Yahoo Tuning Groups Ultimate Backup tuning Using commonalities between Rick's PHI scale and Mike's/mine to make an "ultimate PHI scale"

Ok, so here we go...I am trying to learn about PHI in music rather than prove whose scale is "better".

Here is a fairly good "triad" type chord that works in both Rick's PHI scale and my own that we have both mentioned as good:

1
1.236067 (2/PHI)...this is 1.23589 in 36TET IE the tuning for Rick's scale
1.618034 (PHI)...this is 1.61826 in 36TET IE the tuning for Rick's scale

For starters of issues with both scales, I recognize that the interval 1 and 1.618034 beats a bit too much and beats much less as the rather similar sounding 1.625.

Beyond that, there comes the question how best to derive more possible notes for the scale (since a 3-note scale is rather limited).

One issue is symmetry to the 2/1 octave and not simply the PHI-tave. One possible solution I believe could work is using the chord
1
1.236067 (2/PHI)
1.625 = 1.618034 (PHI) slightly rounded to preserve more overtone consonance
1.85408 = 1.618034 * 1.14589 (where 1.14589 = (1/PHI)^4))
2 (2/PHI * PHI)

There is another gap between 1.236067 and 1.618034 which could easily be filled by another note without violating the critical band and yet another gap between 1.236067.

My scale uses 1.14589 (where 1.14589 = (1/PHI)^4 + 1)) to fill the lower gap and 1.381966 (where 1.381966 = (1/PHI)^2 + 1)) to fill in the upper. That's my proposed solution to get a 6-note per octave scale with quite high consonance.

Without putting words into your mouth...Rick...how would you propose to fill the above gaps with more notes? And/or, do you see a problem with using 1.85408 to fill in the gap between 1.618034 and 2...and what's your proposed alternative?

-Michael

🔗rick_ballan <rick_ballan@...>

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

>Michael,

Below you give the intervals (1/PHI)^4 + 1) and (1/PHI)^2 + 1). In 36TET the PHI parts would be (2^-(25/36))^4 = 2^-(100/36) = 0.14581613 and -50 giving 0.381858, both of which are PHI numbers. Now, if these values +1 are also PHI numbers, we should be able to find something close to a whole number divided by 36 if we take 2^x = and solve for x. This is because we found that these values are ridiculously close to PHI numbers and that applying PHI or its inverse 36 times produces the entire scale (and PHI applied 36 times is by far enough distance from the original to test the results). For example, given -25 (which we know in advance is correct), we have 2^x = 0.617947233, solving for x gives x = ln0.61794/ln2 = -0.694444...and multiplying by 36 gives -25. Further, if we add 1 to this number we know in advance that it gives PHI and will also have a solution for x.

However, x = ln1.14581613/ln2 = 0.19637555, times 36 = 7.069519882 and
x = ln1.381858/ln2 = 0.466609371, times 36 = (around) 16.8 = 84/5. Therefore, both numbers do not seem to be PHI generated. Once again I'll say that just because adding 1 to PHI or it's inverse gives other PHI numbers, we cannot assume that this then applies to those numbers. It is just a fact that the interval PHI produces all 36 notes and, unless there is something I haven't thought of yet, the choice of how to limit this seems up for grabs. But this is where the art might come in.

-Rick

>
> Ok, so here we go...I am trying to learn about PHI in music rather than prove whose scale is "better".
>
> Here is a fairly good "triad" type chord that works in both Rick's PHI scale and my own that we have both mentioned as good:
>
> 1
> 1.236067 (2/PHI)...this is 1.23589 in 36TET IE the tuning for Rick's scale
> 1.618034 (PHI)...this is 1.61826 in 36TET IE the tuning for Rick's scale
>
>
> For starters of issues with both scales, I recognize that the interval 1 and 1.618034 beats a bit too much and beats much less as the rather similar sounding 1.625.
>
>
> Beyond that, there comes the question how best to derive more possible notes for the scale (since a 3-note scale is rather limited).
>
>
> One issue is symmetry to the 2/1 octave and not simply the PHI-tave. One possible solution I believe could work is using the chord
> 1
> 1.236067 (2/PHI)
> 1.625 = 1.618034 (PHI) slightly rounded to preserve more overtone consonance
> 1.85408 = 1.618034 * 1.14589 (where 1.14589 = (1/PHI)^4))
> 2 (2/PHI * PHI)
>
> There is another gap between 1.236067 and 1.618034 which could easily be filled by another note without violating the critical band and yet another gap between 1.236067.
>
> My scale uses 1.14589 (where 1.14589 = (1/PHI)^4 + 1)) to fill the lower gap and 1.381966 (where 1.381966 = (1/PHI)^2 + 1)) to fill in the upper. That's my proposed solution to get a 6-note per octave scale with quite high consonance.
>
> Without putting words into your mouth...Rick...how would you propose to fill the above gaps with more notes? And/or, do you see a problem with using 1.85408 to fill in the gap between 1.618034 and 2...and what's your proposed alternative?
>
>
> -Michael
>

🔗Michael Sheiman <djtrancendance@...>

Rick>"unless there is something I haven't thought of yet, the choice of how
to limit this seems up for grabs. But this is where the art might come
in."

The simple summary, again, is that my theory is based around splitting lines into PHI-th's...rather than making everything fit an exponential of PHI (as yours seems to do). Note that, if you take a parabola and translate it 1 graph unit to the right...it still has to same symmetry: that's the basic geometric principle my scale is based around (again, splitting a line from 1 to 2 using 1.618034 instead of from 0 to 1 by 0.61804...the +1 translation does >not< change the way the line is split at all).

For the record, you did get the numbers and generation method related to my scale right, BTW. :-) And indeed, if you take powers of PHI divided by powers of 2 you end up getting 36TET. Again, I don't think it's a question of better or worse, just different. And, to note, my hold PHI tuning (the exponential one) hit virtually all the notes of the 36TET tuning you use dead-on...so it seems we're simply interpreting PHI symmetry in different ways.

-Michael

--- On Mon, 5/4/09, rick_ballan <rick_ballan@...> wrote:

From: rick_ballan <rick_ballan@...>
Subject: [tuning] Re: Using commonalities between Rick's PHI scale and Mike's/mine to make an "ult
To: tuning@yahoogroups.com
Date: Monday, May 4, 2009, 9:24 PM

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

>Michael,

Below you give the intervals (1/PHI)^4 + 1) and (1/PHI)^2 + 1). In 36TET the PHI parts would be (2^-(25/36)) ^4 = 2^-(100/36) = 0.14581613 and -50 giving 0.381858, both of which are PHI numbers. Now, if these values +1 are also PHI numbers, we should be able to find something close to a whole number divided by 36 if we take 2^x = and solve for x. This is because we found that these values are ridiculously close to PHI numbers and that applying PHI or its inverse 36 times produces the entire scale (and PHI applied 36 times is by far enough distance from the original to test the results). For example, given -25 (which we know in advance is correct), we have 2^x = 0.617947233, solving for x gives x = ln0.61794/ln2 = -0.694444... and multiplying by 36 gives -25. Further, if we add 1 to this number we know in advance that it gives PHI and will also have a solution for x.

However, x = ln1.14581613/ ln2 = 0.19637555, times 36 = 7.069519882 and

x = ln1.381858/ln2 = 0.466609371, times 36 = (around) 16.8 = 84/5. Therefore, both numbers do not seem to be PHI generated. Once again I'll say that just because adding 1 to PHI or it's inverse gives other PHI numbers, we cannot assume that this then applies to those numbers. It is just a fact that the interval PHI produces all 36 notes and, unless there is something I haven't thought of yet, the choice of how to limit this seems up for grabs. But this is where the art might come in.

-Rick

> Ok, so here we go...I am trying to learn about PHI in music rather than prove whose scale is "better".

> Here is a fairly good "triad" type chord that works in both Rick's PHI scale and my own that we have both mentioned as good:

> 1

> 1.236067 (2/PHI)...this is 1.23589 in 36TET IE the tuning for Rick's scale

> 1.618034 (PHI)...this is 1.61826 in 36TET IE the tuning for Rick's scale

> For starters of issues with both scales, I recognize that the interval 1 and 1.618034 beats a bit too much and beats much less as the rather similar sounding 1.625.

> Beyond that, there comes the question how best to derive more possible notes for the scale (since a 3-note scale is rather limited).

> One issue is symmetry to the 2/1 octave and not simply the PHI-tave. One possible solution I believe could work is using the chord

> 1

> 1.236067 (2/PHI)

> 1.625 = 1.618034 (PHI) slightly rounded to preserve more overtone consonance

> 1.85408 = 1.618034 * 1.14589 (where 1.14589 = (1/PHI)^4))

> 2 (2/PHI * PHI)

> There is another gap between 1.236067 and 1.618034 which could easily be filled by another note without violating the critical band and yet another gap between 1.236067.

> My scale uses 1.14589 (where 1.14589 = (1/PHI)^4 + 1)) to fill the lower gap and 1.381966 (where 1.381966 = (1/PHI)^2 + 1)) to fill in the upper. That's my proposed solution to get a 6-note per octave scale with quite high consonance.

> Without putting words into your mouth...Rick. ..how would you propose to fill the above gaps with more notes? And/or, do you see a problem with using 1.85408 to fill in the gap between 1.618034 and 2...and what's your proposed alternative?

> -Michael

🔗martinsj013 <martinsj@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> Rick>"unless there is something I haven't thought of yet, the choice of how
> to limit this seems up for grabs. But this is where the art might come
> in."
>
> The simple summary, again, is that my theory is based around splitting lines into PHI-th's...rather than making everything fit an exponential of PHI (as yours seems to do). Note that, if you take a parabola and translate it 1 graph unit to the right...it still has to same symmetry: that's the basic geometric principle my scale is based around (again, splitting a line from 1 to 2 using 1.618034 instead of from 0 to 1 by 0.61804...the +1 translation does >not< change the way the line is split at all).

Hi Mike,

But splitting lines into PHI'ths and taking exponentials of PHI are the same thing. Starting with a length 1, the larger length, we add 0.618..., the smaller, which equates to multiplying by 1.618...i.e.
1 (larger) + 0.618 (smaller) = 1 x 1.618 = 1.618. Next, we treat 1.618 as the new larger length and add (now) the smaller length 1 to obtain 2.618...But this again is the same as multiplying by 1.618, and therefore PHI^2 = 2.618...That is, 1.618 (larger) + 1 (smaller) = 1.618 x 1.618 = 2.618. Next,2.618 (larger) + 1.618 (smaller) = 4.23703...= PHI^3, and so on. Thus, the series is PHI^N, N = 1,2,3,...And this is pretty much the same as taking (2^(25/36))^N = 2^((25+25+25...N factors)/36). Bringing these into a single 8ve by mod 36 will generate all 36 numbers i.e. N = 1,2,3,...35.

Of course we can go the other way (towards smaller values) by treating the original 1 as the sum of two other PHI numbers, which is the same as now calling 0.618 the larger value. The smaller length is now obtained as 1 - 0.618 = 0.3818... = (1/PHI)^2 = PHI^-2. Continuing this process we get the series PHI^-N, or in 36 TET, (2^(25/36))^ -N, where N = 1,2,3,...35. In other words, bringing it into one 8ve we get all 35 notes once again.

This is mathematics Michael, and the math's doesn't lie. The problem we are now faced with is not finding a few other notes we can add to the original PHI triad because ALL 35 notes are already PHI notes, but rather which notes to leave out. My suggestion is to start by analysing 0:11:25 and begin to add other notes which are the differences between them eg 25 - 11 = 14, so add 14, 14 - 11 = 3, so add 3 etc... Then we can take symmetries, 36 - 14 = 22, 36 -3 = 33, etc...But since all 35 notes will again eventually be included, we need to find some critical cut-off point. Perhaps this can only be decided by the ears, much the same as why we cut-off sequences of major and minor thirds and vice-versa before the second 8ve i.e. maj69(#11) for major and min11 for minor chords?

-Rick
>
> For the record, you did get the numbers and generation method related to my scale right, BTW. :-) And indeed, if you take powers of PHI divided by powers of 2 you end up getting 36TET. Again, I don't think it's a question of better or worse, just different. And, to note, my hold PHI tuning (the exponential one) hit virtually all the notes of the 36TET tuning you use dead-on...so it seems we're simply interpreting PHI symmetry in different ways.
>
> -Michael
>
> --- On Mon, 5/4/09, rick_ballan <rick_ballan@...> wrote:
>
> From: rick_ballan <rick_ballan@...>
> Subject: [tuning] Re: Using commonalities between Rick's PHI scale and Mike's/mine to make an "ult
> To: tuning@yahoogroups.com
> Date: Monday, May 4, 2009, 9:24 PM
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> --- In tuning@yahoogroups. com, djtrancendance@ ... wrote:
>
>
>
> >Michael,
>
>
>
> Below you give the intervals (1/PHI)^4 + 1) and (1/PHI)^2 + 1). In 36TET the PHI parts would be (2^-(25/36)) ^4 = 2^-(100/36) = 0.14581613 and -50 giving 0.381858, both of which are PHI numbers. Now, if these values +1 are also PHI numbers, we should be able to find something close to a whole number divided by 36 if we take 2^x = and solve for x. This is because we found that these values are ridiculously close to PHI numbers and that applying PHI or its inverse 36 times produces the entire scale (and PHI applied 36 times is by far enough distance from the original to test the results). For example, given -25 (which we know in advance is correct), we have 2^x = 0.617947233, solving for x gives x = ln0.61794/ln2 = -0.694444... and multiplying by 36 gives -25. Further, if we add 1 to this number we know in advance that it gives PHI and will also have a solution for x.
>
>
>
> However, x = ln1.14581613/ ln2 = 0.19637555, times 36 = 7.069519882 and
>
> x = ln1.381858/ln2 = 0.466609371, times 36 = (around) 16.8 = 84/5. Therefore, both numbers do not seem to be PHI generated. Once again I'll say that just because adding 1 to PHI or it's inverse gives other PHI numbers, we cannot assume that this then applies to those numbers. It is just a fact that the interval PHI produces all 36 notes and, unless there is something I haven't thought of yet, the choice of how to limit this seems up for grabs. But this is where the art might come in.
>
>
>
> -Rick
>
>
>
> >
>
> > Ok, so here we go...I am trying to learn about PHI in music rather than prove whose scale is "better".
>
> >
>
> > Here is a fairly good "triad" type chord that works in both Rick's PHI scale and my own that we have both mentioned as good:
>
> >
>
> > 1
>
> > 1.236067 (2/PHI)...this is 1.23589 in 36TET IE the tuning for Rick's scale
>
> > 1.618034 (PHI)...this is 1.61826 in 36TET IE the tuning for Rick's scale
>
> >
>
> >
>
> > For starters of issues with both scales, I recognize that the interval 1 and 1.618034 beats a bit too much and beats much less as the rather similar sounding 1.625.
>
> >
>
> >
>
> > Beyond that, there comes the question how best to derive more possible notes for the scale (since a 3-note scale is rather limited).
>
> >
>
> >
>
> > One issue is symmetry to the 2/1 octave and not simply the PHI-tave. One possible solution I believe could work is using the chord
>
> > 1
>
> > 1.236067 (2/PHI)
>
> > 1.625 = 1.618034 (PHI) slightly rounded to preserve more overtone consonance
>
> > 1.85408 = 1.618034 * 1.14589 (where 1.14589 = (1/PHI)^4))
>
> > 2 (2/PHI * PHI)
>
> >
>
> > There is another gap between 1.236067 and 1.618034 which could easily be filled by another note without violating the critical band and yet another gap between 1.236067.
>
> >
>
> > My scale uses 1.14589 (where 1.14589 = (1/PHI)^4 + 1)) to fill the lower gap and 1.381966 (where 1.381966 = (1/PHI)^2 + 1)) to fill in the upper. That's my proposed solution to get a 6-note per octave scale with quite high consonance.
>
> >
>
> > Without putting words into your mouth...Rick. ..how would you propose to fill the above gaps with more notes? And/or, do you see a problem with using 1.85408 to fill in the gap between 1.618034 and 2...and what's your proposed alternative?
>
> >
>
> >
>
> > -Michael
>
> >
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > Below you [Michael] give the intervals (1/PHI)^4 + 1) and (1/PHI)^2 + 1). In 36TET the PHI parts would be (2^-(25/36))^4 = 2^-(100/36) = 0.14581613 and -50 giving 0.381858, both of which are PHI numbers. Now, if these values +1 are also PHI numbers ... these values are ridiculously close to PHI numbers ...
>
> They are close, but not ridiculously close. OTOH, (1/PHI)^3 + 1 is exactly equal to 2/PHI (easy to show using the recurrence relations).
>
> Regards,
> Steve M.
>
Hi Steve,

Yes (1/PHI)^3 + 1 is exactly equal to 2/PHI, unlike powers of 4 and 2.
But concerning closeness, here is where musical/wave theoretical considerations override mathematical exactness. In 36 TET, 2/PHI becomes 2 x 2^-(25/36)= 2^(11/36) = 1.235894, which is inverse PHI brought into one 8ve. PHI is replaced by 2^(25/36) and all notes in a 36TET can be generated via either of these two generating intervals to obtain 2^(1/36) as the new "semitone". Further, tests will show that the differences b/w these and the original PHI intervals really is very negligible (i.e.way below 5 cents. For a more general discussion, you might want to look into harmonic entropy). Samples given by Michael also confirm that they produce all of the beating we would expect from PHI numbers i.e. beats 8ve equivalent to ratios. However, the advantage of 36TET is that it not only preserves 8ve equivalence, but also includes 12TET and ipso facto traditional harmonies i.e. 2^(3/36) = 2^(1/12) so that the standard EDO tuning is 0:3:6:...:33. I really do think that the problem is completely solved.

Cheers

Rick

🔗rick_ballan <rick_ballan@...>

🔗Charles Lucy <lucy@...>

I am very skeptical about these 5 cent thresholds and all the traditional Just Noticeable Difference (JND) claims; it may be fine for rough single notes but for my purposes tuning is also about harmony, which requires hi res granularilty.

BTW .243..... is nearer to a quarter of a cent than a fifth.

How about looking at the beat frequencies instead in centre of range?

I am now beginning to develop my scales database at:

http://www.lucytune.com/scales/

to enable users to discover which tuning codes at:

http://www.lucytune.com/midi_and_keyboard/pitch_bend.html

to use for which scales, for various choices of key (defined by the tonic; C C# ............ B etc.) and which scales in which keys are available using each tuning code.

I'll put the results up when I get a little further with it.

On 6 May 2009, at 11:00, rick_ballan wrote:

>
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > > Below you [Michael] give the intervals (1/PHI)^4 + 1) and (1/> PHI)^2 + 1). In 36TET the PHI parts would be (2^-(25/36))^4 = 2^-> (100/36) = 0.14581613 and -50 giving 0.381858, both of which are PHI > numbers. Now, if these values +1 are also PHI numbers ... these > values are ridiculously close to PHI numbers ...
> >
> > They are close, but not ridiculously close. OTOH, (1/PHI)^3 + 1 is > exactly equal to 2/PHI (easy to show using the recurrence relations).
> >
> > Regards,
> > Steve M.
> >
> Oh and one more thing Steve. Just out of interest I calculated the > difference. PHI in cents is around 833.0902966 while 2^(25/36) is > about 833.33333308. The difference is 0.2430364 cents, that is, > about (1/5)'th of 1 cent, which is way below 5 cents threshold. > Cheers, Rick.
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

🔗Michael Sheiman <djtrancendance@...>

Rick>"But splitting lines into PHI'ths and taking exponentials of PHI are the same thing."
>"The smaller length is now obtained as 1 - 0.618 = 0.3818... = (1/PHI)^2
= PHI^-2. Continuing this process we get the series PHI^-N, or in 36
TET, (2^(25/36))^ -N, where N = 1,2,3,...35."

I agree (and I'm not trying to pit our methods against each other)!
Basically, my scale is almost summarized dead-on by notes in 36TET: and my scale is basically a subset of the 36TET tuning (with notes off by about a mere/negligible few cents at most). In fact, we agreed on this a long time ago...along with the fact
36TET very near-perfectly (but not quite perfectly) includes PHI.

>"But since all 35 notes will again eventually be included, we need to
find some critical cut-off point. Perhaps this can only be decided by
the ears"
One question you asked long ago was (paraphrased) "surely we can't use all 36 notes at once...so which ones should we use?". And now here you are asking it asking.
My scale was proposed as one possible answer to that question (and one I did test by ears vs. several other methods ALL of which used combinations of tones very very close to 36TET notes)...and if you can think of any, others...more power to you. :-)

-Michael

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> I am very skeptical about these 5 cent thresholds and all the
> traditional Just Noticeable Difference (JND) claims; it may be fine
> for rough single notes but for my purposes tuning is also about
> harmony, which requires hi res granularilty.
>
> BTW .243..... is nearer to a quarter of a cent than a fifth.
>
> How about looking at the beat frequencies instead in centre of range?
>
>Hi Charles,

I don't think I've ever done a set where my guitar stayed exactly in tune throughout. And yet nobody ever really notices. Therefore, I'm more inclined to believe that it is precisely because of this uncertainty that harmony can exist in the first place. Hindemith makes this point in craft of musical composition. This is why 5/4 can be substituted by 81/64, 645/512, 2^(1/3) and even the major third in your Lucy tuning (And the differences in these numbers are higher than the phi example I gave). So in my opinion it is not without some irony that mathematical exactitude can actually work against being an accurate model of musical harmony. But as to the width of this uncertainty before it is heard as being out of tune (or becoming another note), or as to which is the "correct" interval to which the others approximate, I don't know.

-Rick
>
>
> I am now beginning to develop my scales database at:
>
> http://www.lucytune.com/scales/
>
> to enable users to discover which tuning codes at:
>
> http://www.lucytune.com/midi_and_keyboard/pitch_bend.html
>
> to use for which scales, for various choices of key (defined by the
> tonic; C C# ............ B etc.) and which scales in which keys are
> available using each tuning code.
>
> I'll put the results up when I get a little further with it.
>
>
>
> On 6 May 2009, at 11:00, rick_ballan wrote:
>
> >
> >
> > --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > > > Below you [Michael] give the intervals (1/PHI)^4 + 1) and (1/
> > PHI)^2 + 1). In 36TET the PHI parts would be (2^-(25/36))^4 = 2^-
> > (100/36) = 0.14581613 and -50 giving 0.381858, both of which are PHI
> > numbers. Now, if these values +1 are also PHI numbers ... these
> > values are ridiculously close to PHI numbers ...
> > >
> > > They are close, but not ridiculously close. OTOH, (1/PHI)^3 + 1 is
> > exactly equal to 2/PHI (easy to show using the recurrence relations).
> > >
> > > Regards,
> > > Steve M.
> > >
> > Oh and one more thing Steve. Just out of interest I calculated the
> > difference. PHI in cents is around 833.0902966 while 2^(25/36) is
> > about 833.33333308. The difference is 0.2430364 cents, that is,
> > about (1/5)'th of 1 cent, which is way below 5 cents threshold.
> > Cheers, Rick.
> >
> >
>
> Charles Lucy
> lucy@...
>
> - Promoting global harmony through LucyTuning -
>
> for information on LucyTuning go to:
> http://www.lucytune.com
>
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk
>

🔗martinsj013 <martinsj@...>

🔗Cameron Bobro <misterbobro@...>

🔗Michael Sheiman <djtrancendance@...>

>"As you will soon discover :-) , the key to packing more "tall chords"
into a small area is to mix Just intervals in with the spectrally
"fuzzy" intervals, preferably intervals with nice appropriate
difference tones."
Cameron, I will attest from my own experimentation that I've found that is true. :-) In fact when experimenting to find both the golden section scale and the equivalent silver section one (which so far as I know really is original so far: I was unable to find an equivalent of with SCALA)...I found rounding each ratio to the nearest JI ratio (if the JI ratio is within about 5 cents of the original value) manages to improve overall consonance without destroying the sense of symmetry or disrupting the relations of tones which bring about "appropriate difference tones". An easy example: that is why when I interpreted the golden sections I rounded PHI to 1.625 (13/8) instead of 1.618034.

On the other hand, I wouldn't go so far as to shove in JI intervals that have nothing to do/are nothing near original values from the golden sections, for example...as they will inevitably clash in several chord formations involving "fuzzy" intervals. One example of this becoming a problem, I've found, is splitting PHI logarithmically (IE 2^(1/PHI^x)): parts of it look profoundly like mean-tone and very much near JI alignment naturally yet many combination of tones in the whole construct sound annoyingly dissonant rather than a combination of assonant and consonant.
Golden sections of PHI may have some notes convenient to convert to JI equivalents...but the entire thing sure is not JI swappable (at least if you care about either aligning overtones and making assonant tones that at least avoid dissonance rather than conflicting overtone matches).

-Michael

🔗rick_ballan <rick_ballan@...>

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

What, I thought that "ridiculously close" was a well-defined mathematical concept! No it's not a problem mate. The reasoning I took was in the first post in this strand I think but I can't find it now. Since its easy I'll do it again just for you: I simply asked "what x gives 2^x = PHI?". My intention was to prove to myself that phi and the 8ve can't live together. However, solving for x gives x = ln1.618.../ln2 = 0.6942...(i.e. ln is log) and searching around a bit I found that this times 36 gives 24.9916 approx 25. Some time ago I was entertaining the idea of a 36TET and knew it included 12TET, which is why I recognised it. Voila!

-Rick

> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > PHI is replaced by 2^(25/36) and all notes in a 36TET can be generated via either of these two generating intervals to obtain 2^(1/36) as the new "semitone". Further, tests will show that the differences b/w these and the original PHI intervals really is very negligible ...
>
> OK, fair cop, I was indeed speaking from a mathematical viewpoint, pointing out that the differences were non-zero (and objecting to the undefined term "ridiculously close"). I do see that they are very close - but have not yet worked out why. (Note: I am not talking about the 25 and the 11, but the 7 and the 17.)
>
> > Samples given by Michael also confirm that they produce all of the beating we would expect from PHI numbers i.e. beats 8ve equivalent to ratios.
>
> This also sounds like a significant point, but I don't fully understand it...
>
> Regards,
> Steve M.
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
Cam said> Personally I'd try to dump the 12-tET stuff as much as possible, except the 500 and 700 cent intervals, keep all the phi intervals, and the (unusually good) 7-limit Just approximations of 8/7, 7/6, 9/7 and their inversions, 7/4, 12/7 and 14/9.

Rick objects> Hang on a minute Cam, haven't you be reading my posts? Mere guesswork won't cut it here and your 12TET prejudice is showing. They are ALL PHI intervals, including 3 which is PHI^3 and is of the third successive application of PHI i.e. (2^(25/36))^3 = 2^(75/36) and 75 = 3(mod36). Now, we can all imagine some nice eight-note JI scale based on PHI - perhaps the "Phionian" would be a good name - but PHI is an irrational number and seems to lend itself to symmetries. I admitted that my intuition about PHI being musically irrelevant might have been...premature, but I have to draw the line at those who want to have their PHI and eat it (boom boom).

-Rick
>
> As you will soon discover :-) , the key to packing more "tall chords" into a small area is to mix Just intervals in with the spectrally "fuzzy" intervals, preferably intervals with nice appropriate difference tones.
>
> 8/7, 9/7, and 4/3 are perfect with Phi. 8/7 and 4/3 are in fact the only "reasonable" intervals that return difference tones with a property similar to the unique diff. tone of Phi, because they return themselves in lower but audible octaves.
>
> All n+1/n intervals return subharmonic intervals, all n+1/n intervals with a power of two in the larger part return themselves in lower octaves, all n+2/n also return subharmonic intervals.
>
> Assuming you keep the 700 cent interval and the 366 cent "major" third and it's "minor" complement of 333 cents, you might want to keep the 933 interval (12/7) because it is an octave of the difference tone produced by the "minor" third of 333 cent.
>

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
> >
> Cam said> Personally I'd try to dump the 12-tET stuff as much as possible, except the 500 and 700 cent intervals, keep all the phi intervals, and the (unusually good) 7-limit Just approximations of 8/7, 7/6, 9/7 and their inversions, 7/4, 12/7 and 14/9.
>
> Rick objects> Hang on a minute Cam, haven't you be reading my posts? Mere guesswork won't cut it here and your 12TET prejudice is showing. They are ALL PHI intervals, including 3 which is PHI^3 and is of the third successive application of PHI i.e. (2^(25/36))^3 = 2^(75/36) and 75 = 3(mod36). Now, we can all imagine some nice eight-note JI scale based on PHI - perhaps the "Phionian" would be a good name - but PHI is an irrational number and seems to lend itself to symmetries. I admitted that my intuition about PHI being musically irrelevant might have been...premature, but I have to draw the line at those who want to have their PHI and eat it (boom boom).
>
> -Rick

Mere guesswork, eh? Hahaha! When you reduce the 36, you're going to have more of some intervals and less of others. I agree some kind of symmetry or MOS scale would be nice. Yes I'd go for scale that emphasises more of the phi division intervals, 833 and 366 for example. Maybe I should say "mere guesswork my butt", as I already implied symmetrical scales:

1/1, 8/7, 366.6, 4/3, 3/2, 833.3, 7/4, 2/1

for example is very nice, kind of a soft "maqam" sort of sound, and it's really just as nice or nicer, in a different flavor, in 36-equal, which isn't a common occurance in my experience. And it completely avoids 12-equal thirds and sixths, in any transposition.
:-P

Notice that although I say 8/7, which is chosen because it shares a similar difference-tone property with Phi, 36-equal actually has a tone right smack in the middle of the 4-cent difference between 8/7 and the third succesive golden section of Phi.

Anyway 36-equal is a great choice for this, good call.

-Cameron Bobro

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
> > >
> > Cam said> Personally I'd try to dump the 12-tET stuff as much as possible, except the 500 and 700 cent intervals, keep all the phi intervals, and the (unusually good) 7-limit Just approximations of 8/7, 7/6, 9/7 and their inversions, 7/4, 12/7 and 14/9.
> >
> > Rick objects> Hang on a minute Cam, haven't you be reading my posts? Mere guesswork won't cut it here and your 12TET prejudice is showing. They are ALL PHI intervals, including 3 which is PHI^3 and is of the third successive application of PHI i.e. (2^(25/36))^3 = 2^(75/36) and 75 = 3(mod36). Now, we can all imagine some nice eight-note JI scale based on PHI - perhaps the "Phionian" would be a good name - but PHI is an irrational number and seems to lend itself to symmetries. I admitted that my intuition about PHI being musically irrelevant might have been...premature, but I have to draw the line at those who want to have their PHI and eat it (boom boom).
> >
> > -Rick
>
> Mere guesswork, eh? Hahaha! When you reduce the 36, you're going to have more of some intervals and less of others. I agree some kind of symmetry or MOS scale would be nice. Yes I'd go for scale that emphasises more of the phi division intervals, 833 and 366 for example. Maybe I should say "mere guesswork my butt", as I already implied symmetrical scales:
>
> 1/1, 8/7, 366.6, 4/3, 3/2, 833.3, 7/4, 2/1
>
> for example is very nice, kind of a soft "maqam" sort of sound, and it's really just as nice or nicer, in a different flavor, in 36-equal, which isn't a common occurance in my experience. And it completely avoids 12-equal thirds and sixths, in any transposition.
> :-P
>
> Notice that although I say 8/7, which is chosen because it shares a similar difference-tone property with Phi, 36-equal actually has a tone right smack in the middle of the 4-cent difference between 8/7 and the third succesive golden section of Phi.
>
> Anyway 36-equal is a great choice for this, good call.
>
> -Cameron Bobro
>
Oh I see, my apologies Cam. Took the word "dump" in the wrong meaning.
You're just trying to thin-out the possibilities. Now the closest interval I could find to 8/7 in the 36TET is 7, 7/6 is very close to 8 and 9/7 to 11. Are these correct? But except for 11 I can't see where PHI comes in so could you backtrack a bit, thanks. (I reread your posts but still couldn't find an entry point).

In the meantime, I suspect this reasoning might have some promise: From PHI 25, take its inverse 11. Squaring both gives 14 and 22 while cubing gives 3 and 33. Consecutively we have 0:3:11:14:22:25:33:8ve. Notice this has successive intervals 3,8,3,8,...(both up and down) and successive 11's from 0 and 3 so it might be useful as some type of 7-note PHI scale?? If so I presume rationals numbers could be found.

-Rick

🔗Cameron Bobro <misterbobro@...>

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

> You're just trying to thin-out the possibilities. Now the closest interval I could find to 8/7 in the 36TET is 7,

Yes, it's only about 2 cents off, and as I mentioned it's right on for splitting the already small difference between one of the successive golden section intervals and 8/7- really a perfect compromise, and we're already talking about small variations.

>7/6 is very close to 8

only .2 cents difference. Busoni advocated 36-equal and I guess he was thinking of Just 7-limit.

>and 9/7 to 11.

degree 13, not 11, and once again, only 1.75 cent difference here.

>But except for 11 I can't see where PHI comes in so could you >backtrack a bit, thanks. (I reread your posts but still couldn't >find an entry point).

degree 7 doubles as both Just and a successive golden section interval (which happens more than one might think, part of the reason the scales sound unexpectedly good), 11 and 25 are Phi intervals of course, 4 appears as an interval within the first 5 golden cuts of Phi, degree 5 appears immediately as well.

>
> In the meantime, I suspect this reasoning might have some promise: >From PHI 25, take its inverse 11. Squaring both gives 14 and 22 >while cubing gives 3 and 33. Consecutively we have >0:3:11:14:22:25:33:8ve. Notice this has successive intervals >3,8,3,8,...(both up and down) and successive 11's from 0 and 3 so it >might be useful as some type of 7-note PHI scale?? If so I presume >rationals numbers could be found.

That looks groovy but I think the 100 cent semitone would sound out of place- 133 cents would be better I suspect. Let's hear...hm yes I think so, try it, degree 4 instead of 3. Rationals could be found very easily if we concentrate on the "7" intervals and the Phi intervals.

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
>
> > You're just trying to thin-out the possibilities. Now the closest interval I could find to 8/7 in the 36TET is 7,
>
> Yes, it's only about 2 cents off, and as I mentioned it's right on for splitting the already small difference between one of the successive golden section intervals and 8/7- really a perfect compromise, and we're already talking about small variations.
>
> >7/6 is very close to 8
>
> only .2 cents difference. Busoni advocated 36-equal and I guess he was thinking of Just 7-limit.
>
> >and 9/7 to 11.
>
> degree 13, not 11, and once again, only 1.75 cent difference here.
>
> >But except for 11 I can't see where PHI comes in so could you >backtrack a bit, thanks. (I reread your posts but still couldn't >find an entry point).
>
> degree 7 doubles as both Just and a successive golden section interval (which happens more than one might think, part of the reason the scales sound unexpectedly good), 11 and 25 are Phi intervals of course, 4 appears as an interval within the first 5 golden cuts of Phi, degree 5 appears immediately as well.
>
> >
> > In the meantime, I suspect this reasoning might have some promise: >From PHI 25, take its inverse 11. Squaring both gives 14 and 22 >while cubing gives 3 and 33. Consecutively we have >0:3:11:14:22:25:33:8ve. Notice this has successive intervals >3,8,3,8,...(both up and down) and successive 11's from 0 and 3 so it >might be useful as some type of 7-note PHI scale?? If so I presume >rationals numbers could be found.
>
> That looks groovy but I think the 100 cent semitone would sound out of place- 133 cents would be better I suspect. Let's hear...hm yes I think so, try it, degree 4 instead of 3. Rationals could be found very easily if we concentrate on the "7" intervals and the Phi intervals.
>

Hi Cam,

After weeks of trying I haven't been able to install Scala on my old Mac so, alas, all my checking still has to be done mathematically.

I have no doubt that Busoni's use of 36EDO to approx JI 7-limit will sound good in this system (a good justification for adopting it), but unfortunately I'm still having trouble seeing where these intervals relate to phi. For eg, "degree 7 doubles as both Just and a successive golden section interval". According to my calculations, we don't get to 7 until 19 successive apps of 25 i.e. 475 = 7(MOD36), so although it is a phi interval, as are all in the system, it is indeed very remote from the root. Also, what do you have in mind when you say "4 appears as an interval within the first 5 golden cuts of Phi"? I imagine that you're taking phi, perhaps in cents, and cutting up the numbers in further phi proportions. Or are you calculating these from inverse phi? From 11, 44=8(MOD36) is the only thing I can think of.

-Rickus Ignoramus

🔗Cameron Bobro <misterbobro@...>

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

> Hi Cam,
>
> After weeks of trying I haven't been able to install Scala on my >old Mac so, alas, all my checking still has to be done >mathematically.

Hopefully you keep trying to install Scala, as it is a fantastic tool.
>
> I have no doubt that Busoni's use of 36EDO to approx JI 7-limit >will sound good in this system (a good justification for adopting >it), but unfortunately I'm still having trouble seeing where these >intervals relate to phi. For eg, "degree 7 doubles as both Just and >a successive golden section interval". According to my calculations, >we don't get to 7 until 19 successive apps of 25 i.e. 475 = >7(MOD36), so although it is a phi interval, as are all in the >system, it is indeed very remote from the root. Also, what do you >have in mind when you say "4 appears as an interval within the first >5 golden cuts of Phi"? I imagine that you're taking phi, perhaps in >cents, and cutting up the numbers in further phi proportions. Or are >you calculating these from inverse phi? From 11, 44=8(MOD36) is the >only thing I can think of.
>
> -Rickus Ignoramus

What I've been doing for the last couple of years is making golden sections of frequencies linearly, not logarithmically. When I mentioned this on the tuning list in 2007, it didn't go over too well :-) so I had to discover the non-numerological properties myself, as well as historical precedents.

Maybe you can imagine how I laughed when the other day Michael said I had "finally" got "his" scale. But there in his numerous experiments Michael really has done something which as far as I know is historically unprecedented: he used the golden cut as a frequency mean, as I've been doing, and I'm sure others have done, and as perhaps the ancient Greeks did as well (we don't know, see Divisions of the Tetrachord), but instead of treating it as a tool to be used with "given" Just ratios, or continuing on until he found where the process "returns" to Just ratios (which happens quite quickly), he simply divided, using the mean, to a certain point and said, there's the scale! Which is very groovy. You could do this with any of the means of the ancients, some dozen altogether, but as far as I know the only similar approach is found in Schleisinger's harmoniai, which are created with succesive harmonic means.

Anyway, the thing is that it's linear divisions, not logarithmic (cents), and there is no generator, but divisions in an ancient style. A symmetrical scale can also be symmetrical from the geometric center of the octave, sqrt2 or 600 cents, whether 600 cents is found in the tuning or not.

I described the process in an earlier post but when I get the time I'll do a step-by-step, which I have to illustrate anyway in order to demonstrate how using phi also produces Just intervals, and a simple geometric/monochord way to make a scale which is both a phi scale and an ancient Greek tuning, etc.

-Cameron Bobro

🔗Michael Sheiman <djtrancendance@...>

>"What I've been doing for the last couple of years is making golden sections
of frequencies linearly, not logarithmically. When I mentioned this on
the tuning list in 2007, it didn't go over too well :-) so I had to
discover the non-numerological properties myself, as well as historical
precedents.

Maybe you can imagine how I laughed when the other day Michael said I had "finally" got "his" scale."

Hehe...so you had gotten the same scale ages before. Which makes me wonder why you only recently mentioned it. :-P
But, perhaps more importantly, I agree with you that taking linear sections of PHI is a quite genuine way to go when creating PHI scales (and can actually make something quite consonant to my ears for an 8-9 tone scale while JI-type scales, I've noticed, tend to have trouble when they have over 7 notes in the scale and/or more than 5 note chords per octave). :-)

>"but instead of treating it as a tool to be used with "given" Just
ratios, or continuing on until he found where the process "returns" to
Just ratios (which happens quite quickly), he simply divided"
Exactly, I just kept taking smaller sections, never trying to round anything the any sort of JI or TET. Except for the ratio of PHI itself which I rounded to 1.625 as it was by far the most sour interval.

However...I wonder...how do the other methods "merge to just intervals"?

>"but as far as I know the only similar approach is found in
Schleisinger' s harmoniai, which are created with succesive harmonic
means."
Interesting...where can I read about this?

>"A symmetrical scale can also be symmetrical from the geometric center
of the octave, sqrt2 or 600 cents, whether 600 cents is found in the
tuning or not."
Funny, that's pretty much exactly how I derived the Silver Ratio scale I posted before using the "universal" (1/noblenumber)^x+1 formula: it turns out (1/silver ratio)+1 = sqrt(2)

-Michael

🔗Cameron Bobro <misterbobro@...>

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

>
> Hehe...so you had gotten the same scale ages before. Which >makes me wonder why you only recently mentioned it. :-P

Really I think you should more carefully read my posts. :-) As I wrote here last June:

"No different than you'd reckon the golden section in the visual arts."

But it never occurred to me to simply take the first successive golden sections, which is the first subset of slicing the harmonic series up with the golden cut, and just call that the scale as a finished product. That's a really groovy idea and as far as I know you're the first to have it, as I said before.

Anyway, you really should explore the Means of ancient Greece, you'll find how these kinds of things were experimented with and even put into practice thousands of years ago.

> But, perhaps more importantly, I agree with you that taking >linear sections of PHI is a quite genuine way to go when creating >PHI scales (and can actually make something quite consonant to my >ears for an 8-9 tone scale while JI-type scales, I've noticed, tend >to have trouble when they have over 7 notes in the scale and/or more >than 5 note chords per octave). :-)

I agree with this principle, although I've found that maximizing possiblities takes a mixture of JI and very irrational intervals together.

But here's something trippy that you might want think about:

take a look at the Mediant of two ratios. It goes like this:

you have a/b and x/y, the Mediant is a+b/x+y. In the harmonic series, this makes "JI" intervals right away of course:

1/1 and 2/1 gives you 3/2. 3/2 is the Mediant of the tonic and the octave. Now let's keep going... the Mediant of 3/2 and 2/1 is 5/3, the Mediant of 5/3 and 3/2 is 8/5, the Mediant of 8/5 and 5/3 is 13/8... we're just creating the Mediant between each ratio and the one before it... the mediant of 13/8 and 8/5 is 21/13...

you see where it's going, right? :-)

Notice that the first coincident partials of each ratio are linked on the way up the harmonic series- in 5/3, the 3d partial of the higher tone coincides with the 5th partial of the lower tone, then in the next ratio the 5th with the 8th, and so on forever.

You might even say, only half-humorously, that phi *is* JI. Hahaha!

There many such converging series, Wilson and Dudon tunings use them.

What is probably "the" characteristic interval of 20th century music both "classical" and jazz, the 12-tET tritone, along with being the geometric Mean of the octave, is also a convergent of one of these series- take 4/3 and 3/2 and keep taking the Mediant forever and there you are. Square root of 2, or the Silver Mean-1 if you want to put it that way.

>
> >"but instead of treating it as a tool to be used with "given" Just
> ratios, or continuing on until he found where the process "returns" >to
> Just ratios (which happens quite quickly), he simply divided"
> Exactly, I just kept taking smaller sections, never trying to >round anything the any sort of JI or TET. Except for the ratio of >PHI itself which I rounded to 1.625 as it was by far the most sour >interval.

This is yet another reason why you should read John Chalmer's Divisions of the Tetrachord (also available online free now). You'll find ancient and modern methods of doing just that- not generating a tuning via a generator and period, but by dividing and adding, and not necessarily with regard to JI or not.

I also rarely use generators and periods, almost always simply dividing things into what I need.

Maybe when you start using more long-sustaining tones you'll find as I do that phi actually isn't a sour interval at all. It's definitely not sour in the wugga-wugga! beating kind of way, as the beating is kind of a big blur, and the unique difference tone contributes to its eerie stability I believe.
>
> However...I wonder...how do the other methods "merge to just >intervals"?

If you make a bunch of successive golden cuts and start poking around in the intervals that are created internally (in Scala you can just type "show intervals" and it will show them all), and you're familiar with the cent values of different JI intervals, you'll find them.

Aside from JI intervals and the very close JI approximations that just pop up, here's an example of how quickly golden sections and JI can be linked:
take two intervals which you already know, and which reapppear in different places- 466 and 560 cents. In typical "hall of mirrors" form, the 560 cent interval lies at a golden section of 466 and 3/2.

So you see how 3/2 in this context is both a JI and phi interval- or is it the other way around?! There's just tons of this.

>
> >"but as far as I know the only similar approach is found in
> Schleisinger' s harmoniai, which are created with succesive harmonic
> means."
> Interesting...where can I read about this?

In Divisions of the Tetrachord, and google around, there's even a Schleisinger society online.

🔗djtrancendance@...

>"No different than you'd reckon the golden section in the visual arts."
Right...and I said that same thing in an earlier post: I'm using the golden section in exactly the way ancient architecture did.

>"But it never occurred to me to simply take the first successive golden
sections, which is the first subset of slicing the harmonic series up
with the golden cut, and just call that the scale as a finished product."
Of course, all it is, in the end of the day, turns out to be taking the traditional artistic splitting of a line from 0 to 1 and applying it to splitting the octave between 1 and 2 (hence why the + 1 remains in the formula).

Me>"I agree with you that taking linear sections of PHI is a quite
genuine way to go when creating PHI scales (and can actually make
something quite consonant to my ears for an 8-9 tone scale while
JI-type scales, I've noticed, tend to have trouble when they have
over 7 notes in the scale and/or more than 5 note chords per
octave). :-)"

Cameron> "I agree with this principle, although I've found that maximizing
possiblities takes a mixture of JI and very irrational intervals
together."

Actually I'm starting to think that may well be step forward from my original idea as well. In fact I created a program to devise the most consonant 8-tone scale and it came up with a combination of splits of the silver ratio (using a similar type of generation/artistic-splitting formula to derive it) and a few tones very close to JI intervals.
So, Cameron: what JI intervals do you think work best with these irrational number-based scales?

>"1/1 and 2/1 gives you 3/2. 3/2 is the Mediant of the tonic and the
octave. Now let's keep going... the Mediant of 3/2 and 2/1 is 5/3, the
Mediant of 5/3 and 3/2 is 8/5, the Mediant of 8/5 and 5/3 is 13/8...
we're just creating the Mediant between each ratio and the one before
it... the mediant of 13/8 and 8/5 is 21/13...you see where it's going, right? :-) "

So you are doing the same kind of splits, only using 1.5 rather than 1.618 as the splitting point. Although 13/8 is >also< what I came across as the best estimation for phi (boom!). :-) I'm going to have to see what making a scale this way sounds like for myself...or do you actually already have a SCALA file of one which is pre-built?

>"Square root of 2, or the Silver Mean-1 if you want to put it that way."
Another bingo. Perhaps this explains why so many of what my program identifies as the ideal ratios are very very close to intervals derived from the Silver Ratio scale I created in the same way I did the "plain old successive golden cuts" phi scale.

>"(In) John Chalmer's Divisions of the Tetrachord (also available online free now) you'll find ancient and modern methods of doing just that-
not generating a tuning via a generator and period, but by dividing and
adding, and not necessarily with regard to JI or not."

Ok, now you definitely have my attention: I'm definitely going to have to grab that (e)-book and try to look for such patterns as it seems to go into methods that reach far beyond standard generators and into the type of stuff I may take ages to end up duplicating by accident (when I could have just learned it straight out the first time).

>"It's definitely not sour in the wugga-wugga! beating kind of way,"
Kind of in the same the way I find too much periodic buzz in a pure harmonic series as an overly mechanical beating...

>" as the beating is kind of a big blur, and the unique difference tone
contributes to its eerie stability I believe."
And, if I have it right, here we go again with assonance vs. dissonance. And I'm all for controlled assonance and "eerie stability" vs. too much mechanical-sounding periodicity buzz. :-)

-Michael

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> >"No different than you'd reckon the golden section in the visual >arts."
> Right...and I said that same thing in an earlier post: I'm >using the golden section in exactly the way ancient architecture did.

The use of the means and extremes ratio in ancient architecture is actually unproven, as there are simply too many points from which to measure, so you can "prove" an astronomical number of different proportions. Which means of course that you can really prove only general proportions.

But various Means can be proven to have been used in the arts, as they were deliberately used and well documented. We use the term Harmonic Mean, a Mean used in all kinds of sciences from aviation to finance, but it was originally called the subcontrary mean until Archytas renamed it Harmonic a couple thousand years ago, as it was (and still is) used in musical tuning.

Far more fascinating than "Da Vinci code" style speculation is phi in nature, and as a deliberate tool.
>

>
> Cameron> "I agree with this principle, although I've found that >maximizing
> possiblities takes a mixture of JI and very irrational intervals
> together."
>
> Actually I'm starting to think that may well be step forward >from my original idea as well. In fact I created a program to >devise the most consonant 8-tone scale and it came up with a >combination of splits of the silver ratio (using a similar type of >generation/artistic-splitting formula to derive it) and a few tones >very close to JI intervals.
> So, Cameron: what JI intervals do you think work best with these >irrational number-based scales?

It completely depends, in my opinion and experience. In the case of phi and harmonic timbres, I find that the more concrete the fuzz of non-coincident partials, the MORE smooth the effect. So although I don't regard the pure octave as a mandatory interval in music at all, with phi it is best to have it. And rich bright harmonic spectra (phi spectra being a whole different world). And long sustained tones are nice to emphasize the effect. 2/1, 3/2 and 4/3 work wonderfully with the first few golden sections of the octave, and then other Just intervals depending on the "mode" so to speak. 8/7 and 7/4 show up next.

Of course it is not necessary to make the golden section on the octave, it can be done on any interval, rational or otherwise.

>
> >"1/1 and 2/1 gives you 3/2. 3/2 is the Mediant of the tonic and the
> octave. Now let's keep going... the Mediant of 3/2 and 2/1 is 5/3, the
> Mediant of 5/3 and 3/2 is 8/5, the Mediant of 8/5 and 5/3 is 13/8...
> we're just creating the Mediant between each ratio and the one before
> it... the mediant of 13/8 and 8/5 is 21/13...you see where it's going, right? :-) "
>
> So you are doing the same kind of splits, only using 1.5 rather >than 1.618 as the splitting point.
>Although 13/8 is >also< what I >came across as the best estimation >for phi (boom!). :-) I'm going >to have to see what making a scale >this way sounds like for >myself...or do you actually already have a >SCALA file of one which is pre-built?

Not making splits by 1.5- simply carrying out the mediant process. Look and see what happens- the next interval is 34/21, then 55/34... it's the Fibonacci series, and it is converging on phi!

Exactly like the famous Fibonacci spiral illustration. The intervals quickly become clashing or indistinguishable as the spiral into the convergence point of phi- even 8/5 and 13/8 in the same tuning is pretty darn microtonal. An example of a fractal kind of use would be to have spirals spirally off of spirals, or convergences within convergences like you did.

>
> >"(In) John Chalmer's Divisions of the Tetrachord (also available >online free now) you'll find ancient and modern methods of doing >just that-
> not generating a tuning via a generator and period, but by >dividing and
> adding, and not necessarily with regard to JI or not."
>
> Ok, now you definitely have my attention: I'm definitely going to >have to grab that (e)-book and try to look for such patterns as it >seems to go into methods that reach far beyond standard generators >and into the type of stuff I may take ages to end up duplicating by >accident (when I could have just learned it straight out the first >time).

>
> >"It's definitely not sour in the wugga-wugga! beating kind of way,"
> Kind of in the same the way I find too much periodic buzz in a >pure harmonic series as an overly mechanical beating...
>
> >" as the beating is kind of a big blur, and the unique difference >tone
> contributes to its eerie stability I believe."
> And, if I have it right, here we go again with assonance vs. >dissonance. And I'm all for controlled assonance and "eerie >stability" vs. too much mechanical-sounding periodicity buzz. :-)