Yahoo Tuning Groups Ultimate Backup tuning Scale using a "1/octave" section instead of a 1/PHI section

>
> Well, I just plugged 1/octave in and
> value = (1/2)^x+1 for x = 1 to 4
> ...and got......
> 1.0625
> 1.125
> 1.25
> 1.5 (1/octave-tave)
> ...sounds kind of familiar, doesn't it?

Man I know for certain that you're just skimming through the replies you get! :-)

For this works out to be the Arithmetic Mean! Hmmmm, didn't I suggest more than once that you should check out the ancient Means?

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> Ironically this comes out as a very very JI-type scale.
> Furthermore taking 1.5 / 1.0625 gives an extra note: 1.4117647 >(exactly 24/17)
> And going the other direction 1.5 / 1.125 = 1.3333333 (4/3)
> 1.5 / 1.25 = 1.2 (6/5)
> ....yet more JI ratios.

Not surprising because it IS JI. Yes 24/17 is 17/16 down from 3/2,
17/16 being the arithmetic mean of 9/8 and 1/1, and 9/8 being the AM of 5/4 and 1/1, and so on...

>
> Perhaps best of all, try the scale for yourself and see how it >sounds like.
> I'm betting even JI aficionados will find a lot of good uses for >this...

Yes they have, for thousands of years, since the ancient Greeks at the very least, and very likely long, long before.

> Or help explain another reason JI works well...and perhaps can >be extended through: use of inverse ratio symmetry.

There's certainly a whole lot going on that makes things work, or not, but these simple proportions must be a part of (physics makes these simple proportions into tangible events that aren't just "numbers" of course).

-Cameron Bobro

🔗djtrancendance@...

Cameron>"For this works out to be the Arithmetic Mean! Hmmmm, didn't I suggest more than once that you should check out the ancient Means?"

I know I often complain of your saying "didn't I/they say you should read" when I/they haven't even mentioned it before but, this time you did.
In an old message you had said
Cameron>"Anyway, you really should explore the Means of ancient Greece, you'll find how these kinds of things were experimented ... harmoniai"
To be fair to myself, though, I see nothing there about the arithmetic mean (nor do I have the time to read 100% of the extensive lists of "to read" items. :-D And of course I know what an arithmetic mean is...but don't see how it translates to the (1/(2/1))^x + 1 formula I used to make the above scale beside that the arithmetic mean identifies the 1/(2/1) part of the formula.

And, quite frankly, it baffles me why so many people make JI so complicated (tonality diamonds and periodicity blocks anyone?) when so much of it can be summed up in an ancient formula which seems to produce the most beautiful JI scales I've ever heard.

And, also to note...those scales don't intersect the octave or get near
it but do intersect (3/2)^x (the octave/2-tave)...yet they don't sound off-center the way many non-octave based scales do. Weird eh?

Or, dare I ask, is the fact that I'm taking the half/octave equivalent of unmodified successive golden sections (but with 0.5 not 0.618) responsible for this (I know you commented before that was actually perhaps the one truly original thing in my PHI scale that wasn't well documented in history)?

>"Not surprising because it IS JI. Yes 24/17 is 17/16 down from 3/2,

17/16 being the arithmetic mean of 9/8 and 1/1, and 9/8 being the AM of 5/4 and 1/1, and so on..."
I was pretty sure of that as well...I just wanted to double check because it seemed pretty uncanny to me that if it did generate JI so well I've never heard of it used much in modern times.

>"There's certainly a whole lot going on that makes things work, or not,
but these simple proportions must be a part of (physics makes these
simple proportions into tangible events that aren't just "numbers" of
course)."

One real test I've been meaning to try out is to take instruments with a "perfect" PHI timbre using the golden ratio section scales (frequency-based not logarithmic) and then compare the result to and arithmetic mean scale with the usual harmonic series based timbre. I have a hunch that, if you match the timbre, the PHI and/or golden ratio based scale will sound as natural as the octave based one.

But, as for accepting harmonic timbres well, I think it is fair to say the arithmetic mean scale wins the contest hands down. Plus, with it, I can get 8 to 9-tone scales as smooth as standard 7-tone JI diatonic...and contain a few mystery JI ratios not commonly used such as 24/17 and 27/16 (that oddly enough still work very well despite having neither o-tonal or u-tonal relationships to other parts of the scale).

Again my question, to you and everyone knowledgeable about JI (and regardless of the fact people far back in history have used it), is why aren't more people using the arithmetic mean (in the same way I and perhaps others have) in generating their scales through the (1/var)^x+1 formula?

Again, a key point (beside that PHI and 0.5 seem to be "magic numbers" in the above formula)...seems to be the (1/var)^x+1 "sections" formula lend itself very well to generating symmetrical and beautiful scales when given the right value for VAR.

-Michael

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> Cameron>"For this works out to be the Arithmetic Mean! Hmmmm, >didn't I suggest more than once that you should check out the >ancient Means?"
>
> I know I often complain of your saying "didn't I/they say you >should read" when I/they haven't even mentioned it before but, this >time you did.
> In an old message you had said
> Cameron>"Anyway, you really should explore the Means of ancient >Greece, you'll find how these kinds of things were experimented ... >harmoniai"
> To be fair to myself, though, I see nothing there about the >arithmetic mean (nor do I have the time to read 100% of the >extensive lists of "to read" items. :-D And of course I know what >an arithmetic mean is...but don't see how it translates to the >(1/(2/1))^x + 1 formula I used to make the above scale beside that the >arithmetic mean identifies the 1/(2/1) part of the formula.

The arithmetic mean adds the figures/quantities you have, then divides them by the number of figures/quantities. Multiplying by (1/(2/1)) is the same as dividing by two: 2/1 is 2, so 1/(2/1) is 1/2, is .5, is, divide by 2.

As you know.

So when you have two figures, add them, then divide by 2, you have the arithmetic mean.

In your case here, working with frequencies between 1/1 and 2/1, you
are adding a frequency, say .5, to zero, taking the arithmetic mean, and adding it back to one. Just like you were doing, correctly, with the golden section.

Using ^ in this case may be throwing you off, using .5 (divide by two): what you are actually doing is taking the arithmetic mean of 2 figures, one of which is zero, then taking the arithmetic mean of that, then of that... 1 divided by 2, add that to one: 1 divided by 2, divided by two, add to one: 1 divided by 2, divided by 2, divided by 2, add to one: and so one.

Do it long hand and you'll see that you're taking successive arithmetic means of the octave.

Then you made a jump into straight old JI manipulations- took the arithmetic mean of 9/8 and 1/1 and subtracted it from 3/2, and so on.

There's certainly nothing wrong with all this of course.

>
>
> And, quite frankly, it baffles me why so many people make JI so >complicated (tonality diamonds and periodicity blocks anyone?) when >so much of it can be summed up in an ancient formula which seems to >produce the most beautiful JI scales I've ever heard.

Tonalities diamonds are incredibly simple- they're basically just spelling out the harmonic series in a row, so to speak. Periodicity blocks are also really simple, and even their use in tempering schemes is actually extremely simple, in concept if not always in practice.

>
> And, also to note...those scales don't intersect the octave or >get near
> it but do intersect (3/2)^x (the octave/2-tave)...yet they don't >sound off-center the way many non-octave based scales do. Weird eh?

Successive arithmetic means of 3/2 and 1/1, then using 3/2 as the interval of repetition? It will sound good, you should make a whole bunch of music with it and see if that fills all your musical needs.
>
> Or, dare I ask, is the fact that I'm taking the half/octave >equivalent of unmodified successive golden sections (but with 0.5 >not 0.618) responsible for this (I know you commented before that >was >actually perhaps the one truly original thing in my PHI scale >that >wasn't well documented in history)?

Taking successive golden sections and simply calling that a scale, I think that's an original idea, and it sounds great. The actual material, the successive golden sections themselves, that's not new at all. If you would go over the replies you've recieved here, put things into Scala and examine them, google around a lot, you'll find just how much phi has been used.

Phi+1 as the "octave" in 1654! That should tell you something. Didn't you notice that Jacques' tunings include your scale and tons of others?

>
> >"Not surprising because it IS JI. Yes 24/17 is 17/16 down from 3/2,
>
> 17/16 being the arithmetic mean of 9/8 and 1/1, and 9/8 being the >AM of 5/4 and 1/1, and so on..."
> I was pretty sure of that as well...I just wanted to double >check because it seemed pretty uncanny to me that if it did generate >JI so well I've never heard of it used much in modern times.

Did you try the Fibonacci scale modulo 2, as I mentioned earlier? That may surprise you as well.
>
> >"There's certainly a whole lot going on that makes things work, or not,
> but these simple proportions must be a part of (physics makes these
> simple proportions into tangible events that aren't just "numbers" of
> course)."
>
> One real test I've been meaning to try out is to take >instruments with a "perfect" PHI timbre using the golden ratio >section scales (frequency-based not logarithmic) and then compare >the result to and arithmetic mean scale with the usual harmonic >>>the PHI and/or golden ratio based scale will sound as natural as >the octave based one.

Arrrrgggg.... you should pay attention when people write replies to you, check out what they've been doing, in some cases for decades....

>
> But, as for accepting harmonic timbres well, I think it is fair >to say the arithmetic mean scale wins the contest hands down.

Maybe you like it best at this point, but there's a lot more out there.

>Plus, with it, I can get 8 to 9-tone scales as smooth as standard >7-tone JI diatonic...and contain a few mystery JI ratios not >commonly used such as 24/17 and 27/16 (that oddly enough still work >very well despite having neither o-tonal or u-tonal relationships to >other parts of the scale).
>
> Again my question, to you and everyone knowledgeable about JI >(and regardless of the fact people far back in history have used it), is why aren't more people using the arithmetic mean (in the same >way I and perhaps others have) in generating their scales through >the (1/var)^x+1 formula?

I guess most people who are knowledgable about JI are just going to be wondering why you're stopping at such a tiny subset of it.

>
>
> Again, a key point (beside that PHI and 0.5 seem to be "magic >numbers" in the above formula)...seems to be the (1/var)^x+1 >"sections" formula lend itself very well to generating symmetrical >and beautiful scales when given the right value for VAR.

Well it's certainly going to give some kind of order to things. As you know, I'm a big fan of divisions and means, and I think you should study them! The harmonic mean is fantastic, and it can also be used in at least one way I've never seen mentioned in any tuning or music theoretical writing, ever. :-)

🔗Michael Sheiman <djtrancendance@...>

>"what you are actually doing is taking the arithmetic mean of 2 figures,
one of which is zero, then taking the arithmetic mean of that, then of
that"

Right...meaning the fact I'm using 1/2 and not 0.618034 as the "midpoint" of the line makes this an "arithmetic mean" operation...taking the mean of consistently smaller sections.

>"Successive arithmetic means of 3/2 and 1/1, then using 3/2 as the
interval of repetition?"
Precisely... Indeed I am using 1.5 as the period.
>"It will sound good, you should make a whole
bunch of music with it and see if that fills all your musical needs."
I believe I will...the whole structure seems to have the very same "assonant symmetrical" sound from the PHI scale in areas where JI is usually rough...and yet matches overtones much better than the PHI scale for standard harmonic timbres...plus I can get the desired flexibility of 8-9 tone scales this way.

>"Taking successive golden sections and simply calling that a scale, I
think that's an original idea, and it sounds great. The actual
material, the successive golden sections themselves, that's not new at
all."
Agreed...it's a new application of an age old idea. Like I said before...I got the idea from taking Golden-sections within a scale to best mimic the way they are used in architecture (and have been for thousands of years).

>"Phi+1 as the "octave" in 1654! That should tell you something. Didn't
you notice that Jacques' tunings include your scale and tons of others?"

I have seen many of Jacques' tunings include parts of my scale or super-sets of my scale (IE my scale plus many extra tones)...but never only successive golden sections. That seems to be a consistent pattern among PHI scales I've read, they include interval within successive golden sections such as est. 1.38 and 1.23...but also include a bunch of ratios in no way related to them (so far as I can see) such as 1.309 which, IMVHO, is many cases sound exponentially more dissonant.

Me> "One real test I've been meaning to try out is to take
>instruments with a "perfect" PHI timbre using the golden ratio"

Cameron>"Arrrrgggg... . you should pay attention when people write replies to
you, check out what they've been doing, in some cases for decades...."
Well...to do that obviously is going to take more than reading material; I was considering the re-test in order to get sound samples of things like a successive golden section scale combined with PHI timbres. In general...remember that no matter how many beautiful equations I see I won't consider a comparison between any two scales final until I listen to sound samples and try composing with the two.

If you have any sound samples of such as experiment, though, I'm always up for good links.

Me> "But, as for accepting harmonic timbres well, I think it is fair
>to say the arithmetic mean scale wins the contest hands down. "

Cameron>"Maybe you like it best at this point, but there's a lot more out there. "
I'm always on the hunt...if there are any SCALA files in particular you'd recommend please feel free to send them my way. Fact is I find my way around most efficiently by actually composing with scales vs. reading huge papers about them...though I usually end up doing the latter to supplement the compositional experience.

>"I guess most people who are knowledgable about JI are just going to be
wondering why you're stopping at such a tiny subset of it."
Well...if you octave-reverse all the interval the same way Lorne Temes did with his scales, you can easily get 10+ tone per octave scales.
From what I've heard so far, taking successive "golden sections" using 0.5 simply gives a list of tones which all work well together, as opposed to diatonic JI, for example, where only certain sets of tones (IE triads) are optimized.
So, apparently, it takes a very optimized subset of JI that, according to my ears, combines 5-limit (or better) consonance despite having 8+ notes per octave (it seems to win on both accounts rather than compromise between high note # vs. good consonance). And, of course, the ability to "have the entire scale be the chord" greatly reduces the amount of music theory needed to play the scale and enhances general ease of adoption of the scale by both existing and new musicians.

So far I haven't heard a JI scale which provides the same level of optimization...but if someone would like to show me some I would be happy to try them.

Cameron>"The harmonic mean is fantastic, and it can also be used in at least one
way I've never seen mentioned in any tuning or music theoretical
writing, ever. :-)"

You see...it's a weird battle that way: open-mindedness makes a huge difference (case in point that these scales I've come across are actually quite simple but, at the same time, seem to be running across a lot of lesser-known historic theories that have a lot of potential and definitely deserve some 'resurrection').
That being, I know a lot of this stuff is far from original, minus the fact I dared to take 'odd routes' in experimentation and ended up uncovering a lot of lost treasure. So the real original concept seems not to be advanced scale creation but plain old open-mindedness and curiosity.

I am far too much of the micro-tonal world seems anything but open-minded: so caught up in Pythagorean tuning and everything that has stemmed from it that many of them stopped looking for other explanations or other possible solutions.

So I hope more people will turn to the idea of "creating their own methods", particular using inspiration from patterns other "un-related" arts and sciences.
True, much if not most of the time they will run into something already made somewhere (perhaps deep) in history but not well known about...but at least then they can make the point, as we seem to be doing now, of saying "look at this...we really should tap into this and try to improve it".

-Michael

🔗Cameron Bobro <misterbobro@...>

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

>
> Right...meaning the fact I'm using 1/2 and not 0.618034 as the >"midpoint" of the line makes this an "arithmetic >mean" >operation...taking the mean of consistently smaller sections.

But, no- you're using 1/2, and one frequency, which is actually two frequencies, (one of which is zero in this case), so it works out to be the same as the arithmetic mean.

1/2, in and of itself, has nothing to do with the arithmetic mean! It is only when you are taking the AM of two things that you divide by two. The arithmetic mean of 3 frequencies would require you to multiply by 1/3, and 99 by 1/99.

It is what most people call the "average", but this isn't a good idea because you can go way off base using the "average" where it is not appropriate. It is not actually the "average" here, for example.

If you do the writing out of harmonics like I suggested earlier, you'll find some interesting things about these Means.

>
> I have seen many of Jacques' tunings include parts of my scale >or super-sets of my scale (IE my scale plus many extra tones)...but >never only successive golden sections. That seems to be a >consistent pattern among PHI scales I've read, they include interval >within successive golden sections such as est. 1.38 and 1.23...but >also include a bunch of ratios in no way related to them (so far as >I can see) such as 1.309 which, IMVHO, is many cases sound >exponentially more dissonant.

What?! 1.309... is one of the first ratios that shows up in phi tunings of all kinds! Including yours- several of the tunings you have posted have it, have you been doing "show intervals" in Scala as I suggested? It is a prime example of an interval that can be "beatless" in the right context, not by being actually beatless, but because the beating becomes a blur.

>
> Me> "One real test I've been meaning to try out is to take
> >instruments with a "perfect" PHI timbre using the golden ratio"
>
> Cameron>"Arrrrgggg... . you should pay attention when people write replies to
> you, check out what they've been doing, in some cases for decades...."
> Well...to do that obviously is going to take more than reading >material; I was considering the re-test in order to get sound >samples of things like a successive golden section scale combined >with PHI timbres. In general...remember that no matter how many >beautiful equations I see I won't consider a comparison between any >two scales final until I listen to sound samples and try composing >with the two.
>
> If you have any sound samples of such as experiment, though, I'm always up for good links.

They're right there under your nose, if you would read and research the replies you have already recieved.

>
> Me> "But, as for accepting harmonic timbres well, I think it is >fair
> >to say the arithmetic mean scale wins the contest hands down. "
>
> Cameron>"Maybe you like it best at this point, but there's a lot more out there. "
> I'm always on the hunt...if there are any SCALA files in >particular you'd recommend please feel free to send them my way. >Fact is I find my way around most efficiently by actually composing >with scales vs. reading huge papers about them...though I usually >end up doing the latter to supplement the compositional experience.

Well we should hear your compositions.

🔗Michael Sheiman <djtrancendance@...>

>"What?! 1.309... is one of the first ratios that shows up in phi tunings
of all kinds! Including yours- several of the tunings you have posted
have it, have you been doing "show intervals" in Scala as I suggested?"

My very old PHI scale PHI^x/2^y had it. But the formula for the new scale I use, (1/PHI)^x + 1, does not produce 1.309. Try it for yourself: where x = 2 the result is 1.38 and where x = 3 it becomes 1.23.

>"They're right there under your nose, if you would read and research the replies you have already recieved."
I'll look deeper into your replies, but I never remember you referencing sound examples. And believe me, when someone posts a sound example or a link to one, even if I'm not generally interested in the type of scale being presented, I at least give it a shot.

>"It is what most people call the "average", but this isn't a good idea"
If I have this right, you are describing the arithmetic mean as a midpoint between two points; 0 and 1...and my formula translates this by +1 to make it between 1 and 2.

>"Well we should hear your compositions. "
I'm still polishing my last one up based on the "arithmetic mean scale". The piece is very minimalistic (a bit like chamber music), but progressive is anything with loads of tonal colors...and no melodic part even exact motif repeats once (IE good luck memorizing it without "sheet music"). Anyhow, I'll be sure to post it on MMM when I'm done. :-)

-Michael

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> >"What?! 1.309... is one of the first ratios that shows up in phi >tunings
> of all kinds! Including yours- several of the tunings you have >posted
> have it, have you been doing "show intervals" in Scala as I >suggested?"
>
> My very old PHI scale PHI^x/2^y had it. But the formula for the new scale I use, (1/PHI)^x + 1, does not produce 1.309. Try it for yourself: where x = 2 the result is 1.38 and where x = 3 it becomes 1.23.

Play your tuning- what's the interval between x=1 and x=3?

>
> >"They're right there under your nose, if you would read and >research the replies you have already recieved."
> I'll look deeper into your replies, but I never remember you >referencing sound examples. And believe me, when someone posts a >sound example or a link to one, even if I'm not generally interested >in the type of scale being presented, I at least give it a shot.

I guess you never got my PMs? But I definitely put up a link to a piece in a "phi" kind of tuning at MMM. Maybe you're not into dense counterpoint or thick chords, but "Ocean Intro" at

http://xenharmonic.ning.com/

is in an 18-tone non-equal, non-Just tuning derived from Pi, which creates phi intervals as well , lessee... yes also some of the golden sections of golden sections. There are also some Just intervals in there, they're byproducts of the process strangely enough.

>
> >"It is what most people call the "average", but this isn't a good >idea"
> If I have this right, you are describing the arithmetic mean as a >midpoint between two points; 0 and 1...and my formula translates >this by +1 to make it between 1 and 2.

Nope, not necessarily two points, and not necessarily a "mid"- maybe you should look it up in different places and go over it.
>
> >"Well we should hear your compositions. "
> I'm still polishing my last one up based on the "arithmetic >mean scale". The piece is very minimalistic (a bit like chamber >music), but progressive is anything with loads of tonal colors...and >no melodic part even exact motif repeats once (IE good luck >memorizing it without "sheet music"). Anyhow, I'll be sure to post >it on MMM when I'm done. :-)

Cool, hopefully you'll use some sustained sounds and tall chords in it so we can really hear how consonance/dissonance/assonance is working out.

🔗Michael Sheiman <djtrancendance@...>

Me>"Try it for yourself: where x = 2 the result is 1.38 and where x = 3 it becomes 1.23."

Cameron>Play your tuning- what's the interval between x=1 and x=3?
As I said clearly above where x=2 it is 1.38 >not< 1.309.

>"But I definitely put up a link to a piece in a "phi" kind of tuning at
MMM. Maybe you're not into dense counterpoint or thick chords, but
"Ocean Intro" at"
I have not kept up with MMM lately: I'll look that one up, thank you. :-)

>Cool, hopefully you'll use some sustained sounds and tall chords in it
so we can really hear how consonance/dissonan ce/assonance is working
out.

Believe me I do...in fact entire lines of my composition are made for sustained string/pad instruments...can't get much more sustained than that.

-Michael

--- On Sat, 5/16/09, Cameron Bobro <misterbobro@...> wrote:

From: Cameron Bobro <misterbobro@...>
Subject: [tuning] Re: Scale using a "1/octave" section instead of a 1/PHI section
To: tuning@yahoogroups.com
Date: Saturday, May 16, 2009, 3:54 PM

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

> >"What?! 1.309... is one of the first ratios that shows up in phi >tunings

> of all kinds! Including yours- several of the tunings you have >posted

> have it, have you been doing "show intervals" in Scala as I >suggested?"

> My very old PHI scale PHI^x/2^y had it. But the formula for the new scale I use, (1/PHI)^x + 1, does not produce 1.309. Try it for yourself: where x = 2 the result is 1.38 and where x = 3 it becomes 1.23.

Play your tuning- what's the interval between x=1 and x=3?

> >"They're right there under your nose, if you would read and >research the replies you have already recieved."

> I'll look deeper into your replies, but I never remember you >referencing sound examples. And believe me, when someone posts a >sound example or a link to one, even if I'm not generally interested >in the type of scale being presented, I at least give it a shot.

I guess you never got my PMs? But I definitely put up a link to a piece in a "phi" kind of tuning at MMM. Maybe you're not into dense counterpoint or thick chords, but "Ocean Intro" at

http://xenharmonic. ning.com/

> >"It is what most people call the "average", but this isn't a good >idea"

> If I have this right, you are describing the arithmetic mean as a >midpoint between two points; 0 and 1...and my formula translates >this by +1 to make it between 1 and 2.

Nope, not necessarily two points, and not necessarily a "mid"- maybe you should look it up in different places and go over it.

> >"Well we should hear your compositions. "

> I'm still polishing my last one up based on the "arithmetic >mean scale". The piece is very minimalistic (a bit like chamber >music), but progressive is anything with loads of tonal colors...and >no melodic part even exact motif repeats once (IE good luck >memorizing it without "sheet music"). Anyhow, I'll be sure to post >it on MMM when I'm done. :-)

Cool, hopefully you'll use some sustained sounds and tall chords in it so we can really hear how consonance/dissonan ce/assonance is working out.

🔗Cameron Bobro <misterbobro@...>

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

>
>
> Cameron>Play your tuning- what's the interval between x=1 and x=3?

>--- Michael Sheiman wrote:
> As I said clearly above where x=2 it is 1.38 >not< 1.309.

Notice that I said "play" your tuning, and if you had, you would have immediately heard that your answer was not correct.

The interval between 1.23606797749979 and 1.618033988749895 is NOT 1.38. It is 1.309016994374947, that very same lovely 466 cent interval we were talking about before.

The frequency distance between then is 0.381966011250105, but that's not the interval- the interval is a proportion, a relation, a ratio.

The interval is 1.618033988749895/1.23606797749979, which is 1.309016994374947.

Look-
1.0
1.25
1.5
1.75
2.0

This is, as you know, 1/1, 5/4, 3/2, 7/4, 2/1. But 1.75-1.5 is .25! Does it make the same interval as the first interval of 1.25? No- it's the same frequency difference, .25, but the ratio, the interval, and what we hear, between 1.5 and 1.75 is 1.75/1.5, which is 7/6. So, a raw frequency amount in one place, .25 above the 1.0, makes a major third, and the same raw frequency amount in another place, between 1.75 and 1.5, makes a subminor third.

And the same amount, .25, between 1.5 and 1.25, makes a 6/5 minor third. And the same amount from 1.75 to 2.0 makes an 8/7.

Now obviously there must be something going on here, and as you know full well, it's a harmonic series. As you yourself have said several times here, the frequency difference between harmonics is a constant. That's actually not a trivial observation, though some people might dismiss it as such. The reason why we hear different intervals is because we percieve the proportions.

And no, I wasn't being sarcastic or silly or anything about making a monochord of some kind, or writing all these things out on a big sheet of paper, I was being dead serious.

🔗Michael Sheiman <djtrancendance@...>

Cameron>"The interval is 1.618033988749895/ 1.23606797749979 , which is 1.309016994374947."

I understand that. Still the fact remains the formula (1/PHI)^x +1 does not produce this number. The formula PHI / ((1/PHI)^x +1) does, but that formula is not a part of the scale I use.

>"Does it make the same interval as the first interval of 1.25? No- it's
the same frequency difference, .25, but the ratio, the interval, and
what we hear, between 1.5 and 1.75 is 1.75/1.5, which is 7/6."

You are making the assumption I am building the scale I use in a multiplicative, ratio based fashion. However I am >not< doing so. Look at the golden sections scale again.
1
1.09
1.14
1.23606
1.38
1.618

Note the relationships of 1.09 + 1.14 ~= 1.23 and 1.14 + 1.23 ~= 1.38 and 1.23 + 1.38 ~= 1.618.

Also note how 1.309 does not fit into this pattern. You simply do not seem to understand how I came up with my scale...and it has nothing to do with dividing 1.618 by anything I get from the (1/PHI)^x formula. I do get a note 1.528...but that is from PHI >minus< 1.09...again no division involved.

There exists an interval 1.309, but it's the ratio between the 1.618 and 1.23 in the scale...and it certainly does not fit between 1.23 and 1.38 as it does in the other PHI scales I've seen, including Temes's own ones (that would give a 1.0642 ratio between those tones...not anything close to the value obtained by taking intervals between the above tones).

>"The reason why we hear different intervals is because we perceive the proportions."
Agreed. However, simply put, if you shove 1.309 into the golden section scale you get the ratio 1.0642, a ratio that does not exist between any two existing notes in the above scale.

Simply put, even on an interval basis, putting 1.309 between 1.23606 and 1.38 disrupts the symmetry of the golden section scale.

The other side of the argument, of course, is.....why/how do you think using 1.309 between 1.23606 and 1.38 would be a good idea? I've tried it before several times, and every time I have my ear jumps at it as sounding off relative to the chord formed by 1.38 and 1.23606.

-Michael

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> Cameron>"The interval is 1.618033988749895/ 1.23606797749979 , >which is 1.309016994374947."
>
> I understand that. Still the fact remains the formula (1/>PHI)^x +1 does not produce this number. The formula PHI / ((1/>PHI)^x +1) does, but that formula is not a part of the scale I use.
>
> >"Does it make the same interval as the first interval of 1.25? No- it's
> the same frequency difference, .25, but the ratio, the interval, and
> what we hear, between 1.5 and 1.75 is 1.75/1.5, which is 7/6."
>
> You are making the assumption I am building the scale I use in a >multiplicative, ratio based fashion. However I am >not< doing so.

I am NOT making that assumption at all- it is I who explained to you that that is not what you're doing, rather that it is golden sections of golden sections.

> Look at the golden sections scale again.
> 1
> 1.09
> 1.14
> 1.23606
> 1.38
> 1.618
>
> Note the relationships of 1.09 + 1.14 ~= 1.23 and 1.14 + 1.23 ~= 1.38 and 1.23 + 1.38 ~= 1.618.

Note the golden sections relationships in a golden section tuning- who would have thought!

Look at the relation between 1.618... and 1.23606... It IS the 1.309...!

you said " Still the fact remains the formula (1/>PHI)^x +1 does not produce this number."

It sure DOES produce this interval, it just doesn't produce it relative to the "tonic" in the first "phiave".

>
> Also note how 1.309 does not fit into this pattern.
> You simply do not seem to understand how I came up with my >scale...and it has nothing to do with dividing 1.618 by anything I >get from the (1/PHI)^x formula. I do get a note 1.528...but that is >from PHI >minus< 1.09...again no division involved.
>
> There exists an interval 1.309, but it's the ratio between the >1.618 and 1.23 in the scale...

that's what I said! You've actually got a neat trick there: I explain something to you in detail, then you repeat it as if you're teaching me something. More entertaining than TV. :-)

>and it certainly does not fit between 1.23 and 1.38 as it does in >the other PHI scales I've seen, including Temes's own ones (that >would give a 1.0642 ratio between those tones...not anything close >to the value obtained by taking intervals between the above tones).

Have you overlooked the fact that 1.618... is your interval of repetition, or "octave substitute" or whatever you want to call it, and so, when the tuning is continued over more than one "phiave", the interval two steps down from the tonic...1.309...

And it's right there in a dense setting exactly as you've been trying to avoid in your orignial "phiave".

>
> >"The reason why we hear different intervals is because we perceive the proportions."
> Agreed. However, simply put, if you shove 1.309 into the >golden section scale you get the ratio 1.0642, a ratio that does not >exist between any two existing notes in the above scale.

Well the 1.309 interval IS in the golden section scale, just not above the "tonic" in the first "phiave". But you're right that if you did have it there, you'd have to add other notes to get it into a chain golden proportion intervals.

.
>
> Simply put, even on an interval basis, putting 1.309 between >1.23606 and 1.38 disrupts the symmetry of the golden section scale.

>
> The other side of the argument, of course, is.....why/how do you >think using 1.309 between 1.23606 and 1.38 would be a good idea? >I've tried it before several times, and every time I have my ear >jumps at it as sounding off relative to the chord formed by 1.38 and >1.23606.

How did you manage to sidetrack this from a correction of your plainly incorrect statement about the 1.309... interval not existing in your tuning into a discussion about different implementations of golden section tunings? :-)

Different people playing different notes in different combinations and orders in different musics, that's the answer to your question.

Anyway, it is clearly NOT the 1.309... interval itself that bothers you after all. What you don't want is, say, "densely packed" areas in your tuning, which is in keeping with your whole concept. In order to have the 1.309... interval in a golden proportion with the next scale step, you'd have to either drop or add notes to your scale- adding them would mean more than a single modality, and you want one modality.

🔗Jacques Dudon <fotosonix@...>

Posted by: "Michael Sheiman" djtrancendance@... djtrancendance
Fri May 15, 2009 9:14 am (PDT)

Cameron > "Phi+1 as the "octave" in 1654! That should tell you something. Didn't
you notice that Jacques' tunings include your scale and tons of others?"

Michael > I have seen many of Jacques' tunings include parts of my scale or super-sets of my scale (IE my scale plus many extra tones)...but never only successive golden sections. That seems to be a consistent pattern among PHI scales I've read, they include interval within successive golden sections such as est. 1.38 and 1.23...but also include a bunch of ratios in no way related to them (so far as I can see) such as 1.309 which, IMVHO, is many cases sound exponentially more dissonant.

Nonsense, Michael ! First take a closer look for example at the tuning I posted on April 1st :
1.0000000
1.05572809
1.0901699
1.145898
1.20162612
1.2360679775
1.29179607
1.32623792
1.381966
1.4376941
1.472135955
1.527864045
1.583592135
1.6018034 (= period)

It is actually ONLY based on what we would call "arithmetical" golden sections, successive and everywhere, and maximally, because it is based on Phi waveform infinite word and is itself a fractal, infinite division.
On the contrary, you should explain the reason to bring your "1.532" here, and what interval would it divide in an arithmetical golden section, since you say your scale does "only" that.

Instead, 1.527864, that initiates the 4F series, divides 1.854102 - 1 in a golden section, and a infinity of others.

One thing you should understand in these tunings I posted, like the Iph-heptaphone and others, is that they're not conceived as modal. "1/1" is only given for indication because you use those ratios, but they contain as many modes as tones they have.

This is why I think the nomenclatura of series is simpler and more adapted to fractals.
Instead of writing :
1/1
1.055728
1.236068
1.309017
1.527864
1.618034
1.854102
(or even the same thing in cents, or approximated ratios),
which is BTW another version and more symmetrical mode of an IPH-Heptaphone,
you may write it :
[2F 4P 4F F 8F 2F 6F]
and that's done, including all octaves/phiaves/iphaves/phidiames etc. up and down.
Because all fractals have always precise ratios between their series, ex. here
P/F = 1.381966 = (5 - sqrt5) /2,
This writing has absolute precision, plus it can be translated in whole number series instantly,
ex. here
110 116 136 144 168 178 204

Which is also another way to write the same scale.

- - - - - - -
Jacques

🔗Jacques Dudon <fotosonix@...>

Posted by: "Michael Sheiman" djtrancendance@... djtrancendance
Sat May 16, 2009 7:12 pm (PDT)

Me > "Try it for yourself: where x = 2 the result is 1.38 and where x
= 3 it becomes 1.23."

Cameron > Play your tuning- what's the interval between x=1 and x=3?

Michael > As I said clearly above where x=2 it is 1.38 >not< 1.309.

A few remarks Michael :

1) do you think it is correct to round 1.236068 to 1.23 ? and even
1.382068 to 1.38 ? please, one more digit at least, thanks !! -
otherwise we don't know what we are talking about...

2) I think you misunderstood Cameron's remark -
Cameron, I believe, meant that "1.309017" = "1.618034" / "1.236068"

3) it is strange you feel that interval "off" ... because it is
omnipresent in your scale ! :

1.309 / 1
1.382 / 1.0557
1.618 / 1.236 (mentionned above)
plus two others approached by the Phi(F/8P) comma :
2 / 1.532 (actually my 1.527864 would be better)
and 1.0557 at the octave / 1.618, if you use it, I do myself.

For the record, 1.309017 = 2.618034 / 2 = (Phi + 1) / 2 = Phi^2 / 2
Personnally I love those indonesian low fourths. You should try using
1.309017 as generator :
It creates a beautiful slendro that is the base of an old piece of
mine, "Fleurs de lumière", originally constructed on selecting one
every two Fibonacci numbers : 3 8 21 55 144 , or back to one
single octave :
32 36 42 48 55 64 which is a Fibonacci version of Lou
Harrisson's school slendro gamelan in Aptos :
9/8 7/6 8/7 8/7 7/6, where sounds based on 1.618 and 2.618
inharmonics will work of course perfectly. I love these coïncidences
between Phi and factor 7.

- - - - - - -
Jacques

> Nice...I tried this but the note 1.309 and 1.708204 sounded "off" > to me. I was just wondering what you'd think of this
> 1/1
> 1.055728
> 1.236068
> 1.309017
> 1.381966
> 1.532 (new tone of 1.618034 / 1.05573)
> 1.618034
> 1.8541 (new tone of 1.618034 * 1.14589 where (1/PHI)^5+1 = 1.14589)

correction : 1.8541 is the phiave of (1/PHI^4) + 1

🔗Michael Sheiman <djtrancendance@...>

Cameron>
"you said 'Still the fact remains the formula (1/>PHI)^x +1 does not produce this number.' It sure DOES produce this interval, it just doesn't produce it relative to the "tonic" in the first "phitave"."

I know, and my scale is based on golden sections from the tonic. So what's wrong with that?

Me> "There exists an interval 1.309, but it's the ratio between the >1.618 and 1.23 in the scale..."

Cameron> "that's what I said! You've actually got a neat trick there: I explain
something to you in detail, then you repeat it as if you're teaching me
something."

I never said it in such a tone. If I was "teaching" I would have said something like "you need to learn about...You see..." before saying "there exists". Anyhow, I don't care about this who teaches who crap...of course this is a tuning forum not a music class; the gist of what I said is "I realize this exists...even though you might not realize I do".

Back to what I said before...I realize there is no 1.309 in my scale but there is a resulting 1.309 interval between 1.618034. But stating my scale as having 1.309 relative to the root is simply not true...you admited this yourself when you said " it just doesn't produce it relative to the "tonic" in the first "phitave".

I have enough difficulty explaining the golden-section-from-the-root scale I ran across without your telling me what it is. In a previous post you actually stated "Mike's PHI scale" (and >not< a mode of the scale or anything that may imply 1.309) as having 1.309 relative to the root...and for all the efforts I have made to clarify how the scale is derived you bet I resent that!

>"And it's right there in a dense setting exactly as you've been trying to avoid in your orignial "phiave". "

I lost you there. The only 1.309 interval I've found within the first phi-tave and 1.23. Even if you cross the octave to 1.09*PHI and 1.145*PHI notice 1.145 / 1.309 = about 1.38 (which is a note in the scale)....but even then the nearest notes are 1.23 and 1.54...not exactly a dense setting in the same way that 1.23 (1.309 away from 1.618) is between 1.14 and 1.38...again, not in a dense setting.

So what's your main example that somehow proves even using the scale (approximated)
1
1.09
1.145
1.23
1.38
1.54
......that I somehow end up with a 1.309 interval landing me in a dense area between tones? I have yet to be convinced...

-Michael

🔗Michael Sheiman <djtrancendance@...>

Jacques>"It is actually ONLY based on what we would call "arithmetical" golden sections, successive and everywhere"

Ok, now I'm seriously confused.
   My scale is simply a result of taking the formula (1/PHI)^x + 1 and setting a limit of having no notes less than the ratio 1.05 apart.
   This gives (approximately)
1.05555
1.09
1.23
1.38
1.618034
...and then I simply subtract 1.09 (taking the series in reverse) to get
1.527864
   ...note: my scale does not include any other notes than the ones above and fits the (1/PHI)^x + 1 formula and its reverse perfectly (which is what makes it different than other larger scales like the one you posted which include it)!

What confuses me is that when I posted it someone said "that's just successive Golden sections" and here you are saying (paraphrased) "that's not just successive golden sections, but much more". Who to believe when so many people are forcing different/conflicting terms upon the same scale.

>--
Maybe I should just call it "the scale generated by (1/PHI) + 1 and PHI - (1/PHI) + 1 and omit the term "golden sections" entirely since the scale apparently does not include all possible "golden sections"? It seems however I display it or what terms I use I piss someone off...it kind of makes me wonder how taking other people's advice often only gets me into even more controversy rather than less!!
--<

>"On the contrary, you should explain the reason to bring your "1.532" here"
As I said before, the values I gave for the scale were >apx.<, meaning approximations or quick estimates off the top off my head since I was posting from work. You're right, the correct value is 1.527864.

When I look at your golden sections scale I realize it contains the notes I posted plus several others. Hence why I said your scales contain supersets of the one I showed. For example
1.00000001.05572809 *
1.0901699 *
1.145898 *
1.201626121.2360679775 *
1.291796071.326237921.381966 *
1.43769411.4721359551.527864045 *
1.5835921351.6018034 (= period)...shown with *'s are the notes of the scale I ran into which just happen to be a subset of your scale. In the same way 12TET is >not< the same as C-major, the (1/PHI)^x + 1 scale does not cover all the values in your above scale.

-Michael

🔗Michael Sheiman <djtrancendance@...>

>"1) do you think it is correct to round 1.236068 to 1.23 ? and even 1.382068 to 1.38 ?"
No but I was writing this from work and would have a tricky time calculating it precisely or remembering it well in that environment.

>"2) I think you misunderstood Cameron's remark -Cameron, I believe, meant that "1.309017" = "1.618034" / "1.236068""

And that's how I understood it. That still doesn't change the fact my scale generation equation of (1/PHI)^x does not generate that interval from the root nor gives him the right to blatantly mis-state that scale and claim I the version I made has that tone from the root of 1/1 (because it does not).

>"3) it is strange you feel that interval "off" ... because it is omnipresent in your scale ! "
I >never< said/meant it was off beyond saying it sounds bad between the 1.05, 1.23 and 1.38 (again, estimated) intervals from the root that are in my scale. Maybe I could have been more clear before, but that was my point.

BTW, below is a >completely< different scale than the one Cameron and I were debating about...and, in the below scale (which is not the aforementioned (1/PHI)^x + 1scale)...obviously the tone 1.309017.
Nice...I tried this but the note 1.309 and 1.708204 sounded "off" to me. I was just wondering what you'd think of this1/11.0557281.2360681.3090171.3819661.532 (new tone of 1.618034 / 1.05573)1.6180341.8541 (new tone of 1.618034 * 1.14589 where (1/PHI)^5+1 = 1.14589)
Hopefully that will help clear things up (since apparently you confused one of my scales not mentioned in the discussion with Cameron with the (1/PHI)^x + 1 based scale we were discussing), Michael

🔗Jacques Dudon <fotosonix@...>

Posted by: "Michael Sheiman" djtrancendance@... djtrancendance
Mon May 18, 2009 4:51 pm (PDT) :

Jacques> "It is actually ONLY based on what we would call "arithmetical" golden sections, successive and everywhere"

Michael> Ok, now I'm seriously confused.
My scale is simply a result of taking the formula (1/PHI)^x + 1 and setting a limit of having no notes less than the ratio 1.05 apart.
This gives (approximately)
1.05555
1.09
1.23
1.38
1.618034
...and then I simply subtract 1.09 (taking the series in reverse) to get
1.527864

Oh yes ? it's not " 1.532 (new tone of 1.618034 / 1.05573)" anymore ? -
(citation from your 14th of may post) - then this is a bit hard to follow.

Jacques> "On the contrary, you should explain the reason to bring your "1.532" here"
Michael> As I said before, the values I gave for the scale were >apx.<, meaning approximations or quick estimates off the top off my head since I was posting from work. You're right, the correct value is 1.527864.

Sounds better indeed - you know what, I tried to figurate what series this 1.532624 = 1.618034 / 1.055728 belonged to, and I had some difficulties... It is close to 17F, but not even exact.
But seriously, if you were using more whole numbers series instead of these truncated ratios you would make things clearer.
Now what's your scale finally ? is it something like 55 : 58 : 60 : 68 : 76 : 84 : 89 ? [modulo Phi] ?
Then you can write it [F 2P J 2F P 4F] - how do you like that ?
(note : P = (5 - sqrt of 5) /2 and J = Phi^5 - 10)

- - - - - - -
Jacques

🔗Michael Sheiman <djtrancendance@...>

🔗Jacques Dudon <fotosonix@...>

Jacques>"Oh yes ? it's not " 1.532 (new tone of 1.618034 / 1.05573)"

Michael> No, it is...I simply rounded 1.05573 to 1.055 in the
calculation. The exact value would indeed be 1.618034 / 1.05573 =
1.532 and not 1.5278. Wow...is about a 0.0042 difference of error
such a big deal?

First, 1.618034 / 1.055728 makes 1.532624, so 1.532 = wrong rounding
again, but never mind.
Then the ratio between 1.532624 and 1.527864 is exactly the Phi(F/8P)
comma I mentionned earlier, ~322/321 or a quarter syntonic comma,
it's up to you if it means nothing, as you want ...
But one day you say your interval is 1.618034 / 1.055728, the next
day it is (1.618 - 1,09017 +1) ("taking the series in reverse" ) -
and alternatively - makes it hard to follow !
Tertio yes, that difference is important in a scale supposed to have
coherent difference tones.
Especially when you claim that your scale is based on "only
successive golden sections" : I asked you how "1,532" (or 1.532624)
had this property, but I had no answer.

Jacques>"Now what's your scale finally ? is it something like 55 :
58 : 60 : 68 : 76 : 84 : 89 ? [modulo Phi] ?"
Michael> Exactly. That's at least close enough to the ear that you
can't tell the difference. :-)
However, I wonder how many people recognize this means 58/55,
60/55, etc.

On this list, I hope everybody should know what it means. Actually it
does means more than 58/55, 60/55, etc. - it means a scale with no
specific 1/1.
But here is a new mystery with your scale. I thought you said it had
"no notes less than the ratio 1.05 apart" - I think the quartertone
60/58 is smaller.

Jacques>"Then you can write it [F 2P J 2F P 4F] - how do you like
that ? (note : P = (5 - sqrt of 5) /2 and J = Phi^5 - 10)"
Michael> Sounds good...the only part that escapes me is...what is F?

Each of those is an infinite Phi series, wether in real fractal
ratios or in series of whole numbers.
F is the "1/1" of the Phi series and this letter is commonly used in
reference to the Fibonacci series,
1 1 2 3 5 8 13 21 34 55,
that contains here f = 8, but it's only a coïncidence.
BTW I just made up the series "J" here because it is a "prime"
series, that is needed by 1.09017 :
1 4 5 9 14 23 37 60 97 ...
(If anyone has heard of specific names for other series than F & P or
L, I am interested).
- - - - - - -
Jacques

🔗Michael Sheiman <djtrancendance@...>

Jacques>"But
one day you say your interval is 1.618034 / 1.055728, the next day it
is (1.618 - 1,09017 +1) ("taking the series in reverse" ) - and
alternatively - makes it hard to follow !"

Agreed, I've changed the definition of the scale once...but the basic pattern has every tone except the one tone around 1.5278 in common. You'll notice that taking 1.618034 / 1.055728 and 1.618 - 1,09017 +1 yield almost exactly the same value within a few cents...it is really up to the composer which one they want to use (since they sound just about exactly the same).

>"Especially when you claim that your scale is based on "only
successive golden sections" : I asked you how "1,532" (or 1.532624) had
this property, but I had no answer."
True, it's an exception to the other tones and an "extra" note, as I said before. But it is basically just a golden section going in the opposite direction.

>"Actually it does means more than 58/55, 60/55, etc. - it means a scale with no specific 1/1."
Yes, I need to clarify that. In practical usage you must choose between 60/55 (about 1.09) and 58/55 (about 1.055)...but never both at once. That way, when compared to 63/55...all possible ratios are 1.05 or greater (both 58/55 and 60/55 are further than 1.05 from the next tone of 63/55).

>"BTW I just made up the series "J" here because it is a "prime" series, that is needed by 1.09017 :1 4 5 9 14 23 37 60 97 ..."
Ah, ok, thank you...now I understand.

-Michael

Scale using a "1/octave" section instead of a 1/PHI section

🔗djtrancendance@...

🔗Cameron Bobro <misterbobro@...>

🔗djtrancendance@...

🔗Cameron Bobro <misterbobro@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Cameron Bobro <misterbobro@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Cameron Bobro <misterbobro@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Cameron Bobro <misterbobro@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Cameron Bobro <misterbobro@...>

🔗Jacques Dudon <fotosonix@...>

🔗Jacques Dudon <fotosonix@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Jacques Dudon <fotosonix@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Jacques Dudon <fotosonix@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Kraig Grady <kraiggrady@...>