you misunderstood. I said the 7-limit pentads you
asked for don't exist, that's all.

>As I recall you also gave me some 13-limit examples of 7-note
>chords...but those you gave me sounded far more dissonant than
>what I was aiming for.

I told you about harmonic series segments, which you praised
on many occasions, while expressing certain reservations that
were never clearly explained.

>> How do you know your methods do anything like this?
>
> Rick and I have both been discussing this. Easy example: take
> 1.23606 from my scale and multiply it by PHI and you get 2/1:
> it literally nails the octave despite not being generated by it
> or tempered by the octave any way.

What does that have to do with anything? Obviously, you divided
2 by phi if you can multiply by phi to get 2.

-Carl

🔗Carl Lumma <carl@...>

🔗djtrancendance@...

>"I have no idea what you're talking about. The harmonic
series doesn't do what you said earlier."

Ugh...here we go again on a bizarre off-topic tangent.

Back to basics: this is about having a similar emotional feel/sound, and not similar math.
An obvious example: just because an artificial medicine and a vegetable are composed of completely different things does >not< mean they lead toward a different effect/goal (health).

1/PHI = 0.618 (btw, my mistake in saying -1/PHI rather than 1/PHI).
And the chord 0.618 1 1.618, for example, is the three tones
262hz*0.618= 161.916hz, 1*262 = 262hz , and 1.618 * 262 hz = 423hz.

>"You mean tetrads? I've never given you any tetrachords."
Slip of the tongue, yes I meant those 4-note tetrads you gave as examples, my mistake. However, I was close enough that you realized what I was talking about.

>"I told you about harmonic series segments, which you praised
on many occasions, while expressing certain reservations that
were never clearly explained."

I recall saying they sounded good up to about 5-tone-per-octave chords, but not beyond that...and that I want 7+ note per 2/1 interval chords. What reservations are you talking about that I mentioned?

BTW...I recall you dropped the above discussion because of family business...and I respected you were busy & we never really finished the discussion: I wasn't "ignoring you".

>"What does that have to do with anything? Obviously, you divided
2 by phi if you can multiply by phi to get 2."

Here we go with the habit of "assuming Mike did the most stupid thing possible".
BTW, that's >>not<< how I came up with 1.23606!

My scale is created by the formula (1/PHI)^x+1 (and how many times do I have to state this formula before people actually bother to read it...this is about the 25th time I've posted the formula?!)
And, guess what, (1/PHI)^3+1 = 1.23606. Meaning, >>no<<, I did not simply force 1.23606 in there for the sake of argument...it is (and has always been) a part of my scale...only recently Rick and I picked up on the fact...that multiplying it by 1.618034 produces the octave.

I realize my theory may not be perfect, but it sure as hell isn't base-less...unless you really are trying that hard to ignore my repeated explanations.

-Michael

--- On Tue, 5/5/09, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: PHI interval tuning (for Michael S): with "Phicicles" song example :-)
To: tuning@yahoogroups.com
Date: Tuesday, May 5, 2009, 3:12 PM

--- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ ...> wrote:

>>There are lot of ways to get symmetry... what makes you

>>think any of these methods will help you solve the goal

>>you just stated?"

> Easy example (that Rick and I worked on)...try playing the

> chord -1/PHI,1/1,PHI.

-1/phi ??

> Now play 1/1,2/PHI,PHI. See how they

> seem to point to the same virtual root tone and beat as if they

> were overtones of the same root tone? That's rather like a PHI

> corollary to the harmonic series.

I have no idea what you're talking about. The harmonic

series doesn't do what you said earlier.

>>> As you've said yourself, it is hard to get anything more

>>> than a 4-note chord (per 2/1 period) to sound fairly consonant

>>> with JI...and, in other words, I am looking to push past that

>>> limitation.

>> I never said that and it certainly isn't true.

> Well you seemed to point to it very strongly when you gave all

> those examples about tetra-chords.

You mean tetrads? I've never given you any tetrachords.

I think you misunderstood. I said the 7-limit pentads you

asked for don't exist, that's all.

>As I recall you also gave me some 13-limit examples of 7-note

>chords...but those you gave me sounded far more dissonant than

>what I was aiming for.

I told you about harmonic series segments, which you praised

on many occasions, while expressing certain reservations that

were never clearly explained.

>> How do you know your methods do anything like this?

> Rick and I have both been discussing this. Easy example: take

> 1.23606 from my scale and multiply it by PHI and you get 2/1:

> it literally nails the octave despite not being generated by it

> or tempered by the octave any way.

What does that have to do with anything? Obviously, you divided

2 by phi if you can multiply by phi to get 2.

-Carl

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>> I have no idea what you're talking about. The harmonic
>> series doesn't do what you said earlier."
>
> Ugh...here we go again on a bizarre off-topic tangent.
>
> Back to basics: this is about having a similar emotional
> feel/sound, and not similar math.

Well then, just say that. Instead of "proportional beating".

> 1/PHI = 0.618 (btw, my mistake in saying -1/PHI rather
> than 1/PHI). And the chord 0.618 1 1.618, for example,

Ok, got it.

> >"You mean tetrads? I've never given you any tetrachords."
>
> Slip of the tongue, yes I meant those 4-note tetrads you
> gave as examples, my mistake. However, I was close enough
> that you realized what I was talking about.

Close enough that I apparently guessed correctly, but
not close enough that I didn't have to ask.

>> I told you about harmonic series segments, which you praised
>> on many occasions, while expressing certain reservations that
>> were never clearly explained.
>
> I recall saying they sounded good up to about
> 5-tone-per-octave chords, but not beyond that...and that I
> want 7+ note per 2/1 interval chords. What reservations are
> you talking about that I mentioned?

Don't know. But you stopped using them and started on the
phi stuff, so I assumed they weren't doing it for you.

> BTW...I recall you dropped the above discussion because of
> family business...and I respected you were busy & we never
> really finished the discussion: I wasn't "ignoring you".

Don't remember doing that but my wife had a baby a couple
weeks ago.

>>What does that have to do with anything? Obviously, you divided
>>2 by phi if you can multiply by phi to get 2."
>
> My scale is created by the formula (1/PHI)^x+1 (and how many
> times do I have to state this formula before people actually
> bother to read it...this is about the 25th time I've posted
> the formula?!)
> And, guess what, (1/PHI)^3+1 = 1.23606. Meaning, >>no<<,
> I did not simply force 1.23606 in there for the sake of
> argument...it is (and has always been) a part of my scale...
> only recently Rick and I picked up on the fact...that
> multiplying it by 1.618034 produces the octave.

You poor, poor man. I am >>so<< sorry I didn't read your mind
or your previous messages about your formula and inadvertently
hurt your feelings. Will you be OK?

OK, 1/phi^3+1 ~ 2/phi. So what?

>I realize my theory may not be perfect, but it sure as hell isn't
>base-less...unless you really are trying that hard to ignore my
>repeated explanations.

But you haven't explained anything!

-Carl

🔗djtrancendance@...

Carl>"Well then, just say that. Instead of "proportional beating"."
Agreed...if that works for you then great: I don't really have a die-hard preference for either term.
To recap for people looking at this for the first time: the point is that tones related to PHI (IE 1/PHI 2/PHI PHI etc.)...all come together to form a similar "overtones pointing to a root tone" feeling/vibe that the harmonic series produces.

Carl>"Don't know. But you stopped using them and started on the

phi stuff, so I assumed they weren't doing it for you."

Well, to put it this way, the pentads your showed me seemed consonant enough...but the hexads and 7 note chords you showed me did not. On one hand, no they didn't quite do it for me...on the other hand, we never really finished that discussion and you said you would show me more examples just before you said you were too busy. So...it's certainly not case closed yet.

>"Don't remember doing that but my wife had a baby a couple

weeks ago."
That was the time the discussion was put on hold and, come to think of it, that was the issue you brought up that made me understand why you had to throw your hands up and continue things later.

>"You poor, poor man. I am >>so<< sorry I didn't read your mind

or your previous messages"
Hmm...well all it would have taken was a combination of your reading 1 of the 25 or so times posting the formula and actually trying it...to come to the conclusion that 2/PHI is a natural part of my scale. Haha...well (enough drama), teasing aside at least you seem to get it now, I hope...

>"OK, 1/phi^3+1 ~ 2/phi. So what?"
You made the rather outrageous claim that all I had done to get 2/PHI is multiply 2 times 1/PHI. And, guess what, the above equation is practical proof that there is nothing superficial or forces about the way I derived 2/PHI in my scale and hit the octave dead-on. Again, the "so what?" goes back to the point you were whining about before: (paraphrased) "how the H.E.-double-hockey-sticks does the octave relate to PHI?!"
And now I'm re-telling this to you for about the third time: 1/phi^3+1 = 2/phi which, times the
PHI-tave, equals 2/1. Does that make it a bit more than obvious how my scale achieves a degree of symmetry to both PHI and the octave?

>"But you haven't explained anything!"
It's amazing what lengths you will go to ignore all my explanations. You know, Rick, Chris, several others...didn't get my 'crap' at all at first...but then got it to a very large extent. Then again, they regularly read my posts...rather than jump in and make assumptions and say random things like 'It's obvious you simply took 2 * 1/PHI to get 2/PHI' that seem to indicate more interest in getting my goat than actually looking at realistic improvements for my scale(s) (not to mention that they are blatantly false).

-Michael

🔗Carl Lumma <carl@...>

🔗djtrancendance@...

🔗Carl Lumma <carl@...>

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

> To recap for people looking at this for the first time: the
> point is that tones related to PHI (IE 1/PHI 2/PHI PHI etc.)...
> all come together to form a similar "overtones pointing to a
> root tone" feeling/vibe that the harmonic series produces.

1/anything 2/samething 3/samething 4/samething...

is a subharmonic series, and subharmonics do share common
overtones. But they don't reinforce a single fundamental
like harmonics do. Certainly I do not hear your scales
that way.

> Well, to put it this way, the pentads your showed me seemed
> consonant enough...but the hexads and 7 note chords you showed
> me did not.

If you enforce the somewhat artificial restriction that all
tones have to be in a single octave, then there are triads
more consonant than any tetrad, tetrads more consonant than
any pentad, and so on. If your goal is to find the most
consonant 7-ads, it hardly matters if the ones I sent aren't
as consonant as you'd like. All that matters is whether you
can find more consonant ones?

>on the other hand, we never really finished that discussion
>and you said you would show me more examples just before you
>said you were too busy. So...it's certainly not case closed yet.

I don't recall leaving this loose end, but if you can find
that message or remember what I promised you I'd be happy
to send it now.

>reading 1 of the 25 or so times posting the formula and actually
>trying it...to come to the conclusion that 2/PHI is a natural
>part of my scale. Haha...well (enough drama), teasing aside at
>least you seem to get it now, I hope...

I read more of this list than almost anyone, I suspect.
But even I can't read it all.

>the way I derived 2/PHI in my scale and hit the octave dead-on.
>Again, the "so what?" goes back to the point you were whining
>about before: (paraphrased) "how the H.E.-double-hockey-sticks
>does the octave relate to PHI?!"

I don't recall saying that, but OK, interesting. So what?

>And now I'm re-telling this to you for about the third time:
>1/phi^3+1 = 2/phi which, times the PHI-tave, equals 2/1. Does
>that make it a bit more than obvious how my scale achieves a
>degree of symmetry to both PHI and the octave?

It shows that successive cuts of phi will eventually get
you octaves. The significance of this isn't clear.

> >"But you haven't explained anything!"
>
> It's amazing what lengths you will go to ignore all my
> explanations.

What you're doing is nice recreational number play, but
it doesn't explain anything, at all, and the sooner you
realize this the better off you'll be. Explanations are
models that predict past and future observations. You
haven't described any observations that anyone else seems
to agree with, and haven't shared any models with us either.
The fact that you continue to fail to understand this is
pretty grim if you want to contribute to music theory.
Recreational number play, applied to microtonal scales,
and coupled with some skill in music making (which you
seem to have) are a fine combination, and there's even a
mailing list devoted to it --- MakeMicroMusic.

Chris knows at least as much as you about this stuff, but
compare his attitude and the frequency and content of his
posts. If you want to do music theory, though, your browser
is currently at the right website but your attitude is going
to prevent you from making progress.

> You know, Rick, Chris, several others...didn't get my 'crap'

What do you think it is that Rick and Chris understand?

>Then again, they regularly read my posts...rather than jump
>in and make assumptions and say random things like 'It's
>obvious you simply took 2 * 1/PHI to get 2/PHI' that seem to
>indicate more interest in getting my goat than actually
>looking at realistic improvements for my scale(s) (not to
>mention that they are blatantly false).

I was wrong about phi and octaves, and I'm sorry about that.

-Carl

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@> wrote:
> > Take a line going for 0 to 1. Now take 1 and keep multiplying >it by 1.618034 (IE splitting it into PHI-ths). That's how you get >my 1.3819, 1.23...values. Now take that same line, but move it >between the values 1 and 2. Artistically/in architecture, of >course, the way PHI splits the line in the exact same way my scale >does.
>
> I do not think that this analogy works for pitches. Yes, the line >intervals 0-1, 1-2 and 2-3 are equivalent on a piece of graph paper, >or indeed on an artist's canvas, but not in the context of pitches. >The human ear perceives pitches according to a logarithmic scale, so >that 1-2 is equivalent to 2-4 (not 2-3) and to 0.5-1 (not 0-1). >That is why multiplying and dividing are much more usual than adding >and subtracting, when dealing with pitches. That's not to say your >way cannot work, but it is unusual.

What Michael is talking about is the Golden Section applied to the frequency realm. I don't know if it is unusual, but it certainly isn't new. Last year I mentioned here that I use the golden sections of frequencies rather than the noble mediant, and I'm sure this has been tried in various ways for hundreds of years.

AKJ posted a big list of Phi tunings that are already in the Scala archive- in stark contrast to Michael's idea that it is a little-explored territory, it is a greatly explored territory.

At any rate, taking golden sections of frequencies is not necessarily numerology at all. But you have to understand the golden section in the first place.

Using the golden section of frequencies, you will discover a strange hall of mirrors in the tuning and difference tones- and the reflections appear in the logarithmic realm as well.

Check it out:

let's take the octave. Phi in the frequency realm, 1.618033988749895, marks a golden section of the octave. Now if we play the dyad of Phi and the 1/1, we get a curious difference tone- it is Phi mirrored below the 1/1, in the logarithmic realm, ie. it's 833.090 cents below 1/1 just as Phi is 833.090 cents above the 1/1.

The idea of a "Phi-tave" might start looking less "numerological" already, but let's not forget the golden proportion.

A golden proportion requires 3 quantities such that A is divided into B and C, and A is to B as B is to C. In order to hear our golden proportion here, we need to play A,B and C, which are 1/1, Phi, and 2/1, together. It's a strange and interesting sound, but maybe it's still kind of "so what?" at this point, so let's keep going.

Notice that above, I said "a" golden section, not "the" golden section. Let's call the first golden section of the octave in the frequency realm the upper section, for just as you can paint a saint at the golden cut of a canvas reading from the left, you can also make the cut from the right. This will not be the logarithmic reverse of Phi in the octave. The logarithmic reverse is already present in our original cut of course, being the logarithmic difference between Phi and the octave. We'll drop that interval, the logarithmic mirror of our original Phi within the octave, into the tuning for future reference.

Tuning so far:

1/1
(366.910 cents)
833.090 cents
2/1

Now let's make our lower golden cut of the octave, in the frequency realm. This looks at first like a naive failure to understand the logarithmic nature of pitch perception, but as you will see, with Phi it works out a bit more interesting than that.

Taking freqency as a simple quantity, we subtract as much from the octave as we added earlier. This is 2-phi, just as Phi is 1+phi (phi being 1/Phi).

This gives us a frequency of 1.381966011250105, which works out to
560.06656 cents.

Tuning so far:

1/1
(366.910 cents)
560.06656
833.090 cents
2/1

This doesn't look too promising at first, but if you play it as a single sonority you may find it far less dissonant than it looks on paper, even very euphonic.

Now, as we looked at the difference tone of the first, upper, golden cut and the 1/1, let's look at the difference tone of the second, lower, cut and the 1/1.

-1666.181 cents

But hang on a second- we've actually seen, or rather heard, that same difference tone before. In our original, upper, golden section, if we play B and C as a dyad, rather than A and B, there it is.

Now let's bring our difference tones up by octaves into the tuning. Our -833.090 appears as 366.910 cents, which we already have, and our -1666.181 appears as 733.81941 cents. Which we also already have as an interval, in mirror form: it is the same interval, logarithmically, from 2/1 as our original Phi cut is from it's logarithmic mirror (log. interval between 833.090 and 366.910), and it is also 366.910 above 366.910.

Our tuning so far:

1/1
(366.910 cents )
560.067 cents
(733.819 cents)
833.090 cents
2/1

Playing this as a single sonority, once again we get what some might percieve as a very pleasant sound with a kind of "oneness" to it, only at this point we can demonstrate that we really are getting a hall of mirrors effect. Try the same thing with, say, 13/8, and you'll find that it is radically different. Right off the bat, the first difference tone is an 8/5 below the 1/1, and has the complete opposite implications of the Phi interval as far as tonality.

A little detour: if we take our two golden cuts, in the frequency realm, of the octave, and strike the geometric mean between them, we find the tempered fifth of quarter-comma meantone. Exactly.

Anyway, let's go on:

Inspired by the visual arts, how about let's take the golden section, in the frequency realm, of our first, higher, golden section. This works out to 1.381966011250105... hang on, it is exactly the same as the lower golden section of the octave. How about the lower golden section of our original Phi ("B") section? Hm, we've already got that interval, too, it's the B-C, logarithmically, of our original golden section.

And you can just keep on going: a hall of mirrors. For example, at this point you might as well throw in Phi modulo 2, 466.18100 cents, in the interests of symmetry, as it has already a number of times in the difference tones, octaved, and in the tuning itself.

1/1
366.910 cents
466.181 cents
560.067 cents
733.819 cents
833.090 cents
2/1

Now how about the golden sections of our original B:C?

1068.86470 is the upper, 1.76393202250021, or 982.55396 cents, the lower. To no surprise, both have difference tones whose mirrors against the 1/1 already appears elsewhere in the tuning.

1/1
366.910 cents
466.181 cents
560.067 cents
733.819 cents
833.090 cents
982.554 cents
1068.865 cents
2/1

This is quite beautiful IMO, and it is going to look very familiar to Michael- hopefully he'll have figured out by now that he's not the first to do explore the golden section, and that his explorations are only a beginning.

The tuning at this point now exhibts a surprising characteristic: the interval between degree 1 and degree 7 is a Just fifth of 3/2, and you will find 7-limit Just intervals within microscopic differences in the tuning and difference tones. And so maybe you can see how this tuning leads "naturally" to what I call shadow tunings, which use Just and very non-Just intervals together. Another more blatant example of this- take a listen to the interval between the square root of 3 (another surd, like Phi) and its logarithmic inversion against the octave.

>
> AKJ's method, using 2^p (where p=0.618, or a Fibonacci-number >approximation to it) is in fact splitting the octave geometrically >(rather than arithmetically) and it seems to me to be the closest >aural analogy to the visual splitting of a line segment.

There are many, many methods, and I'm always happy when I discover that they've been done before- the Fibonacci series modulo 2 (William Burt already did this), Phi as a member of the simple surds (John Chalmers already did this), and so on. I think my "abs2cos(chop up circumference in different ways)", approach might be original in its different implementations, but Jacques Dudon already had the idea years ago. It's well explored but ever-fruitful territory.

🔗djtrancendance@...

>"If you enforce the somewhat artificial restriction that all
tones have to be in a single octave, then there are triads
more consonant than any tetrad, tetrads more consonant than
any pentad, "

Of course, the less notes (IE the further a sub-set you take) the more consonance in general. And a subset of my scale is always going to have more consonance than the entire scale played at once. But the impression I get (so far) is that Just Intonation itself implies certain consonance limits that step from the harmonic series.

And when I tried the harmonic series based scale 9/8 10/8 11/8 12/8 13/8 14/8 15/8 16/8 18/8 20/8 to make 8-tone per octave chords...I found out that my ears/mind actually objected to having so many tones beating against each other at the same rate (which actually sounds a bit mechanical and forced, even with pure sine waves). And, if I have it right, such a construct should sound more pure than any JI scale...yet it does not sound so good to my ears. And so started my train of thought "let's try looking beyond the harmonic series..."

>"I read more of this list than almost anyone, I suspect.
But even I can't read it all."

And that's understandable...the only thing I didn't understand is why you suddenly jumped into this discussion making very very strong statements...as if you had been there watching the whole thing.

>"It shows that successive cuts of phi will eventually get
you octaves."
Correct...

>"The significance of this isn't clear."
"...but, OK, interesting. So what?"

So far, by ear, I have tested several scales based on fractal numbers. And, out of the ones I've made, those that intersect relatively well with 2/1, 3/1, 4/1, 5/1 6/1...virtually always sound more confident/centered. You said yourself aligning tones with the first six harmonics is a good idea...well, turns out (if I interpreted your point correctly) I agree with you.

>"You haven't described any observations that anyone else seems
to agree with"

In what sense? It baffles me that I can, say, release songs using the Silver Ratio scale which people seem to thing are very good and consonant...and then have them go "uh uh I didn't say that...the numbers are wrong" when it comes down to my mathematical explanations (such as those above). Perhaps people's ears agree with how it works in music, but can't figure out why the numbers behind it are so weird.
>", and haven't shared any models with us either."
Models of what exactly? I told you how I derive the scale from a continuous function, how I mirror certain notes, how the whole process represents how PHI works in architecture, why taking 0.618^x represents a construct with both exponential and additive (IE each number in the scale = the number before it plus the number after it), the intersection of the octave by my scales (without using an existing theory like MOS to do
so)...

It seems to me you are frustrated with me because you can't find any obvious common ties with existing music theory. Minus, perhaps, the fact my scale intersects the octave.

>"The fact that you continue to fail to understand this is

pretty grim if you want to contribute to music theory."
Well then what on earth constitutes being contributing to music theory? Aren't my extensive references to psychoacoustics enough?
Or is a key requirement "references existing literature" (including, perhaps, things like the Noble Mediant which have absolutely nothing to do with the construction of my noble/fractal scale theories)?
I realize even "weird tune-iks" like Sethares seem to go back and use parts of there theories to explain "why diatonic JI works" just like Wilson's MOS explains the construction of pentatonic and diatonic scales. But, if people like the sound of the tunings in the end of the day...what's so wrong about making something without a huge deal of references to existing music theory?

>"Chris knows at least as much as you about this stuff, but
compare his attitude and the frequency and content of his
posts."

Chris obviously knows more about formal music theory (he's fully degreed)...but I wouldn't go so far as to say he knows more about things like Psychoacoustics proportionate beating (which are the main thrusts of my scales).

>"What do you think it is that Rick and Chris understand?"
Well one main thing they seem to get...is that the PHI-tave has
emotional significance relative to all other tones, just like the octave and 5th do. They may disagree with certain things I do beyond the chord
1/PHI 1 PHI/2 PHI...but they at least appear to agree things based directly on the PHI-tave are both unique and musically thesible. It was at least enough, for example, to get Rick to build his own PHI scale based on 36TET which, I agree, really is based truly on the PHI-tave (unlike most so-called PHI scales in history).

That and, they both appear to have taken very well
to how music based on my theories actually sounds (which is the end goal of any promising music theory, no matter how mathematically clever, no?)

>"I was wrong about phi and octaves, and I'm sorry about that."
Thank you, and I'm not going to hold a grudge on that I think we can both acknowledge such grudges don't help us any with progress on creating scales. :-)
. However, I do wish, in general, you regularly gave me more credit for the huge amounts of time I put into developing these scales. You have the right to think I am a bit nuts for trying (hahaha)... A recent comment on this scale said something like "2^(1/PHI) would work because it's logarithmic...but I don't see how (1/PHI)^x would work because it's additive and the ear works exponentially"...which is a fair criticism plus ties directly to psychoacoustics (an issue fundamental to why I created the scale), but it certainly didn't say anything
horridly assuming like (paraphrased) "so it's obvious you just picked random numbers and used (1/PHI)^x to approximate them". My point is, in general, please at least give me credit for the fact I build these constructs very carefully and >not< at all randomly, even if you don't agree with the goals I build them to reach.

-Michael

🔗Carl Lumma <carl@...>

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

>the impression I get (so far) is that Just Intonation itself
>implies certain consonance limits that step from the harmonic
>series.

If you can explain this further, we can discuss it.

> And when I tried the harmonic series based scale 9/8 10/8
> 11/8 12/8 13/8 14/8 15/8 16/8 18/8 20/8 to make 8-tone per
> octave chords...I found out that my ears/mind actually objected
> to having so many tones beating against each other at the same
> rate (which actually sounds a bit mechanical and forced, even
> with pure sine waves).

I don't know what you mean by 'beating together at the same rate'.
As I've said several times, this doesn't occur in the way
you suggested. Maybe you're referring to the fact that the beat
_ratios_ (called "brats") tend to be small whole number ratios?

> And, if I have it right, such a construct should sound more
> pure than any JI scale...

_Which_ construct?

>>It shows that successive cuts of phi will eventually get
>>you octaves. The significance of this isn't clear.
>
>So far, by ear, I have tested several scales based on fractal
>numbers.

What's a fractal number?

>And, out of the ones I've made, those that intersect relatively
>well with 2/1, 3/1, 4/1, 5/1 6/1...virtually always sound more
>confident/centered.

Such an observation hardly seems surprising, and hardly seems
to have anything to do with phi.

>You said yourself aligning tones with the first six harmonics
>is a good idea...

It's a good idea if you want consonance.

>>You haven't described any observations that anyone else seems
>>to agree with
>
>In what sense?

When you've posted chords and said, this phi-based one beats
less than this JI-based one, almost always the reverse has
been true. I'm not saying you aren't hearing _something_ you
like better, but you incorrectly describe it as 'less beating'
or 'more proportional beating' or some other quasi-technical
phrase.

>It baffles me that I can, say, release songs using the
>Silver Ratio scale which people seem to thing are very good and
>consonant...and then have them go "uh uh I didn't say that...the
>numbers are wrong"

When did this happen?

>I told you how I derive the scale from a continuous function,
>how I mirror certain notes, how the whole process represents
>how PHI works in architecture,

A continuous function??

Yes, you've told us how you construct your scale, and I understand
it now. But _why_ would you ever construct such a thing?
What observable effects does such a construction engender?

>It seems to me you are frustrated with me because you can't find
>any obvious common ties with existing music theory.

Nothing could be further from the truth.

>>"The fact that you continue to fail to understand this is
>>pretty grim if you want to contribute to music theory."
>
> Well then what on earth constitutes being contributing to
> music theory?

Read Helmholtz. That's an example of a contribution to
music theory. Or any of Paul Erlich's papers. Have you taken
the time to read any of the stuff we've referred you to over
the past year?

Read it yet? Good. Now go through your outbox and read some
of your posts. See a difference? Not in the content, but
in the format.

> Aren't my extensive references to psychoacoustics enough?

No, because they don't make any sense. Because you apparently
don't have enough experience with basic quantitative methods
to apply any knowledge of psychoacoustics you may have.

> Or is a key requirement "references existing literature"

Key requirements are high-school level algebra, English
literacy, and loose application of the scientific method.

>(including, perhaps, things like the Noble Mediant which have
>absolutely nothing to do with the construction of my
>noble/fractal scale theories)?

I see you're still under the misapprehension that you've
presented one or more theories. Wikipedia says,

""A theory, in the general sense of the word, is an analytic
structure designed to explain a set of observations. A theory
does two things:
1. it identifies this set of distinct observations as a class
of phenomena, and
2. makes assertions about the underlying reality that brings
about or affects this class.""

Google offers:

""theory- A hypothesis that has withstood extensive testing by
a variety of methods, and in which a higher degree of certainty
may be placed. A theory is NEVER a fact, but instead is an attempt
to explain one or more facts.""

If you have something that fits this description, go ahead and
post it in a new thread and we'll have a look. Be sure to say
exactly what hypotheses were tested, how they were tested, and
what results you got, and how your theory explains the results.
An "explanation" is generally something that summarizes
results using fewer words than it took to list the results.

>what's so wrong about making something without a huge deal of
>references to existing music theory?

I've tried and tried, but I can't seem to make you understand
that existing references are irrelevant. Above, I suggest you
read Helmholtz and Erlich to see what constitutes a contribution
to music theory, because you seem to have trouble imagining
what one looks like. But one could presumably discover a great
theory of music working alone in a cave in India.

> but I wouldn't go so far as to say he knows more about things
> like Psychoacoustics proportionate beating (which are the main
> thrusts of my scales).

You can't explain "proportionate beating". I've asked you.
You're welcome to do so now and prove me wrong, but I have
tentatively concluded that not only don't you have an
explanation called "proportionate beating", you apparently
are also ignorant of the fact that you lack it.

> >"What do you think it is that Rick and Chris understand?"
>
> Well one main thing they seem to get...is that the PHI-tave
> has emotional significance relative to all other tones, just
> like the octave and 5th do.

If you're saying that the phi-tave can function as an interval
of equivalence better than any random interval can, you're wrong,
and I highly doubt Rick or Chris agree with such a statement,
but I welcome them to chime in.

>they at least appear to agree things based directly on the
>PHI-tave are both unique and musically thesible.

Well anything could be unique, couldn't it? Musically... feasible?

>It was at least enough, for example, to get Rick to build his
>own PHI scale based on 36TET which, I agree, really is based
>truly on the PHI-tave (unlike most so-called PHI scales
>in history).

Cool. Means nothing in terms of music theory, but it's cool.

> That and, they both appear to have taken very well to how music
> based on my theories actually sounds (which is the end goal of
> any promising music theory, no matter how mathematically clever,
> no?)

No. I can make good-sounding music with a djembe. It doesn't
even produce pitched sounds. So obviously, intonation theory
has a poor correlation with good-sounding music.

>However, I do wish, in general, you regularly gave me more
>credit for the huge amounts of time I put into developing these
>scales.

Anyone can develop scales. I'm glad you're interested and it's
a great thing that you're sharing your work. Keep it up.
But like many in tuning (including myself at various points
in the past), you seem to have a greatly inflated view of the
significance of your work.

>A recent comment on this scale said something like "2^(1/PHI)
>would work because it's logarithmic...but I don't see how
>(1/PHI)^x would work because it's additive and the ear works
>exponentially"...which is a fair criticism plus ties directly

Who said that? It makes no sense, because the ear doesn't
"work exponentially".

>please at least give me credit for the fact I build these
>constructs very carefully and >not< at all randomly, even if
>you don't agree with the goals I build them to reach.

I don't think I've ever accused you of building scales
randomly! I know you write software (and I tried your
javascript page) and so on and so forth. It's all fine
and well and it'll all be *excellent* inspiration for
making music.

-Carl

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Carl Lumma <carl@...>

🔗djtrancendance@...

Mike>"Just Intonation itself implies certain consonance limits that stem from the harmonic series."

Carl> "If you can explain this further, we can discuss it."
In other words, if the harmonic series does not sound good with, say, more than 7-tones per octave...it's very doubtful JI will sound any more consonant. In fact, so far I have found the opposite: the most tones I have fit into an octave while sounding relaxed (to my ear) is about 5 well, with the harmonic series (6/6 7/6 8/6 9/6 10/6 11/6 12/6)...I could manage up to about 6 notes per octave with a fair sense of relaxation but any more notes and my ears prefer a 7-tone subset of my PHI or Silver Ratio scales, for example.

Note: I'm now using the term "relaxed" rather than "consonant" because, I figure, the formal definition of consonance implies harmonic-series-style periodicity, which certain combinations of notes in my scale do not have.

Mike> "And when I tried the harmonic series based scale 9/8 10/8 11/8 12/8 13/8 14/8 15/8 16/8 18/8 20/8"

In other words, if the numbers in the harmonic series are
100,200,300,400,500(hz)....
....then 100 is going to beat against 200 at the same rate 200 does against 300. Same old simple phenomena.

Mike> "And, if I have it right, such a construct should sound more
pure than any JI scale..."

Carl> "_Which_ construct?"
The overtones/"harmonics" of the harmonic series itself.

Carl> "What's a fractal number?"
Some examples of fractal numbers include PHI and the Silver Ratio IE http://en.wikipedia.org/wiki/Silver_ratio.
If you take the equation x = 1/x + b where b is any constant whole number and solve for x you can get several fractal numbers, including the Silver and Golden ratio plus many many more (Jacques Dudon went over this in a previous post and corrected me on my use of the term 'noble numbers' when I really should have said 'fractal numbers').

Mike>"Aren't my extensive references to psychoacoustics enough?"
Carl>"No, because they don't make any sense. Because you apparently

don't have enough experience with basic quantitative methods

to apply any knowledge of psychoacoustics you may have."

There you go whining at me without giving examples. If I really didn't know "basic quantitative methods", how on earth do you think I managed to figure out that solving x = 1/x + 1 defines the golden ratio and then have Jacques Dudon agree with me such a proof is true and is a very good point. BTW, just for reference, I aced calculus and received an A- (boohoo?!) in graduate school statistics...if you think I never studied math, that perhaps says a lot more about your ego than it does of my skills.
Also, for crying out loud, this is just like what you did with trying to tell me I got 2/PHI by dividing two by PHI and only admitting I had successfully proved it by (1/PHI)^3 + 1 and not just randomly forced it in my scale. I learn b/c I study this extensively...and not only if/when you agree with what I've studied. So please stop picking fights
about things you obviously don't know yet about noble numbered scales...it's not helping anyone.

>"Yes, you've told us how you construct your scale, and I understand

it now. But _why_ would you ever construct such a thing?

What observable effects does such a construction engender?"
At least now you have admitted...that I >do< explain how I come up with my scales.
BTW, as I've said here many many times...the observable effects are that the beating which occurs does so in a very predictable and soothing manner.

Carl>"Read Helmholtz. That's an example of a contribution to

music theory."
Don't be an @$$. I have read Helmhotz. And Plompt and Llevelt. And Sethares. Those three I had read even before I found this group...and I had talked to Bill Sethares personally about it in the process of building a DSP program also made long before I joined this list.

Notice how my scales never have notes closer than about the interval 1.055 together and how I either try to hit whole number intervals like 2/1 3/1...or avoid getting them completely? The reason I do that...is observing P&L and Helmholtz's theories...for example.

Carl>"A theory is NEVER a fact, but instead is an attempt

to explain one or more facts."
Beside obeying the psychoacoustic theories mentioned above I also explain things like why people I've tested my scales onto, for example, often manage to mistake PHI for the octave. Not to mention how the same formula to create scales with PHI can make consonant scales with >any< fractal number, including the Silver Ratio and beyond. If that doesn't count as "universal proof" to you, Lord knows what does...

>"I've tried and tried, but I can't seem to make you understand

that existing references are irrelevant. Above, I suggest you

read Helmholtz and Erlich..."
    So you say "existing references are irrelevant" and then say "now go read some existing references to become more relevant in what you do". Kind of a contradicting standard, don't you think?
    Also, if you go back to a theory being an attempt to explain more than one fact, you can say that 'all Helmholtz did was try to explain how consonance is a function of beating...and the less beating the more consonance" and then whine 'therefore Helmholtz is not a true theoretician as he only really explains one abstract fact'.
   Translation: if you are enough of a bastard about it...I'm pretty sure you could find a way to say all contributors are not good enough for you. But, furthermore, none of this "sensei-ism" is actually going to help us make more beautiful scales...so I would really appreciate it if you cut it out.

>"You can't explain
"proportionate beating". I've asked you."
And I've answered and will again. Here is a dead simple example involving the definition of the Golden Ratio.
Take the frequencies 1000 1618 and 2000. The ratio of 1618-1000 to 2000-1618 is the same as the ratio of 2000-1000 to 2000-1618. The whole definition of fractal numbers implies proportionate beating. Do you think, for example, Jacques Dudon is also a whack job because he believes in the advantage of such symmetries as well? Come on...

Carl>'Who said that? It makes no sense, because the ear doesn't

"work exponentially" .'
I meant works in the form of 2^x, 1.618034^x, etc. You seem to have this thing of only accepting the 100% perfect technical terms...so maybe I should just restate it as "logarithmically" also most people would just, for example, refer to the octave as 2^x and parts of it as, for example, 2^(x/12) or 2^(x/22) (for 12 and 22tet).

>"I don't think I've ever accused you of building scales

randomly! "

Hmm...well then why do you seem to randomly accuse me of things like ignoring P&L and Helmholtz's theories when I obey them 'religiously' or talk about how you "know" I derived 2/PHI (and give the most dumb and elementary method you can think of) when you have no clue?
If you're not accusing me of being 'random'...then you sure as heck seem to be accusing me of being ignorant...and of things not only I follow but we've discussed in detail (especially...we've discussed P&L's theories in detail).

Please...enough with the assumptions...and no one here that I have seen is deficient in things like basic quantitative methods, including myself...give me a freaking break.

-Michael

🔗Chris Vaisvil <chrisvaisvil@...>

🔗rick_ballan <rick_ballan@...>

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

Just one thing Michael,

You said below <A recent comment on this scale said something like "2^(1/PHI) would work because it's logarithmic...">. I think you were referring to Steve's last post which said:
"AKJ's method, using 2^p (where p=0.618, or a Fibonacci-number approximation to it) is in fact splitting the octave geometrically (rather than arithmetically) and it seems to me to be the closest aural analogy to the visual splitting of a line segment."

I don't know who AKJ is or whether Steve's comment is truly representative, but this is some very faulty logic. Because exponentials are added, this seems at face value to be interesting. eg (2^phi) x 2 = 2^(phi+1) = 2^(phi squared), or taking lower 8ve and dividing 2 gives -1 in exponent. But whoever thought of this "method" answer me this: given 2^(1/phi), then we reach the octave at (2^(1/phi))^phi = 2. How on earth are we supposed to have a phi number of frequencies/notes within an octave? Perhaps we can cut down a piano string and call it (1/phi)'th of an entire frequency! THIS is why the exponents cannot be irrational, only the resultant frequencies can be irrational. Besides, there is a great deal of difference b/w 2^(1/phi) and phi. It is like saying that 2^3 = 8 can somehow substitute the number 3.

-Rick
>
> >"If you enforce the somewhat artificial restriction that all
> tones have to be in a single octave, then there are triads
> more consonant than any tetrad, tetrads more consonant than
> any pentad, "
>
> Of course, the less notes (IE the further a sub-set you take) the more consonance in general. And a subset of my scale is always going to have more consonance than the entire scale played at once. But the impression I get (so far) is that Just Intonation itself implies certain consonance limits that step from the harmonic series.
>
> And when I tried the harmonic series based scale 9/8 10/8 11/8 12/8 13/8 14/8 15/8 16/8 18/8 20/8 to make 8-tone per octave chords...I found out that my ears/mind actually objected to having so many tones beating against each other at the same rate (which actually sounds a bit mechanical and forced, even with pure sine waves). And, if I have it right, such a construct should sound more pure than any JI scale...yet it does not sound so good to my ears. And so started my train of thought "let's try looking beyond the harmonic series..."
>
>
> >"I read more of this list than almost anyone, I suspect.
> But even I can't read it all."
>
> And that's understandable...the only thing I didn't understand is why you suddenly jumped into this discussion making very very strong statements...as if you had been there watching the whole thing.
>
>
> >"It shows that successive cuts of phi will eventually get
> you octaves."
> Correct...
>
> >"The significance of this isn't clear."
> "...but, OK, interesting. So what?"
>
>
> So far, by ear, I have tested several scales based on fractal numbers. And, out of the ones I've made, those that intersect relatively well with 2/1, 3/1, 4/1, 5/1 6/1...virtually always sound more confident/centered. You said yourself aligning tones with the first six harmonics is a good idea...well, turns out (if I interpreted your point correctly) I agree with you.
>
> >"You haven't described any observations that anyone else seems
> to agree with"
>
> In what sense? It baffles me that I can, say, release songs using the Silver Ratio scale which people seem to thing are very good and consonant...and then have them go "uh uh I didn't say that...the numbers are wrong" when it comes down to my mathematical explanations (such as those above). Perhaps people's ears agree with how it works in music, but can't figure out why the numbers behind it are so weird.
> >", and haven't shared any models with us either."
> Models of what exactly? I told you how I derive the scale from a continuous function, how I mirror certain notes, how the whole process represents how PHI works in architecture, why taking 0.618^x represents a construct with both exponential and additive (IE each number in the scale = the number before it plus the number after it), the intersection of the octave by my scales (without using an existing theory like MOS to do
> so)...
>
> It seems to me you are frustrated with me because you can't find any obvious common ties with existing music theory. Minus, perhaps, the fact my scale intersects the octave.
>
> >"The fact that you continue to fail to understand this is
>
> pretty grim if you want to contribute to music theory."
> Well then what on earth constitutes being contributing to music theory? Aren't my extensive references to psychoacoustics enough?
> Or is a key requirement "references existing literature" (including, perhaps, things like the Noble Mediant which have absolutely nothing to do with the construction of my noble/fractal scale theories)?
> I realize even "weird tune-iks" like Sethares seem to go back and use parts of there theories to explain "why diatonic JI works" just like Wilson's MOS explains the construction of pentatonic and diatonic scales. But, if people like the sound of the tunings in the end of the day...what's so wrong about making something without a huge deal of references to existing music theory?
>
> >"Chris knows at least as much as you about this stuff, but
> compare his attitude and the frequency and content of his
> posts."
>
> Chris obviously knows more about formal music theory (he's fully degreed)...but I wouldn't go so far as to say he knows more about things like Psychoacoustics proportionate beating (which are the main thrusts of my scales).
>
> >"What do you think it is that Rick and Chris understand?"
> Well one main thing they seem to get...is that the PHI-tave has
> emotional significance relative to all other tones, just like the octave and 5th do. They may disagree with certain things I do beyond the chord
> 1/PHI 1 PHI/2 PHI...but they at least appear to agree things based directly on the PHI-tave are both unique and musically thesible. It was at least enough, for example, to get Rick to build his own PHI scale based on 36TET which, I agree, really is based truly on the PHI-tave (unlike most so-called PHI scales in history).
>
> That and, they both appear to have taken very well
> to how music based on my theories actually sounds (which is the end goal of any promising music theory, no matter how mathematically clever, no?)
>
> >"I was wrong about phi and octaves, and I'm sorry about that."
> Thank you, and I'm not going to hold a grudge on that I think we can both acknowledge such grudges don't help us any with progress on creating scales. :-)
> . However, I do wish, in general, you regularly gave me more credit for the huge amounts of time I put into developing these scales. You have the right to think I am a bit nuts for trying (hahaha)... A recent comment on this scale said something like "2^(1/PHI) would work because it's logarithmic...but I don't see how (1/PHI)^x would work because it's additive and the ear works exponentially"...which is a fair criticism plus ties directly to psychoacoustics (an issue fundamental to why I created the scale), but it certainly didn't say anything
> horridly assuming like (paraphrased) "so it's obvious you just picked random numbers and used (1/PHI)^x to approximate them". My point is, in general, please at least give me credit for the fact I build these constructs very carefully and >not< at all randomly, even if you don't agree with the goals I build them to reach.
>
> -Michael
>

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

Hello Michael,

I'm still not sure that the rate at which you hear beating has
anything at all to do with the subjective "quality" of any chords you
might construct. When you turn the volume on and off from a signal in
a sinusoidal fashion you create a form of "beating", and it doesn't
seem to lead to any observable phenomenon even close to something like
the perception of a "chord quality" or a "tonal color."

Nonetheless, I can't ignore the simple fact that your phi scale does
sound great. Is it that you're taking equal divisions of phi and using
that as a period instead of 2/1? If so, I think the unique sound that
it has stems from the fact the phi ratio itself is extremely high
entropy, but when you stack a bunch of copies of phi on top of each
other you eventually come close to familiar small-integer ratios.

I do notice that it does seem to have a lot of maximally different
"tone colors" as you mentioned above, or tone chroma, as I prefer to
call it - I think it's due to that characteristic of it mentioned
above. The pattern I notice is that the farther harmonically "out" you
get from a tone, the more the tone chroma changes. Like how 3/2 has a
pretty close chroma to 1/1. And you can't get much farther out from
1/1 than phi, after all.

But it is an interesting concept. A lot scales talked about here work
from the other way around - a consonant interval, such as some kind of
a fifth, will be used as a generator that when stacked on top of
itself enough will approximate more complex and dissonant intervals.
The fact that this scale uses the opposite approach is something I
especially like. Perhaps the fact that the period itself is so
dissonant has something to do with it as well.

-Mike

On Thu, Apr 30, 2009 at 12:09 AM, <djtrancendance@...> wrote:
>
>
>
> Chris.
> I made a mistake with my PHI scale...once of the notes (at about 443.17
> cents) was a mis-copy and sound dissonant vs. virtually any other note in
> the scale. :-(
>
> I'll re-do the PHI scale tomorrow and re-send you the scala file, in the
> mean-time, trying listening to this
> http://www.geocities.com/djtrancendance/PHI/phicicles.mp3
>
> It was made with my PHI-scale, minus the sour note mentioned above. I did
> notice a problem where changing the root to certain notes causes issues,
> unlike with the silver-ratio scale, so I think you may have a point
> still...that my silver ratio scale is easier to compose with.
>
> -Michael
>
>

🔗Chris Vaisvil <chrisvaisvil@...>

You are mixing two different things in the conversation I'm afraid.

This is the context and the question I'm trying to answer.

Chris > Any random interval may very well get you something that
> at least at first glance will seem to be less useful
> musically, for me anyway.

Carl spoke thusly "Have you tried it?

Note, I'm not saying there isn't. In fact, I've said there
*is* something special about it, regarding its dissonance.

But you should try it. It would be a great exercise."

----------------------------------

So... indeed I have tried it. And no I don't think it is particularly
interesting. But I do *love* metallic sounds.

The instrument were a row of oil pipes in a oil / chemical storage facility.
I'm pretty sure the striker was a 6 inch brass plump bob. My job at the time
was to take samples from storage tanks and bring them back into the lab for
analysis. I brought a portable cassette (lugable - 4 D cells) with me.
Somewhere I should have a tape of my singing inside a 30 or 40 foot high by
20 or so foot wide storage tank. - The echo was so strong and good that I
was able to build up 3 and 4 part chords - singing harmony to my echoing
voice. That was *very* cool.

This should give you a satellite view of the terminal

http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=GATX&sll=41.778609,-87.811832&sspn=0.246295,0.617981&gl=us&ie=UTF8&t=h&safe=active&ll=41.766655,-87.831166&spn=0.007698,0.019312&z=16

On Thu, May 7, 2009 at 2:19 AM, Carl Lumma <carl@...> wrote:

>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > Ok Carl,
> >
> > I've been there and done that....
> >
> > This was a live performance recorded 1891-ish on cassette - just a touch
> of
> > noise reduction after digitalization.
> >
> > http://clones.soonlabel.com/public/micro/random-interval-challenge
> > /tf-excerpt.mp3
> >
> > your challenge is to tell us the instrument - and any other
> > particulars - one clue - it was performed outside.
>
> I give. Sounds like a metallophone. Isn't cleanly pitched, so
> if you explore intervals with it you won't be able to compare
> them to intervals played on pitched instruments.
>
> But here, we were talking about whether phi can be an
> equivalence interval, better than other intervals. Are you
> saying it can?
>
> -Carl
>
>
>

🔗djtrancendance@...

🔗Michael Sheiman <djtrancendance@...>

Carl>"?? The harmonic series is JI. What did you mean by "JI" then?"
By JI I mean ANY JI scale.
A really obvious example, 7-tone diatonic JI is
1/1 9/8 5/4 4/3 3/2 5/3 15/8 (2/1: octave)
...whereas a 7-note per octave scale in the pure harmonic series is
7/7 8/7 9/7 10/7 11/7 12/7 13/7 (14/7: octave)
My point is...the 7-tone-per-octave "pure harmonic series" scale should always sound more pure than any JI scale (that's not the harmonic series exactly) with the same # of tone, right? (and if that pure scale won't cut mustard, neither will any other JI scale, right?)

>"The string "fractal" does not occur on that Wikipedia page.

I've never heard this term. I believe the correct term is

noble number. Care to point us to Jacques' post?"
Shoot, it's going to take me a while to dig that up...I always thought it was "noble number" as well until Jacques corrected me. I'll try to find it but, in the meantime, you might want to ask him yourself...

>"I'm trying to point out to you that your statements are vacuous

pseudoscientific garbage."
If they were then isn't it kind of hard to explain why Jacques thought I had some very good points and Rick (who used to slam me all the time for doing 'irrelevant PHI-based methods' is now making them himself and likes the sound of them? At this point, I'm pretty sure this has some substance and is not a 'cult'...though you could argue things like my old tetra-chordal scales were a bit cult-ish as many of them turned out to be nothing more than not-so-polished versions of existing tuning theory (but note, those scales did not have any followers and, dare I say it, I think I have improved much since then).

>"Oh, it's soothing. My mistake! What was I thinking?"
Ugh...now you are just being a bastard for no apparent reason. If every-time I bring up consonance you give me some lecture about how consonance can only occur when harmonics/overtones match the same way they do in the harmonic series or how "harmony" is exclusive to the harmonic series rather than representative of a higher general state of relaxation, I'm going to use more abstract terms. And now I use the abstract terms, and you still get pissed off. What do you expect me to do far as terminology anyhow, part the Red Sea? I am just trying to avoid what you claim to be conflicting terms...but you can't even seem to give any respect to my efforts to do so.

>"Or, you can ask nicely how an observation you've made might be put on firm technical footing."
That becomes tricky as
A) >None< of the paper you've handed me have anything to do with true noble numbered scales generated in a way anyway vaguely like my own. You've given me countless documents and advice on MOS scales (all those Mt. Meru scales are, are MOS scales), noble mediants, Just Intonation and tetra/penta...chords that simply don't bear relevance to my system (not to mention I've read a vast majority of them anyhow).

It is like you've given me very well made plans for a hydrogen blimp and are wondering why I don't want to use it to make alternative-style fire trucks/blimps (lol). Just because it's valid information doesn't make it automatically fit the intended purpose.

>"None of these examples will beat at all with sine tones.

What beat rates are you expecting?"

Very very slow ones as beating becomes faster as you approach a tone with another tone and slower as you move away. Even if you put two sine waves a whole 5th apart, for example, they are going to slowly fluctuate.

-Michael

🔗caleb morgan <calebmrgn@...>

🔗Carl Lumma <carl@...>

🔗chrisvaisvil@...

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> >"I'm saying your _explanations_ of their psychoacoustic
> >properties are bad."
> Well, so far, it seems to me your definition of bad
> is "anything that doesn't agree perfectly with the harmonic
> series".

Why do you think that? Didn't I point you to the metastable
intervals paper? What about the MOS scales?

>You actually had the chutzpah to >tell< me that the term I was
>coining "proportionate beating" must mean the beating in the
>harmonic series even though you also admitted the term didn't
>exist yet.

If you're describing the _most irrational_ beating possible
as proportionate and saying you like it better than periodicity
buzz (which is based on rational proportions), you're living on
a strange planet indeed.

> And about 80% of the phenomena in this scale have nothing to
> do with the harmonic series.

Cool. Which phenomena are those?

> For the record, a huge deal of scales look great on paper but
> sound terrible in reality.

Which ones?

> >"I've read nothing here to suggest that the claims you're
> > making about the scale are being understood or corroborated
> > by Rick or Jacques or anyone else."
>
> Jacques verified the methods

I said understood and corroborated the psychoacoustic claims.
Not verified the scale construction method.

> Rick surely understands what properties I'm going for as
> his choice of the 36TET tuning, for example, almost perfectly
> captures all the notes in my scale within a few cents of
> accuracy.
> Well, what kind of proof are you looking for?

You would have to make a clear, falsifiable claim, and stick
to it for longer than 30 seconds. Such as "phi functions as
an equivalence interval". Then we can ask Rick if he agrees,
making sure Rick understands what you meant first. Sound like
a plan?

-Carl

🔗Carl Lumma <carl@...>

🔗Daniel Forro <dan.for@...>

🔗Mike Battaglia <battaglia01@...>

🔗djtrancendance@...

Carl>"Why do you think that? Didn't I point you to the metastable
intervals paper? What about the MOS scales?"
For crying out loud are you even listening, I've said about 50 times my scale are not based nor have anything to do with either of the two!
Relate MOS scale construction to my (1/PHI)^x + 1 formula and I'll give you a free lollipop. Oh wait, my formula doesn't produce two distinct interval sizes...guess that kind of screws you over. And how about taking the noble mediant to find the maximum point of dissonance between rational intervals? Oh, wait, my scale does not use any rational intervals nor does it take the mediant between any of the irrational intervals it uses...
>>
If anyone here beside Carl has some twisted notion my scale and the formula (1/PHI)^x + 1 can miraculously be twisted to match either the concept of the Noble Mediant or MOS scales please speak up. I, for one, seriously doubt the connection is there...
<<

Carl> "If you're describing the _most irrational_ beating possible"
There you go comparing everything to the rational number beating of the harmonic series again...apparently to you anything that differs from that is automatically wrong. Enough!

Simply listen and compare:
1) 8-tone per octave harmonic series
http://www.geocities.com/djtrancendance/PHI/seriesharmonic8.wav.mp3
2)
http://www.geocities.com/djtrancendance/PHI/PHIharmonic8.wav.mp3

I realize some people will likely pick #1 vs. #2...but to say without doing any sort of survey that the harmonic series automatically MUST win the contest simply by stating a bunch of mathematics rather than actually having people listen and compare....is just pure ignorance: it has nothing to do with "everyone who likes this must have alien ears b/c it doesn't fit both rational math and the 2/1 octave".
BTW, I dare you to try and fit 8-tones within an octave (each tone being at least a ratio of 1.04 away from the last tone) and generate something that sounds smoother and more string-section-like than my PHI-harmonic example.

BTW...you know, if music were about math every famous musician would have to also be mathematician.

I'm starting to agree with Mike B here...these nasty arguments you launch seem far more about your stubborn shtick against non-octave scales then it does about actually making scales (figures...I haven't heard you posting any of your own new scales here lately, yet you seem more than happy to bash me for lack of productivity).

Meanwhile...I'm back to actually working on making new scales...perhaps something you should be posting more about (your own new scales) before you whine or bully around the bush about my efforts or anyone elses....

-Michael

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

> If anyone here beside Carl has some twisted notion my scale and the formula
> (1/PHI)^x + 1 can miraculously be twisted to match either the concept of the
> Noble Mediant or MOS scales please speak up. I, for one, seriously doubt the
> connection is there...

The phi interval is the noble mediant between 1/1 and 2/1. That means
that as your brain tries to lock the "phi" interval in and figure out
what it is, it's going to be interpreted as being as far from both 1/1
and 2/1 as possible.

Remember how I mentioned that 1/1 and 2/1 (and 4/1, and 8/1, and so
on) are all going to have the same chroma? Well, the metastable
interval approach would dictate that the phi interval is as
harmonically far from being interpreted as 1/1 and 2/1 as possible.
This logically means that the phi interval is also going to have as
"different" a tone chroma as possible from your starting note. So
according to that theory, it means that if 1/1 is C, phi times C is
going to have the "polar opposite" chroma. This is actually one of the
properties of this scale that I like, and why I said that I thought it
was exceptionally "colorful" - I'm hearing different pitch colors all
over the place. The first scale of yours sounded a bit more colorful
though - what were the specifications for it?

Keep in mind though that from an AP standpoint, chroma is such a
strange and understudied concept for me that I'm not sure if
metastable ratios can really apply to chroma labels just like that.
Chroma perception, from my observations, seems to be highly, highly
listener dependent; much more so than fundamental placement. It's also
an unresolved question at least for me what part of the overall
"chroma" perception is the fundamental percept and what parts are
rather rooted in associations from psychological pitch categories.
Nonetheless, that's the idea behind why the metastable concept applies
particularly to the phi scale.

Keep in mind that this only applies to two note dyads. As of right
now, we don't have HE curves for triads or tetrads or anything more
than two-note chords. As you stated earlier, if you play an entire
harmonic series and replace 13 with phi, which is 7 cents flat, I
would be extremely surprised if the timbre didn't "fuse" nonetheless.

It is certainly possible that in this new system of tonality you're
creating, certain properties of the scale would lead to you hearing
notes a phi ratio apart as being "equivalent" in some sense. What is
likely is that the scale structure repeating at phi is an easily
recognizable pattern that leads to some perceptual nature of
"equivalence," although it's not the same type of equivalence as 2/1
chroma equivalence.

I recommend you read the metastable ratio paper, it's quite an
interesting read. It might also spark some new ideas for scales for
you.

-Mike

🔗djtrancendance@...

Carl>"I recommended scales that distinctly go against the harmonic series..and a paper that suggests that 'JI is not the only way'."

Hmm...really (meaning ones that go against the harmonic series and still promote consonance)? Which ones, for example?

>"I take it you hadn't read his latest when you wrote this.
He thinks you argue, he doesn't know which scale he was
commenting on, etc."

Considering I've been talking about the (1/PHI)^x + 1 scale for well over a month, I figure it would be obvious that is the scale the PHIcicles example was made with. I've posted that scale well over 5 times and the formula well over 15 times. And I've clearly called it the "new phi scale" and the phi^y/2^x one the "old phi scale". And, if he'd asked me directly to re-post the scale again or asked which version of the scale I used, you bet I would answer quickly with a copy of the new scale printed in my reply.

Also, for the record I am very busy working on a mathematical proof that goes deep into psychoacoustics. It turns out no harmonic (meaning 1-times, 2-times, 3-times) are root note of my scales is either
1) more than the ratio 1.013 away from the nearest root note on a higher period
OR
2) less than the ratio 1.05 away from the nearest root note on a higher period

For example, one of the notes in my PHI scale is 1.23606, which times 3 (third overtone) is 3.70821...which is very near 3.72 (another note on a higher octave of the PHI scale). Meanwhile 1.09017 (on the PHI scale) * 3 = 3.4377, which is right in between the notes 3.25 and 3.72 on a higher octave in the PHI scale (and 3.4377 / 3.25 = 1.0577...far enough away to stop violent beating).

So...the way that harmonics either hit the notes dead-on or slip right in-between them is an obvious example of mathematically why the funky noble numbered scales exhibit consonance in the traditional sense of the word. The way tones exclusively fit either very very closely (chorus-like effect) or fairly far away (eliminating beating for the most part) goes hand-in-hand with Sethares' consonance curves.

>>
Does that sort of thing count as a psycho-acoustic explanation...or what else is missing?
<<

Far as explaining "proportionate beating"...there are too many other theories about noble numbered scales that having nothing to do with the pleasant-ness of beating. Would "symmetric beating" be a better title?

-Michael

🔗Daniel Forró <dan.for@...>

🔗Daniel Forro <dan.for@...>

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> Carl>"Why do you think that? Didn't I point you to the metastable
> intervals paper? What about the MOS scales?"
> For crying out loud are you even listening, I've said about 50 >times my scale are not based nor have anything to do with either of >the two!

"For crying out loud", please stop saying "my scale" when referring to (1/PHI)^x + 1, also known as succesive golden sections! If you were earnest about this, you would have already discovered, as I did a couple of years ago, that Lorne Temes did this 40 years ago- and it's documented in the Scala library!

> Relate MOS scale construction to my (1/PHI)^x + 1 formula and >I'll give you a free lollipop.
>Oh wait, my formula doesn't produce two distinct interval >sizes...guess that kind of screws you over.

You've got to be kidding.

! Golden_MOS_02.scl
!
period Phi, generator second successive golden section of Phi, Bobro
9
!
69.09700
168.36790
267.63880
366.90970
436.00670
535.27760
634.54850
733.81940
833.09030

! Golden_MOS_03.scl
!
period Phi, generator third successive golden section of Phi, Bobro
7
!
110.00734
235.77441
345.78175
471.54882
581.55616
707.32323
833.09030

! Golden_MOS_04.scl
!
period Phi, generator 4th successive golden section of Phi, Bobro
11
!
63.69166
149.46366
213.15532
298.92732
362.61898
448.39098
512.08264
597.85464
661.54630
747.31830
833.09030

Shall I go on, and on, and on?

If you were seriously working with Phi and the golden section you'd have already guessed, by instinct as it were, that it is a most egregiously MOSalicious proportion!

>And how about taking the noble mediant to find the maximum point of >dissonance between rational intervals?

Please actually read the Noble Mediant paper so that you'll stop using the word "dissonance" in this context.

>Oh, wait, my scale does not >use any rational intervals

Explore more deeply and you may find some real surprises.

> >>
> If anyone here beside Carl has some twisted notion my scale and >the formula (1/PHI)^x + 1 can miraculously be twisted to match >either the concept of the Noble Mediant or MOS scales please speak >up. I, for one, seriously doubt the connection is there...

Yee Olde (not "yours" or "my") Golden Section tunings, don't need any miraculous twisting to be used for MOS scales, quite radically the opposite, as I demonstrated above- and that is just scratching the surface.

And (1/PHI)^x + 1 tunings are EXACTLY like Noble Mediant tunings in terms of achieving, and I quote from the Noble Mediant paper, "maximum complexity or
"ambiguity"" (NOT "dissonance" in a conventional sense!). Plot out the spectra and you will see, although you should already recognize this from the sound!

Anyway, take care

-Cameron Bobro

🔗Carl Lumma <carl@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> >"Lorne Temes did this 40 years ago- and it's documented in the Scala library! "
>
> I have not seen that/his scale yet...actually a few weeks ago I >got and list of all the supposed scales in the SCALA library related >which did not include this one. Carl hasn't mentioned it, you >haven't...in fact within the year or so I've been I haven't heard >Lorne Temes' name mentioned once...

>
> So It's not like you told me and I actively/ignorantly refused >to acknowledge Lorne. Again this is the first time his name has >been brought up on this entire list (not even Jacques Dudon managed >to bring his name up)...

It's been mentioned, linked to anaphoria from many times- Temes wrote what's maybe one of "the" paper on phi/golden ratio tuning ideas, though phi has been used deliberately in tuning hundreds of years ago (have you found that one yet?) and just by chance (as far as I know), a couple of thousand years ago. If you'd read the papers and books that keep being referred to here again and again, you'd already know these things.

>
> Anyhow, I did find the following link http://anaphoria.com>>>/temes.PDF...though I am very interested in finding more. He seems >to include 1.309 in his scale (a number not in mine)...so if I saw >it right I wonder why as (1 + (1/PHI)^x) does not produce this >ratio. If you have any more links though: of course I would be >interested to look...
> More importantly, where as an exact copy of Lorne's tuning...I >want to see if it indeed matches mine note-by-note.

Read his paper carefully, and check out the "temes" tunings in the Scala archive.
>
> >"And (1/PHI)^x + 1 tunings are EXACTLY like Noble Mediant tunings in
> terms of achieving, and I quote from the Noble Mediant paper, >"maximum
> complexity"
> In that case, I wonder why the tunings mentioned in the noble >mediant paper sound nothing like the (1 + (1/PHI)^x) tuning >(regardless of who created it).

The tunings mentioned merely incorporate, in a functional way, intervals of this "fuzziness" or whatever you want to call it. If you played them as functionally intended, you'd hear where it happens. A golden section tuning on the other hand tends to generate piles of ambiguous intervals.

They ARE related to each other in many subtle and curious ways- just look how there are MOS's left and right using successive golden sections, look at the difference and combination tones (as mentioned in Temes' paper).

First you must understand what a difference tone is- google it with "Tartini" or "terzo suono" and figure it out.

> However, I do understand and appreciate your point of >mentioning that tonal-ambiguity does not mean maximal dissonance. >Several others on this list seemed to be implying it was "the >opposite of consonance", therefore indirectly implying it was >dissonance.

"The opposite of love isn't hate, it's indifference." :-) I think Margo Schulter has used the term "asonance", which is very good because once you get certain amount of "noise" in the partials, you can get a strangely calm effect, like wind or ocean.

🔗Cameron Bobro <misterbobro@...>

Oh, and you owe me a lollipop for demonstrating how (1/PHI)^x + 1
are related to MOS! Funny how you made no mention of this in your reply, LOL.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@> wrote:
> >
> > >"Lorne Temes did this 40 years ago- and it's documented in the Scala library! "
> >
> >     I have not seen that/his scale yet...actually a few weeks ago I >got and list of all the supposed scales in the SCALA library related >which did not include this one. Carl hasn't mentioned it, you >haven't...in fact within the year or so I've been I haven't heard >Lorne Temes' name mentioned once...
>
> >
> > So It's not like you told me and I actively/ignorantly refused >to acknowledge Lorne. Again this is the first time his name has >been brought up on this entire list (not even Jacques Dudon managed >to bring his name up)...
>
> It's been mentioned, linked to anaphoria from many times- Temes wrote what's maybe one of "the" paper on phi/golden ratio tuning ideas, though phi has been used deliberately in tuning hundreds of years ago (have you found that one yet?) and just by chance (as far as I know), a couple of thousand years ago. If you'd read the papers and books that keep being referred to here again and again, you'd already know these things.
>
> >
> >     Anyhow, I did find the following link http://anaphoria.com>>>/temes.PDF...though I am very interested in finding more.   He seems >to include 1.309 in his scale (a number not in mine)...so if I saw >it right I wonder why as (1 + (1/PHI)^x) does not produce this >ratio. If you have any more links though: of course I would be >interested to look...
> >    More importantly, where as an exact copy of Lorne's tuning...I >want to see if it indeed matches mine note-by-note.
>
> Read his paper carefully, and check out the "temes" tunings in the Scala archive.
> >
> > >"And (1/PHI)^x + 1 tunings are EXACTLY like Noble Mediant tunings in
> > terms of achieving, and I quote from the Noble Mediant paper, >"maximum
> > complexity"
> >    In that case, I wonder why the tunings mentioned in the noble >mediant paper sound nothing like the (1 + (1/PHI)^x) tuning >(regardless of who created it).
>
> The tunings mentioned merely incorporate, in a functional way, intervals of this "fuzziness" or whatever you want to call it. If you played them as functionally intended, you'd hear where it happens. A golden section tuning on the other hand tends to generate piles of ambiguous intervals.
>
> They ARE related to each other in many subtle and curious ways- just look how there are MOS's left and right using successive golden sections, look at the difference and combination tones (as mentioned in Temes' paper).
>
> First you must understand what a difference tone is- google it with "Tartini" or "terzo suono" and figure it out.
>
>
>
> >    However, I do understand and appreciate your point of >mentioning that tonal-ambiguity does not mean maximal dissonance. >Several others on this list seemed to be implying it was "the >opposite of consonance", therefore indirectly implying it was >dissonance.
>
> "The opposite of love isn't hate, it's indifference." :-) I think Margo Schulter has used the term "asonance", which is very good because once you get certain amount of "noise" in the partials, you can get a strangely calm effect, like wind or ocean.
>

🔗Daniel Forro <dan.for@...>

🔗Michael Sheiman <djtrancendance@...>

I still don't get how Temes' scales are MOS. They have more than 2 interval sizes...in fact, many have more than 4 interval sizes.

--- On Fri, 5/8/09, Cameron Bobro <misterbobro@...> wrote:

From: Cameron Bobro <misterbobro@...>
Subject: [tuning] Re: PHI interval tuning (for Michael S): with "Phicicles" song example :-)
To: tuning@yahoogroups.com
Date: Friday, May 8, 2009, 6:56 AM

Oh, and you owe me a lollipop for demonstrating how (1/PHI)^x + 1

are related to MOS! Funny how you made no mention of this in your reply, LOL.

--- In tuning@yahoogroups. com, "Cameron Bobro" <misterbobro@ ...> wrote:

> --- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ > wrote:

> >

> > >"Lorne Temes did this 40 years ago- and it's documented in the Scala library! "

> >

> > I have not seen that/his scale yet...actually a few weeks ago I >got and list of all the supposed scales in the SCALA library related >which did not include this one. Carl hasn't mentioned it, you >haven't...in fact within the year or so I've been I haven't heard >Lorne Temes' name mentioned once...

> >

> > So It's not like you told me and I actively/ignorantly refused >to acknowledge Lorne. Again this is the first time his name has >been brought up on this entire list (not even Jacques Dudon managed >to bring his name up)...

> It's been mentioned, linked to anaphoria from many times- Temes wrote what's maybe one of "the" paper on phi/golden ratio tuning ideas, though phi has been used deliberately in tuning hundreds of years ago (have you found that one yet?) and just by chance (as far as I know), a couple of thousand years ago. If you'd read the papers and books that keep being referred to here again and again, you'd already know these things.

> >

> > Anyhow, I did find the following link http://anaphoria. com>>>/temes.PDF. ..though I am very interested in finding more. He seems >to include 1.309 in his scale (a number not in mine)...so if I saw >it right I wonder why as (1 + (1/PHI)^x) does not produce this >ratio. If you have any more links though: of course I would be >interested to look...

> > More importantly, where as an exact copy of Lorne's tuning...I >want to see if it indeed matches mine note-by-note.

> Read his paper carefully, and check out the "temes" tunings in the Scala archive.

> >

> > >"And (1/PHI)^x + 1 tunings are EXACTLY like Noble Mediant tunings in

> > terms of achieving, and I quote from the Noble Mediant paper, >"maximum

> > complexity"

> > In that case, I wonder why the tunings mentioned in the noble >mediant paper sound nothing like the (1 + (1/PHI)^x) tuning >(regardless of who created it).

> The tunings mentioned merely incorporate, in a functional way, intervals of this "fuzziness" or whatever you want to call it. If you played them as functionally intended, you'd hear where it happens. A golden section tuning on the other hand tends to generate piles of ambiguous intervals.

> They ARE related to each other in many subtle and curious ways- just look how there are MOS's left and right using successive golden sections, look at the difference and combination tones (as mentioned in Temes' paper).

> First you must understand what a difference tone is- google it with "Tartini" or "terzo suono" and figure it out.

> > However, I do understand and appreciate your point of >mentioning that tonal-ambiguity does not mean maximal dissonance. >Several others on this list seemed to be implying it was "the >opposite of consonance", therefore indirectly implying it was >dissonance.

> "The opposite of love isn't hate, it's indifference. " :-) I think Margo Schulter has used the term "asonance", which is very good because once you get certain amount of "noise" in the partials, you can get a strangely calm effect, like wind or ocean.

🔗Cameron Bobro <misterbobro@...>

🔗djtrancendance@...

>"for you're still only make the section in one direction... "
Not even true, I get the est. 1.52 in my scale from taking 1.618 - 1.0901...that's taking the golden section in the other direction.

>"for you're still only make the section in one direction... and only in the frequency realm."
Well...what realm is there beside the frequency realm you think is possible within music?

>"And these golden cuts in the frequency realm are NOT NEW, they are hundreds of years old!"

I knew they golden-section existed in the artistic realm (and have since high school), but not in the frequency realm.

And, you know, what the he**...I just posted a message before your reply below acknowledging Temes work and admited (truthfully) also that I never have heard Temes work mentioned
before you mentioned it yesterday... And yet here you are >still< slamming me for it as if I had said "yes I've read it and it's irrelevant"...when the fact is I read it and wrote a huge message about why it is relevant. And I certainly don't deserve this "you are trying to steal someone else's work" tone of voice you keep dishing out.
So, please get a life and stop insulting me for "avoiding giving references" crimes I am not guilty of attempting to propogate...
It's not my fault Temes is not a household name among tuning enthusiasts or that I had not heard of his before you mentioned him. His tuning are also listed as "Temes" and not "PHI-something-or-other", so it's not like the name would make it blantantly obvious his scale was any more PHI than the other 5000 or so scales on first glance. It's not like I looked through the scales and said "aha, this guy has the same kind of scale structure as
myself...best to try and keep this unknown". If you really do think that, you truely are being a bastard.

Now...enough of this freaking "why Michael is supposedly guilty of taking credit for something he knew existed" BS...let's talk about what I'm interested in in the first place: further developing work relating to the Golden Section (for the record, I don't care who created the idea or gets credit for it, but, rather, I care if it can be applied to music and if I can help develop it further...and I strongly believe the answer to both is a resounding "Yes".

Now let's get back to talking to something musicians can actually use, namely how to improve the use of golden sections to create beautiful scales and music.

For the record, actual musicians don't give who created 12TET, for example, more over they care how it sounds and how it can be used...and same goes with Golden Section scales. I am
here to improve scales, not to discuss ownership...so please do me a favor and concentrate on improving scales when you're talking with me: it simply isn't productive and can be destructive (including in the way that it may cause people, for example, to take Temes work less seriously as you are bashing it as being based on something too old and archaic (IE treating the concept like a dead-horse), regardless of if you are intending to do something more positive eventually with such statements or not).

-Michael

🔗Michael Sheiman <djtrancendance@...>

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> >"for you're still only make the section in one direction... "
> Not even true, I get the est. 1.52 in my scale from taking 1.618 - 1.0901...that's taking the golden section in the other direction.

Good! I had overlooked that, carry on!
>
> >"for you're still only make the section in one direction... and only in the frequency realm."
> Well...what realm is there beside the frequency realm you think is possible within music?

You can make the golden section logarithmically as well as linearly, check it out!
>
> >"And these golden cuts in the frequency realm are NOT NEW, they are hundreds of years old!"
>
> I knew they golden-section existed in the artistic realm (and have since high school), but not in the frequency realm.

Hunt some more, it's pretty trippy to see how long ago these things were explored.
>
> And, you know, what the he**...I just posted a message before >your reply below acknowledging Temes work and admited (truthfully) >also that I never have heard Temes work mentioned
> before you mentioned it yesterday... And yet here you are >still< >slamming me for it as if I had said "yes I've read it and it's >irrelevant"...when the fact is I read it and wrote a huge message >about why it is relevant. And I certainly don't deserve this "you >are trying to steal someone else's work" tone of voice you keep >dishing out.

Your reply sounds, let me be blunt, a bit wacko. I'm not accusing you of stealing others' work at all, I'm just showing you things that have already been done. I've done tons of "reinventing the wheel" myself, not realizing it had been done before, so what?

> So, please get a life and stop insulting me for "avoiding giving >references" crimes I am not guilty of attempting to propogate...
> It's not my fault Temes is not a household name among tuning >enthusiasts or that I had not heard of his before you mentioned >him. His tuning are also listed as "Temes" and not "PHI-something->or-other", so it's not like the name would make it >blantantly obvious his scale was any more PHI than the other 5000 or >so scales on first glance. It's not like I looked through the >scales and said "aha, this guy has the same kind of scale structure a>s
> myself...best to try and keep this unknown". If you really do >think that, you truely are being a bastard.

Your reply is way out of hand.

Do you use the "compare scale" function in Scala? set to subsets and supersets and make sure the directory is set right.
>
> Now...enough of this freaking "why Michael is supposedly guilty >of taking credit for something he knew existed" BS...let's talk >about what I'm interested in in the first place: further developing >work relating to the Golden Section (for the record, I don't care >w>ho created the idea or gets credit for it, but, rather, I care if i>t can be applied to music and if I can help develop it >further...and I strongly believe the answer to both is a resounding >"Yes".
>
> Now let's get back to talking to something musicians can actually >use, namely how to improve the use of golden sections to create >beautiful scales and music.

Certainly- you can use golden-section MOS of Phi, for example, as I do.

Speaking of credit, where's my credit for demonstrating the connection between the golden cut in the frequency realm and MOS?
>
> For the record, actual musicians don't give who created 12TET, >for example, more over they care how it sounds and how it can be >used...and same goes with Golden Section scales.

I make my living as an "actual musician", and I think such generalizations about musicians are bogus.

>I am
> here to improve scales, not to discuss ownership...so please do me >a favor and concentrate on improving scales when you're talking with >me: it simply isn't productive and can be destructive (including in >the way that it may cause people, for example, to take Temes work >less seriously as you are bashing it as being based on something too >old and archaic (IE treating the concept like a dead-horse), >regardless of if you are intending to do something more positive >eventually with such statements or not).

I agree that you should stop harping on the concept of "ownership" by using such terms as "my scale..." so often. hahahahaha! Didn't that sound just like Carl? :-)

Seriously- whatever. Read my ealier post about difference tones, then do the golden sections in the logarithmic realm and check out beat freqs. and so on and so on.

🔗djtrancendance@...

Cameron>"You can make the golden section logarithmically as well as linearly, check it out!"
Do you mean 2^(0.618034*x) ('phi-ths' of an octave) or something else? I have tried 2^(0.618034*x) and the sound came out terribly...but I realize that may not be what you meant...

>"I'm not accusing you of stealing others' work at all, I'm just showing you things that have already been done. "
In that case, why are you going on and on, using caps, using extreme phrasings like "that has been done for thousands of years!!!" and then putting multiple exclamation points behind them.
Your tone of voice in the past few messages seems to indirectly imply you think either I am stealing or have been purposefully ignorant and not tried to look for other PHI theories at all (turns out I've looked at a good few PHI theories, including Golden Meantone and the Keenan paper about the Noble Mediant...not to mention the other PHI scales posted on this list).

>"Do you use the "compare scale" function in Scala? set to subsets and supersets and make sure the directory is set right."
I don't but, you're right, I should and will in the future...I figure now that function would have pointed me to Temes' work.

>"Certainly- you can use golden-section MOS of Phi, for example, as I do."

You can, but it's neither my scale not Temes...and I was asking the question how does MOS relate to my scale not how does MOS relate to the golden section. But, yes, you did prove the golden section can related to MOS...though the method you use to split sections obviously differs from mine (are you splitting the golden section logarithmically...or what is your method exactly)? For sure, your scale seems to have many tones not in common with the scale I made or Temes own scales...

>"I make my living as an "actual musician", and I think such generalizations about musicians are bogus."
Maybe I went too far, but I meant to say, on the whole from experience, I've seen composing musicians as musicians first and historians second (if at all). I've asked 7 musicians I know who Temes is and who invented the 12TET temperament and they all scratched their heads...including my brother and two of my friends who make a living
doing the stuff (professional jazz guitarists).

>"I agree that you should stop harping on the concept of "ownership" by using such terms as "my scale..."
If I were really into ownership I would say something like "the scale I invented" or even "and am applying for a patent on".
"My scale" is analoguous to "the scale I came across" or "the scale I found when working with golden sections".
I never claimed "I'm sure no one has made a scale anything like this" though I have said "I haven't seen one yet"...which was honest until I found Temes work.

I mean, do you really expect me to write "the scale I came across which was originally made by Lorne Temes" every single time I mention it, making everything I write sound awkwardly long?
Give me a break...most people can figure out they mean it's a scale simply I came across with working with
numbers. And considering my scale has several different notes not in Temes scale, it should be called my scale at least to the point people won't assume it's exactly the same thing.

So far as the tonal theory behind it, I will refer to it generally as Temes golden-section theory. Because my scale does not have any fundamental basis that differ from Temes, minus the fact my scale is built to allow for coinciding overtone partials plus heavily non-coinciding ones and his is designed to not allow coinciding partials. And even that isn't that large a difference, more a matter of opinion that I allow golden section tone which he acknowledges, but purposefully avoids.

>"Read my ealier post about difference tones,"
I understand difference tones, where the difference between two notes gives the illusion of a third note with a base frequency of the difference between the two tones.
It's also explained here
http://www.phys.unsw.edu.au/jw/beats.html

>"then do the golden sections
in the logarithmic realm and check out beat freqs. and so on and so on."

Again, what do you mean by logarithmic realm? I've already tried the 2^((1/PHI)*x) scale and, frankly it sounds like garbage to me. Or did you mean something else...if so, please explain: I certainly didn't hear anything about logarithmic golden sections in Temes' paper... I'm all ears so long as I can find example links, do you have any?

Far as beats, I've been using scales with root tones at least a ratio of 1.05 away from each other for years. One look at the sound examples on http://www.phys.unsw.edu.au/jw/beats.html
with 420/400 (1.05) and ones beyond it like 430/400 and 440/400 show that I've been following a pattern of avoiding overly noticable interferance beating all along when making my scales all along.

-Michael

🔗Daniel Forro <dan.for@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Carl Lumma <carl@...>

🔗martinsj013 <martinsj@...>

🔗djtrancendance@...

🔗martinsj013 <martinsj@...>

🔗djtrancendance@...

🔗martinsj013 <martinsj@...>

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
> > What Michael is talking about is the Golden Section applied to the frequency realm. ...
> > 1/1
> > 366.910 cents
> > 466.181 cents
> > 560.067 cents
> > 733.819 cents
> > 833.090 cents
> > 982.554 cents
> > 1068.865 cents
> > 2/1
>
> Cameron,
> Many thanks for this very full answer, which makes me realise I was missing the point about difference tones. At the same time it reinforced my point, in that I can fully understand your scale and your explanation of it, whereas I am still having difficulty with Michael's (sorry, Michael! - I'll respond to you separately).
>
> Steve M.
>

You're welcome- it's a fascinating area, but of course, like Just Intonation, you can get very "New Age" or "speculative fiction" or whatever very quickly.

But there's nothing wrong with that either, "speculative fiction" is great in the arts. It's only hazardous here at the tuning list because you have to put warning stickers all over everything: "This part here isn't hard science!"

For example, it's a safe bet that someone is going to pooh-pooh difference tone, because they're not the "explaination for everything", or something like that. Well they aren't, but they are something substantial, and anyone can "prove" to themselves on a personal level how important or unimportant they are in actual practice.

Temes predicted, correctly in my experience, that using phi in tuning would create all kinds of ""unexpected"" (his quote-marks there) "combinatorial properties and striking relationships", and that the tones would be "maximally dissonant" in a special way- he called the effect "white". Here on the tuning list the effect has been called "gray", "ambiguous"; I've used the expressions "far-away consonance" and "otherness" as well. Basically we're talking about combined spectra that tend to be both "noisey" and yet strangely "ordered", at the same time.

The scale above was made to quickly demonstrate the "hall of mirrors" nature of Phi in this context.

These are the first five golden sections of Phi, which is in turn a golden section (in the frequency realm) of the octave.

!First_Five_Golden_Cuts_of_Phi.scl
!

6
!
93.88597
149.46366
235.77441
366.90970
560.06656
833.09030

The MOS scales I posted earlier use Phi as the period, and an "internal" golden cut as the period.

So perhaps Michael will see that
they are indeed very much related to what he is doing, certainly he'll recognize the tuning laid out like this. Michael, remember that
you started using the golden section of the octave (that is, Phi) as the generator. The Golden MOSs are simply using golden sections of golden sections as period and generator.

Any of the "nested" golden sections will produce many Modes of Symmetry within Phi- they will also do this within each other, ie., a smaller cut becomes the period. Phi will also make MOS with the octave, cf. Wilson.

The natural question here would be, how about a golden section of other intervals? Well, not surprisingly this also makes MOSs, here's an MOS of 7/4, with 7/4 as the period and the first golden section of 7/4 (in the frequency, that is, linear, realm):

!Golden_MOS_Just_7.scl
!

7
!
40.36054
80.72107
349.84899
390.20953
659.33745
699.69799
7/4

If you drop it into Scala, you'll find that it is an MOS, and this works with any damn interval, as far as I know.

I must say that it may be that these tunings, in spite of being comprised of two intervals, don't necessarily all fulfill all the requirements of being MOS.

The high percentage of aurally delightful tunings that pop out this way makes my take on that point to be "whatever". :-)

What is unique about Phi as the period is the singular logarithmically "mirrored" difference tone with the 1/1, and that the golden-sections-of-golden-sections period/generator approach always, as far as I know, produces at least some degree of differential coherence (difference tones appearing, octaved, in the tuning as intervals, and vice versa).

Obviously this is all just tooling down a path found by Erv Wilson,
beautifully explored by Kraig Grady. And Jacques Dudon is a pioneer of Phi and differentially-coherent tuning work.

Now I wonder if anyone else has dug up some old and ancient Phi tunings? Surely someone has following this discussion has found Brouncker, 17th century, but has anyone else found the documented Phi tuning from nearly 2 and a half millenia ago? :-)

-Cameron Bobro

🔗Michael Sheiman <djtrancendance@...>

>"These are the first five golden sections of Phi, which is in turn a golden section (in the frequency realm) of the octave.

!First_Five_ Golden_Cuts_ of_Phi.scl

93.88597

149.46366

235.77441

366.90970

560.06656

833.09030
"
Aha! You finally have it, meaning about 90% of the scale I discovered when tooling with PHI. And the only note missing vs. the scale I found (not necessarily "made") is 733 cents, which is the equivalent of 833 cents minus the interval 1.09 (IE the one note that represents the octave inverse). Congrats...finally this is a correct derivation of my scale.

>"The MOS scales I posted earlier use Phi as the period, and an "internal" golden cut as the period."
I understand this, yet they do >not< form the same tones at all as those mentioned above, or even close estimates for the most part.

>"The natural question here would be, how about a golden section of other
intervals? Well, not surprisingly this also makes MOSs, here's an MOS
of 7/4, with 7/4 as the period"
Right, but that's basically taking the golden section of an interval which does not fit the equation interval = 1/interval + b. Hence why I make a point of plugging in PHI and the Silver Ratio into such a construct, but not any other intervals.

>"So perhaps Michael will see that they are indeed very much related to what he is doing, certainly he'll recognize the tuning laid out like this. "
Again...indeed Cameron, this time you really have derived the scale I came up with: the first 5 golden cuts of phi derives a huge majority of my scale (minus the one mirrored tone). No complaints from me on this point. :-)

Can we all agree though....that the study of use of the golden-section type of tuning formula (as opposed to the mean-tone generator construct) is especially useful for use with any "noble number" generators and is well worth study and not just a random formula that works no better than any other random formula?

🔗martinsj013 <martinsj@...>

🔗Aaron Krister Johnson <aaron@...>

Hi,

To clarify for all, including Rick, Steve, and Michael:

I was using a logarithmic (geometric) division of the octave into a phi-ratio as a generator, and the octave as a period, in my example...
2^(0.618..*x) as has been mentioned by Michael, I think. I was NOT using phi as an acoustic ratio....so I'm talking about approximations with edos to the interval corresponding to "1/phi octaves (~0.618 octaves)", or 741.64078649987357 cents, and not phi itself, which is 833.09029635674085 cents. The 741.64 cent interval, or it's nearest approximation in an appropriate edo is then used as a generator, where the period is the good ol' octave.

The 'justification', should one need it, is that the octave is THE established interval of equivalence to the ear, and thus metaphorically serves as a 'unit circle' with which we can explore angles. Form a phyllotaxic point-of-view, the way certain plants use the 'golden angle', or 2PI/phi radians, or ~0.618*360 degrees, as an angular 'generator' is interesting in the way that it maximizes the amount of sunlight the various layers of the plant can receive. By analogy, one would expect that the 'spectral space' of the octave would also be so divide as to give a kind of maximal richness or complexity...

Now, since Phi itself is approximated by neighboring terms of the Fibonacci sequence in ratio form, it stands to reason that it can be approximated by higher and higher numbered "Fibonacci edos" (8,13,21,34,55 etc.). If 'n' is a Fibonnaci number representing an edo, pick the 'n-1' Fibonacci number as an index to use as a generator. You will find that this pattern forms MOS scales, the number of notes in the MOS scale always being itself a Fibonnacci number.

So, for instance, 34-edo using 21deg34 as a generator, we have a MOS scale when we get to 8 notes, 13 notes, etc...I agree with Michael that 8 is a nice, medium size scale--not too small to be uninteresting, not too large to be unweildy, chaotic, or hard to grasp in its totality.

Now, mind you, I think one's milage with this scale depends on the timbre...good for inharmonic spectra like gongs, FM inharmonic patches, samples of kitchen pots and pans, or in general, any timbre which has a decay on the order of 1 second or less. Then the scale will have an appealing 'alien otherness'...but I wouldn't recommend sustaining big chords on a church organ or hammond organ or string sound or fat sawtooth synth sound, unless you are using it as a torture technique at Gitmo.

Best,
Aaron.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > I think you were referring to Steve's last post which said:
> > "AKJ's method, using 2^p (where p=0.618, or a Fibonacci-number approximation to it) is in fact splitting the octave geometrically (rather than arithmetically) and it seems to me to be the closest aural analogy to the visual splitting of a line segment."
> > I don't know who AKJ is or whether Steve's comment is truly representative, but this is some very faulty logic. ... whoever thought of this "method" answer me this: given 2^(1/phi), then we reach the octave at (2^(1/phi))^phi = 2. How on earth are we supposed to have a phi number of frequencies/notes within an octave? ... THIS is why the exponents cannot be irrational, only the resultant frequencies can be irrational. Besides, there is a great deal of difference b/w 2^(1/phi) and phi. It is like saying that 2^3 = 8 can somehow substitute the number 3.
>
> Hello Rick,
> Sorry for the imprecise reference, which was to Aaron Krister Johnson's post #83263. A few points:
> 1) I only said I thought it "the closest aural analogy ...", not that it would necessarily make a good tuning;
> 2) Aaron actually uses a rational approximation (21/34) to phi, so it does reach a whole number of octaves (just as your 25/36 does);
> 3) but OTOH why insist on it reaching a whole number of octaves - the Pythagorean system doesn't;
> 4) neither Aaron or I is saying that 2^(1/phi) is the same as phi - it is a different use of phi from yours and Michael's.
>
> Regards,
> Steve M.
>

🔗Michael Sheiman <djtrancendance@...>

>"I was using a logarithmic (geometric) division of the octave into a
phi-ratio as a generator, and the octave as a period, in my example...

2^(0.618..*x)"

Right...so then you have to take
2^(0.618*x) where x = -1,-2,-3... or 1,2,3,4...and then divide or multiply by two to bring everything back to the octave. Then you get values like 1.177, 1.8075, 1.387, 1.0643, 1.6335, and 1.5347...right?

--- On Sat, 5/9/09, Aaron Krister Johnson <aaron@...> wrote:

From: Aaron Krister Johnson <aaron@...>
Subject: [tuning] PHI tuning---further explanation
To: tuning@yahoogroups.com
Date: Saturday, May 9, 2009, 12:03 PM

Hi,

To clarify for all, including Rick, Steve, and Michael:

I was using a logarithmic (geometric) division of the octave into a phi-ratio as a generator, and the octave as a period, in my example...

2^(0.618..*x) as has been mentioned by Michael, I think. I was NOT using phi as an acoustic ratio....so I'm talking about approximations with edos to the interval corresponding to "1/phi octaves (~0.618 octaves)", or 741.64078649987357 cents, and not phi itself, which is 833.09029635674085 cents. The 741.64 cent interval, or it's nearest approximation in an appropriate edo is then used as a generator, where the period is the good ol' octave.

The 'justification' , should one need it, is that the octave is THE established interval of equivalence to the ear, and thus metaphorically serves as a 'unit circle' with which we can explore angles. Form a phyllotaxic point-of-view, the way certain plants use the 'golden angle', or 2PI/phi radians, or ~0.618*360 degrees, as an angular 'generator' is interesting in the way that it maximizes the amount of sunlight the various layers of the plant can receive. By analogy, one would expect that the 'spectral space' of the octave would also be so divide as to give a kind of maximal richness or complexity.. .

Now, since Phi itself is approximated by neighboring terms of the Fibonacci sequence in ratio form, it stands to reason that it can be approximated by higher and higher numbered "Fibonacci edos" (8,13,21,34, 55 etc.). If 'n' is a Fibonnaci number representing an edo, pick the 'n-1' Fibonacci number as an index to use as a generator. You will find that this pattern forms MOS scales, the number of notes in the MOS scale always being itself a Fibonnacci number.

Now, mind you, I think one's milage with this scale depends on the timbre...good for inharmonic spectra like gongs, FM inharmonic patches, samples of kitchen pots and pans, or in general, any timbre which has a decay on the order of 1 second or less. Then the scale will have an appealing 'alien otherness'.. .but I wouldn't recommend sustaining big chords on a church organ or hammond organ or string sound or fat sawtooth synth sound, unless you are using it as a torture technique at Gitmo.

Best,

Aaron.

--- In tuning@yahoogroups. com, "martinsj013" <martinsj@.. .> wrote:

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

> > I think you were referring to Steve's last post which said:

> > "AKJ's method, using 2^p (where p=0.618, or a Fibonacci-number approximation to it) is in fact splitting the octave geometrically (rather than arithmetically) and it seems to me to be the closest aural analogy to the visual splitting of a line segment."

> > I don't know who AKJ is or whether Steve's comment is truly representative, but this is some very faulty logic. ... whoever thought of this "method" answer me this: given 2^(1/phi), then we reach the octave at (2^(1/phi))^ phi = 2. How on earth are we supposed to have a phi number of frequencies/ notes within an octave? ... THIS is why the exponents cannot be irrational, only the resultant frequencies can be irrational. Besides, there is a great deal of difference b/w 2^(1/phi) and phi. It is like saying that 2^3 = 8 can somehow substitute the number 3.

> Hello Rick,

> Sorry for the imprecise reference, which was to Aaron Krister Johnson's post #83263. A few points:

> 1) I only said I thought it "the closest aural analogy ...", not that it would necessarily make a good tuning;

> 2) Aaron actually uses a rational approximation (21/34) to phi, so it does reach a whole number of octaves (just as your 25/36 does);

> 3) but OTOH why insist on it reaching a whole number of octaves - the Pythagorean system doesn't;

> 4) neither Aaron or I is saying that 2^(1/phi) is the same as phi - it is a different use of phi from yours and Michael's.

> Regards,

> Steve M.

🔗djtrancendance@...

🔗Mike Battaglia <battaglia01@...>

> I was using a logarithmic (geometric) division of the octave into a
> phi-ratio as a generator, and the octave as a period, in my example...
> 2^(0.618..*x) as has been mentioned by Michael, I think. I was NOT using phi
> as an acoustic ratio....so I'm talking about approximations with edos to the
> interval corresponding to "1/phi octaves (~0.618 octaves)", or
> 741.64078649987357 cents, and not phi itself, which is 833.09029635674085
> cents.
The 2^(0.618..*x) equation IS the "phith" of an octave. If instead you
did want to use successive multiples of phi on top of one another, you
would be using (0.618..)^x instead, I think.

> The 'justification', should one need it, is that the octave is THE
> established interval of equivalence to the ear, and thus metaphorically
> serves as a 'unit circle' with which we can explore angles. Form a
> phyllotaxic point-of-view, the way certain plants use the 'golden angle', or
> 2PI/phi radians, or ~0.618*360 degrees, as an angular 'generator' is
> interesting in the way that it maximizes the amount of sunlight the various
> layers of the plant can receive. By analogy, one would expect that the
> 'spectral space' of the octave would also be so divide as to give a kind of
> maximal richness or complexity...

That's an interesting idea. When you say maximal richness, do you mean
that as successive tones from the generator are included, they will be
placed "maximally far apart", similar to the arrangement shown at
http://www.abc.net.au/science/photos/mathsinnature/img/13.jpg?

That seems to logically make sense, but what about the successive
multiples of phi on top of each other approach? Each multiple of phi
should be, chroma-wise, "maximally far apart" from the previous note,
since phi is the noble mediant between 1/1 and 2/1, both of which have
the same chroma. Thus if the noble mediant approach is successful, phi
should have a maximally different chroma from 1/1.

I wonder which approach works better? If the second approach really
works better to give "maximally opposite chromas", then what
perceptual, idiomatic, or cognitive effects would the first one have?

> Now, mind you, I think one's milage with this scale depends on the
> timbre...good for inharmonic spectra like gongs, FM inharmonic patches,
> samples of kitchen pots and pans, or in general, any timbre which has a
> decay on the order of 1 second or less. Then the scale will have an
> appealing 'alien otherness'...but I wouldn't recommend sustaining big chords
> on a church organ or hammond organ or string sound or fat sawtooth synth
> sound, unless you are using it as a torture technique at Gitmo.
>
> Best,
> Aaron.

Let's not forget samples with detuned overtones. I'd love to come up
with a quick program to do this - does anyone have any ideas as to
what format the output samples should be in? Would folks be able to do
anything useful if I just made it output a wav file in the end?

-Mike

🔗rick_ballan <rick_ballan@...>

I see, thanks. Looks like a case of the old Chinese whispers. I'll look into it (what date is Aaron's post?). [Just to clear up any confusion, you mean that Aaron's 21/34 is a rational Fibonnachi number and mine is irrational 2^(25/36) which replaces PHI itself, not rational 25/36?] Regarding 3), whatever else we do, I do think that 8ve equivalence should be preserved nowadays. The fact that 12 successive fifths didn't reach the 8ve was a problem that had to be solved via comma's and ultimately 12 TET (and besides, even the Pythagoreans themselves held 1:2, 2:3 and 3:4 as the most important, their sum equal to the holy number 10, so they would have liked their theories to add-up too).

Cheers

-Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
>
> Hi,
>
> To clarify for all, including Rick, Steve, and Michael:
>
> I was using a logarithmic (geometric) division of the octave into a phi-ratio as a generator, and the octave as a period, in my example...
> 2^(0.618..*x) as has been mentioned by Michael, I think. I was NOT using phi as an acoustic ratio....so I'm talking about approximations with edos to the interval corresponding to "1/phi octaves (~0.618 octaves)", or 741.64078649987357 cents, and not phi itself, which is 833.09029635674085 cents. The 741.64 cent interval, or it's nearest approximation in an appropriate edo is then used as a generator, where the period is the good ol' octave.
>
> The 'justification', should one need it, is that the octave is THE established interval of equivalence to the ear, and thus metaphorically serves as a 'unit circle' with which we can explore angles. Form a phyllotaxic point-of-view, the way certain plants use the 'golden angle', or 2PI/phi radians, or ~0.618*360 degrees, as an angular 'generator' is interesting in the way that it maximizes the amount of sunlight the various layers of the plant can receive. By analogy, one would expect that the 'spectral space' of the octave would also be so divide as to give a kind of maximal richness or complexity...
>
> Now, since Phi itself is approximated by neighboring terms of the Fibonacci sequence in ratio form, it stands to reason that it can be approximated by higher and higher numbered "Fibonacci edos" (8,13,21,34,55 etc.). If 'n' is a Fibonnaci number representing an edo, pick the 'n-1' Fibonacci number as an index to use as a generator. You will find that this pattern forms MOS scales, the number of notes in the MOS scale always being itself a Fibonnacci number.
>
> So, for instance, 34-edo using 21deg34 as a generator, we have a MOS scale when we get to 8 notes, 13 notes, etc...I agree with Michael that 8 is a nice, medium size scale--not too small to be uninteresting, not too large to be unweildy, chaotic, or hard to grasp in its totality.
>
> Now, mind you, I think one's milage with this scale depends on the timbre...good for inharmonic spectra like gongs, FM inharmonic patches, samples of kitchen pots and pans, or in general, any timbre which has a decay on the order of 1 second or less. Then the scale will have an appealing 'alien otherness'...but I wouldn't recommend sustaining big chords on a church organ or hammond organ or string sound or fat sawtooth synth sound, unless you are using it as a torture technique at Gitmo.
>
> Best,
> Aaron.
>
> Hi Aaron,

Thought there must be more to it. Thanks for clearing this up. As a matter of general interest, are you saying that the phyllotaxic concept behind PHI applied to music could give something like the most 'distant' or 'foreign' intervals from a key? Interesting. I was thinking more along the lines of intervals which have beats equal to ratios.

I agree with you that 8ve equivalence must be self-evident so no justification needed there. Forgive my ignorance but I'm still not sure what you mean by a logarithmic division of the 8ve as opposed to an acoustic ratio and therefore find it hard to keep the distinction in my head without reverting back to what I'm used to. Given 2^(0.618..*x), you said "I'm talking about approximations with edos to the interval corresponding to "1/phi octaves (~0.618 octaves)", which seems to make perfect sense, but I can't yet follow it through mathematically.

Regards

Rick

> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
> >
> > --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > > I think you were referring to Steve's last post which said:
> > > "AKJ's method, using 2^p (where p=0.618, or a Fibonacci-number approximation to it) is in fact splitting the octave geometrically (rather than arithmetically) and it seems to me to be the closest aural analogy to the visual splitting of a line segment."
> > > I don't know who AKJ is or whether Steve's comment is truly representative, but this is some very faulty logic. ... whoever thought of this "method" answer me this: given 2^(1/phi), then we reach the octave at (2^(1/phi))^phi = 2. How on earth are we supposed to have a phi number of frequencies/notes within an octave? ... THIS is why the exponents cannot be irrational, only the resultant frequencies can be irrational. Besides, there is a great deal of difference b/w 2^(1/phi) and phi. It is like saying that 2^3 = 8 can somehow substitute the number 3.
> >
> > Hello Rick,
> > Sorry for the imprecise reference, which was to Aaron Krister Johnson's post #83263. A few points:
> > 1) I only said I thought it "the closest aural analogy ...", not that it would necessarily make a good tuning;
> > 2) Aaron actually uses a rational approximation (21/34) to phi, so it does reach a whole number of octaves (just as your 25/36 does);
> > 3) but OTOH why insist on it reaching a whole number of octaves - the Pythagorean system doesn't;
> > 4) neither Aaron or I is saying that 2^(1/phi) is the same as phi - it is a different use of phi from yours and Michael's.
> >
> > Regards,
> > Steve M.
> >
>