The Mathematics of Music book

Hi all, John here.

I have worked out a mathematical formula for the consonance value of a musical interval x/y (where x is less then or equal to y)...

(2 + 1/x + 1/y - diss)/2 where x<=y.

If x/y is between 0 and 0.9375 then 'diss' = x/y
if x/y is between 0.9375 and 1.0 then 'diss' = (1 - x/y)*15

If the result is 1.0 or higher then the interval is Major.
If the result is between 0.75 and 0.9999 then the interval is Minor.
If the reult is less than 0.75 then the interval is no good.

The formula is based on a few educated guesses but it seems to work consistently when tested.

How I worked it out is explained in my book "The Mathematics of Music" which you can download from the "Files" link on the left.

You can hear a tune I wrote and recorded using a "Just" guitar that I built myself at my web site... www.johnsmusic7.com

The web site has a few other bells and whistles as well: you can download a chord and scale dictionary for my "Just" guitar and for luthiers, a spec for fret placement is also available.

My 12-key tuning system is...
1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.
It is identical to the Just scale but the second note is 15/14 instead of 16/15. 3/2 can also be used as a tonic with equal strength.

Check out the book, any and all comments/feedback welcome.

John.

On 30 March 2010 22:21, jfos777 <jfos777@...> wrote:

> How I worked it out is explained in my book "The Mathematics of Music" which you can download from the "Files" link on the left.

From that,

"There is an organisation based in San Francisco called The Just
Intonation Network which deals with other “just” tuning systems. Their
web address is: www.justintonation.net . I think that their system is
based on prime numbers. My own system uses a different approach."

If you've discovered a new kind of integer that isn't based on prime
numbers, you should write up your proof and claim your Fields Medal.

"In physics, two notes with the same frequency, and an
amplitude (loudness value) of 1.0 each, when played simultaneously,
produce a sound wave with an amplitude of 2.0."

Only if they're in phase. The amplitude could be as little as zero if
they aren't.

"There is simply not enough room on a guitar fretboard to
implement a temperament with more than 12 keys [sic] to the octave."

There is.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 30 March 2010 22:21, jfos777 <jfos777@...> wrote:
>
> "There is an organisation based in San Francisco called The Just
> Intonation Network which deals with other “just” tuning systems.
> Their web address is: www.justintonation.net . I think that their
> system is based on prime numbers. My own system uses a different
> approach."
>
> If you've discovered a new kind of integer that isn't based on prime
> numbers, you should write up your proof and claim your Fields Medal.
>

Maybe he meant that his formula does not determine consonance based on prime numbers. Indeed the "classical" JI approach (if there is a thing like that) is much focused on primes, as the term "p-limit" suggests, which sort of constitues a hierarchy of intervals. Euler's gradus suavitatis function is also based on prime numbers.

John's formula, OTOH, depends only on the sizes of the numbers in question. It is also not independent on prime numbers, though, since the numbers are supposed to be in their most simplified form.

John, I haven't studied your formula in detail - but it appears to me it has two of the major flaws that many consonance formulae have: 1) it does not take the timbre into account (such as the theories of Helmholtz and Bill Sethares do) 2) it does not take into account that two ratios x/y can have neasrly the same value with quite different sizes of the x and the y. E.G., I highly doubt that the ratios 5/4 and 500/401 will sound much different.
--
Hans Straub

I'm not trying to be unkind, I'm trying to be a straight shooter, here.

This sounds exactly like every advanced-beginner guitar-jam there ever was, with approximately one year of experience.

-The lead is ahead of the beat or free-floating, running up and down the same blues-ey E minor-ish pentatonic scale over and over and over -- E,G,A,B,D (with some slight blues-y bending on the B, without being 'melodic'-sounding (because it sounds just like running up and down--hand habits instead of melodies), or being rhythmic--because it doesn't 'lock in' or use rests, or silence, or even syncopations.

-The chords go Em G A B C, D E major or some such, over and over and over

Is this a joke?

To me, and I'm not trying to be unkind, it sounds like every thrasher you ever hear in a music store when you're in there to buy something.

The recording is hissy, and there are digital artifacts 'pops' or 'clicks' on the loud notes.

Is this how you want to sell your ideas?

If it's a significant departure from 12ET, it doesn't make much difference to my ears--which are expert-level according to tests, and also given how many years I've been doing this kind of thing.

I'm genuinely puzzled--I think you're either joking or are as naive as some beginners.

What it doesn't sound like is 'strange' or 'wrong' or 'primitive' or 'passionate' or 'exotic' or original.

Caleb

> You can hear a tune I wrote and recorded using a "Just" guitar that > I built myself at my web site... www.johnsmusic7.com
>

Thanks for the comments Graham

With regard to Prime Numbers, I worked out my formula without consideration of these. In fact I'm not quite sure how prime numbers work in JI. If my formula is correct then prime numbers are, perhaps, irrelevant.

With regard to out of phase tones, have you ever played a unison on a guitar that is 100% out of phase? The unison always sounds the same no matter what the timing of each each note is.

As regards the 12 fret per octave limit on a guitar, I will concede that there is room for extra frets but I believe that extra frets would greatly impair the "playability" of the guitar. Also the extra frets are only viable on the nut end of the neck. From around the tenth fret upwards extra frets would clearly be too close to others.

John

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 30 March 2010 22:21, jfos777 <jfos777@...> wrote:
>
> > How I worked it out is explained in my book "The Mathematics of Music" which you can download from the "Files" link on the left.
>
> From that,
>
> "There is an organisation based in San Francisco called The Just
> Intonation Network which deals with other “just” tuning systems. Their
> web address is: www.justintonation.net . I think that their system is
> based on prime numbers. My own system uses a different approach."
>
> If you've discovered a new kind of integer that isn't based on prime
> numbers, you should write up your proof and claim your Fields Medal.
>
> "In physics, two notes with the same frequency, and an
> amplitude (loudness value) of 1.0 each, when played simultaneously,
> produce a sound wave with an amplitude of 2.0."
>
> Only if they're in phase. The amplitude could be as little as zero if
> they aren't.
>
> "There is simply not enough room on a guitar fretboard to
> implement a temperament with more than 12 keys [sic] to the octave."
>
> There is.
>
>
> Graham
>

Thanks Hans,

you are correct when you say my formula depends only on the size of the numbers in the "simplified" ratio. Primes are not considered.

With regard to timbre, if my understanding of the word is correct, I suspect that my formula is still good, regardless of the timbre. This has to do with repeating points. If the sound waves of two notes are plotted (in phase) then repeating points occur at equal intervals (e.g. the plotted waves look the same every, say, 4 inches). It doesn't matter how the waves look (timbre), all that counts is the distance between the "repeating points". The smaller the distance, the greater the consonance.

The 500/401 interval is 4.3227 cents out of tune with the 5/4 interval. If you play this interval (500/401) on a tuned midi keyboard the mistuning is audible if you listen carefully. The following is not in my book but I have a notion that the maximum allowable deviation from a "perfect" interval is 6.776 cents (256/255). In this case the "perfect" interval is 5/4 and the "imperfect" interval is 500/401. If the strength of the 5/4 interval is 'z' then the strength of the imperfect interval (500/401) is perhaps ((6.776 - 4.3227)/6.776)*z

In any case, it doesn't matter to me much because for me it's perfection or bust.

Thanks again,

John.

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > On 30 March 2010 22:21, jfos777 <jfos777@> wrote:
> >
> > "There is an organisation based in San Francisco called The Just
> > Intonation Network which deals with other “just” tuning systems.
> > Their web address is: www.justintonation.net . I think that their
> > system is based on prime numbers. My own system uses a different
> > approach."
> >
> > If you've discovered a new kind of integer that isn't based on prime
> > numbers, you should write up your proof and claim your Fields Medal.
> >
>
> Maybe he meant that his formula does not determine consonance based on prime numbers. Indeed the "classical" JI approach (if there is a thing like that) is much focused on primes, as the term "p-limit" suggests, which sort of constitues a hierarchy of intervals. Euler's gradus suavitatis function is also based on prime numbers.
>
> John's formula, OTOH, depends only on the sizes of the numbers in question. It is also not independent on prime numbers, though, since the numbers are supposed to be in their most simplified form.
>
> John, I haven't studied your formula in detail - but it appears to me it has two of the major flaws that many consonance formulae have: 1) it does not take the timbre into account (such as the theories of Helmholtz and Bill Sethares do) 2) it does not take into account that two ratios x/y can have neasrly the same value with quite different sizes of the x and the y. E.G., I highly doubt that the ratios 5/4 and 500/401 will sound much different.
> --
> Hans Straub
>

John mentioned, via Graham...

> "There is simply not enough room on a guitar fretboard to
> implement a temperament with more than 12 keys [sic] to the octave."

Hmmm.

You *do* need to get out more. Just from within the small environs of these lists, one could see some very fine guitar playing and music making by Neil Haverstick, Jon Catler, and Dante Rosati, all playing on guitars with very many more frets per octave, including unequal divisions. Of course, I cut my teeth on some of Partch's guitars, which date back to the late 30's.

Homework. You have some.

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> I'm not trying to be unkind, I'm trying to be a straight
> shooter, here.

Heh- I kinda liked it! Everything you say is true about it,
but I found it endearing.

-Carl

--- In tuning@yahoogroups.com, "jfos777" <jfos777@...> wrote:
>
> With regard to timbre, if my understanding of the word is correct,
> I suspect that my formula is still good, regardless of the timbre.
> This has to do with repeating points. If the sound waves of two
> notes are plotted (in phase) then repeating points occur at equal
> intervals (e.g. the plotted waves look the same every, say, 4
> inches). It doesn't matter how the waves look (timbre), all that
> counts is the distance between the "repeating points". The smaller
> the distance, the greater the consonance.

It is definitely not regardless of the timbre - but most, say, "naturally occurring" timbres will work because they have their overtones at places close to the harmonic series, which makes many "simple" interval ratios sound good. But there exist timbres where the case is dramatically different - as Bill Sethares has shown.
I recommend the following article on Bill Sethares:

http://www.nzzfolio.ch/www/21b625ad-36bc-48ea-b615-1c30cd0b472d/showarticle/c2dd2d93-cfb9-442c-9ef2-58b5241f6da1.aspx

>
> The 500/401 interval is 4.3227 cents out of tune with the 5/4
> interval. If you play this interval (500/401) on a tuned midi
> keyboard the mistuning is audible if you listen carefully.

Alright, the interval I gave, yes. But I can give you another interval that is less than 1 cent away from 5/4, even one that is less than a hundredth of a cent away - and there I am almost sure no one will hear the difference. And if there would be somebody, I can go even closer - arbitrarily close to any "simple" interval ratio there is another, arbitrarily "complex" one.

It depends on what you want to do. As a tool for you to make better music, your formula is sure well-suited - you should just be aware it cannot be a general theory of consonance or even "the mathematics of music".
--
Hans Straub

Hans,

I've had a think about timbre and I now see that my idea on "repeating points" is wrong. Because the frequency of the first overtone is usually slightly more than twice the frequency of the fundamental then the repeating points are blurred. What can be said is that with "ideal notes" (where the frequencies of the harmonics are exactly x, 2x, 3x, 4x etc) then my formula might be spot on. The formula should be good enough however to serve as a "rough guide" to working out the strength values of musical intervals with different timbres.

With regard to slightly out of tune intervals you said that...

""""Alright, the interval I gave, yes. But I can give you another interval that is less than 1 cent away from 5/4, even one that is less than a hundredth of a cent away - and there I am almost sure no one will hear the difference. And if there would be somebody, I can go even closer - arbitrarily close to any "simple" interval ratio there is another, arbitrarily "complex" one.""""

As I said in my last reply to you, if a complex interval is close to a simple interval (say within 6.78 cents, 256/255) then the strength of the complex interval is the strength of the simple interval multiplied by (6.78 - x)/6.78 where x is how out of tune, in cents, the complex interval is with the simple interval.

John.

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
>
> --- In tuning@yahoogroups.com, "jfos777" <jfos777@> wrote:
> >
> > With regard to timbre, if my understanding of the word is correct,
> > I suspect that my formula is still good, regardless of the timbre.
> > This has to do with repeating points. If the sound waves of two
> > notes are plotted (in phase) then repeating points occur at equal
> > intervals (e.g. the plotted waves look the same every, say, 4
> > inches). It doesn't matter how the waves look (timbre), all that
> > counts is the distance between the "repeating points". The smaller
> > the distance, the greater the consonance.
>
> It is definitely not regardless of the timbre - but most, say, "naturally occurring" timbres will work because they have their overtones at places close to the harmonic series, which makes many "simple" interval ratios sound good. But there exist timbres where the case is dramatically different - as Bill Sethares has shown.
> I recommend the following article on Bill Sethares:
>
> http://www.nzzfolio.ch/www/21b625ad-36bc-48ea-b615-1c30cd0b472d/showarticle/c2dd2d93-cfb9-442c-9ef2-58b5241f6da1.aspx
>
> >
> > The 500/401 interval is 4.3227 cents out of tune with the 5/4
> > interval. If you play this interval (500/401) on a tuned midi
> > keyboard the mistuning is audible if you listen carefully.
>
> Alright, the interval I gave, yes. But I can give you another interval that is less than 1 cent away from 5/4, even one that is less than a hundredth of a cent away - and there I am almost sure no one will hear the difference. And if there would be somebody, I can go even closer - arbitrarily close to any "simple" interval ratio there is another, arbitrarily "complex" one.
>
> It depends on what you want to do. As a tool for you to make better music, your formula is sure well-suited - you should just be aware it cannot be a general theory of consonance or even "the mathematics of music".
> --
> Hans Straub
>

Thanks Caleb,

I posted a reply twice already but they're not getting through. 3rd attempt...

I never thought that I was Jimi Handrix and my friends have often commented on the monotony of my lead playing, the same Blues scale with the same licks repeated over and over. I just wanted something basic that showed that at the very least my guitar was viable.

There are plenty of chords that sound very unusual (but still are good) with my guitar but I haven't used them because to most people, because they are different, they sound out of tune. They aren't, it just takes time to get used to them.

I played my tune to a (proper) musician acquaintance and his comment was: "it sounds very in tune".

John.

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> I'm not trying to be unkind, I'm trying to be a straight shooter, here.
>
> This sounds exactly like every advanced-beginner guitar-jam there ever
> was, with approximately one year of experience.
>
> -The lead is ahead of the beat or free-floating, running up and down
> the same blues-ey E minor-ish pentatonic scale over and over and over
> -- E,G,A,B,D (with some slight blues-y bending on the B, without
> being 'melodic'-sounding (because it sounds just like running up and
> down--hand habits instead of melodies), or being rhythmic--because it
> doesn't 'lock in' or use rests, or silence, or even syncopations.
>
> -The chords go Em G A B C, D E major or some such, over and over and
> over
>
> Is this a joke?
>
> To me, and I'm not trying to be unkind, it sounds like every thrasher
> you ever hear in a music store when you're in there to buy something.
>
> The recording is hissy, and there are digital artifacts 'pops' or
> 'clicks' on the loud notes.
>
> Is this how you want to sell your ideas?
>
> If it's a significant departure from 12ET, it doesn't make much
> difference to my ears--which are expert-level according to tests, and
> also given how many years I've been doing this kind of thing.
>
> I'm genuinely puzzled--I think you're either joking or are as naive as
> some beginners.
>
> What it doesn't sound like is 'strange' or 'wrong' or 'primitive' or
> 'passionate' or 'exotic' or original.
>
> Caleb
>
>
>
>
>
>
> > You can hear a tune I wrote and recorded using a "Just" guitar that
> > I built myself at my web site... www.johnsmusic7.com
> >
>

On 31 March 2010 20:15, jfos777 <jfos777@...> wrote:

> I never thought that I was Jimi Handrix and my friends have
> often commented on the monotony of my lead playing, the
> same Blues scale with the same licks repeated over and over.
> I just wanted something basic that showed that at the very
> least my guitar was viable.

So how long *have* you been playing? It sounds like something that
should improve with practice. And like you put it out a day early.
But we could all be accused of that. We could proclaim this
International Microtonalists' Day.

Similarly with the theory. Read a lot more, experiment, and give it
another 20 years. Maybe then you'll have a book worth selling. For
now, your tuning is probably fine, so keep at it.

Graham

p.s. doesn't your formula predict that any interval written as 1/n
should be great for melody?

John,
 
I would like to print out your book The Mathematics of Music, but where can the link be found?
 
-- Mark Rankin

--- On Tue, 3/30/10, jfos777 <jfos777@...> wrote:

From: jfos777 <jfos777@...m>
Subject: [tuning] The Mathematics of Music book
To: tuning@yahoogroups.com
Date: Tuesday, March 30, 2010, 11:21 AM

Hi all, John here.

I have worked out a mathematical formula for the consonance value of a musical interval x/y (where x is less then or equal to y)...

(2 + 1/x + 1/y - diss)/2 where x<=y.

If x/y is between 0 and 0.9375 then 'diss' = x/y
if x/y is between 0.9375 and 1.0 then 'diss' = (1 - x/y)*15

If the result is 1.0 or higher then the interval is Major.
If the result is between 0.75 and 0.9999 then the interval is Minor.
If the reult is less than 0.75 then the interval is no good.

The formula is based on a few educated guesses but it seems to work consistently when tested.

How I worked it out is explained in my book "The Mathematics of Music" which you can download from the "Files" link on the left.

You can hear a tune I wrote and recorded using a "Just" guitar that I built myself at my web site... www.johnsmusic7. com

The web site has a few other bells and whistles as well: you can download a chord and scale dictionary for my "Just" guitar and for luthiers, a spec for fret placement is also available.

My 12-key tuning system is...
1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.
It is identical to the Just scale but the second note is 15/14 instead of 16/15. 3/2 can also be used as a tonic with equal strength.

Check out the book, any and all comments/feedback welcome.

John.

Mark,

to open/download the book click the "Files" link on the left of your screen and my book is near the bottom of the list of files. Alternatively, go to www.johnsmusic7.com and scroll down towards the bottom of the page where you will find a link to my book.

John.

--- In tuning@...m, Mark Rankin <markrankin95511@...> wrote:
>
> John,
>  
> I would like to print out your book The Mathematics of Music, but where can the link be found?
>  
> -- Mark Rankin
>
> --- On Tue, 3/30/10, jfos777 <jfos777@...> wrote:
>
>
> From: jfos777 <jfos777@...>
> Subject: [tuning] The Mathematics of Music book
> To: tuning@yahoogroups.com
> Date: Tuesday, March 30, 2010, 11:21 AM
>
>
>  
>
>
>
> Hi all, John here.
>
> I have worked out a mathematical formula for the consonance value of a musical interval x/y (where x is less then or equal to y)...
>
> (2 + 1/x + 1/y - diss)/2 where x<=y.
>
> If x/y is between 0 and 0.9375 then 'diss' = x/y
> if x/y is between 0.9375 and 1.0 then 'diss' = (1 - x/y)*15
>
> If the result is 1.0 or higher then the interval is Major.
> If the result is between 0.75 and 0.9999 then the interval is Minor.
> If the reult is less than 0.75 then the interval is no good.
>
> The formula is based on a few educated guesses but it seems to work consistently when tested.
>
> How I worked it out is explained in my book "The Mathematics of Music" which you can download from the "Files" link on the left.
>
> You can hear a tune I wrote and recorded using a "Just" guitar that I built myself at my web site... www.johnsmusic7. com
>
> The web site has a few other bells and whistles as well: you can download a chord and scale dictionary for my "Just" guitar and for luthiers, a spec for fret placement is also available.
>
> My 12-key tuning system is...
> 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.
> It is identical to the Just scale but the second note is 15/14 instead of 16/15. 3/2 can also be used as a tonic with equal strength.
>
> Check out the book, any and all comments/feedback welcome.
>
> John.
>

Michael,

go to the "Files" link on the left of your screen and you should find the book (and a tune) there. If you can't download it from here go to www.johnsmusic7.com and scroll down towards the bottom of the page and click the appropriate link.

John.

>"My formula fro the strength of a
harmonic (two notes played simultaneously) is:(2 + 1/x + 1/y
- diss) / 2"

Here it seems (for even 6/5) is: (2 + 1/6 + 1/5 - (2/6 + 2/5) )/ 2 = about 1.62...makes sense. Meanwhile 12/11 is (2 + 1/12 + 1/11 = 0.72 (a tad less than 0.75)...so I'd guess your "threshold of harmonic acceptability" is around 11/10. This sounds about right to me...independent tests I've run put the smallest interval for "harmonic acceptability" around 11/10-12/11.

*********************************
>"1/1
15/14
9/8 6/5
5/4 4/3 7/5
3/2
8/5 5/3 9/5
15/8
2/1"

The only thing here that really jumps out at me is so many notes are x/5 o-tonal and others are x/8 o-tonal....and the 15/14 doesn't match o-tonally with anything except 3/2 which also = 21/14. It makes it look on the surface like the chords best optimized are those starting from the root (IE 1/1 9/8 5/4 3/2 15/8 OR 1/1 6/5 7/5 8/5 9/5), but not other notes.

Michael,

you've read the formula`wrong. First of all I write all intervals so that the x (numerator) is equal to or less than y (denominator). So with the 5/6 or 6/5 interval, whichever way you want to write it, then the 'diss' = 5/6 and not 6/5. Most intervals are wider than 15/16 so the formula could be simply written:

(2 + 1/x + 1/y - x/y)/2 ! remember x<=y !

If the interval is narrower than 15/16 then

'diss' = (1 - x/y)*15 ! again x<=y !

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"My formula fro the strength of a
> harmonic (two notes played simultaneously) is:(2 + 1/x + 1/y
> - diss) / 2"
>
> Here it seems (for even 6/5) is: (2 + 1/6 + 1/5 - (2/6 + 2/5) )/ 2 = about 1.62...makes sense. Meanwhile 12/11 is (2 + 1/12 + 1/11 = 0.72 (a tad less than 0.75)...so I'd guess your "threshold of harmonic acceptability" is around 11/10. This sounds about right to me...independent tests I've run put the smallest interval for "harmonic acceptability" around 11/10-12/11.
>
> *********************************
> >"1/1
> 15/14
> 9/8 6/5
> 5/4 4/3 7/5
> 3/2
> 8/5 5/3 9/5
> 15/8
> 2/1"
>
> The only thing here that really jumps out at me is so many notes are x/5 o-tonal and others are x/8 o-tonal....and the 15/14 doesn't match o-tonally with anything except 3/2 which also = 21/14. It makes it look on the surface like the chords best optimized are those starting from the root (IE 1/1 9/8 5/4 3/2 15/8 OR 1/1 6/5 7/5 8/5 9/5), but not other notes.
>

Jihn>"you've read the formula`wrong. First of all I write all intervals so
that the x (numerator) is equal to or less than y (denominator) . So
with the 5/6 or 6/5 interval, whichever way you want to write it, then
the 'diss' = 5/6 and not 6/5. Most intervals are wider than 15/16 so the formula could be simply written: "

(2 + 1/x + 1/y - x/y)/2 ! remember x<=y

Ah, ok....but then (in that case):

(2 + 1/5 + 1/6 - (5/6)) / 2 = 0.76666666666666.
So the Just Minor 3rd would be the about the smallest "legal" interval, correct?

Michael,

if my formula is correct then yes, it would seem that 5/6 is the narrowest "legal" interval in harmony. I should point out at this stage, to you and to anyone else who has been reading my messages that I am not 100% sure that my formula is correct, it's a "best guess" after sifting through hundreds of various combinations and permutations. It seems reasonable when tested and I have spent months testing it. What I am fairly sure of though, for several reasons, is that my 12 key tuning system (NPT) is the "best" 12 key tuning system that has two tonics of equal strength, one tonic works better for ascending music and the other tonic for descending music.

A better tuning system has 16/15 as the second note instead of the 15/14 in my system. This system works better for music that ascends from and then descends to 1/1 (low tonic). However, for music that descends from, and then ascends to 2/1 (high tonic) this system is weaker. My NPT system has the best of both worlds with regard to ascending or descending music, both tonics (1/1 and 3/2) have equal strength, one better for ascending music and the other for descending music. The "Pan" in "Natural Pan Tuning" means "including everything".

It seems to me that it is impossible to have a "just" 12 key tuning system where both 1/1 and 2/1 can be used as the "key notes" or tonics with equal strength.

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Jihn>"you've read the formula`wrong. First of all I write all intervals so
> that the x (numerator) is equal to or less than y (denominator) . So
> with the 5/6 or 6/5 interval, whichever way you want to write it, then
> the 'diss' = 5/6 and not 6/5. Most intervals are wider than 15/16 so the formula could be simply written: "
>
> (2 + 1/x + 1/y - x/y)/2 ! remember x<=y
>
> Ah, ok....but then (in that case):
>
> (2 + 1/5 + 1/6 - (5/6)) / 2 = 0.76666666666666.
> So the Just Minor 3rd would be the about the smallest "legal" interval, correct?
>

>"A better tuning system has 16/15 as the second note instead of the
15/14 in my system. This system works better for music that ascends
from and then descends to 1/1 (low tonic)."
That's about what I was thinking...because then it harmonizes well with all your x/5 o-tonal tones.

>"However, for music that
descends from, and then ascends to 2/1 (high tonic) this system is
weaker."
Still trying to figure out why that would be since the octave can be summarized as 10/5 (IE still along the same harmonic series)...how can it be weaker?

>"both tonics (1/1 and 3/2) have equal
strength"
For 3/2 that makes better sense to me since 14 is 15/14 has the common factor of 2 with 3/2 and 15 in 16/15 does not.

>"The "Pan" in "Natural Pan Tuning" means "including everything".
I had faced a similar struggle in my latest 7-tone BSP (balanced super particular) scale. I thought 10/9 was the best tone since it shared an x/18 GCD with every note in the scale and had further spacing from the root than 13/12 (thus avoiding critical band dissonance issues). However later I realize it wasn't balanced since that relationship failed when building a 4-note chord between two octaves, so I replaced it with 13/12 to make it match.
I tried my best to make "pan"/"balanced" between the "very simple" intervals of 1/1,4/3,3/2,5/3, and 2/1...all I can say (not sure if it helps you) is that trying to make things symmetrical about those intervals, and not just 1/1 or 2/1 really helped me fine-tune the scale.

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

>Of course, I cut my teeth on some of Partch's guitars

Did it hurt?

Ah, we so missed the notorious clashes between Jon Stanzo and Gene
Ward Smith. Welcome back Gene!

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Apr 6, 2010, at 8:06 AM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, "jonszanto" <jszanto@...> wrote:
>
>> Of course, I cut my teeth on some of Partch's guitars
>
> Did it hurt?
>
>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "jonszanto" <jszanto@> wrote:
>
> >Of course, I cut my teeth on some of Partch's guitars
>
> Did it hurt?

Nope. I was already brandy'd up! The high 'e' strings are great for getting pieces of mint out from between your teeth. The juleps are worth the trouble.

Very good to see you post, Gene. Get back in here and straighten people out! :)

Cheers,
Jon

--- In tuning@yahoogroups.com, "jfos777" <jfos777@...> wrote:

What I am fairly sure of though, for several reasons, is that my 12 key tuning system (NPT) is the "best" 12 key tuning system that has two tonics of equal strength, one tonic works better for ascending music and the other tonic for descending music.

In order to make that claim meaningful, you need to define "best". Just giving your formula will not suffice, and in any case you've suggested you will allow close approximations to just intervals, which changes the nature of the game.

Two questions to start out with are whether you are trying to maximize the number of intervals or chords in some set of chords, and how close to just an interval must be. Does getting within 256/255 suffice?

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

> Very good to see you post, Gene. Get back in here and straighten people out! :)

Now that I can see again I may even compose, but please don't take that as a threat.

Gene (to John)>"In order to make that claim meaningful, you need to define "best". Just
giving your formula will not suffice, and in any case you've suggested
you will allow close approximations to just intervals, which changes the nature of the game."

John and Gene,
I would define "best" by how many notes in how many chords are possible while sounding "sweet/relaxed" to the average listener (as shown in a random survey of, say, 20 people).
For example, a major 7th and minor 7th chord would add 8 notes to the total. One easy way to get a list of most of the chords that satisfy JI fairly well is simply to do an automatic chord count in Scala...but for more non-standard chords IE 15:17:20, you are on your own to find them.

>"Two questions to start out with are whether you are trying to maximize
the number of intervals or chords in some set of chords, and how close
to just an interval must be."
Again, I'd say chords here are the frontier...more or more unique intervals may be interesting to some people...but I'd be hard pressed to find a musician who wouldn't love having more chords accessible.

The interesting part is that enabling virtually all "common theory" chords to sound decent is more-or-less already accomplished by 12TET. So "beat" 12TET IMVHO one would have to enable chords with intervals NOT possible in 12TET. Things like intervals set between the major and minor second or major and minor third far enough apart from the JI-standard versions of those chords that they have their own character and yet still feel resolved and consonant enough to sound confident.
-----------------------------------------------------------------------------
The most obvious "flaw" in 12TET, IMVHO, is that the semi-tone is not usable in most chords while the whole tone is and it begs for a scale with some sort of interval between the two that can be used more freely in chords thus enabling more chords. Yes, 12TET is obviously less pure than JI, but not so much that many untrained listeners would notice without listening hard for the difference.

John,
In your scale the interval between 9/5 and 5/3 is around 1.08....making it a not-so-bad tone for chords and still better than the usual JI 15/14=1.07 and certainly better than 12TET's tiny (and somewhat useless in chords, IMVHO) 1.06 ratio semitone. You also have a couple of 1.07-ish ratio semi-tones in there, but (at least to my ear) those seem a bit too close for chord use...unless you are going for 13th chords of something where the semi-tones are spread out across multiple octaves.

But other than that....my main question is what new "magical" intervals in your scale supposedly enable many new chords that aren't already possible (or possible in variations not noticeably different to the average listener)?

________________________________
From: genewardsmith <genewardsmith@...>
To: tuning@yahoogroups.com
Sent: Tue, April 6, 2010 1:08:50 AM
Subject: [tuning] Re: (2 + 1/x + 1/y - diss)/2

--- In tuning@yahoogroups. com, "jfos777" <jfos777@... > wrote:

What I am fairly sure of though, for several reasons, is that my 12 key tuning system (NPT) is the "best" 12 key tuning system that has two tonics of equal strength, one tonic works better for ascending music and the other tonic for descending music.

In order to make that claim meaningful, you need to define "best". Just giving your formula will not suffice, and in any case you've suggested you will allow close approximations to just intervals, which changes the nature of the game.

Two questions to start out with are whether you are trying to maximize the number of intervals or chords in some set of chords, and how close to just an interval must be. Does getting within 256/255 suffice?

Gene,

why I think my system is the best is outlined in chapter 7 of my book which can be found in "Files" on the left of your screen.

With regard to how close to "just" an interval should be, I'm going for perfection or bust, perfection meaning that the fundamentals of two notes in an interval correspond exactly to a simple integer ratio and the "value" of the interval (in harmony) is 0.75 or higher according to my formula. My idea on the 6.78 cents or 255/256 tolerance was an afterthought.

When working out my 12 key scale I identified all of the "obvious" notes that "go" with 1/1. In most cases, for each key, I chose the strongest notes that go with 1/1 but I exchanged 8/7 for 9/8 and 7/4 for 9/5 because (i) the intervals containing a seven "go" with relatively few other notes and (ii) with the inverse temperament (tonic is 3/2) 9/8 goes better with 3/2 (3/4) than 8/7, and 9/5 goes better with 3/2 (5/6, a "good" harmonic interval) than 7/4.
Also 9/8 goes better with 3/2 than 7/6.

To all, the answers to a lot of the questions I've been asked can be found in my book. At the very least you should read chapters 4, 6 ,7, 10 (chords) and 11 (scales) before asking me a question. The answers to many of your questions are probably in the book.

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "jfos777" <jfos777@> wrote:
>
> What I am fairly sure of though, for several reasons, is that my 12 key tuning system (NPT) is the "best" 12 key tuning system that has two tonics of equal strength, one tonic works better for ascending music and the other tonic for descending music.
>
> In order to make that claim meaningful, you need to define "best". Just giving your formula will not suffice, and in any case you've suggested you will allow close approximations to just intervals, which changes the nature of the game.
>
> Two questions to start out with are whether you are trying to maximize the number of intervals or chords in some set of chords, and how close to just an interval must be. Does getting within 256/255 suffice?
>

Michael,

the following "good" harmonic intervals (and how much they differ, in cents, from Equal Temperament) occur in my system. Note that many of these intervals do not occur among intervals that contain either tonic (1/1 or 3/2)...

5/7 17.5c
5/9 17.6c
9/14 35.1c
4/7 31.2c
15/28 19.4c
21/40 15.5c
14/25 3.8c
7/12 33.1c
25/48 29.3c
16/27 5.9c
7/10 17.5c
27/50 33.2c

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gene (to John)>"In order to make that claim meaningful, you need to define "best". Just
> giving your formula will not suffice, and in any case you've suggested
> you will allow close approximations to just intervals, which changes the nature of the game."
>
>
> John and Gene,
> I would define "best" by how many notes in how many chords are possible while sounding "sweet/relaxed" to the average listener (as shown in a random survey of, say, 20 people).
> For example, a major 7th and minor 7th chord would add 8 notes to the total. One easy way to get a list of most of the chords that satisfy JI fairly well is simply to do an automatic chord count in Scala...but for more non-standard chords IE 15:17:20, you are on your own to find them.
>
> >"Two questions to start out with are whether you are trying to maximize
> the number of intervals or chords in some set of chords, and how close
> to just an interval must be."
> Again, I'd say chords here are the frontier...more or more unique intervals may be interesting to some people...but I'd be hard pressed to find a musician who wouldn't love having more chords accessible.
>
> The interesting part is that enabling virtually all "common theory" chords to sound decent is more-or-less already accomplished by 12TET. So "beat" 12TET IMVHO one would have to enable chords with intervals NOT possible in 12TET. Things like intervals set between the major and minor second or major and minor third far enough apart from the JI-standard versions of those chords that they have their own character and yet still feel resolved and consonant enough to sound confident.
> -----------------------------------------------------------------------------
> The most obvious "flaw" in 12TET, IMVHO, is that the semi-tone is not usable in most chords while the whole tone is and it begs for a scale with some sort of interval between the two that can be used more freely in chords thus enabling more chords. Yes, 12TET is obviously less pure than JI, but not so much that many untrained listeners would notice without listening hard for the difference.
>
> John,
> In your scale the interval between 9/5 and 5/3 is around 1.08....making it a not-so-bad tone for chords and still better than the usual JI 15/14=1.07 and certainly better than 12TET's tiny (and somewhat useless in chords, IMVHO) 1.06 ratio semitone. You also have a couple of 1.07-ish ratio semi-tones in there, but (at least to my ear) those seem a bit too close for chord use...unless you are going for 13th chords of something where the semi-tones are spread out across multiple octaves.
>
> But other than that....my main question is what new "magical" intervals in your scale supposedly enable many new chords that aren't already possible (or possible in variations not noticeably different to the average listener)?
>
>
>
>
>
>
> ________________________________
> From: genewardsmith <genewardsmith@...>
> To: tuning@yahoogroups.com
> Sent: Tue, April 6, 2010 1:08:50 AM
> Subject: [tuning] Re: (2 + 1/x + 1/y - diss)/2
>
>
>
>
> --- In tuning@yahoogroups. com, "jfos777" <jfos777@ > wrote:
>
> What I am fairly sure of though, for several reasons, is that my 12 key tuning system (NPT) is the "best" 12 key tuning system that has two tonics of equal strength, one tonic works better for ascending music and the other tonic for descending music.
>
> In order to make that claim meaningful, you need to define "best". Just giving your formula will not suffice, and in any case you've suggested you will allow close approximations to just intervals, which changes the nature of the game.
>
> Two questions to start out with are whether you are trying to maximize the number of intervals or chords in some set of chords, and how close to just an interval must be. Does getting within 256/255 suffice?
>

Especially wondering how this relates to the works of (or compared to the works of) Marcel, John, and others intent on making a replacement for 12TET using virtually all Just intervals....

I just noticed that many of harmonics 12-24 in the harmonic series match almost perfectly with 12TET...and the only tones that really get in any way "scrambled" in that is the first half step IE the ratio note of 1.05999 and the next note of 1.12246.

Using 25/24 to replace that gives the perhaps overly narrow 1.046666 semi-tone. Other than that you get a bunch of near matches of 12TET that all fit the x/12 harmonic series, making a more-or-less "perfectly just" scale of

25/24 = 1.0416666
13/12 = 1.0833333
14/12 = 1.1666666
15/12 = 1.25
16/12 = 1.3333333
17/12 = 1.4166666
18/12 = 1.5
20/12 = 1.6666666
21/12 = 1.75
23/12 = 1.9166666
24/12 = 2 (octave)

Just wondering what research has been done on this (surely there has been some....it's such an obvious scale)?

Have you actually looked at the intervals within the scale? Some of the "major thirds" are wider than perfect fourths.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Especially wondering how this relates to the works of (or compared to the works of) Marcel, John, and others intent on making a replacement for 12TET using virtually all Just intervals....
>
> I just noticed that many of harmonics 12-24 in the harmonic series match almost perfectly with 12TET...and the only tones that really get in any way "scrambled" in that is the first half step IE the ratio note of 1.05999 and the next note of 1.12246.
>
> Using 25/24 to replace that gives the perhaps overly narrow 1.046666 semi-tone. Other than that you get a bunch of near matches of 12TET that all fit the x/12 harmonic series, making a more-or-less "perfectly just" scale of
>
> 25/24 = 1.0416666
> 13/12 = 1.0833333
> 14/12 = 1.1666666
> 15/12 = 1.25
> 16/12 = 1.3333333
> 17/12 = 1.4166666
> 18/12 = 1.5
> 20/12 = 1.6666666
> 21/12 = 1.75
> 23/12 = 1.9166666
> 24/12 = 2 (octave)
>
> Just wondering what research has been done on this (surely there has been some....it's such an obvious scale)?
>

Me> I just noticed that many of harmonics 12-24 in the harmonic
series match almost perfectly with 12TET...and the only tones that
really get in any way "scrambled" in that is the first half step IE the
ratio note of 1.05999 and the next note of 1.12246.

Tony>"Have you actually looked at the intervals within the scale? Some of the "major thirds" are wider than perfect fourths."
Not every single combination, obviously, but I did find most of the intervals starting from the "warped" area I discussed before....the area around 25/24 and 13/12 (which stand-in for 1.05999 and 1.12246 in 12TET) still sound much like the original intervals in general.
For example, from D (13/12) to F# (17/12) sounds a bit different and "lower" than in 12TET, but still sounds just as harmonic IE I couldn't find a way to key in a chord-fingering from 12TET that doesn't also produce an equally sweet-sounding chord in this system. Practical summary: 12TET theory appears to work hear and modulation appears to also work in a similar fashion.

BTW...where (exactly) are the "thirds which are wider than 4ths" you are referencing? I saw a few 3rds which approach the width of 4ths, but none which go past them.

Oops, previous message was a mistake...

(23:12):(17:12) gives a "major third" of 23:17, or 436 millioctaves.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Me> I just noticed that many of harmonics 12-24 in the harmonic
> series match almost perfectly with 12TET...and the only tones that
> really get in any way "scrambled" in that is the first half step IE the
> ratio note of 1.05999 and the next note of 1.12246.
>
>
> Tony>"Have you actually looked at the intervals within the scale? Some of the "major thirds" are wider than perfect fourths."
> Not every single combination, obviously, but I did find most of the intervals starting from the "warped" area I discussed before....the area around 25/24 and 13/12 (which stand-in for 1.05999 and 1.12246 in 12TET) still sound much like the original intervals in general.
> For example, from D (13/12) to F# (17/12) sounds a bit different and "lower" than in 12TET, but still sounds just as harmonic IE I couldn't find a way to key in a chord-fingering from 12TET that doesn't also produce an equally sweet-sounding chord in this system. Practical summary: 12TET theory appears to work hear and modulation appears to also work in a similar fashion.
>
> BTW...where (exactly) are the "thirds which are wider than 4ths" you are referencing? I saw a few 3rds which approach the width of 4ths, but none which go past them.
>

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

> Now that I can see again I may even compose, but please don't
> take that as a threat.

Hi Gene, great to see you here. If you're looking for inspiration,
try this radio show:

http://lumma.org/stuff/TheArchMusician.mp3

I found both the history and the music intoxicating. YMMV.

-Carl

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

> John and Gene,
> I would define "best" by how many notes in how many chords are possible while sounding "sweet/relaxed" to the average listener (as shown in a random survey of, say, 20 people).

That pretty well confines you to the 5-limit. 7-limit JI is more in the nature of bluesy or jazzy in a general sense.

> Again, I'd say chords here are the frontier...more or more unique intervals may be interesting to some people...but I'd be hard pressed to find a musician who wouldn't love having more chords accessible.

If you are dealing with 12 notes to the octave, aside from equal temperament there is at least also meantone, circulating temperaments, and tempering out 225/224 to consider; the last is for people who want near JI results and deserves more attention than it seems to get, I think.

If we take the Ellis Duodene

! duodene.scl
!
Ellis's Duodene : genus [33355]
12
!
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2

and temper it by (for example) 72 equal, we get 7, 12, 19, 23, 30, 35, 42, 49, 53, 61, 65, 72. If we take another theoretically interesting scale, the 7-limit dwarf:

! dwarf12_7.scl
Dwarf(<12 19 28 34|) five major triads, four minor triads two otonal pentads
12
!
16/15
9/8
6/5
5/4
4/3
7/5
3/2
8/5
5/3
9/5
28/15
2

And temper it in the same way, we get exactly the same scale. But wait, there's more! If you take John's scale and temper out 225/224, for instance by using 72 equal, the same scale turns up yet again. Now how much would you pay?

This is the scale I call the 12 note, 7-limit marvelous dwarf, and it has among other attributes four complete pentads, all with errors well within 256/255.

> The interesting part is that enabling virtually all "common theory" chords to sound decent is more-or-less already accomplished by 12TET.

Playing Palestrina in 12et is a vile idea which totally misses the point of the music. Even rock music is not immune to the problem, consider for instance:

http://www.mediafire.com/?z2hcwvj2mei

Do you think this would sound as good in 12et? Vocal harmony tends to be either simply bad and out of tune, or with talented performers, not very close to 12et.

>So "beat" 12TET IMVHO one would have to enable chords with intervals >NOT possible in 12TET.

As for example, 7-limit chords?

Thanks Carl.

On Tue, Apr 6, 2010 at 6:07 PM, Carl Lumma <carl@...> wrote:

>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "genewardsmith"
> <genewardsmith@...> wrote:
>
> > Now that I can see again I may even compose, but please don't
> > take that as a threat.
>
> Hi Gene, great to see you here. If you're looking for inspiration,
> try this radio show:
>
> http://lumma.org/stuff/TheArchMusician.mp3
>
> I found both the history and the music intoxicating. YMMV.
>
> -Carl
>
>
>
>

Me> John and Gene, I would define "best" by how many notes in how many chords are
possible while sounding "sweet/relaxed" to the average listener (as
shown in a random survey of, say, 20 people).

Gene>"That pretty well confines you to the 5-limit. 7-limit JI is more in the nature of bluesy or jazzy in a general sense."
While I agree 5-limit is a good "sure fire" metric to use to get there...I strongly believe there are many other ways to arrive there, especially when you consider the possibility that periodicity is not the only determinant of consonance. Otherwise, there would be very little justifiable push to bring popular music beyond 5-limit...right?
For example, if you take the "Ptolemy Homalon" scale of of 18:20:22:24:26:30:33:36...I'm pretty sure it's 11 "odd" limit...yet many people I've shown the scales (blind tested) to say it sounds just as consonant as "5-limit" JI, just different in that many historically backed 5-limit intervals don't work in it.

>"If you are dealing with 12 notes to the octave, aside from equal
temperament there is at least also meantone, circulating temperaments,
and tempering out 225/224 to consider;"
I'm interested in those last two as I know little about them...unless, for example, anything with a "circle of xth's" (IE like meantone, but not limited to 5ths) counts as a circular temperament.

>"The last is for people who want
near JI results and deserves more attention than it seems to get, I
think."
I'm a huge fan of tempering within 7 cents or so of an octave...I often finds it provides a sense of "whole/together-ness" missing from un-tempered scales that highly favor certain root notes and/or chord types while harming others.

>"If you take John's scale and temper out 225/224, for instance by using
72 equal, the same scale turns up yet again. Now how much would you pay?"
Interesting, so that would imply many of these experimental JI scales merge toward a tempered scale that combines the best of all their qualities?

>"This is the scale I call the 12 note, 7-limit marvelous dwarf, and it
has among other attributes four complete pentads, all with errors well
within 256/255."
I like the scale and (as I argued before)...nothing wrong with the so called lack-of-sweetness in 7-limit...in fact, that scale (so far as I just now played around with it) sounds "5-limit-sweet" to me. :-)

Me>>"The interesting part is that enabling virtually all "common
theory" chords to sound decent is more-or-less already accomplished by
12TET. "
>"Playing Palestrina in 12et is a vile idea which totally misses the
point of the music. Even rock music is not immune to the problem,
consider for instance: http://www.mediafire.com/?z2hcwvj2mei"...
Sadly, I barely think it would sound any different even in something like Adaptive JI to a musically untrained ear for almost all intervals (minus the major 3rd and the few other intervals 11+ cents off "just"). I do also assume it would make a difference in the special case of vocals and guitars since they match the harmonic series so accurately, but not so much for other instruments.

Could I see it marketed to musicians with a good ear and especially those with "perfect pitch"?...of course...but sadly ( I fear) not enough difference for the general public to care.

>So "beat" 12TET IMVHO one would have to enable chords with intervals NOT possible in 12TET. As for example, 7-limit chords?
Yes, but I would even go so far as 13-limit IE the 13/12 interval and chords like 9:10:12:13...anything closer than that (at least to me) begins to sound very sour in chords due to critical band issues. And I agree that anything with 7 cents or so of a "pure" interval ("pure" meaning for conforming to whatever limit you choose) is also fair game.

Hi Michael,

> Gene>"That pretty well confines you to the 5-limit. 7-limit JI is more in the nature of bluesy or jazzy in a general sense."
>    While I agree 5-limit is a good "sure fire" metric to use to get there...I strongly believe there are many other ways to arrive there, especially when you consider the possibility that periodicity is not the only determinant of consonance.  Otherwise, there would be very little justifiable push to bring popular music beyond 5-limit...right?
>      For example, if you take the "Ptolemy Homalon" scale of of 18:20:22:24:26:30:33:36...I'm pretty sure it's 11 "odd" limit...yet many people I've shown the scales (blind tested) to say it sounds just as consonant as "5-limit" JI, just different in that many historically backed 5-limit intervals don't work in it.

The way I see it, Gene is pretty much right about 5-limit.

There's some logic behind 5-limit. I'll try to explain how.

Take the harmonic series till the 5th harmonic.
1/1 2/1 3/1 4/1 5/1

The is the most basic of chords right?
And one wouldn't want to tune it any other way and have it still be as
"consonant".
.
And there's a basic rule for microtonal music according to many ears.
A held note does not change it pitch slightly, it does not "comma
shift". (it doesn't just sound bad, but there are other logical
problems with comma shifting held notes)

Now if we take this simple rule, and that basic 5th-harmonic-limit
major triad chord, and add one other simple thought:
Looking at the intervals making up the chord, and allow to rearange these.
This is a very musical thing. If one accepts 1/1 +2/1 +3/2, the it
seems logical that one also accepts 1/1 +3/2 +2/1.
If one accepts 1/1 +2/1 +3/2 +4/3, then it seems logical one also
accepts 1/1 +4/3 +3/2 +2/1 right?
Try playing it, it keeps the simplicity of the intervals and the
chords formed this way make perfect musical sense.
Why tune it any other way?

This thinking leads to many things.
One could rearange / permutate the intervals of the 1/1 2/1 3/1 4/1
5/1 (1/1 + 2/1 +3/2 +4/3 +5/4) chord.
For instance to 1/1 3/2 3/1 15/4 5/1 (1/1 + 3/2 +2/1 +5/4 +4/3), or
1/1 5/4 5/3 10/3 5/1 (1/1 +5/4 +4/3 +2/1 +3/2)

Now if one takes all permutations of this 5th-harmonic-limit 1/1 2/1
3/1 4/1 5/1 chord we get:
1/1 5/4 4/3 3/2 5/3 15/8 2/1 (when reduced to one octave)
After some experimenting, and some thoughts about the logic behind
music, and remembering the rule that held notes do not "comma shift"..
We can then say, hey it should be possible to hit every note in the
scale in combination with any other note of this scale.
So one should be able to hit 1/1 5/4 3/2 15/8, or 1/1 4/3 3/2 5/3 etc,
even though these chords are not a single permutation of the original
5th-harmonic-limit chord.
It is for instance because one can play one permutation, and then
another but for instance hold some notes of the previous permutation
chord.
So the main structure is still the permuation chords, but because of
these chords more complex chords can also arrise.
These chords do allways have one single tone in common though. The
1/1. This is the true "root" of the chord.

But uptill now, we've centered all the permutations of the 1/1 2/1 3/1
4/1 5/1 chord on 1/1.
Why not permutate this chord, and not have it have 1/1 in common with
the original chord, but have it have 2/1, or 3/1 or 4/1 in common?
(with 5/1 as the center of permutation we get the exact same results
as with 1/1 as the center)
So a permutation of 1/1 2/1 3/1 4/1 5/1 could also be 8/5 2/1 3/1 4/1
8/1 (1/1 + 5/4 +3/2 +4/3 +2/1 centered on 2/1) etc.
(note that when centering on 2/1, there's allways one interval below
and 3 above. When centering on 3/1 there's allways 2 intervals below
and 2 above. It would make no sense to shift the entire chord up or
something like that)
The above, non 1/1 centered permutations gives completely new
permutations (centered on 2/1, 3/1, 4/1. Note I say "centered", not
"rooted" )
These do not share the same 1/1 root. The root that gives the harmonic
series of 1/1 2/1 3/1 4/1 5/1.
However, they are related to the 1/1 2/1 3/1 4/1 5/1 chord directly.
If one combines all permutations of the 1/1 2/1 3/1 4/1 5/1 chord,
centered on all the intervals of this chord, one gets the following
scale:

1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1 (when reduced
to one octave)

Note, this is the 12 tone Just Intonation scale! :)
We've just showen that when following basic musical logic of the
harmonic series limited to the 5th harmonic will give this scale.
To change a single pitch in this 12-tone scale will limit your musical
possiblities and put things seriously out of tune.
However, we've also shown it's somewhat limited what one can do in this scale.
We need more pitches/tones!

Now to the next most logical step.
The harmonic series limited to the 6th harmonic.
1/1 2/1 3/1 4/1 5/1 6/1
All 6-limit harmonic permutations centered on 1/1 will give the
following scale (when reduced to one octave):
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
(note that this happends to be the major and minor just scales laid on
top of eachother)

And when we combine all permutation of the 6th-harmonic-limit chord,
centered on the root + each of its harmonics (till the 6th harmonic),
we get the following scale: (reduced to one octave)

1/1 25/14 16/15 10/9 9/8 6/5 5/4 4/3 27/20 25/18 45/32 3/2 25/16 8/5
5/3 27/16 16/9 9/5 15/8 2/1

This 19-tone scale isn't very limiting anymore.
Note that all these pitches are directly related to the original 1/1
2/1 3/1 4/1 5/1 6/1 chord.
And that even though one finds intervals like 81/80 between for
instance 10/9 and 9/8, these cannot happen in one chord, as the "chord
model" is 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 (for chords
there has to be a single 1/1 root for the permutations).
And that the permutations of the original 6-limit chord will tell us
perfectly every single possible way the chords can move, and that not
a single one of these movements will ever give a "comma shift" in any
of it's tones.

Now to go yet further, lets take the harmonic series till the 7th harmonic:
1/1 2/1 3/1 4/1 5/1 6/1 7/1
All 7-limit permutations centered on 1/1 will give the following scale
(when reduced to one octave):
1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9 8/5 5/3
7/4 9/5 28/15 15/8 35/18 2/1
There are 20 tones per octave in this scale.
I've yet to calculate the scale that results from all the 7-limit
permutations centered on all of the harmonics of the 7-limit chord,
but I can tell you the resulting scale will be very huge...

Now what does this all tell us?
That the above examples give the scales with the maximum musical
possibilities in pure tuning (progresseing from most simple to more
complex).
So Michael, I think this shows your quest to find the "best scale" for
JI consonance is either over because it has been described above in my
post, or that my above post shows that what you're looking for can't
be done in the way you think and that the higher harmonics only have a
place in very huge scales (not in a 12tone JI scale).

Now for some deeper insights into these scales / this system, and some
of the questions I'm still working with:

In 5-limit harmonic permutation:

I see 1/1 5/4 4/3 3/2 5/3 15/8 2/1 (reduced to one octave for
simplicity of writing) as the 5-limit "harmonic model".
That is, all chords can only be made up of a combination of these
tones in order to be a 5-limit chord.

I see 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1
(reduced to one octave for simplicity of writing) as the 5-limit
"tonality model".
The tonic beeing 1/1 (or more accurately, the tonic beeing 1/1 2/1 3/1
4/1 5/1 actually)

The 5-limit tonality has 2 harmonic model "roots". (that is 1/1 points
for the 1/1 5/4 4/3 3/2 5/3 15/8 2/1 harmonic model).
These root points are 1/1 and 3/2.
This is because every tone of the 5-limit (12-tone) tonality scale is
related to the tonic (1/1 2/1 3/1 4/1 5/1).
Therefore, only harmonies can be made within this tonality that both
adhere to the 5-limit tonality model, and include all the tones of the
tonic (original 1/1 2/1 3/1 4/1 5/1 chord).
On 1/1 this is the case with 1/1 5/4 4/3 3/2 5/3 15/8 2/1 harmonic model.
On 3/2 this is the case with 3/2 15/8 2/1 9/4 5/2 45/16 3/1
The other tones (16/15 6/5 45/32 8/5 9/5) cannot be reached in 5-limit
tonality with the 5-limit harmonic model.
One would have to either "5-limit modulate" to reach these tones, or
(much more likely and better way to see it) use the 6-limit harmonic
model to reach these.

Now in 6-limit things start to really take off :)

I see 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 (reduced to one
octave for simplicity of writing) as the 6-limit "harmonic model".
That is, all chords can only be made up of a combination of these
tones in order to be a 5-limit chord.

I see 1/1 25/14 16/15 10/9 9/8 6/5 5/4 4/3 27/20 25/18 45/32 3/2 25/16
8/5 5/3 27/16 16/9 9/5 15/8 2/1 (reduced to one octave for simplicity
of writing) as the 6-limit "tonality model".
The tonic beeing 1/1 (or more accurately, the tonic beeing a 1/1 2/1
3/1 4/1 5/1 6/1 chord)

The 6-limit tonality has 4 harmonic model "roots". (that is 1/1 points
for the 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 harmonic model)
These points are 1/1, 4/3, 3/2 and 5/3.
1/1>  1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
|
4/3>  1/1 16/15 10/9 6/5 5/4 4/3 3/2 8/5 5/3 16/9 2/1
|
3/2>  1/1 9/8 6/5 5/4 27/20 45/32 3/2 27/16 9/5 15/8 2/1
|
5/3>  1/1 25/24 10/9 5/4 4/3 25/18 3/2 25/16 5/3 15/8 2/1
|

What's nice is that with the 6-limit harmonic model on these points we
can reach every single tone of the 6-limit tonality model :)

I'm not sure yet if one can also see a tone as a harmonic model root
when not all of the harmonic model's tones are found in the tonality
scale relevant to the root.
For instance with 5/4 as harmonic root we get the following:
5/4> 1/1 25/24 9/8 (75/64) 5/4 45/32 3/2 25/16 5/3 15/8 2/1
As you can see, 75/64 is not an interval of the 6-limit tonality
model, so I'm guessing it's not so wise to see 5/4 as a true 6-limit
harmonic root.

Anyhow many many more beautifull things in this system.
It's pure art. Wish I could take credit for inventing it, but I think
I can only take credit for finding it :)

What I've further found is that in common practice music, only a small
portion of the most simple music will fit the 5-limit harmonic model
and 5-limit tonality model.
A slightly larger, but still small section of music will fit the
5-limit tonality model but with 6-limit harmonic model.
And it seems to me all common practice music will fit the 6-limit
tonality model with 6-limit harmonic model! (though offcourse for some
music this will involve modulations, chance of tonic, but this isn't a
problem at all. And this happends much less than one would think I
think. For instance beethovens drei equale is in one tonic as a other
pieces I've looked at so far by bach and beethoven)
So, as far as I'm concerned. Normal common practice / tonal music as
we know is 6-limit.
And looking at the 6-limit tonality and harmonic models, 6-limit seems
far from depleted to me.

Ok pfew this was a long mail to write :)
I'm going to save it and put it online lol

Anyhow, I hope you can see the logic in my thinking and agree.
But even if not, best of luck with your tuning :)

Marcel
www.develde.net

On 7 April 2010 06:06, Michael <djtrancendance@...> wrote:

>      For example, if you take the "Ptolemy Homalon" scale of of
> 18:20:22:24:26:30:33:36...I'm pretty sure it's 11 "odd" limit...yet
> many people I've shown the scales (blind tested) to say it sounds
> just as consonant as "5-limit" JI, just different in that many
> historically backed 5-limit intervals don't work in it.

As it has a 26 in it, it must be at least 13-limit. The interval
26:33 can't be reduced, so it isn't even a 13-limit chord. You could
call it a scale for 13-limit music.

Gene:
> >"If you are dealing with 12 notes to the octave, aside from
> > equal temperament there is at least also meantone,
> > circulating temperaments, and tempering out 225/224 to consider;"

Michael:
>     I'm interested in those last two as I know little about them...
> unless, for example, anything with a "circle of xth's" (IE like
> meantone, but not limited to 5ths) counts as a circular temperament.

A circulating temperament is one where each key should be playable.
Historically, these have 12 notes to the octave.

Tempering out 225/224 gives a rank 3 temperament. That is, you have
the same number of distinct intervals as 5-limit JI, but some of them
also have a 7-limit interpretation. It gets mentioned here sometimes.
You can call it "marvel" if you don't like the numbers.

One way of thinking about it is that it allows augmented triads with
two intervals of 5:4 and one of 9:7 to add up to a 2:1 octave. You
can then split that 9:7 to give 7:8:9 so that both 5:4 and 7:4 are
relative to the tonic. Somebody mentioned that chord recently.

Another way of defining it: 16:15 and 15:14 are identical.

>    I'm a huge fan of tempering within 7 cents or so of an octave...
> I often finds it provides a sense of "whole/together-ness" missing
> from un-tempered scales that highly favor certain root notes and/or
> chord types while harming others.

What does "within 7 cents or so of an octave" mean?

> >"If you take John's scale and temper out 225/224, for instance
> by using 72 equal, the same scale turns up yet again. Now how
> much would you pay?"
>     Interesting, so that would imply many of these experimental
> JI scales merge toward a tempered scale that combines the best
> of all their qualities?

As the choice between 16:15 and 15:14 is the main arbitrary part of
John's scale, then yes, different scales will come out the same when
you ignore this distinction.

>    Sadly, I barely think it would sound any different even in
> something like Adaptive JI to a musically untrained ear for
> almost all intervals (minus the major 3rd and the few other
> intervals 11+ cents off "just").  I do also assume it would
> make a difference in the special case of vocals and guitars
> since they match the harmonic series so accurately, but
> not so much for other instruments.

Palestrina wrote for voices, and used a lot of major thirds.

Graham

>"And there's a basic rule for microtonal music according to many ears.A held note does not change it pitch slightly, it does not "comma
shift". (it doesn't just sound bad, but there are other logical problems with comma shifting held notes)"
I figure the periodicity would become slightly longer/less-desirable. But I don't see how comma-shifting is so bad...especially when you consider that most instruments themselves fluctuate in tone slightly (thus making the overtones themselves already "shifted" to some extent). The other issue is periodicity buzz...especially with chords using tones of 9/8 interval or smaller it becomes almost grating despite being "in strict JI".

>"The is the most basic of chords right? And one wouldn't want to tune it any other way and have it still be as "consonant".
Right, but the expression is IMVHO fairly limited and, I'm betting, most people could digest much higher-limit chords IF they don't violate the critical band so much AKA have many overly close notes.
Take, for example, ways to summarize the 12TET chord C E F A (which, mostly due to the fact the "overly close" E-F are surrounded by compensating wide intervals, I can show to many people without them saying it's "too tense")...I'm pretty sure you'll have a tough time doing that in strict 5-limit.

Simply put, I think higher limits can be used with fairly minimal losses of consonance IF people are careful about not putting too many notes close together...and same goes with tempering of "pure" chords (too much temperament on too many notes in a row in a chord can distort the sense of wholeness of the series, but some on occasional notes I find is harmless).

>"Why tune it any other way?"
AKA why would I want to bother with higher limits in the first place? Many many more chords and emotional possibilities...it's like comparing the musical of someone who knows Debussy's progressions to someone who knows standard pop progressions.
---------------------------------------------------
>"Now if one takes all permutations of this 5th-harmonic- limit 1/1 2/1
3/1 4/1 5/1 chord we get:
1/1 5/4 4/3 3/2 5/3 15/8 2/1 (when reduced to one octave)"
Where is the 15/8 calculated from (maybe 3/2 * 5/4)?

>"1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1 (when reduced
to one octave) Note, this is the 12 tone Just Intonation scale! :)"
So you are pretty much using the above ratios and multiplying them by each other to get new ones?
------------------------------------------------------------------------------
>"And when we combine all permutation of the 6th-harmonic- limit chord,
centered on the root + each of its harmonics (till the 6th harmonic),
we get the following scale: (reduced to one octave)
1/1 25/14 16/15 10/9 9/8 6/5 5/4 4/3 27/20 25/18 45/32 3/2 25/16 8/5
5/3 27/16 16/9 9/5 15/8 2/1"

Now this looks a lot more interesting, minus the fact so many keys would be needed for it. The only real issue I see is there are so many different (at least at first glance) common LCD's that it would take a good deal of theory to find all the sweet spots (thus making it harder to catch on among 12TET musicians). For example 16/15 and 9/8 near having an LCD at 18 (the interval formed is around 19/18 AKA is a "tempered 19/18") while 25/18 and 4/3 give an LCD of 24 (25/24).

The way I see it (please tell me if you find a loop-hole in this) is that even 7-tone JI...the common denominator of all fractions lies somewhere around 24 with the clearer chords having LCDs of mostly 12, 3,6, and 4...and there are way to cut through and find those chords, but it takes a lot of theory. For example, the chord 1/1 10/9 4/3 25/18.

Here is a challenge: find a 7-tone scale within that 19-tone scale that has many 4 note chords which sound as or more consonant than C E F A does in 12TET. I agree with you the 19-tone is a flexible enough scale, but the other question becomes can we push it to the average musician without them thinking it's far too hard to learn?
-----------------------
>"or that my above post shows that what you're looking for can't be done in the way you think and that the higher harmonics only have a place in very huge scales (not in a 12tone JI scale)."
Very huge scales? I have my doubts...although I see that using your method it requires a lot of notes before such a scale can be formed (though I realize your method could be of great use to those who require scales that allow transposition).
Again, using even just Ptolemy's Homalon ratios (completely unmodified be myself) can give a 7-tone scale of 18:20:22:24:27:30:33:36 which is loaded with very consonant chords of over 5-limit. And again I'll point out, even the 7-tone JI diatonic rounds down to x/24 as it can be written in lowest form as 24:27:30:32:36:40:45:48 (yes, that means it has a higher LCD/lower-full-scale-periodicity than the Ptolemy-based scale)!

I think the rift between most JI theory I've seen (in many cases, including yours) is and what I'm trying to do is

A) People appear to shy away from using close ratios like 11/10, 12/11, and 13/12 or ones anywhere near quarter tones when they decide what chords to use in their scales, presumably because these intervals just "don't fit common practice"...and indirectly assume they must be "bad" because they are "wrong" in common practice.
Trying even the basic Homalon-based interval scale (which uses both 11/10 and 12/11) and you'll find plenty of chords that can be made with the above intervals. Again I'll bring up the 12TET C E F A example...there is often nothing terribly wrong with chords that use close spaces or "have" to be summarized at higher limits.

B) So many people making JI scales say things like "2/1,3/2,4/3,5/4 are the most consonant ratios because they are the most periodic ones" and ignore the fact they also succeed much because they are among the intervals that avoid critical band dissonance the best since they are so far apart.
I've seen this causes people to avoid chords like 10:13:16 because they jump to the conclusion it must be bad because it's relatively high-limit, but they forget there is a fair amount of good in the form of good critical band dissonance avoidance (which it shares in common with high-spaced ratios like 3:4) balancing it out and, IMVHO, making it quite listenable regardless of "limit".
True, there is a point where periodicity gets so bad chords using such periodicity will sound bad matter how far apart the tones are and how well they "take critical band into account"...but I think that limit is around odd-limit 11 or 13....and certainly not stuck at 5.

>"And it seems to me all common practice music will fit the 6-limit
tonality model with 6-limit harmonic model! "
I'll say this much...I hope you continue looking into "6-limit"...you previous theories seem to just polish existing musical possibilities (though pretty well!)...but I'm really look forward to seeing you open the door to never-before-heard chords and such with 6-limit. :-)

-Michael

Graham>"The interval 26:33 can't be reduced, so it isn't even a 13-limit chord"
Got it...I suppose you could modify it to 33/27, but that would make 27/22 and 27/20 non-reduce-able.
Meanwhile you could take 34/26 and just make the scale 9:10:11:12:13:15:17:18....but then you'd be stuck at 17-limit and get that nastily close 18/17 interval which screams "critical band dissonance".

Unless you have any smart hacks for this one...I think I'm going to have to stick to the idea that 9:10:11:12:13 is the only large 13-limit chord I can get using consecutive notes from this scale since 10:11:12:13:15 is 15-limit.

Then again compared to 7-tone JI diatonic IE 24:27:30:32:36:40:45:48...the one 5-note chord I can find is 24:27:30:36:45 AKA 8:9:10:12:15...which even at "only" 5 notes isn't even 13-limit.
Same goes for 24:30:32:36:40 AKA 12:15:16:18:20.
---------------
>"One way of thinking about it is that it allows augmented triads with two intervals of 5:4 and one of 9:7 to add up to a 2:1 octave. You can then split that 9:7 to give 7:8:9 so that both 5:4 and 7:4 are
relative to the tonic. Somebody mentioned that chord recently."
Right, 5/4 * 9/7 * 5/4 = just a tad over 2/1.
The result I get from breaking this into triads (though) is.....
1/1 (???) 5/4 (4/3) (3/2) 25/16 (25/14)
........now how do I get/obtain that 7th note?

>"Another way of defining it: 16:15 and 15:14 are identical."
Cool...I don't see that as a problem at all...in fact (at least to me) for anything above around 14/13 IE 14/13 vs. 15/14 I can barely tell a difference in "feel".

>"What does "within 7 cents or so of an octave" mean?"
Bad wording on my part...I meant to say 7 cents of the original JI tone.

>"Palestrina wrote for voices, and used a lot of major thirds."
Didn't know anything about Palestrina but, now that you've explain it, between the timbre of voices and 12TET's somewhat lousy ability to imitate true major thirds, I certainly agree 12TET would be notably worse than JI even to an untrained ear.

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

> If we take the Ellis Duodene
>
> ! duodene.scl
> !
> Ellis's Duodene : genus >[33355]
> 12
> !
> 16/15
> 9/8
> 6/5
> 5/4
> 4/3
> 45/32
> 3/2
> 8/5
> 5/3
> 9/5
> 15/8
> 2
>
> and temper it by (for example) 72 equal, we get 7, 12, 19, 23, 30, >35, 42, 49, 53, 61, 65, 72.

Are these: "7, 12, 19, 23, 30, >35, 42, 49, 53, 61, 65, 72" equal temperaments that can be used to approximate the scale you're referring to? Or the counts where you'd get an MOS?

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "genewardsmith" genewardsmith@ wrote:
>
> > If we take the Ellis Duodene
> >
> > ! duodene.scl
> > !
> > Ellis's Duodene : genus >[33355]
> > 12
> > !
> > 16/15
> > 9/8
> > 6/5
> > 5/4
> > 4/3
> > 45/32
> > 3/2
> > 8/5
> > 5/3
> > 9/5
> > 15/8
> > 2
> >
> > and temper it by (for example) 72 equal, we get 7, 12, 19, 23, 30,
>35, 42, 49, 53, 61, 65, 72.
>
> Are these: "7, 12, 19, 23, 30, >35, 42, 49, 53, 61, 65, 72" equal
temperaments that can be used to approximate the scale you're referring
to? Or the counts where you'd get an MOS?

Neither, they are scale steps in 72 equal, Multiply the numbers by 50/3
and you get values in cents. This is not, of course, the only way to
tune marvel. Aside from TOP, another interesting plan is to make 2, 5/3
and 7 all exactly pure, and flatten 3 and 5 by 1/4 of a septimal
kleisma, that is, lower by (225/224)^(1/4)

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:

> This is a very musical thing. If one accepts 1/1 +2/1 +3/2, the it
> seems logical that one also accepts 1/1 +3/2 +2/1.
> If one accepts 1/1 +2/1 +3/2 +4/3, then it seems logical one also
> accepts 1/1 +4/3 +3/2 +2/1 right?
> Try playing it, it keeps the simplicity of the intervals and the
> chords formed this way make perfect musical sense.
> Why tune it any other way?
>
> This thinking leads to many things.
> One could rearange / permutate the intervals of the 1/1 2/1 3/1 4/1
> 5/1 (1/1 + 2/1 +3/2 +4/3 +5/4) chord.
> For instance to 1/1 3/2 3/1 15/4 5/1 (1/1 + 3/2 +2/1 +5/4 +4/3), or
> 1/1 5/4 5/3 10/3 5/1 (1/1 +5/4 +4/3 +2/1 +3/2)
>
> Now if one takes all permutations of this 5th-harmonic-limit 1/1 2/1
> 3/1 4/1 5/1 chord we get:
> 1/1 5/4 4/3 3/2 5/3 15/8 2/1 (when reduced to one octave)
> After some experimenting, and some thoughts about the logic behind
> music, and remembering the rule that held notes do not "comma shift"..

You are writing multiplications as if they were additions here, but without introducing logarithms. What you seem to be saying is "If you take the intervals of the chord 1-2-3-4-5, which are 2, 3/2, 4/3, and 5/4, permute them in every possible way, and reduces the intervals thus obtained to the octave, you obtain the scale 5/4-4/3-3/2-5/3-15/8-2". While I don't find your justification for this procedure very convincing, it seems to me the idea is worth exploring. I'd like to know if I've gotten your idea correctly, though.

Hi Gene,

You are writing multiplications as if they were additions here, but without
> introducing logarithms.
>
Yes I am, but I find it perfectly ok to do it that way.
It's easily understood, both by our mind and our ear.

> What you seem to be saying is "If you take the intervals of the chord
> 1-2-3-4-5, which are 2, 3/2, 4/3, and 5/4, permute them in every possible
> way, and reduces the intervals thus obtained to the octave, you obtain the
> scale 5/4-4/3-3/2-5/3-15/8-2".
>
Yes, that is completely correct. (you have put all permutations relative to
/ centered on 1/1)
I call the resulting scale the 5-limit harmonic scale / model.
The full scale is: 1/1 5/4 4/3 3/2 5/3 15/8 2/1 5/2 8/3 3/1 10/3 15/4 4/1
5/1 (note it's perfectly symetrical between 1/1 and 5/1)
Reduced to one octave: 1/1 5/4 4/3 3/2 5/3 15/8 2/1

One can also center the permutations not only on 1/1, but on 2/1, or on 3/1,
or on 4/1 or on 5/1.
The scale that results from doing this I call the 5-limit tonality scale /
model.
The full scale is:
1/1 16/15 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 32/15 9/4 12/5 5/2 8/3
3/1 16/5 10/3 15/4 4/1 9/2 5/1 16/3 45/8 6/1 20/3 15/2 8/1 9/1
When reduced to one octave: 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5
15/8 2/1 (the 12 tone JI scale).
The permutations will show you all simple chords in this scale, and how
they're related, and how they're all connected to the tonic 1/1 2/1 3/1 4/1
5/1 chord.

But as I said, 5-limit is very "limited" for actual music.
6-limit will cover just about everything.
6-limit harmonic scale / model is:
1/1 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 9/4 12/5 5/2 8/3 3/1 16/5 10/3 18/5
15/4 4/1 9/2 24/5 5/1 6/1 (offcourse symetrical between 1/1 and 6/1)
When reduced to one octave: 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

6-limit tonality scale / model is:
(not yet calculated all pitches, but it's between 1/1 and 10/1 from the top
of my head)
When reduced to one octave: 1/1 25/24 16/15 10/9 9/8 6/5 5/4 4/3 27/20 25/18
45/32 3/2 25/16 8/5 5/3 27/16 16/9 9/5 15/8 2/1 (19 tones per octave)

To hear an example of 5-limit permutations (not real 5-limit chords
according to the harmonic model though, only the base permutations), that
are played connected at random, and do not feature the tonality model yet
(so it modulates at random and is fairly hard to follow because of this)
listen to this:
http://sites.google.com/site/develdenet/mp3/9-12-2009_5limit_harmonic-permutation-ji_choir.mp3
It's not "real music" and doesn't show all the beauty of this theory etc,
but in the most basic form you can hear the permutations (again, without the
tonality model)

For a good 6-limit Tonal-JI example listen and look to this tuning:
http://sites.google.com/site/develdenet/mp3/Drei_Equale_No1_%28Tonal-JI%29.mp3
http://sites.google.com/site/develdenet/mp3/Drei_Equale_No1_%28Tonal-JI%29.png
I have been testing this (and previous theories) on this piece as it
combines several problems and is hard to get right for the ear.
Only about a week ago when I've understood the tonality part of my theory
have I been able to complete it satisfactory.
I'll be retuning many more pieces to 6-limit Tonal-JI soon, but for now this
is the best practical example.
Btw also note the wolfs, and wolf major triads. They're 100% correct and "in
tune" according to my theory, they're not seen as a compromise or anything
like that. And they function well withing the piece. (what I mean, that if
you'd put a 1/1 5/4 3/2 inplace of a wolf major triad etc, this would be
making it out of tune. It's not what the music does.)

While I don't find your justification for this procedure very convincing, it
> seems to me the idea is worth exploring. I'd like to know if I've gotten
> your idea correctly, though.
>
Ah there are many ways to justify this procedure of permutations.
I originaly found it by thinking I wanted to categorise simple to more
complex, perfect to less perfect.
I see the harmonic series as beeing perfect and progressing from simple, to
more complex the higher you go up the harmonics.
Then I said, if the harmonic series is perfect, the the intervals making up
the harmonic series must be perfect too.
And I will work with the perfect intervals, only disturbing their perfect
order.
Another way to see it is that nature tells us how to divide the 4/1, or 5/1,
or 6/1 etc. And I'm simply stating all possible comfigurations of this
perfect division.

But I'm glad it seems worth exploting to you!
I hope you will get involved in this theory.
I call it Tonal Just Intonation.
And I feel like I'm onto somthing very special here with a great future, but
I can't do everything alone.
Too much work and thinking, so the more people working on this the better.
Btw, I hope to have a short version of the theory online on my website
within a week.

Kind regards,
Marcel

> I figure the periodicity would become slightly longer/less-desirable.
> But I don't see how comma-shifting is so bad...especially when you consider
> that most instruments themselves fluctuate in tone slightly (thus making the
> overtones themselves already "shifted" to some extent).
>

Ah we differ of opinion here.
But then fi you don't mind comma shifts.. the why not use adaptive-ji or
something like that?
You can have every chord as a harmonic series for instance and comma shift
all over the place.
I think it gives terribly out of tune results, but you may think otherwise
(several list members seem to like it)

> The other issue is periodicity buzz...especially with chords using tones of
> 9/8 interval or smaller it becomes almost grating despite being "in strict
> JI".
>

I don't see that.
A semitone is used a lot in chords in even simple common practice music.
Sure 15/8 2/1 will sound better as 1/1 15/8 or 15/8 4/1, but one can use
15/8 2/1 perfectly well depending on the musical context etc.
This even goes for 25/24.

Right, but the expression is IMVHO fairly limited
>

I don't think you understand the number of possible chords within 1/1 9/8
6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
I'm better just about every single chord in common practice music and beyong
is in this scale.
The expression possiblity in it is huuuuge. And besides this, expression is
not a single chord, it's a sequence of chords, melodies, and place relative
to the tonic etc.

> and, I'm betting, most people could digest much higher-limit chords IF they
> don't violate the critical band so much AKA have many overly close notes.
> Take, for example, ways to summarize the 12TET chord C E F A (which,
> mostly due to the fact the "overly close" E-F are surrounded by compensating
> wide intervals, I can show to many people without them saying it's "too
> tense")...I'm pretty sure you'll have a tough time doing that in strict
> 5-limit.
>

Digest much higher limit chords is not easy.
I've ran a test making simple 7-limit chords (based on permutations, not
even according to the much more complex and dissonant 7-limit harmonic
model), and even those chords are often extremely difficult and dissonant.

As for the C E F A. If the tonic is C the it would most likely be 1/1 5/4
4/3 5/3, nice chord.
(it could in extreme circumstances also be 1/1 5/4 27/20 27/16 in the tonic
of C, long story when this happends)
So where's the 5-limit problem? The E-F is even much wider in 5/4 4/3 than
it is in 12tet (and severely wider in 5/4 27/20)

Simply put, I think higher limits can be used with fairly minimal losses
> of consonance IF people are careful about not putting too many notes close
> together...and same goes with tempering of "pure" chords (too much
> temperament on too many notes in a row in a chord can distort the sense of
> wholeness of the series, but some on occasional notes I find is harmless).
>

No, higher limits can't be used with fairly minimal loss of consonance with
a 12 tone scale.
Unless one knows exactly what to play, and it is extremely simple, and one
constructs a 12 tone scale with higher limits especially for this.

As for temperaments, sure! 12-tone temperaments will work great! :)
But then don't kid yourself into doing Just Intonation.

AKA why would I want to bother with higher limits in the first place?
>

Well I ment why bother with higher limits in 12-tone scales.

> Many many more chords and emotional possibilities...it's like comparing
> the musical of someone who knows Debussy's progressions to someone who knows
> standard pop progressions.
>

Debussy's progressions are 6-limit progressions according to my ears.
But perhaps you ment it only as a comparison. And are meaning never heard
chords etc.
Well yes I agree, I would love to be able to use 7-limit in the future.
But in order to do this I think I must understand 6-limit well first, and
that's still some way off.
But 7-limit and higher limits in 12-tone scales and call it JI? I don't get
it. And up till now, my ear hasn't gotten it either, I've never come across
a practical music example that sounded right to me.

> So you are pretty much using the above ratios and multiplying them by each
> other to get new ones?
>

No, I'm working out all possible permutations of the harmonic series till a
certain limit, and then not multiply them but simply combine them.

>"And when we combine all permutation of the 6th-harmonic- limit chord,
> centered on the root + each of its harmonics (till the 6th harmonic),
> we get the following scale: (reduced to one octave)
> 1/1 25/14 16/15 10/9 9/8 6/5 5/4 4/3 27/20 25/18 45/32 3/2 25/16 8/5
> 5/3 27/16 16/9 9/5 15/8 2/1"
>
> Now this looks a lot more interesting, minus the fact so many keys would
> be needed for it. The only real issue I see is there are so many different
> (at least at first glance) common LCD's that it would take a good deal of
> theory to find all the sweet spots (thus making it harder to catch on among
> 12TET musicians). For example 16/15 and 9/8 near having an LCD at 18 (the
> interval formed is around 19/18 AKA is a "tempered 19/18") while 25/18 and
> 4/3 give an LCD of 24 (25/24).
>

Oh please don't call anything in my scale "tempered" :)
As for the LCD, that's not the way to look at it. The harmonic model, the
tonic, and the underlying permuation structures take care of meaning of
relations of 2 pitches. I've explained those in the previous mail to some
degree.

> I agree with you the 19-tone is a flexible enough scale, but the other
> question becomes can we push it to the average musician without them
> thinking it's far too hard to learn?
>

No I don't see it that way.
I see it as beeing in future music theory books for students of composition
and music theory.
I see it beeing developed into clever algorithms for computer programs.
Things like that.

Very huge scales? I have my doubts...although I see that using your
> method it requires a lot of notes before such a scale can be formed (though
> I realize your method could be of great use to those who require scales that
> allow transposition).
> Again, using even just Ptolemy's Homalon ratios (completely unmodified be
> myself) can give a 7-tone scale of 18:20:22:24:27:30:33:36 which is loaded
> with very consonant chords of over 5-limit.
>

Consonant in a tempered kind of way.
It says nothing new musically. Only gives a bit of color in my opinion.

> And again I'll point out, even the 7-tone JI diatonic rounds down to x/24
> as it can be written in lowest form as 24:27:30:32:36:40:45:48 (yes, that
> means it has a higher LCD/lower-full-scale-periodicity than the
> Ptolemy-based scale)!
>

I don't see why you have to look at all scales as x/x
If you see my tonality scale and see the way it's constructed it's all
related to the tonic 1/1 2/1 3/1 4/1 5/1 chord. Every single note.
x/24 isn't something magical indicating tonic. One can play 6/5 from the
tonic perfectly well aswell etc.

>"And it seems to me all common practice music will fit the 6-limit
> tonality model with 6-limit harmonic model! "
> I'll say this much...I hope you continue looking into "6-limit"...you
> previous theories seem to just polish existing musical possibilities (though
> pretty well!)...but I'm really look forward to seeing you open the door to
> never-before-heard chords and such with 6-limit. :-)
>

Thanks :)
I think it's not so long off before that happends.
But I have to do some more retuning of existing music to get a better grip
on things.

Would you like to join in that?
Simply grab some common practice music you like, find the tonic, and try to
retune it to JI using 6-limit harmonic and tonality models.
This still seems like the best way to get a deeper understanding of JI and
how some parts of music works etc.

Marcel

>"But then fi you don't mind comma shifts.. the why not use adaptive-ji or something like that?"
A few reasons
A) At least in the implementations I've seen, it "only" supports standard practice intervals. If I want to use, say, the 12/11 interval in a non-standard chord like 9:11:12:15, I can't. And if I want a huge and dense chord like 18:20:24:27:30:33...Adaptive JI will simply try and pull a simple chord IE 18:24:27 or 6:8:9 and leave the other tones as neighboring tones, thus mangling the feel/emotion of the original chord.

B) It makes chords that create periodicity buzz due to "overly perfect" matching...often to the point of sounding quite rough/mechanical and, to a degree, not human.

C) It's virtually impossible to implement on acoustic instruments, and thus near-impossible to market to people who use such instruments.

D) The theory is hard to market as it involves teaching musicians music theory from scratch to fit whatever scale you are feeding in to the adaptive JI algorithm. That is, unless you use 12TET as the basis for it...in which case you don't get any extra dimensions of emotion (which is half the fun of micro-tonal music to me), but instead just more purity.

To me Adaptive JI is a great solution for those who tolerate comma shifts and periodicity buzz well, work with common practice music, compose music on electronic instruments only, and don't mind being stuck with mostly the same intervals/emotions they'd get from diatonic scales under 12TET so long as they get extra purity. For those who want to get past any of those limitations, however, there must be a better option.
***************************************************************.
Marcel>"I don't see that. A semitone is used a lot in chords in even simple common practice music."...."This even goes for 25/24."
I actually agree (for the most part)...but, in that case, why do you seem so intent on sticking with only low limit chords?

>"I don't think you understand the number of possible chords within 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 I'm better just about every single chord in common practice music and beyond is in this scale."
Here are some challenges
A) How would the 12TET chord C E F A B work in that scale?
B) How about C D G A B?
C) And C E G A B?
D) Now try F G C D E
E) And F A C D E
These are some of the "beyond" chords I'm eager to see perfected in a single (and not huge!) JI scale. They (at least to me) gives unique moods but I doubt any Adaptive JI software, for example, would successfully attempt to turn them into full chords well.
----------------------------------------
>"I've ran a test making simple 7-limit chords (based on permutations,
not even according to the much more complex and dissonant 7-limit
harmonic model), and even those chords are often extremely difficult
and dissonant."
I think it may be fair to say you have a natural bias (perhaps partly due to musical training) far toward 5-limit. I hear a consonance difference between 5 and 7 in general...but I don't think they are by any means immensely different so far as consonance just 7 is (in general)...a bit weirder.

>"As for the C E F A. If the tonic is C the it would most likely be 1/1 5/4 4/3 5/3, nice chord."
Bizarre, because it certainly doesn't seem to simplify down to any low-limit chord. Using the LCD of 12 you get 12/12 15/12 16/12 20/12........AKA 12:15:16:20...doesn't seem very "low limit" to me...although I agree it's a very nice chord.
----------------------------------------------------
>"So where's the 5-limit problem? The E-F is even much wider in 5/4 4/3 than it is in 12tet (and severely wider in 5/4 27/20)"
Come to think of it...the 27/20 actually works quite well to my ears...it's barely outside of the 13/12 limit where my ears start telling me two tones are pushing the critical band too much. Then again, I was using C E F A as a fairly "easy" example. If you want hard, try chords A) B) C) D) and E) above. :-)
--------------------------------------------------------
>"No, higher limits can't be used with fairly minimal loss of consonance with a 12 tone scale.
Unless
one knows exactly what to play, and it is extremely simple, and one
constructs a 12 tone scale with higher limits especially for this."
Well how about a scale (try to use as few notes as possible) that enables A) B) C) D) and E) given above? Try to do it keeping a strict 6-limit standard and then let yourself stretch that scale to as many limits or temper "as needed" to best fit all the chords.
------------------------------------------------------------
>"Well I meant why bother with higher limits in 12-tone scales."
Again, try the above challenge. I'm convinced it can be done with 12 tones or less...but only if you free yourself from this so-called absolute necessity of staying within very low limits of JI.
----------------------------------------------------------
Me>"(the interval formed is around 19/18 AKA is a "tempered 19/18") while 25/18 and 4/3 give an LCD of 24 (25/24)."
Marcel>"Oh please don't call anything in my scale "tempered" :)"
Why not?...I actually meant it as a complement meaning the interval between the tones comes across to my ears as more periodic than it actually is.
-------------------------------------------
>"As for temperaments, sure! 12-tone temperaments will work great! :) But then don't kid yourself into doing Just Intonation."
What's so alien about the idea of slight temperaments from JI, particularly when they enable many more chords with incredibly small loss of accuracy regarding periodicity?

Me>"I
agree with you the 19-tone is a flexible enough scale, but the other
question becomes can we push it to the average musician without them
thinking it's far too hard to learn?"
Marcel>"No I don't see it that way. I see it as being in future music theory books for students of composition and music theory. I see it being developed into clever algorithms for computer programs.
Things like that."
Sounds good only my question becomes...how do you plan to realistically get, say, music teachers to adapt this theory and, furthermore, for students to have the patience to learn to understand it?
------------------------------------------------------
>"Again,
using even just Ptolemy's Homalon ratios (completely unmodified be
myself) can give a 7-tone scale of 18:20:22:24: 27:30:33: 36 which is
loaded with very consonant chords of over 5-limit.
Consonant in a tempered kind of way. It says nothing new musically. Only gives a bit of color in my opinion."
And new/different color isn't something new? Considering just how much work is done programming new sounds and timbres by electronic musicians and even use of different orchestral arrangements to get new "colors" into music, I'm still betting there's a demand for that. That and (until/unless you and others manage to disprove it), I'm still sticking by the idea that such scales as that have more possible chords than in 7-tone diatonic JI and that musicians will find the extra chords give them inspiration for fresh new songs.
--------------------------------------
>"I don't see why you have to look at all scales as x/x
If you see my
tonality scale and see the way it's constructed it's all related to the
tonic 1/1 2/1 3/1 4/1 5/1 chord. Every single note. x/24 isn't something magical indicating tonic."
True, but my point is if the whole scale reduces to something reasonable and with many possible common denominators (IE an x/24 scale has 8 (IE 8:9:11),6 (IE 6:7:8) , 4 (IE 4:5:6), 3 (IE 3:4:5) and 2 (IE 2:3:4) as possible chord structures.
>"One can play 6/5 from the tonic perfectly well as well etc."
Right, but that just summarizes the starting point for a dyad. For a triad you could be thinking "x/5" if you were lucky enough to stumble upon a 5:6:7 chord. More likely though (and especially with large chords), you could be stuck with an "x/10" chord IE 1/1 6/5 3/2. Hence if you are thinking of a quick way to look at a JI scale and get an idea of which parts of the series all possible chords are rooted at, you can take factors of the LCD for the whole scale.

(not shouting, just stressing the below as important)
>>>
IN SHORT, THE WHOLE X/18 OR X/24 TYPE NOTATION OF SCALES IS MY OWN PERSONAL SYSTEM FOR EVALUATING THE GENERAL LEVEL OF PERIODICITY PLUS THE CRITICAL BAND COMPLIANCE IN A SCALE.
<<<
So if a scale is x/18 IE 18:20:22:24:27:30:33:36, it's more likely to be periodic than one that's x/24. And, in that form, you can quickly find the smallest interval and match it to where you think critical band dissonance becomes too harsh (for me the closest interval possible to avoid this is around 1.08333).
So this way I can see clearly that 22/20 and 36/33 are the closest intervals in the scale and quickly judge the associated critical band dissonance.
So far as I can see the "limit" notation system has no easy way of doing this...you're stuck with calculating the closest intervals from scratch. Although sadly, I don't see many people doing this because (as I've said before) it seems far too many people don't seem to understand the critical band's role in consonance (particularly in enabling chords with many consecutive closely-spaced intervals)

And with tempering you can do cool things like get the benefits of an "x/12" and an "x/18" scale in one scale, IE getting possible not-so-high-limit chords based on x/2,x/3, x/4, x/6, x/9, x/12, and x/18 (all factors of x/12 and x/18) in one scale and, at worst, get about "x/18" level periodicity.

Far as the level of periodicity you appear to be going for, I'd recommend "x/12" scales...that way you're guaranteed to get no worse/higher-up-the-series than 23/24 and far more likely to get something more like 13/12 (not far at all up the harmonic series, and better that way than 15/14).
An easy x/12 scale that has good periodicity yet isn't just a straight harmonic series is
12:14:15:16:18:20:22:24
...how does that scale work for you?

---------------------------------------------------------------------------
>"Would you like to join in that?
Simply grab some common practice music you like, find the tonic, and
try to retune it to JI using 6-limit harmonic and tonality models.
This still seems like the best way to get a deeper understanding of JI and how some parts of music works etc."
Sure thing, I'll try it with some of my old 12TET tunes: I was notorious for using "not-quite-legal" 6+ note chords before I discovered micro-tonal music. :-)

_,_._,___

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Hi Gene,
>
> You are writing multiplications as if they were additions here, but without
> > introducing logarithms.
> >
> Yes I am, but I find it perfectly ok to do it that way.
> It's easily understood, both by our mind and our ear.

I'm a mathematician, and it confused me. Why not write
"3/2 * 4/3" rather than "3/2 + 4/3" if multiplication is what you mean?

> The scale that results from doing this I call the 5-limit tonality scale /
> model.
> The full scale is:
> 1/1 16/15 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 32/15 9/4 12/5 5/2 8/3
> 3/1 16/5 10/3 15/4 4/1 9/2 5/1 16/3 45/8 6/1 20/3 15/2 8/1 9/1
> When reduced to one octave: 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5
> 15/8 2/1 (the 12 tone JI scale).

This looks more interesting than your original version, so I will write more Maple programs to see what it does.

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:

> One can also center the permutations not only on 1/1, but on 2/1, or on 3/1,
> or on 4/1 or on 5/1.

Could you show the result of "centering" the permutation on, for example, 3?

I see- the "temper by 72" isn't clear, though if you'd said "temper so that if fits into 72-edo" or something along those lines it would have been clear I think.

I guess the "61" is a typo, shouldn't it be "60"? Then you'd have all steps either 7 or 5 steps of 72. If they're all 7 or 5 except one 8 and one 4, practical implementation is surely going to take a hit.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "genewardsmith" genewardsmith@ wrote:
> >
> > > If we take the Ellis Duodene
> > >
> > > ! duodene.scl
> > > !
> > > Ellis's Duodene : genus >[33355]
> > > 12
> > > !
> > > 16/15
> > > 9/8
> > > 6/5
> > > 5/4
> > > 4/3
> > > 45/32
> > > 3/2
> > > 8/5
> > > 5/3
> > > 9/5
> > > 15/8
> > > 2
> > >
> > > and temper it by (for example) 72 equal, we get 7, 12, 19, 23, 30,
> >35, 42, 49, 53, 61, 65, 72.
> >
> > Are these: "7, 12, 19, 23, 30, >35, 42, 49, 53, 61, 65, 72" equal
> temperaments that can be used to approximate the scale you're referring
> to? Or the counts where you'd get an MOS?
>
> Neither, they are scale steps in 72 equal, Multiply the numbers by 50/3
> and you get values in cents. This is not, of course, the only way to
> tune marvel. Aside from TOP, another interesting plan is to make 2, 5/3
> and 7 all exactly pure, and flatten 3 and 5 by 1/4 of a septimal
> kleisma, that is, lower by (225/224)^(1/4)
>

> Could you show the result of "centering" the permutation on, for example,
> 3?
>

Here for 5-limit:

1/1 2/1 [3/1] 4/1 5/1
1/1 2/1 [3/1] 15/4 5/1
9/8 9/4 [3/1] 9/2 45/8
9/8 9/4 [3/1] 15/4 45/8
6/5 12/5 [3/1] 9/2 6/1
6/5 12/5 [3/1] 4/1 6/1

1/1 3/2 [3/1] 4/1 5/1
1/1 3/2 [3/1] 15/4 5/1
3/2 9/4 [3/1] 6/1 15/2
3/2 9/4 [3/1] 15/4 15/2
8/5 12/5 [3/1] 6/1 8/1
8/5 12/5 [3/1] 4/1 8/1

9/8 3/2 [3/1] 9/2 45/8
9/8 3/2 [3/1] 15/4 45/8
3/2 2/1 [3/1] 6/1 15/2
3/2 2/1 [3/1] 15/4 15/2
9/5 12/5 [3/1] 6/1 9/1
9/5 12/5 [3/1] 9/2 9/1

6/5 3/2 [3/1] 9/2 6/1
6/5 3/2 [3/1] 4/1 6/1
8/5 2/1 [3/1] 6/1 8/1
8/5 2/1 [3/1] 4/1 8/1
9/5 9/4 [3/1] 6/1 9/1
9/5 9/4 [3/1] 9/2 9/1

All permutations from 3/1 combined:
1/1 9/8 6/5 3/2 8/5 9/5 2/1 9/4 12/5 3/1 15/4 4/1 9/2 5/1 45/8 6/1 15/2 8/1
9/1

In one octave:
1/1 9/8 6/5 5/4 45/32 3/2 8/5 9/5 15/8 2/1

I don't have 6-limit written out like this on hand.
But I belief 6-limit centered on 3/1 reduced to one octave gives:
1/1 9/8 6/5 5/4 27/20 45/32 [3/1] 8/5 5/3 27/16 9/5 15/8

Note that the 27/16 occurs only higher than 3/1 here (by for instance 3/1
+3/2 +5/4 +6/5) and the 5/3 only below 3/1 (by for instance 3/1 -3/2 -6/5).
And that 5/3 and 27/16 may never occur in the same harmony and it does not
work the same as the harmonic model.
The 6-limit harmonic model is rooted /centered on 1/1 only. The 6-limit
harmonic model occurs in several places in the full 6-limit tonality scale.
In the 6-limit harmonic model there's never a comma.

> I'm a mathematician, and it confused me. Why not write
> "3/2 * 4/3" rather than "3/2 + 4/3" if multiplication is what you mean?

Well, for most people it seems easyest to understand by visualizing it in
the following way.
Take a piano keyboard, and draw a mental line from for instance C1 till E3.
Call C1 1/1 and E3 5/1
Then cut the line into pieces at C2 2/1, G2 3/1 and C3 4/1.
Then simply rearange the line segments into an alternative configuration of
your choice.
And you can take C1 1/1 as the place to start rearanging, or C2, or G2 or C3
or E3 (centering on E3 will offcourse give thesame results as C1)
I've had some trouble in the past explaining it to people in an easy to
understand way (partly because I'm not the greatest writer), but this way of
telling it seems to get across.

> The scale that results from doing this I call the 5-limit tonality scale /
> > model.
> > The full scale is:
> > 1/1 16/15 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 32/15 9/4 12/5 5/2 8/3
> > 3/1 16/5 10/3 15/4 4/1 9/2 5/1 16/3 45/8 6/1 20/3 15/2 8/1 9/1
> > When reduced to one octave: 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3
> 9/5
> > 15/8 2/1 (the 12 tone JI scale).
>
> This looks more interesting than your original version, so I will write
> more Maple programs to see what it does.

You mean the non octave reduced version looks more interesting than the
octave reduced version?

What's a Maple program btw? And how can it make you see what this scale
does?

Marcel

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
> What's a Maple program btw?

http://en.wikipedia.org/wiki/Maple_%28software%29

There is no la from sol in your 5-limit system, and you've got an almost "Middle-Eastern" sound in the upper tetrachord because the le is an augmented third from mi, ie., it's G#. Not to mention both a minor and a major seventh. Sounds great, but it's wildly off from any historical interpretations of "common practice".

Your large sets of what you call "6-limit" should cover a lot of simple "common practice" music either without problems, or by distributing the syntonic comma in places where the composition anticipates it, that is, pretty well hidden. You can't say there's "no comma" just because the 27/16 and 5/3 won't occur in the same harmony. I believe that you are correct in the assessment that they won't occur together, such an occurance would almost certainly be the result of an unlikely deliberate compositional choice, but if they occur in the same piece of music, that is considered, and heard, as syntonic movement, a bitterly disputed thing among choral folks by the way.

And also by the way, I do not disagree with you that syntonic variation of an interval within a piece is unsingable, impossible, blah blah, as some claim (do you have access to JSTOR?). Nor do I dismiss your solution as inviable. I just don't buy it as "THE" answer (nor ANY answer as "THE" answer for that matter). The principle sources of the idea that there can be a "the" answer is the bogus universality of 12-tET and the whacko cartoon of "common practice" it has helped propagate. Replacing one false hegemony with another is not the answer.

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> > Could you show the result of "centering" the permutation on, for example,
> > 3?
> >
>
> Here for 5-limit:
>
> 1/1 2/1 [3/1] 4/1 5/1
> 1/1 2/1 [3/1] 15/4 5/1
> 9/8 9/4 [3/1] 9/2 45/8
> 9/8 9/4 [3/1] 15/4 45/8
> 6/5 12/5 [3/1] 9/2 6/1
> 6/5 12/5 [3/1] 4/1 6/1
>
> 1/1 3/2 [3/1] 4/1 5/1
> 1/1 3/2 [3/1] 15/4 5/1
> 3/2 9/4 [3/1] 6/1 15/2
> 3/2 9/4 [3/1] 15/4 15/2
> 8/5 12/5 [3/1] 6/1 8/1
> 8/5 12/5 [3/1] 4/1 8/1
>
> 9/8 3/2 [3/1] 9/2 45/8
> 9/8 3/2 [3/1] 15/4 45/8
> 3/2 2/1 [3/1] 6/1 15/2
> 3/2 2/1 [3/1] 15/4 15/2
> 9/5 12/5 [3/1] 6/1 9/1
> 9/5 12/5 [3/1] 9/2 9/1
>
> 6/5 3/2 [3/1] 9/2 6/1
> 6/5 3/2 [3/1] 4/1 6/1
> 8/5 2/1 [3/1] 6/1 8/1
> 8/5 2/1 [3/1] 4/1 8/1
> 9/5 9/4 [3/1] 6/1 9/1
> 9/5 9/4 [3/1] 9/2 9/1
>
> All permutations from 3/1 combined:
> 1/1 9/8 6/5 3/2 8/5 9/5 2/1 9/4 12/5 3/1 15/4 4/1 9/2 5/1 45/8 6/1 15/2 8/1
> 9/1
>
> In one octave:
> 1/1 9/8 6/5 5/4 45/32 3/2 8/5 9/5 15/8 2/1
>
>
> I don't have 6-limit written out like this on hand.
> But I belief 6-limit centered on 3/1 reduced to one octave gives:
> 1/1 9/8 6/5 5/4 27/20 45/32 [3/1] 8/5 5/3 27/16 9/5 15/8
>
> Note that the 27/16 occurs only higher than 3/1 here (by for instance 3/1
> +3/2 +5/4 +6/5) and the 5/3 only below 3/1 (by for instance 3/1 -3/2 -6/5).
> And that 5/3 and 27/16 may never occur in the same harmony and it does not
> work the same as the harmonic model.
> The 6-limit harmonic model is rooted /centered on 1/1 only. The 6-limit
> harmonic model occurs in several places in the full 6-limit tonality scale.
> In the 6-limit harmonic model there's never a comma.
>

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

> I guess the "61" is a typo, shouldn't it be "60"? Then you'd have all steps either 7 or 5 steps of 72. If they're all 7 or 5 except one 8 and one 4, practical implementation is surely going to take a hit.

Nope; 9/5 maps to 61, not 60. You can get 60 by mapping from 16/9 instead, but the scale as it is is better in terms of harmony.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
>
> > I guess the "61" is a typo, shouldn't it be "60"? Then you'd have all steps either 7 or 5 steps of 72. If they're all 7 or 5 except one 8 and one 4, practical implementation is surely going to take a hit.
>
> Nope; 9/5 maps to 61, not 60. You can get 60 by mapping from 16/9 instead, but the scale as it is is better in terms of harmony.
>

Well I'm all for 9/5 but practical implementation, whether generalized keyboard, code (algorithm or table-reading), zither implementation (qanun, citra, etc) and even just plain keeping track of things all would prefer two consistent intervals of construction and transposition rather than 4, especially considering the fact that it is almost-but-not-quite constructed that way.

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> There is no la from sol in your 5-limit system, and you've got an >almost "Middle-Eastern" sound in the upper tetrachord because the le >is an augmented third from mi, ie., it's G#. Not to mention both a >minor and a major seventh. Sounds great, but it's wildly off from >any >historical interpretations of "common practice".

Sorry, I shouldn't have said "augmented third", though it is "augmented" in terms of 5/4, just listening to it. Anyway, it's almost a 9/7 off Mi, and there's no La, which was the point.

>"The principle sources of the idea that there can be a "the" answer are
the bogus universality of 12-tET and the whacko cartoon of "common
practice" it has helped propagate. Replacing one false hegemony with
another is not the answer."

As many of you can imagine (and perhaps even agree with), I feel the idea of "common practice" is a curse that keeps us from experiencing many of the dimensions of tonality that micro-tonality has to offer.

Another thing just about anyone could argue is that the ultimate 12-tone scale is 12:13:14:15:16...24 AKA simply harmonics number 1-12 of the series or that the ultimate 7-tone scale is undoubtedly harmonics 7 to 14. After all, those are the scales which best follow the harmonic series, right?
When you start trying to maximize the consonance of, say, triads though or simply want to establish a sense of different root tones to add (gasp) tonal color to your music. You may quickly figure out that harmonics 9:11:14 from the 7-14th harmonics might not be the chord with the best periodicity you can imagine. Thus you may split parts of the series up into different types of triads and stack them together and get something like 7-tone diatonic JI. But then you realize there are intervals in there which are less periodic than those in the 7-14 harmonic series.
Want to optimize the periodicity of 4-tone chords (tetra-chords or otherwise) instead? Be prepared for at least one completely different scale system which will conflict in ways with both diatonic JI and the 7-14 harmonic series. You might even have to deal with (gasp) higher limit chords.

And if you abandon the idea of "common practice" and allow for the idea of chords using much higher harmonics with a fair amount of spacing (IE a 12:13:16 chord) and then optimize triads and 4-tone chords, you'll get at least two more completely different answers. BTW (show of hands), how many people hear thing higher harmonics can be used very consonantly if the chords they are used to create are spaced widely enough apart (to avoid critical band dissonance)?

And if you want to make a tuning that allows modulation between of any of the above scales with near-consistent accurately in different keys, be prepared to sacrifice other things such as use of "pure JI" and deal with "ghastly phenomena" such as comma shifts and temperament (as you/Marcel and so many others have run into). ;-)
**************************************************
The point is...there is no ideal system. The system I'm working with now (and using in my song entry to Sevish's micro-tonal competition) is built to handle the next to last situation I mentioned (IE it doesn't allow modulation). In giving itself that flexibility, it aims to allow for more chords and larger chords (IE often 5-6 notes per octave) that sound fairly though not extremely consonant. It sacrifices JI "purity" of many smaller chords and does some de-tuning off JI intervals to accomplish this, but gains certain advantages such as the smallest interval being very near the less harsh 13/12 instead of the usual 15/14 of diatonic JI.

In the end >>>(vote of confidence here anyone?)<<<<<<< micro-tonality is a "game" of <<<different compromises (and not just a few!...that would lead us on a road to 12TET)>>>>. For everything you gain you lose something else and you have to decide what's important to you when designing or choosing scales.

I will say more respected and/or "better" scales tend to reap more advantages with less sacrifices. But I'm sure I'm not the only one who thinks the idea of a perfect scale with "all benefits and no sacrifices" is not possible nor would it be a good thing (it would basically kill all the fun we have here)...such would be like having a formula to produce beautiful music that "always worked" and then having machines compose everything.

__,_._,__

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"The principle sources of the idea that there can be a "the" >answer are
> the bogus universality of 12-tET and the whacko cartoon of "common
> practice" it has helped propagate. Replacing one false hegemony with
> another is not the answer."
>
> As many of you can imagine (and perhaps even agree with), I >feel the idea of "common practice" is a curse that keeps us from >experiencing many of the dimensions of tonality that micro-tonality >has to offer.

"Common practice" itself, understood, encourages microtonality. It's the cartoon understandings (C#=Db, Western music is 5-limit, etc) that are the problem. And of course being locked to triadic harmony, V-I cadences, diatonic scales, major-minor duality, etc., but those are more "frightened consumer sheep" problems rather than actual problems in the current musical state of affairs. :-)

>
> I will say more respected and/or "better" scales tend to reap more advantages with less sacrifices. But I'm sure I'm not the only one who thinks the idea of a perfect scale with "all benefits and no sacrifices" is not possible nor would it be a good thing (it would basically kill all the fun we have here)...such would be like having a formula to produce beautiful music that "always worked" and then having machines compose everything.
>
>
>
> __,_._,__
>

Me> As many of you can imagine (and perhaps even agree with), I feel the idea of "common practice" is a curse that keeps us from experiencing many of the dimensions of tonality that micro-tonality has to offer.

Cameron>"Common practice" itself, understood, encourages microtonality. It's the cartoon understandings (C#=Db, Western music is 5-limit, etc) that are
the problem. And of course being locked to triadic harmony, V-I
cadences, diatonic scales, major-minor duality, etc., but those are more "frightened consumer sheep" problems rather than actual problems in the current musical state of affairs. :-)

Perhaps I have the wrong idea of exactly what "common practice" is then. I always thought it was mainly said "cartoon understandings"...with things like Pythagorean intervals and mean-tone intervals being another interpretation of "common practice". Even Wikipedia seems to rate diatonic JI by how pure the triads are, thus tying back to the triadic harmony "cartoon understanding".
I will say the whole major-minor (AKA o-tonal and u-tonal?) duality and focus on triadic harmony while "forgetting" about larger chords (especially 5+ note) and randomly assuming they will "always be fairly dissonant and limited to practices like jazz...and things like jazz will always be to weird to be widely understandable/pop" are two of the most greatly harmful and limiting stereotypes about "ideal" music theory.

Actually if you look at the bottom of this page ->
...you'll see a large list of "Western Twelve Semitone" intervals IE

Perfect unison (0) · fourth (5) · fifth (7) · octave (12)

Major second (2) · third (4) · sixth (9) · seventh (11)

Minor second (1) · third (3) · sixth (8) · seventh (10)

Augmented unison (1) · second (3) · third (5) · fourth (6) · fifth (8) · sixth (10) · seventh (12)

Diminished unison (-1) · second (0) · third (2) · fourth (4) · fifth (6) · sixth (7) · seventh (9) · octave (11) ....and click on any of those and look at the "Just Intonation" ratio to get the pure versions of those intervals. Would I be wrong in saying that a scale's containing nothing but either those just intervals or intervals within about 13 cents of so of them would be "common practice"...and what is not included in that list would be "non common practice"?

I am just under the impression things like super-major, sub-minor, septimal, di-tone, and other intervals are not in common practice...and am fairly fascinated with said "wrong" intervals...which can (of course) be made "just" as well.

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

> Well I'm all for 9/5 but practical implementation, whether generalized keyboard, code (algorithm or table-reading), zither implementation (qanun, citra, etc) and even just plain keeping track of things all would prefer two consistent intervals of construction and transposition rather than 4, especially considering the fact that it is almost-but-not-quite constructed that way.

If one takes the Ellis duodene and repalces 9/5 with 16/9, one obtains the following scale:

! justchromatic.scl
!
5-limit just chromatic scale
12
!
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
16/9
15/8
2

This isn't in the Scala scl files under any name, and it aeems that it ought to be, since when you approximate it to mode of et, you find it listed as "Just" chromatic scale next to 53, 65, etc. If you compare it with the duodene on Scala, you will find it is more regular, but less harmonic, than the duodene.

I'm sorry but I'm completely dumping my tonality model as I described it before.
I've been doing some more thinking and it just doesn't make enough
sense, and an alternative I've had in the back of my mind has started
to make great sense.
I had rejected the model of centering on intervals other than 1/1
before, but had some more insight into tonality about a week ago and
used my old model again as it seemed to work.
Sorry for the attention some of you allready gave this model that is now lost.

Ok here for the new tonality model:
The permutation model is correct when permutating from 1/1.
The harmonic model resulting from the permutation model is correct.
5-limit harmonic model is: 1/1 5/4 4/3 3/2 5/3 15/8 2/1 (when reduced
to one octave)
6-limit harmonic model is: 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8
2/1 (reduced to octave)
7-limit harmonic model is: 1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 4/3
7/5 35/24 3/2 14/9 8/5 5/3 7/4 9/5 28/15 15/8 35/18 2/1 (reduced to
octave)
etc
(btw reducing to octave is offcourse ok because of ocave equivalence)

Now at the basis of tonality is the tonal chord.
The 5-limit tonal chord is 1/1 2/1 3/1 4/1 5/1 (or simply 1/1 5/4 3/2
when reduced to one octave)
The 6-limit tonal chord is 1/1 2/1 3/1 4/1 5/1 6/1 (or simply 1/1 5/4 3/2)
The 7-limit tonal chord is 1/1 2/1 3/1 4/1 5/1 6/1 7/1 (or simply 1/1
5/4 3/2 7/4)

Tonality means that all notes beeing played in a certain tonality/key
must harmonize with the tonal chord.
The harmonic model combined with the tonal chord will now tell all the
possibilities
The way to do this is as follows.

In the 5-limit harmonic model
1/1 5/4 4/3 3/2 5/3 15/8 2/1
We look how many times and where the 5-limit tonic chord can exist.
We see it can exist at 1/1 5/4 3/2, and at 4/3 5/3 2/1
So we get 1/1 5/4 4/3 3/2 5/3 15/8 2/1
And 1/1 9/8 5/4 45/32 3/2 15/8 2/1 (which is 4/3 5/3 2/1 transposed to
1/1 5/4 3/2)
Combine those, and we get the 5-limit tonality model/scale:
1/1 9/8 5/4 4/3 45/32 3/2 5/3 15/8 2/1
All these tones can be reached while holding the 5-limit tonic chord
and complying to the 5-limit harmonic model.

In the 6-limit harmonic model
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
We look how many times and where the 6-limit tonic chord can exist.
We see it can exist at 1/1 5/4 3/2, at 6/5 3/2 8/5, at 4/3 5/3 2/1, at
3/2 15/8 9/4 and at 8/5 2/1 12/5.
So we get 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 (harmonic model
root at 1/1)
and 1/1 25/24 10/9 5/4 4/3 25/18 3/2 25/16 5/3 15/8 2/1 (6/5 to 1/1,
harmonic model root at 5/3)
and 1/1 9/8 6/5 5/4 27/20 45/32 3/2 27/16 9/5 15/8 2/1 (4/3 to 1/1,
harmonic model root at 3/2)
and 1/1 16/15 10/9 6/5 5/4 4/3 3/2 8/5 5/3 16/9 2/1 (3/2 to 1/1,
harmonic model root at 4/3)
and 1/1 25/24 9/8 75/64 5/4 45/32 3/2 25/16 5/3 15/8 2/1 (8/5 to 1/1,
harmonic model root at 5/4)
See all the above tranpositions of the harmonic model have the 6-limit
tonic chord at 1/1.
Meaning every tone in the above scales will harmonize with the tonic
chord according to the 6-limit harmonic model.
Combine all these scales and you get the 6-limit tonality model / scale:

1/1 25/24 16/15 10/9 9/8 75/64 6/5 5/4 4/3 27/20 25/18 45/32 3/2 25/16
8/5 5/3 27/16 16/9 9/5 15/8 2/1

It is a 20 tone per octave scale.
It is almost equal to my previous 6-limit tonality scale, it has one
extra tone, 75/64 (which will finally give a consonant major triad on
15/8 which I was missing in my previous tonality model (and a
consonant minor traid on 25/16))
(Btw luckily this doesn't seem to affect my Drei Equale tuning which
still appears correct to me, pfew..)

As for 7-limit:
Tonic 1/1 5/4 3/2 7/4 is possible in the following locations in the
7-limit harmonic model:
1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9 8/5 5/3
7/4 9/5 28/15 15/8 35/18 2/1 (harmonic model root at 1/1)
1/1 25/24 35/32 10/9 7/6 175/144 5/4 35/27 4/3 25/18 35/24 3/2 14/9
25/16 175/108 5/3 7/4 175/96 15/8 35/18 2/1 (6/5 to 1/1, harmonic
model root at 5/3)
1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 27/20 7/5 45/32 35/24 3/2 63/40
105/64 27/16 7/4 9/5 15/8 63/32 2/1 (4/3 to 1/1, harmonic model root
at 3/2)
1/1 28/27 16/15 10/9 7/6 6/5 56/45 5/4 35/27 4/3 7/5 35/24 3/2 14/9
8/5 5/3 7/4 16/9 28/15 35/18 2/1 (3/2 to 1/1, harmonic model root at
4/3)
1/1 25/24 35/32 9/8 7/6 75/64 175/144 5/4 21/16 175/128 45/32 35/24
3/2 25/16 105/64 5/3 7/4 175/96 15/8 35/18 2/1 (8/5 to 1/1, harmonic
model root at 5/4)
Giving the following 7-limit tonality model / scale:
1/1 28/27 25/24 21/20 16/15 35/32 10/9 9/8 7/6 75/64 6/5 175/144 56/45
5/4 35/27 21/16 4/3 27/20 175/128 25/18 7/5 45/32 35/24 3/2 14/9 25/16
63/40 8/5 175/108 105/64 5/3 27/16 7/4 16/9 9/5
175/96 28/15 15/8 35/18 63/32 2/1
It's 40 tones per octave large.

And yes it has the same 1/1, 5/4, 4/3, 3/2 and 5/3 harmonic roots of
6-limit tonality :)
I think only in 8-limit will we see 7-limit harmonic roots (haven't
checked though) but I don't think we'll ever need to go that far,
7-limit should give cutting edge extreme microtonal music undepletable
in my lifetime and probably far beyond.

Marcel

On 8 April 2010 06:58, Marcel de Velde <m.develde@...> wrote:
>
>> Could you show the result of "centering" the permutation on, for example,
>> 3?
>
> Here for 5-limit:
>
> 1/1    2/1    [3/1]    4/1    5/1
> 1/1    2/1    [3/1]    15/4    5/1
> 9/8    9/4    [3/1]    9/2    45/8
> 9/8    9/4    [3/1]    15/4    45/8
> 6/5    12/5    [3/1]    9/2    6/1
> 6/5    12/5    [3/1]    4/1    6/1
>
> 1/1    3/2    [3/1]    4/1    5/1
> 1/1    3/2    [3/1]    15/4    5/1
> 3/2    9/4    [3/1]    6/1    15/2
> 3/2    9/4    [3/1]    15/4    15/2
> 8/5    12/5    [3/1]    6/1    8/1
> 8/5    12/5    [3/1]    4/1    8/1
>
> 9/8    3/2    [3/1]    9/2    45/8
> 9/8    3/2    [3/1]    15/4    45/8
> 3/2    2/1    [3/1]    6/1    15/2
> 3/2    2/1    [3/1]    15/4    15/2
> 9/5    12/5    [3/1]    6/1    9/1
> 9/5    12/5    [3/1]    9/2    9/1
>
> 6/5    3/2    [3/1]    9/2    6/1
> 6/5    3/2    [3/1]    4/1    6/1
> 8/5    2/1    [3/1]    6/1    8/1
> 8/5    2/1    [3/1]    4/1    8/1
> 9/5    9/4    [3/1]    6/1    9/1
> 9/5    9/4    [3/1]    9/2    9/1
>
> All permutations from 3/1 combined:
> 1/1 9/8 6/5 3/2 8/5 9/5 2/1 9/4 12/5 3/1 15/4 4/1 9/2 5/1 45/8 6/1 15/2 8/1
> 9/1
>
> In one octave:
> 1/1 9/8 6/5 5/4 45/32 3/2 8/5 9/5 15/8 2/1
>
>
> I don't have 6-limit written out like this on hand.
> But I belief 6-limit centered on 3/1 reduced to one octave gives:
> 1/1 9/8 6/5 5/4 27/20 45/32 [3/1] 8/5 5/3 27/16 9/5 15/8
>
> Note that the 27/16 occurs only higher than 3/1 here (by for instance 3/1
> +3/2 +5/4 +6/5) and the 5/3 only below 3/1 (by for instance 3/1 -3/2 -6/5).
> And that 5/3 and 27/16 may never occur in the same harmony and it does not
> work the same as the harmonic model.
> The 6-limit harmonic model is rooted /centered on 1/1 only. The 6-limit
> harmonic model occurs in several places in the full 6-limit tonality scale.
> In the 6-limit harmonic model there's never a comma.
>
>

On 7 April 2010 18:00, Michael <djtrancendance@...> wrote:

> >"One way of thinking about it is that it allows
> >augmented triads with two intervals of 5:4 and
> > one of 9:7 to add up to a 2:1 octave. You can then
> > split that 9:7 to give 7:8:9 so that both 5:4 and 7:4 are
> > relative to the tonic. Somebody mentioned that
> > chord recently."
> Right, 5/4 * 9/7 * 5/4 = just a tad over 2/1.
> The result I get from breaking this into triads (though) is.....
> 1/1 (???) 5/4 (4/3) (3/2) 25/16 (25/14)
> ........now how do I get/obtain that 7th note?

25/16 is the same as 14/9. 14/9 * 9/8 = 7/4. So the whole chord would be

1/1 5/4 14/9 7/4 2/1

Each interval is in the 9-limit, but only because of the temperament.
That's how I remember it but it wasn't my idea.

> Bad wording on my part...I meant to say 7 cents
> of the original JI tone.

If you temper everything,16:15 and 15:14 can each take half of that
mistuning, so all important intervals are within 4 cents of just. The
7-limit's within 2 cents IIRC.

Graham

> >"I don't think you understand the number of possible chords within 1/1 9/8
> 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 I'm better just about every single
> chord in common practice music and beyond is in this scale."
> Here are some challenges
> A) How would the 12TET chord C E F A B work in that scale?
> B) How about C D G A B?
> C) And C E G A B?
> D) Now try F G C D E
> E) And F A C D E
>

Well you can easily try this yourself.
You can simply take the 6-limit harmonic model and find all the positions
these chords can exist in it.
You'll get several possibilities for each chord.
Which one of tuning a chord will get in actual music depends on the tonic of
that music, and on the context (chords / melodies comming before and after).
But to give a simple example for your chords in the tonic of C, they could
be:
A) 1/1 5/4 4/3 5/3 15/8
B) 1/1 9/8 3/2 5/3 15/8
C) 1/1 5/4 3/2 5/3 15/8
D) 4/3 3/2 2/1 9/4 5/2
E) 4/3 5/3 2/1 9/4 5/2
Again, these are just some of the possiblities. Also in the tonic of C there
are many other possiblities, and all chords could be played without wolfs,
it would depend for each chord on the context as I mentioned earlyer.

>"Would you like to join in that?
> Simply grab some common practice music you like, find the tonic, and try to
> retune it to JI using 6-limit harmonic and tonality models.
> This still seems like the best way to get a deeper understanding of JI and
> how some parts of music works etc."
> Sure thing, I'll try it with some of my old 12TET tunes: I was notorious
> for using "not-quite-legal" 6+ note chords before I discovered micro-tonal
> music. :-)
>

Ok cool :)
Let me know how it goes.

Marcel

> Sorry, I shouldn't have said "augmented third", though it is "augmented" in
> terms of 5/4, just listening to it. Anyway, it's almost a 9/7 off Mi, and
> there's no La, which was the point.
>

The 8/5 is gone in the 5-limit tonality model. (and wasn't in the 5-limit
harmonic model to begin with)
But I don't understand the La not beeing there? La is 5/3 right? Do you mean
there's no 27/16 in 5-limit like there is in 6-limit?

Marcel

Hi Cameron,

There is no la from sol in your 5-limit system,
>
I don't understand, do you mean 3/2 to 5/3?
And do you mean in the harmonic model or in the tonality model? (note I've
just sacked the old tonality model and replaced it with a better one. Makes
large difference in 5-limit, less so in 6-limit)

> and you've got an almost "Middle-Eastern" sound in the upper tetrachord
> because the le is an augmented third from mi, ie., it's G#.
>
Aah you're talking about the tonality model.
First of all, the old tonality model is sacked. New one is nice and doesn't
have what you describe.
But even in the old model. In 5-limit harmonic model you can never get a
diminished fourth (i think you ment) so it doesn't matter if it's in the
tonality model, it's never used as such in 5-limit.
But again, sorry it's even out of the 5-limit tonality model now

> Not to mention both a minor and a major seventh. Sounds great, but it's
> wildly off from any historical interpretations of "common practice".
>
> Your large sets of what you call "6-limit" should cover a lot of simple
> "common practice" music either without problems, or by distributing the
> syntonic comma in places where the composition anticipates it, that is,
> pretty well hidden.
>

Aah I'm glad you see this! Thanks!

> You can't say there's "no comma" just because the 27/16 and 5/3 won't occur
> in the same harmony. I believe that you are correct in the assessment that
> they won't occur together, such an occurance would almost certainly be the
> result of an unlikely deliberate compositional choice, but if they occur in
> the same piece of music, that is considered, and heard, as syntonic
> movement, a bitterly disputed thing among choral folks by the way.
>
> And also by the way, I do not disagree with you that syntonic variation of
> an interval within a piece is unsingable, impossible, blah blah, as some
> claim (do you have access to JSTOR?).
>

I don't have access to JSTOR.
But we seem in agreement here that syntonic (and other comma variations) are
possible withing a piece. (just not in held notes etc).

> Nor do I dismiss your solution as inviable. I just don't buy it as "THE"
> answer (nor ANY answer as "THE" answer for that matter). The principle
> sources of the idea that there can be a "the" answer is the bogus
> universality of 12-tET and the whacko cartoon of "common practice" it has
> helped propagate. Replacing one false hegemony with another is not the
> answer.
>
Well, my approach is simply trying to make sense of things that I feel
should make sense (if that makes sense ;)
My system doesn't say anything is impossible. Infact anything is possible,
it's just that it categorizes the possiblities.
Infact not even syntonic comma shifts in repeating notes are impossible, but
one would have to go to 8-limit interpretations in order to get those.
And that's simply not very likely, and a very difficult interpretation.
And in other ways, my theory offers the possiblity to work mathematically
with music, in a musical way.
And many other things 12tet doesn't offer (+ my theory produces tuning
results that sound much more consonant and emotional etc)
And think our brain and nature are doing something similar to my theory.
So in these and other aspects there's a lot I think is "right" here. And I
think it will give new insights into music.
So by these grounds I don't think it's right to call it a false hegemony, or
compare it on equal grounds with false hegemonies.
But I do get your point :) And the future will tell how usefull my theory
will end up to be.
(it's just my personal drive to understand things, that I have to think
there's logic and perfection in music, why else bother to try to understand)

Marcel

Btw a few more things I'd like to add about 7-limit.
While I think just about all common practice music is 6-limit (with
the very rare exceptions probably which I have not encountered yet).
7-limit is ok too. I've perhaps been hammering a bit too much on
6-limit for the people on this list :)
It seems to me some modern music, and some arabic music is 7-limit.

To give a very easy example (I've given this one before), even with a
single simple chord, of 7-limit:
C D E F# G# A# C.
The whole tone chord.
It simply can't even be done in the 6-limit harmonic model, there is 0
possiblities for this chord.
And if one tries to do it in another way with only prime-5-limit
intervals, it's a mess.
It is a perfectly normal chord in the 7-limit harmonic model though.
I has several possiblities, for instance 1/1 9/8 5/4 7/5 8/5 9/5 2/1
which sounds absolutely fantastic.

As for arabic music.
Lets take the neutral second and neutral thirds as an example.
Yes there is a 27/25 neutral third like interval in 6-limit (but never
relative to the tonic), and perhaps some could see 6/5 as a kind of
neutral third, and most arabic music does sound like 6-limit music to
me if I'd have to take a guess. But also some arabic music defenately
doesn't sound 6-limit to me.

In the 7-limit harmonic model we find the following neutral seconds:
2: 2 27/25 133.238 cents large limma, BP small semitone
3: 1 175/162 133.633 cents
2: 4 35/32 155.140 cents septimal neutral second
3: 2 192/175 160.502 cents
Of which 35/32 can occur relative to the tonic 1/1.

And the following neutral thirds:
6: 2 135/112 323.353 cents
*: 3 175/144 337.543 cents
6: 2 128/105 342.905 cents septimal neutral third
6: 2 216/175 364.412 cents
6: 4 56/45 378.602 cents
Of which 175/144 and 56/45 can occur relative to the tonic 1/1.

I wish I could tell you in detail how 7-limit music works and how to
make great sounding music with it.
But while I belief my theory is on a speedtrain to that destination,
it's not there yet.
And I'll try to understand 6-limit better first.
Though I'll soon give a new attempt at permutation algorithms that
make automatic 6-limit and 7-limit music. I think the results will be
better this time allready.

Marcel

On 8 April 2010 17:03, Marcel de Velde <m.develde@gmail.com> wrote:
> As for 7-limit:
> Tonic 1/1 5/4 3/2 7/4 is possible in the following locations in the
> 7-limit harmonic model:
> 1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9 8/5 5/3
> 7/4 9/5 28/15 15/8 35/18 2/1 (harmonic model root at 1/1)
> 1/1 25/24 35/32 10/9 7/6 175/144 5/4 35/27 4/3 25/18 35/24 3/2 14/9
> 25/16 175/108 5/3 7/4 175/96 15/8 35/18 2/1 (6/5 to 1/1, harmonic
> model root at 5/3)
> 1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 27/20 7/5 45/32 35/24 3/2 63/40
> 105/64 27/16 7/4 9/5 15/8 63/32 2/1 (4/3 to 1/1, harmonic model root
> at 3/2)
> 1/1 28/27 16/15 10/9 7/6 6/5 56/45 5/4 35/27 4/3 7/5 35/24 3/2 14/9
> 8/5 5/3 7/4 16/9 28/15 35/18 2/1 (3/2 to 1/1, harmonic model root at
> 4/3)
> 1/1 25/24 35/32 9/8 7/6 75/64 175/144 5/4 21/16 175/128 45/32 35/24
> 3/2 25/16 105/64 5/3 7/4 175/96 15/8 35/18 2/1 (8/5 to 1/1, harmonic
> model root at 5/4)
> Giving the following 7-limit tonality model / scale:
> 1/1 28/27 25/24 21/20 16/15 35/32 10/9 9/8 7/6 75/64 6/5 175/144 56/45
> 5/4 35/27 21/16 4/3 27/20 175/128 25/18 7/5 45/32 35/24 3/2 14/9 25/16
> 63/40 8/5 175/108 105/64 5/3 27/16 7/4 16/9 9/5
>  175/96 28/15 15/8 35/18 63/32 2/1
> It's 40 tones per octave large.
>
> And yes it has the same 1/1, 5/4, 4/3, 3/2 and 5/3 harmonic roots of
> 6-limit tonality :)

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>...
> Another thing just about anyone could argue is that the ultimate 12-tone scale is 12:13:14:15:16...24 AKA simply harmonics number 1-12 of the series or that the ultimate 7-tone scale is undoubtedly harmonics 7 to 14. After all, those are the scales which best follow the harmonic series, right?

Your "ultimate" 12-tone scale is not a constant structure, since you have intervals of the same size (e.g., 12:15 and 16:20) subtending a different number of scale degrees (3 and 4 degrees, respectively). Instead, I would propose:
16:17:18:19:20:21:22:24:25:26:28:30:32
To arrive at this, skip over prime harmonic 23 and include the next odd harmonic (25).

There is also a constant-structure decatonic scale that follows the above principle:
16:17:18:20:21:22:24:26:28:30:32
Skip over prime harmonic 19 and include the next odd harmonic (21).

An interesting feature of this decatonic scale is that you can build a chain of triads (and also 7th and 9th chords, etc.) by starting on any scale tone and repeatedly going up 3 scale degrees to get the next tone in the chord (e.g., triads 8:10:12, 10:12:15, 12:15:18, 15:18:22, 9:11:14, etc.). When you return to your starting tone (3 octaves higher), you'll have gone through all 10 tones of the scale. This is not only analogous to building chords by stacking 3rds in conventional harmony, but it also has the property that major and minor 3rds (4:5 and 5:6) subtend the same number of scale degrees (3).

--George

>"Your "ultimate" 12-tone scale is not a constant structure, since you
have intervals of the same size (e.g., 12:15 and 16:20) subtending a
different number of scale degrees (3 and 4 degrees, respectively)"

I get what you are saying, the same number of keys in different places cover a different number of "steps" on the keyboard.
As I said in a thread after the one you responded to, there seems to be a bit of a paradox in that the same people pushing for "common practice" (IE constant structure scales using familiar Western interval types) are saying entire scales should be more "JI-sound"...and I figure what's more "JI-sound" than an exact (and fairly low limit) harmonic series segment to approximate 12TET?

>"Instead, I would propose: 16:17:18:19: 20:21:22: 24:25:26: 28:30:32 To arrive at this, skip over prime harmonic 23 and include the next odd
harmonic (25)."
Right, but that's considerably higher up the harmonic series (and apparently has to be in order to qualify as a "constant structure"). I do see how intervals like 5/4, 9/8, 3/2, and 15/8 fit just about perfectly into this scale from each root and intervals like 4/3 are just a tad off.

>"There is also a constant-structure decatonic scale that follows the
above principle: 16:17:18:20: 21:22:24: 26:28:30: 32"
>"An interesting feature of this decatonic scale is that you can build a
chain of triads (and also 7th and 9th chords, etc.) by starting on any
scale tone and repeatedly going up 3 scale degrees to get the next tone
in the chord (e.g., triads 8:10:12, 10:12:15, 12:15:18, 15:18:22,
9:11:14, etc.)"

Seems like a great idea for not-so-closely-spaced chords created with repeating interval....though I wonder how it works for things like diminished chords. It's actually yet another scale that makes me wonder why it didn't work as an alternative to 12TET in popular music...it seems similarly pure to diatonic JI plus allows modulations.

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:

> Combine all these scales and you get the 6-limit tonality model / scale:
>
> 1/1 25/24 16/15 10/9 9/8 75/64 6/5 5/4 4/3 27/20 25/18 45/32 3/2 25/16
> 8/5 5/3 27/16 16/9 9/5 15/8 2/1
>
> It is a 20 tone per octave scale.

If you look at it as a linear scale it isn't proper and generally fails to be very regular. If you look at it harmonically, it makes a lot more sense. Using Scala's lattice command, we get the following diagram, where you have to digest the notation (E53) which makes the C minor triad to be C D# G and the C major triad to be C E\ G:

Gb\ Db\ Ab\ Eb\
D\ A\ E\ B\ Gb
Bb F C G D A
C# G# D# A# F/

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:

> Giving the following 7-limit tonality model / scale:
> 1/1 28/27 25/24 21/20 16/15 35/32 10/9 9/8 7/6 75/64 6/5 175/144 56/45
> 5/4 35/27 21/16 4/3 27/20 175/128 25/18 7/5 45/32 35/24 3/2 14/9 25/16
> 63/40 8/5 175/108 105/64 5/3 27/16 7/4 16/9 9/5
> 175/96 28/15 15/8 35/18 63/32 2/1
> It's 40 tones per octave large.

Once again this scale is irregular but makes sense harmonically. It can be seen as two stacked "six limit" (not terminology anypne else uses or understands, by the way) scales. Here it is in E171 notation:

Gb\ Db\ Ab\ Eb\ ! G#/ D#/ A#/ F)
D\ A\ E\ B\ Gb ! E) B) GL\ DL\ AL\
Bb F C G D A ! DbL AbL EbL BbL FL CL
C# G# D# A# F/ ! EL BL Gb< Db< Ab<

If you are trying to play this on some sort of souped up guitar, you could save wear and tear on your fingers, not to mention your brain, by marvel tempering. That reduces it to 34 notes to the octave, in the following 5-limit pattern:

A(\ E(\ B(\ F)
F( C( G( D( A(
Gb\ Db\ Ab\ Eb\ Bb\ F\ C\
D\ A\ E\ B\ Gb Db Ab
Bb F C G D A
C# G# D# A# F/

Here the fifth and third should be very slightly (about two cents) flat. It's probably clearer to leave off the note names and just show the diagram:

* * * *
* * * * *
* * * * * * *
* * * * * * *
* * 0 * * *
* * * * *

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
I see Yahoo is still screwing up diagrams. :(

Gene wrote:
> I see Yahoo is still screwing up diagrams. :(

You probably already know about the View Message Option >
Use Fixed Width Font workaround (on the right). But just
in case... -Carl

Hi Gene,

> Giving the following 7-limit tonality model / scale:
> > 1/1 28/27 25/24 21/20 16/15 35/32 10/9 9/8 7/6 75/64 6/5 175/144 56/45
> > 5/4 35/27 21/16 4/3 27/20 175/128 25/18 7/5 45/32 35/24 3/2 14/9 25/16
> > 63/40 8/5 175/108 105/64 5/3 27/16 7/4 16/9 9/5
> > 175/96 28/15 15/8 35/18 63/32 2/1
> > It's 40 tones per octave large.
>
> Once again this scale is irregular but makes sense harmonically. It can be
> seen as two stacked "six limit" (not terminology anypne else uses or
> understands, by the way) scales.

Yes indeed, good for noticing!
I had allready noticed this for the harmonic models, but hadn't noticed it
yet in the 6-limit vs 7-limit tonality models.

There also a trick to easily make the harmonic models (and subsequently the
tonality models by the method I described earlyer)
Take 1/1.
Shift it by 2/1 and combine with 1/1 to get 1/1 2/1, the 2-limit harmonic
model.
Take the 2-limit harmonic model 1/1 2/1 and shit it by 3/2 and combine to
get the 3-limit harmonic model: 1/1 3/2 2/1 3/1
Take the 3-limit harmonic model and shift it by 4/3 and combine to get the
4-limit harmonic model: 1/1 4/3 3/2 2/1 8/3 3/1 4/1
Take the 4-limit harmonic model and shift it by 5/4 and combine to get the
5-limit harmonic model: 1/1 5/4 4/3 3/2 5/3 15/8 2/1 (reduced to one octave)
Take the 5-limit harmonic model, shift by 6/5, combine -> the 6-limit
harmonic model: 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
Take the 6-limit harmonic model, shift by 7/6, combine -> the 7-limit
harmonic model.
etc etc

This trick doesn't work exactly like this with the tonality models.
As for instance the 5-limit tonality model (1/1 9/8 5/4 4/3 45/32 3/2 5/3
15/8 2/1) doesn't give the 6-limit tonality model when shifted.
But it does work in the instance of 6-limit tonality model to 7-limit
tonality model indeed.
It is because the 7-limit tonality model has the exact same harmonic model
roots (at 1/1, 5/4, 4/3, 3/2 and 5/3) as the 6-limit tonality model.
This isn't the case in 5-limit vs 6-limit, or 7-limit vs 8-limit etc.

Btw a few more nice things about this system.
For instance the 6-limit harmonic model, it is built out of the following
permutations:

6-limit permutations:
(permutations without 2/1)

Major:
1/1 3/2 2/1 5/2 3/1
1/1 3/2 15/8 5/2 3/1
1/1 4/3 2/1 5/2 3/1
1/1 4/3 5/3 5/2 3/1
1/1 5/4 15/8 5/2 3/1
1/1 5/4 5/3 5/2 3/1
1/1 3/2 15/8 9/4 3/1
1/1 5/4 15/8 9/4 3/1
1/1 5/4 3/2 9/4 3/1
1/1 4/3 5/3 2/1 3/1
1/1 5/4 5/3 2/1 3/1
1/1 5/4 3/2 2/1 3/1

Minor:
1/1 3/2 2/1 12/5 3/1
1/1 3/2 9/5 12/5 3/1
1/1 4/3 2/1 12/5 3/1
1/1 4/3 8/5 12/5 3/1
1/1 6/5 9/5 12/5 3/1
1/1 6/5 8/5 12/5 3/1
1/1 3/2 9/5 9/4 3/1
1/1 6/5 9/5 9/4 3/1
1/1 6/5 3/2 9/4 3/1
1/1 4/3 8/5 2/1 3/1
1/1 6/5 8/5 2/1 3/1
1/1 6/5 3/2 2/1 3/1

Combined they form the 6-limit harmonic model:
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

Now if we take the 6-limit tonality model (6-limit as an example, the things
below hold for any limit):
1/1 25/24 16/15 10/9 9/8 75/64 6/5 5/4 4/3 27/20 25/18 45/32 3/2 25/16 8/5
5/3 27/16 16/9 9/5 15/8 2/1

We find that every single instance of any of the 6-limit permutation chords
is linked to the 6-limit tonic chord (1/1 2/1 3/1 4/1 5/1 6/1) of the
6-limit tonality model through the 6-limit harmonic model.
In other words, the combining of the 6-limit harmonic models to form the
6-limit tonality does not produce interval combinations that are "unusable"
in it's own system. The system is 100% efficient here.

We also find that if we were to add any interval to the tonality model (any,
doesn't matter what, lets say for instance 225/128), that this extra
interval is "unusable" in this system as it will not harmonize according to
the 6-limit harmonic model with the 6-limit tonic chord.
So the Tonal-JI system is maximally "saturated" to it's limit.

Just like to add one more thing about seeing 7-limit tonality as 2 6-limit
tonality scales 7/6 apart (not that you suggested using it as such).
But only 1 in 7 7-limit permutations have the 7/6 as the first interval from
1/1. If one is to look at 7-limit tonality as 2 6-limit scales, that is to
look at the "7/6 shifted, 2nd" 6-limit tonality scale as continuesly having
the 7/6 as the first interval in the 7-limit harmonic model.
I think that's a bit confusing way to look at it, and would probably would
also be not so easy to understand by our ear (infact using part of the scale
as a 7/6 shifted 6-limit scale would probably be so confusing it would
probably sound horribly out of tune).
7-limit gives a full and huge range of 7-limit chords (according to the
7-limit harmonic model), and the true root of those chords is never shifted
by a 7-limit interval in 7-limit tonality, as all the 7-limit harmonic model
roots are 6-limit relative to the tonic chord. (roots are 1/1, 5/4, 4/3,
3/2, and 5/3).

> Here it is in E171 notation:
>
> Gb\ Db\ Ab\ Eb\ ! G#/ D#/ A#/ F)
> D\ A\ E\ B\ Gb ! E) B) GL\ DL\ AL\
> Bb F C G D A ! DbL AbL EbL BbL FL CL
> C# G# D# A# F/ ! EL BL Gb< Db< Ab<
>
>
Ok thanks!
I have no experience with microtonal notation, allways do everything in
ratios.
But perhaps something that would be usefull for me to learn somewhere in the
future.

> If you are trying to play this on some sort of souped up guitar, you could
> save wear and tear on your fingers, not to mention your brain, by marvel
> tempering.

Ah the T word.. ;-)
Not for me, not just because it's no longer in perfect tune (though sounds
like the practical difference is so small it would be unnoticable with most
sounds), but because it obscures the logic that I see in JI (making it
harder on the brain for me, not easyer).
Btw the graphs look very nice when displayed in fixed width font, thanks.

Marcel

>"Ah the T word.. ;-)
Not for me, not just because it's no longer in
perfect tune (though sounds like the practical difference is so small
it would be unnoticable with most sounds)."
Dare I admit, I double-dare you to take one of your "infamous" JI scales and find a few chords you think are sour and then temper the scale to make them work without being far enough away to ruin other chords. I think you'd probably be amused just how many extra chords you can make that way.

Another thought...if we were going for purity we'd probably get rid of things like diminished chords, 11th/13th chords, and add2 chords altogether. A very impure major 7th chord may still be much more pure than, say, an as-pure-as-possible diminished 7th. My point is that you're always going to have a range of purity no matter how "just" your scale is...and you might want to take into more consideration the idea of making intolerably dissonant chords into "decent" ones over ideas like purifying things like already far-more-consonant-than-needed-to-be-acceptable 7th chords.

>"but because it obscures the
logic that I see in JI"
I will easily agree to that. If I make a tempered tuning, I always keep the original JI scale it is based on (or sometimes also, a list of chords that I tempered off the original JI version to match) to explain how I came up with it. JI is great that way (and, hey, so are things like the circle of 5ths)...they make it much easier to explain music quickly.

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:

> Aah you're talking about the tonality model.
> First of all, the old tonality model is sacked. New one is nice and doesn't
> have what you describe.
> But even in the old model. In 5-limit harmonic model you can never get a
> diminished fourth (i think you ment) so it doesn't matter if it's in the
> tonality model, it's never used as such in 5-limit.
> But again, sorry it's even out of the 5-limit tonality model now

Aaaaaaaaaaaa! You've changed it again! Yes I was referring to the diminshed fourth but I should take more time (which I don't have) to reply, because I had meant to point out that you have an augmented second and a diminished fourth. What I meant was, when that particular set of intervals was rotated to the fifth, the major sixth disappeared and several quasi enharmonic intervals appeared.

In Gene's analysis it is very clear to see that you are doing something that is related to temperament, enharmonic "unison vectors". You've got an augmented second as a minor third right at the center of things, D# functioning as Eb.

If you'd do a 5-limit Fokker periodicity block with 128/125 (the diesis) as a "unison vector" I bet your system will pop right out at you. The block probably requires a couple of other commas, the major diesis maybe, but not the syntonic comma. I don't have the time to figure them out but Gene probably knows them off the top of his head.

(BTW I consider "unison vector" a poor term, it should be enharmonic vector or homophonic vector, but maybe unison vector is best AFTER temperament is applied. But anyway.)

> >
> > Your large sets of what you call "6-limit" should cover a lot of simple
> > "common practice" music either without problems, or by distributing the
> > syntonic comma in places where the composition anticipates it, that is,
> > pretty well hidden.
> >
>
> Aah I'm glad you see this! Thanks!

Well I think almost everyone recognizes a great big grid of
traditional triads! And it is a fact that tons of music doesn't stray beyond things that can be easily Just- remember that the foundation of "common practice" is singing, and that prior to the Industrial Revolution, most instruments for most people had flexible intonation and limited range. So your set of intervals is bound to work for a great deal of music- the question is whether you'll strike the minimum set for maximum use in the kinds of music you want.

>
> I don't have access to JSTOR.

University libraries usually do. I think you should find access (I think it is immensely expensive to have as an individual), because then you get an immense library of scholarly literature for free.
You might also worry or delight in how weak "institutionalized" tuning theory is- plenty of approximation of Just via "equal temperaments", as is done on this list, but glaringly missing evaluation of some very basic and vital bits of information such as whether the approximations provide the third or the ditone at the fourth fifth modulo 2 (aka. tempering out the syntonic comma, "meantone", etc.).

> But we seem in agreement here that syntonic (and other comma >variations) are
> possible withing a piece. (just not in held notes etc).

Sure, it's an approach. I bet there were composers who had it in mind and wrote it in that way too- stick your syntonic comma on a passing sonority or on the word "evil"- and in those cases you could say it's even the correct historical approach.

-Cameron Bobro

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

> If you'd do a 5-limit Fokker periodicity block with 128/125 (the diesis) as a "unison vector" I bet your system will pop right out at you. The block probably requires a couple of other commas, the major diesis maybe, but not the syntonic comma. I don't have the time to figure them out but Gene probably knows them off the top of his head.

I've got the complete set for 5-limit and 12 notes to the octave, but sadly Scala's scale search feature isn't working for me. I await what Manuel has to say.

> (BTW I consider "unison vector" a poor term, it should be enharmonic vector or homophonic vector, but maybe unison vector is best AFTER temperament is applied. But anyway.)

I just call it a comma; I don't much like the term "vector" here, since it is an element of a free abelian group but not of a vector space unless you extend it to one by tensoring with the rationals. Which undoubted only bothers me...

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
>
> > If you'd do a 5-limit Fokker periodicity block with 128/125 (the >diesis) as a "unison vector" I bet your system will pop right out at >you. The block probably requires a couple of other commas, the major >diesis maybe, but not the syntonic comma. I don't have the time to >figure them out but Gene probably knows them off the top of his head.
>
> I've got the complete set for 5-limit and 12 notes to the octave, >but sadly Scala's scale search feature isn't working for me. I await >what Manuel has to say.

Bet you've got Marcel's system there, then. But there may very well be consistency issues, with the comma tempered out in some places and not in others. Not that there's anything wrong with that, dumping commas in "dark" places is downright traditional.

I had it backwards- he's does NOT have got the small diesis 128/125 tempered out, rather, the aug. 2nd functions as m3 in some keys. So it's functionally it's temperament via enharmonic equivalence, aka "wrong note", but the diesis is still there.

I think he does have the watchamacallit, difference between the 5-limit tritones (2048/2025), tempered out, but... just a sec...

"Now if we take the 6-limit tonality model (6-limit as an example, the things below hold for any limit):
1/1 25/24 16/15 10/9 9/8 75/64 6/5 5/4 4/3 27/20 25/18 45/32 3/2 25/16 8/5 5/3 27/16 16/9 9/5 15/8 2/1"

That's Tanaka minus 6 tones. They (Marcel's and Tanaka) go down to 2048/2025 which I now see is the diaschisma, but do not include any smaller commas.

A Fokker block would have to be pretty big I suspect... lessee, yes, Marcel's is a subset of Fokker 41b in the Scala archive.

Marcel, I suggest you read Tanaka (late 19th Century, quite renowned, IIRC it's in German if not English) and see if you're on the same track. If I get a chance to look it up some in the library I'll tell you. The system is under "Tanaka" in the Scala archive.

>
> > (BTW I consider "unison vector" a poor term, it should be enharmonic vector or homophonic vector, but maybe unison vector is best AFTER temperament is applied. But anyway.)
>
> I just call it a comma; I don't much like the term "vector" here, since it is an element of a free abelian group but not of a vector space unless you extend it to one by tensoring with the rationals. Which undoubted only bothers me...
>

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

>
>
> >
> > > (BTW I consider "unison vector" a poor term, it should be enharmonic vector or homophonic vector, but maybe unison vector is best AFTER temperament is applied. But anyway.)
> >
> > I just call it a comma; I don't much like the term "vector" here, since it is an element of a free abelian group but not of a vector space unless you extend it to one by tensoring with the rationals. Which undoubted only bothers me...
> >
>

I don't know what a free abelian group or tensoring is, but I did recognize that they aren't proper vectors just by looking at the pictures. :-)

31: 3 2 3 2 3 2 3 3 2 3 2 3 SP M ME SD: 6.2663 c. Genus diatonico-chromaticum
34: 3 3 3 2 3 3 3 3 2 3 3 3 SP SD: 5.2731 c. "Just" Chromatic
41: 4 3 4 2 4 3 4 4 2 4 3 4 SP SD: 4.3830 c. "Just" Chromatic
53: 5 4 5 3 5 4 5 5 3 5 4 5 SP SD: 1.0934 c. "Just" Chromatic
65: 6 5 6 4 6 5 6 6 4 6 5 6 SP SD: 1.0851 c. "Just" Chromatic

You're right, it's 3 and 2 steps in both 31 and 34, and
could be taken as a subset with steps of 3 and 2 in 41, 53, and 65.

65 is an underrated and underused division IMO, but I've come
across it via combining maximum dissonance with simple Just, where it
really would shine.
--- In tuning@...m, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
>
> > Well I'm all for 9/5 but practical implementation, whether generalized keyboard, code (algorithm or table-reading), zither implementation (qanun, citra, etc) and even just plain keeping track of things all would prefer two consistent intervals of construction and transposition rather than 4, especially considering the fact that it is almost-but-not-quite constructed that way.
>
> If one takes the Ellis duodene and repalces 9/5 with 16/9, one obtains the following scale:
>
> ! justchromatic.scl
> !
> 5-limit just chromatic scale
> 12
> !
> 16/15
> 9/8
> 6/5
> 5/4
> 4/3
> 45/32
> 3/2
> 8/5
> 5/3
> 16/9
> 15/8
> 2
>
> This isn't in the Scala scl files under any name, and it aeems that it ought to be, since when you approximate it to mode of et, you find it listed as "Just" chromatic scale next to 53, 65, etc. If you compare it with the duodene on Scala, you will find it is more regular, but less harmonic, than the duodene.
>

Oh, and this "just chromatic" is also a subset of Tanaka.

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> 31: 3 2 3 2 3 2 3 3 2 3 2 3 SP M ME SD: 6.2663 c. Genus diatonico-chromaticum
> 34: 3 3 3 2 3 3 3 3 2 3 3 3 SP SD: 5.2731 c. "Just" Chromatic
> 41: 4 3 4 2 4 3 4 4 2 4 3 4 SP SD: 4.3830 c. "Just" Chromatic
> 53: 5 4 5 3 5 4 5 5 3 5 4 5 SP SD: 1.0934 c. "Just" Chromatic
> 65: 6 5 6 4 6 5 6 6 4 6 5 6 SP SD: 1.0851 c. "Just" Chromatic
>
>
> You're right, it's 3 and 2 steps in both 31 and 34, and
> could be taken as a subset with steps of 3 and 2 in 41, 53, and 65.
>
> 65 is an underrated and underused division IMO, but I've come
> across it via combining maximum dissonance with simple Just, where it
> really would shine.
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > > Well I'm all for 9/5 but practical implementation, whether generalized keyboard, code (algorithm or table-reading), zither implementation (qanun, citra, etc) and even just plain keeping track of things all would prefer two consistent intervals of construction and transposition rather than 4, especially considering the fact that it is almost-but-not-quite constructed that way.
> >
> > If one takes the Ellis duodene and repalces 9/5 with 16/9, one obtains the following scale:
> >
> > ! justchromatic.scl
> > !
> > 5-limit just chromatic scale
> > 12
> > !
> > 16/15
> > 9/8
> > 6/5
> > 5/4
> > 4/3
> > 45/32
> > 3/2
> > 8/5
> > 5/3
> > 16/9
> > 15/8
> > 2
> >
> > This isn't in the Scala scl files under any name, and it aeems that it ought to be, since when you approximate it to mode of et, you find it listed as "Just" chromatic scale next to 53, 65, etc. If you compare it with the duodene on Scala, you will find it is more regular, but less harmonic, than the duodene.
> >
>

Well strictly speaking 12 tET is neither of these scales but the irrational set 2^n/12, n = 0,1,2,...11, the only two rationals being unison and the 8ve. But I think its important not to turn to conspiracy theories while forgetting both the problems that it solved and its many practical advantages. One of the effects of detuning all notes equally is that each note now serves a double function as both intervals between keys and within chords, which makes it very practical and efficient. Sure C# = Db might be a lack of distinction for the chord based intervals but it doesn't take into account this second function as intervals between keys. At any rate, maybe you shouldn't make up your own version and then make yourself angry about it (lol). Poor old 12 tET!

-Rick

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Your "ultimate" 12-tone scale is not a constant structure, since you
> have intervals of the same size (e.g., 12:15 and 16:20) subtending a
> different number of scale degrees (3 and 4 degrees, respectively)"
>
> I get what you are saying, the same number of keys in different places cover a different number of "steps" on the keyboard.
> As I said in a thread after the one you responded to, there seems to be a bit of a paradox in that the same people pushing for "common practice" (IE constant structure scales using familiar Western interval types) are saying entire scales should be more "JI-sound"...and I figure what's more "JI-sound" than an exact (and fairly low limit) harmonic series segment to approximate 12TET?
>
> >"Instead, I would propose: 16:17:18:19: 20:21:22: 24:25:26: 28:30:32 To arrive at this, skip over prime harmonic 23 and include the next odd
> harmonic (25)."
> Right, but that's considerably higher up the harmonic series (and apparently has to be in order to qualify as a "constant structure"). I do see how intervals like 5/4, 9/8, 3/2, and 15/8 fit just about perfectly into this scale from each root and intervals like 4/3 are just a tad off.
>
>
> >"There is also a constant-structure decatonic scale that follows the
> above principle: 16:17:18:20: 21:22:24: 26:28:30: 32"
> >"An interesting feature of this decatonic scale is that you can build a
> chain of triads (and also 7th and 9th chords, etc.) by starting on any
> scale tone and repeatedly going up 3 scale degrees to get the next tone
> in the chord (e.g., triads 8:10:12, 10:12:15, 12:15:18, 15:18:22,
> 9:11:14, etc.)"
>
> Seems like a great idea for not-so-closely-spaced chords created with repeating interval....though I wonder how it works for things like diminished chords. It's actually yet another scale that makes me wonder why it didn't work as an alternative to 12TET in popular music...it seems similarly pure to diatonic JI plus allows modulations.
>

Rick>"But I think its important not to turn to conspiracy theories while
forgetting both the problems that it solved and its many practical
advantages."
Both scales we came up with allow transposition/modulation with very little change in feel/error between keys...not to far from 12TET transpositions nature of being without any change in interval size from the root (which is obviously attained as it is an equal tempered tuning).
My version of the scale is not a constant structure, so some may consider it more difficult to play on the keyboard melodically as certain sections are "squeezed" or "expanded"...but the other scale mentioned avoids that problem completely.

However, perhaps more importantly, both of these scales avoid things like the nasty 13 cents error between 12TET thirds and pure ones and are closer to the root of the harmonic series than scales under 12TET, supposedly making them more pure for both melody and harmony. I would not call them new concepts but, rather, I'm trying to point out that there are much more ways to solve the "12 tone modulation" problem than 12TET...and some of them, like those below, are very simple.

So Rick, what do you believe 12TET supposedly solve so well while that the two
scales just mentioned or Marcel's 12-tone scale or John's....don't?

There's no "conspiracy theory", it's just that 12-tET simply doesn't do justice or properly explain the musical history from which it (d)evolved. This is not internet cabal stuff or out-dated thinking, it's just normal "classically educated" stuff. I learned "expressive intonation" and "don't play the notes, play the music", just like countless others have and hopefully still do. A pretty young friend recognized the differences between "in tune" (Just) and 12-tET with ease the other day- he didn't know the terminology at all but he sang in the choir. So, he was taught the difference (in a very real way) quite recently. No secrets or conspiracies. Normal everyday musicianship for tons of people.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Rick>"But I think its important not to turn to conspiracy theories while
> forgetting both the problems that it solved and its many practical
> advantages."
> Both scales we came up with allow transposition/modulation with very little change in feel/error between keys...not to far from 12TET transpositions nature of being without any change in interval size from the root (which is obviously attained as it is an equal tempered tuning).
> My version of the scale is not a constant structure, so some may consider it more difficult to play on the keyboard melodically as certain sections are "squeezed" or "expanded"...but the other scale mentioned avoids that problem completely.
>
> However, perhaps more importantly, both of these scales avoid things like the nasty 13 cents error between 12TET thirds and pure ones and are closer to the root of the harmonic series than scales under 12TET, supposedly making them more pure for both melody and harmony. I would not call them new concepts but, rather, I'm trying to point out that there are much more ways to solve the "12 tone modulation" problem than 12TET...and some of them, like those below, are very simple.
>
> So Rick, what do you believe 12TET supposedly solve so well while that the two
> scales just mentioned or Marcel's 12-tone scale or John's....don't?
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Oh, and this "just chromatic" is also a subset of Tanaka.

Due to my current problem with Scala, my claim that it isn't in the scl directory is almost certainly false, but I don't propose to try and find it by hand.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Your "ultimate" 12-tone scale is not a constant structure, since you
> have intervals of the same size (e.g., 12:15 and 16:20) subtending a
> different number of scale degrees (3 and 4 degrees, respectively)"
>
> I get what you are saying, the same number of keys in different places cover a different number of "steps" on the keyboard.
> As I said in a thread after the one you responded to, there seems to be a bit of a paradox in that the same people pushing for "common practice" (IE constant structure scales using familiar Western interval types) are saying entire scales should be more "JI-sound"...and I figure what's more "JI-sound" than an exact (and fairly low limit) harmonic series segment to approximate 12TET?
>
> >"Instead, I would propose: 16:17:18:19: 20:21:22: 24:25:26: 28:30:32 To arrive at this, skip over prime harmonic 23 and include the next odd harmonic (25)."
>
> Right, but that's considerably higher up the harmonic series (and apparently has to be in order to qualify as a "constant structure").

Actually, it isn't considerably higher. I wrote it as:
16:17:18:19:20:21:22:24:25:26:28:30:32
starting it on 16, because that's an octave of the fundamental, but if I had written it using the lowest numbers it would be:
13:14:15:16:17:18:19:20:21:22:24:25:26
which starts only 1 harmonic higher than your 12 thru 24. Another consideration is that your 12 thru 24 has a prime limit of 23, while what I proposed has a lower prime limit (19).

> I do see how intervals like 5/4, 9/8, 3/2, and 15/8 fit just about perfectly into this scale from each root and intervals like 4/3 are just a tad off.

I don't understand what you mean by 4/3 being 'just a tad off'. Off of what? Do you mean 21/16 is not a perfect 4th (3:4) above 1/1? So what? If you can choose another mode of this scale, starting on harmonic 12, then you'll have a tone that's an exact 4/3 above the tonic note.

> >"There is also a constant-structure decatonic scale that follows the
> above principle: 16:17:18:20: 21:22:24: 26:28:30: 32"
> >"An interesting feature of this decatonic scale is that you can build a
> chain of triads (and also 7th and 9th chords, etc.) by starting on any
> scale tone and repeatedly going up 3 scale degrees to get the next tone
> in the chord (e.g., triads 8:10:12, 10:12:15, 12:15:18, 15:18:22,
> 9:11:14, etc.)"
>
> Seems like a great idea for not-so-closely-spaced chords created with repeating interval....though I wonder how it works for things like diminished chords.

It contains 10:12:14:17 (and its 3 inversions), which is a long-accepted ("classic") JI tuning for a diminished 7th chord. Admittedly, you won't get diminished 7ths on every tone of the scale, but at least you have them on 4 tones.

Did you notice that there are two reasonably good rational approximations of 5-equal by taking every other tone of the decatonic scale?

> It's actually yet another scale that makes me wonder why it didn't work as an alternative to 12TET in popular music...it seems similarly pure to diatonic JI plus allows modulations.

It allows modulations, but not free transposition. In 12-equal you can modulate from a given key into 23 other keys (counting modulations to both major & minor keys, 11 of which will be essentially transpositions), but everything is heavily tempered. In the 10- or 12-tone JI scale examples I gave, you have very few major & minor triads, but a great variety of intervals, chords, scales, and modes that you don't have in 12-equal; you can't transpose them (unless you add more tones), but you can modulate if you'll allow your melody and chords to be "modally" transformed.

BTW, if you're willing to accept some temperament, you might want to consider this 11-limit heptatonic scale subset of 31-equal:

C D E F^ G A Bbv C note names
8 9 10 11 12 27/2 14 16 harmonic approimation

where ^ and v represent semi-sharp & semi-flat. This scale contains the following triads: major (4:5:6 on C), minor (10:12:15, on A), subminor (6:7:9, on G), and neutral (18:22:27, on D), plus the isoharmonic triads 5:6:7 (on E), 7:9:11 (on Bbv), and 22:27:32 (on F^).

--George

It's not in the Scala archive as far as I can tell, it just shows up as a subset of a number of systems, most of them enormous (so, trivial inclusion) and none of them "circulting" or built specifically around this scale.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > Oh, and this "just chromatic" is also a subset of Tanaka.
>
> Due to my current problem with Scala, my claim that it isn't in the scl directory is almost certainly false, but I don't propose to try and find it by hand.
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> It's not in the Scala archive as far as I can tell, it just shows up as a subset of a number of systems, most of them enormous (so, trivial inclusion) and none of them "circulting" or built specifically around this scale.

You may have the same problem I did, that you should reset rh default directory to point to scl, or else move there before searching. I just got things set up, and found:

! malcolm.scl
!
Alexander Malcolm's Monochord (1721), and C major in Yamaha synths, Wilkinson: Tuning In
12
!

and

! verdi2.scl
Verdi 2
12

As expected, it's too well-known a scale not to be in the directory.

Is this tuning suitable for accidentals, modulations?

On Fri, Apr 9, 2010 at 1:53 PM, cameron <misterbobro@...> wrote:

>
>
> It's not in the Scala archive as far as I can tell, it just shows up as a
> subset of a number of systems, most of them enormous (so, trivial inclusion)
> and none of them "circulting" or built specifically around this scale.
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "genewardsmith"
> <genewardsmith@...> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "cameron"
> <misterbobro@> wrote:
> > >
> > > Oh, and this "just chromatic" is also a subset of Tanaka.
> >
> > Due to my current problem with Scala, my claim that it isn't in the scl
> directory is almost certainly false, but I don't propose to try and find it
> by hand.
> >
>
>
>

>"Actually, it isn't considerably higher. I wrote it as:
16:17:18:19: 20:21:22: 24:25:26: 28:30:32
starting it on 16, because that's an octave of the fundamental, but if I had written it using the lowest numbers it would be:
13:14:15:16: 17:18:19: 20:21:22: 24:25:26"
Interesting...how did you get to "lowest numbers" form? Just a thought I noticed the scale wraps around the octave to
.... 16: 17:18:19: 20:21:22: 24:25 and then 28 30
and if you take 28 and 30 over two you get
14,15.
...in that case...your scale I agree is the better alternative and I agree few would care about everything just being a single harmonic up.
********************************************
>Off of
what? Do you mean 21/16 is not a perfect 4th (3:4) above 1/1?"
Right...and for the record I don't think it's a big deal at all...just one of the weaker points of the scale (even though it's still fairly strong).
>"If you can choose another mode of this scale, starting on harmonic 12"
Right, it's just another thing I figure 12TET buffs would like pick on...that the scale (even if not much) has different levels of purity in different modes, even if not much different.

>"It contains 10:12:14:17 (and its 3 inversions), which is a long-accepted ("classic") JI tuning for a diminished 7th chord. Admittedly, you
won't get diminished 7ths on every tone of the scale, but at least you
have them on 4 tones."
...which is quite good in my book for a non-tempered scale. Again we have then on only certain modes (4 of 12). Just wondering, can you think of a clever way to tempered this scale so you can get them on all 12 with 7 cents accuracy?
I'm really eager to take some old 12 tone equal temperament songs I've written (often with things like huge 7-9 note chords) and re-tune them to "your" 16-32 scale, Marcel's scale, and a few others I've seen here and then post the results here.

>"you can't transpose them (unless you add more tones), but you can
modulate if you'll allow your melody and chords to be "modally"
transformed."
Which is virtually no problem to my ears. The only thing I worry about involves making such a scale "marketed" as an alternative to 12TET and get, say, keyboard manufacturers to be more likely to adopt it. And that's where I want to minimize the notice-ability of such "modal" transformations such that they "feel" almost like exact transpositions.

>"In the 10- or 12-tone JI scale examples I gave, you have very few major
& minor triads, but a great variety of intervals, chords, scales,
and modes that you don't have in 12-equal"
Which sounds good to me because I think it's an issue with so many JI scales that the sweetest chords (IE 3-note major and minor triads) get sweetened even more while more "sour" chords like diminished and 13th chords get left behind and end up getting labels like "jazz chords" and ousted from most modern popular music IMVHO much because of this.

>"BTW, if you're willing to accept some temperament, you might want to
consider this 11-limit heptatonic scale subset of 31-equal:"
Temperament is just fine by me (I'm pretty sure only Marcel and a handful of others are "anti-temperament"). :-) I'm really more concerned about what the temperament does so far as making borderline sour AKA "jazz chords" (in both common practice and chords that don't exist in common practice) sweet enough sounding so musicians will be inclined to use them more.

George>"Actually, it isn't considerably higher. I wrote it as:
16:17:18:19: 20:21:22: 24:25:26: 28:30:32"

I'm pretty sure your scale is one of my favorite 12-tone scales I've seen. Another question becomes what (within the scale) do you think is the most consonant 7-tone mode that still (feels) like the traditional 7-tone diatonic mode so far as being a "constant structure"?

I'd think 16:18:20:22:24:26:28:32 AKA 8:9:10:11:12:13:14:16 (in lowest form I believe it's simply 7:8:9:10:11:12:13:14) would be a pretty clear mode...wondering if
A) you have any mode you think would work better
B) you have any way to temper that mode to bring as many fairly dissonant "jazzy-sounding" chords to sound more consonant?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Actually, it isn't considerably higher. I wrote it as:
> 16:17:18:19: 20:21:22: 24:25:26: 28:30:32
> starting it on 16, because that's an octave of the fundamental, but if I had written it using the lowest numbers it would be:
> 13:14:15:16: 17:18:19: 20:21:22: 24:25:26"
> Interesting...how did you get to "lowest numbers" form? Just a thought I noticed the scale wraps around the octave to
> .... 16: 17:18:19: 20:21:22: 24:25 and then 28 30
> and if you take 28 and 30 over two you get
> 14,15.

Yep, you figured it out!

> ...in that case...your scale I agree is the better alternative and I agree few would care about everything just being a single harmonic up.

> ********************************************
> ...
> >"It contains 10:12:14:17 (and its 3 inversions), which is a long-accepted ("classic") JI tuning for a diminished 7th chord. Admittedly, you
> won't get diminished 7ths on every tone of the scale, but at least you
> have them on 4 tones."
> ...which is quite good in my book for a non-tempered scale. Again we have then on only certain modes (4 of 12).

Sorry, but we're talking about the same scale. I was referring to the decatonic scale, which makes it 4 out of 10. If we're talking about the dodecatonic scale, then it's 8 out 12: 10:12:14:17 and 15:18:21:25 (another well-accepted "classic" JI tuning), together with their inversions.

> Just wondering, can you think of a clever way to tempered this scale so you can get them on all 12 with 7 cents accuracy?

No. You'd have to alter the 11 and 13 in 11:13:16:19 by such a large amount that they would completely lose their identity. I noticed, however, that this is a rather interesting chord in its own right in the 13:16:19:22 inversion, which is isoharmonic: the difference of 3 between adjacent tones makes it more consonant that you might expect, considering the relatively high prime limit.

> I'm really eager to take some old 12 tone equal temperament songs I've written (often with things like huge 7-9 note chords) and re-tune them to "your" 16-32 scale,

Heh, heh! Unless they're very, very old (at least early renaissance, and even then so), you may be in for a bit of a shock. I think you'd have a better time with with this high-contrast circulating temperament that I just posted:
/tuning/topicId_87190.html#87190

> Marcel's scale, and a few others I've seen here and then post the results here.

> >"you can't transpose them (unless you add more tones), but you can
> modulate if you'll allow your melody and chords to be "modally"
> transformed."
> Which is virtually no problem to my ears. The only thing I worry about involves making such a scale "marketed" as an alternative to 12TET and get, say, keyboard manufacturers to be more likely to adopt it. And that's where I want to minimize the notice-ability of such "modal" transformations such that they "feel" almost like exact transpositions.

Sorry, but it ain't gonna happen! :-( This is the land of xenharmony: strange, but hopefully wonderful, where the establishment fears to tread.

Maybe if we created overtly xenharmonic music boxes and put them in crib toys, the next generation would grow up to appreciate our music more. ;-)

--George

Hm, my archive IS a mess- there are two whole archives nested, one seems incomplete and that's the one I've been searching. My own tunings are mostly in their own folders so I won't lose much if I clean up and consolidate, should download the latest archive anyway.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > It's not in the Scala archive as far as I can tell, it just shows up as a subset of a number of systems, most of them enormous (so, trivial inclusion) and none of them "circulting" or built specifically around this scale.
>
> You may have the same problem I did, that you should reset rh default directory to point to scl, or else move there before searching. I just got things set up, and found:
>
> ! malcolm.scl
> !
> Alexander Malcolm's Monochord (1721), and C major in Yamaha synths, Wilkinson: Tuning In
> 12
> !
>
> and
>
> ! verdi2.scl
> Verdi 2
> 12
>
> As expected, it's too well-known a scale not to be in the directory.
>

Perhaps "conspiracy" was too strong a word? What I meant was that by taking the n'th root of 2 we are solving the problem of equalisation between notes-as-keys. 2^(n/12) is *that* value which equals 2 when multiplied by itself n times. It's analogous to the creation of complex numbers to reach where the rationals can't get too. Once there it can take on a life of its own. The sqrt2 for eg is the only value which is the same under inversion 1:sqrt2 = sqrt2:2. In a manner of speaking this is a 2-tET by its own right. If we take a rational version, say 45/32, then this never reaches the 8ve exactly: (45/32)^2 = 1.977...Same with the diminished 7 chord which is also invariant to inversion and is a 4-tET. And for *these* types of symmetries, even-numbered roots of the 8ve, 12, 24, 36 for eg, seem to excel. There exist no better numbers which meet this demand, not even in principle. Even Helmholtz didn't realise the types of harmonies that would come out later with Thelonius Monk etc...saying that the Meantone system was better than 12-tET. Not for these harmonies it isn't.

Now these symmetries sound very much in tune to my ears and I notice when they have be detuned (Marcel's Beethoven would clash whenever the diminished chords were hit, while the rest would sound very beautiful). Perhaps what I'm getting at is that my 'ideal' system would not be an adaptive JI but an adaptive JI nested inside an ET tuning. To move from C maj to C#dim7, the notes of the C maj are JI but the interval between the C and C#, and the notes of C#dim7, are 12-tET or what have you. In short, I feel that to make everything JI or rational is to fail to distinguish between tonality = rational and atonality = irrational, and that these atonalities have their own place in history and tuning in their own right. Sure, other cultures use other tunings but westerners too need to bring something to the table. To undermine this under the banner of 'cultural harmony' is kind of defeating the purpose. (Whenever I play with muso's from other places we show each other stuff and then jam).

-Rick

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
> There's no "conspiracy theory", it's just that 12-tET simply doesn't do justice or properly explain the musical history from which it (d)evolved. This is not internet cabal stuff or out-dated thinking, it's just normal "classically educated" stuff. I learned "expressive intonation" and "don't play the notes, play the music", just like countless others have and hopefully still do. A pretty young friend recognized the differences between "in tune" (Just) and 12-tET with ease the other day- he didn't know the terminology at all but he sang in the choir. So, he was taught the difference (in a very real way) quite recently. No secrets or conspiracies. Normal everyday musicianship for tons of people.
>
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> >
> > Rick>"But I think its important not to turn to conspiracy theories while
> > forgetting both the problems that it solved and its many practical
> > advantages."
> > Both scales we came up with allow transposition/modulation with very little change in feel/error between keys...not to far from 12TET transpositions nature of being without any change in interval size from the root (which is obviously attained as it is an equal tempered tuning).
> > My version of the scale is not a constant structure, so some may consider it more difficult to play on the keyboard melodically as certain sections are "squeezed" or "expanded"...but the other scale mentioned avoids that problem completely.
> >
> > However, perhaps more importantly, both of these scales avoid things like the nasty 13 cents error between 12TET thirds and pure ones and are closer to the root of the harmonic series than scales under 12TET, supposedly making them more pure for both melody and harmony. I would not call them new concepts but, rather, I'm trying to point out that there are much more ways to solve the "12 tone modulation" problem than 12TET...and some of them, like those below, are very simple.
> >
> > So Rick, what do you believe 12TET supposedly solve so well while that the two
> > scales just mentioned or Marcel's 12-tone scale or John's....don't?
> >
>

On 10 Apr 2010, at 1:10 PM, rick wrote:
> . Even Helmholtz didn't realise the types of harmonies that would > come out later with Thelo nius Monk etc...
>

What's so special with Monk which was not invented before him?

Daniel Forro

Well Shakespeare's original. But he didn't invent English!

-Rick

--- In tuning@yahoogroups.com, Daniel Forró <dan.for@...> wrote:
>
>
> On 10 Apr 2010, at 1:10 PM, rick wrote:
> > . Even Helmholtz didn't realise the types of harmonies that would
> > come out later with Thelo nius Monk etc...
> >
>
> What's so special with Monk which was not invented before him?
>
> Daniel Forro
>

>"The sqrt2 for eg is the only value which is the same under inversion 1:sqrt2 = sqrt2:2."
This is a huge reason I used that as part of the generator for my Silver Sections scale.

>"If we take a rational version, say 45/32, then this never reaches the 8ve exactly: (45/32)^2 = 1.977"
Right, that's why we call it a temperament. :-) Any TET tuning has that same property though...it's certainly not exclusive to 12TET,

>"Perhaps what I'm getting at is that my 'ideal' system would not be an adaptive JI but an adaptive JI nested inside an ET tuning."
Right, then you get those "perfect transposition" properties. The problem is that often times a huge ET tuning is needed for this task. It sounds to me like you're
A) Really more a fan of equal temperament, not 12TET
B) Furthermore...more a fans of tempering notes than a fan of equal temperaments (and you need not use ET's to temper slightly off JI intervals). ET's just happen to do this automatically, but other tunings can be made to do this as well.

>"I notice when they have be detuned (Marcel's Beethoven would clash
whenever the diminished chords were hit, while the rest would sound
very beautiful)."
At the same time, when 12TET hits a major 3rd (which is very detuned vs. just), it has a similar disadvantage. One side note...I fear many on this list are trying to purify things like triads and 7th chords at the expense of things like diminished chords and should be doing the opposite (IE making "jazzy" chords sound more relaxed/accessible at the expense (though hopefully not much) of the consonance of things like triads). Almost all of my scales follow the "make the gap between the most consonant and dissonant chords smaller" credo.

>"In short, I feel that to make everything JI or rational is to fail to
distinguish between tonality = rational and atonality = irrational, and
that these atonalities have their own place in history and tuning in
their own right."
For the record, I don't even believe "religiously" in the whole tonality = rational and atonality = irrational bit. I'm a huge fan of de-tuning a tone from a JI scale in order to make it act like 2 JI intervals while falling with 7 cents or so of each (making the difference barely-noticeable).

I agree, they have their own history, but I feel you may be confusing the symptom with the cause. The "symptom" (as far as many people know it) is just 12TET, but the cause (as I see it) is the concept of temperament and IMVHO, there are many far better ways to get the advantages of temperament without JI.

Hi Mike,

Yeah I think you are right about tonal = rational, atonal = irrational. I used to think this because I knew that GCD's -> tonic. Since this requires rationals then it follows that irrationals cannot have a GCD and are therefore 'atonal'. The b5 = sqrt2 seemed to confirm this since its the most atonal interval i.e. the only interval which is the same under 8ve inversion.

But my recent discovery of approximate GCD's puts an end to this strict version. If we look at the wave of say the irrational minor third as 2^0.25 and solve for the first significant near period it gives ~GCD = (1 + 2^0.25)/(1 + (6/5)). IOW the formula seems to work for irrationals as well. (Though technically speaking the wave will never exactly repeat, I think what's going on is that it's a good approx over a short time length).

Even so, this doesn't really change what I said about equal roots of the 8ve, what I call symmetries, being atonal. It is here where the rationals can never reach but only ever approximate. For 12-tET, the diminished 7 or root 4 is essential to much western music as a modulation chord.

-Rick

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"The sqrt2 for eg is the only value which is the same under inversion 1:sqrt2 = sqrt2:2."
> This is a huge reason I used that as part of the generator for my Silver Sections scale.
>
> >"If we take a rational version, say 45/32, then this never reaches the 8ve exactly: (45/32)^2 = 1.977"
> Right, that's why we call it a temperament. :-) Any TET tuning has that same property though...it's certainly not exclusive to 12TET,
>
> >"Perhaps what I'm getting at is that my 'ideal' system would not be an adaptive JI but an adaptive JI nested inside an ET tuning."
> Right, then you get those "perfect transposition" properties. The problem is that often times a huge ET tuning is needed for this task. It sounds to me like you're
> A) Really more a fan of equal temperament, not 12TET
> B) Furthermore...more a fans of tempering notes than a fan of equal temperaments (and you need not use ET's to temper slightly off JI intervals). ET's just happen to do this automatically, but other tunings can be made to do this as well.
>
> >"I notice when they have be detuned (Marcel's Beethoven would clash
> whenever the diminished chords were hit, while the rest would sound
> very beautiful)."
> At the same time, when 12TET hits a major 3rd (which is very detuned vs. just), it has a similar disadvantage. One side note...I fear many on this list are trying to purify things like triads and 7th chords at the expense of things like diminished chords and should be doing the opposite (IE making "jazzy" chords sound more relaxed/accessible at the expense (though hopefully not much) of the consonance of things like triads). Almost all of my scales follow the "make the gap between the most consonant and dissonant chords smaller" credo.
>
> >"In short, I feel that to make everything JI or rational is to fail to
> distinguish between tonality = rational and atonality = irrational, and
> that these atonalities have their own place in history and tuning in
> their own right."
> For the record, I don't even believe "religiously" in the whole tonality = rational and atonality = irrational bit. I'm a huge fan of de-tuning a tone from a JI scale in order to make it act like 2 JI intervals while falling with 7 cents or so of each (making the difference barely-noticeable).
>
> I agree, they have their own history, but I feel you may be confusing the symptom with the cause. The "symptom" (as far as many people know it) is just 12TET, but the cause (as I see it) is the concept of temperament and IMVHO, there are many far better ways to get the advantages of temperament without JI.
>

Rick>"Yeah I think you are right about tonal = rational, atonal = irrational."
..."GCD's -> tonic (so) it follows that irrationals cannot have a GCD and are therefore 'atonal'. "

I would agree with this statement minus the fact that anything (including irrationals) within 7 cents or so of a rational interval is rounded to it by the brain and thus "made rational". Thus a "near GCD" can act as a GCD.

>"If we look at the wave of say the irrational minor third as 2^0.25 and
solve for the first significant near period it gives ~GCD = (1 +
2^0.25)/(1 + (6/5)). IOW the formula seems to work for irrationals as
well."
Cool, so it does allow for "temperament/de-tune/near-GCD" error and accounts for the brain's "7 cent" rounding process.

>"Even so, this doesn't really change what I said about equal roots of
the 8ve, what I call symmetries, being atonal."
...........
>"It is here where the
rationals can never reach but only ever approximate. For 12-tET, the
diminished 7 or root 4 is essential to much western music as a
modulation chord."
So let me get this right...you are implying that a-tonal essentially in this case means "between tonalities" and that the dim 7 chord essentially falls between two keys?

If so, makes sense but, again, my question is why does this kind of "in between GCD/tonality" partern have to be in only 12TET? Take the x/12 and x/18 harmonic series (two of my favorites because of in how many places they overlap).

If you, for example, treat the two series as two different keys you'll see they overlap exactly in many places and, with some de-tuning/temperament between two tones thrown in (noted as "averages at"), near intersect by about 12 cents error of each note or less (about the same error range as 12TET) on different notes:
x/12__________________x/18 (intersection or near intersection with x/12)
13/12 = 1.0833333
14/12 = 1.1666666______21/18 (exact intersection)
15/12 = 1.25___________23/18 = 1.2777777 average at 1.2636
16/12 = 1.333333333____24/18 (exact intersection)
17/12 = 1.41666666
18/12 = 1.5____________27/18(exact intersection)
19/12 = 1.583333333____28/18 = 1.555555 average at 1.57
20/12 = 1.666666666666 _30/18 (exact intersection)
21/12 = 1.75
22/12 = 1.833333333333_33/18 (exact intersection)
23/12 = 1.916666666666_34/18 = 1.8888888 average at 1.9
Also note, both 12 and 18 share the common factors of 2,3, and 6 which are well within the kind of low-denominator results your GCD algorithm shoots for (if I have it right).

While it is true that you're going to need to use different chords than the usual 12TET ones IE the diminished 7th to modulate in the same way, my point is the chords are still there and there are many ways to find intersections or near-intersections between two different GCD series that don't involve equal temperaments (and, again, certainly aren't limited to 12TET).

-Michael

Musical frequencies are percieved as rates (they are rates- measured in cylcles per second). So the proper average is not the arithmetic mean but the harmonic mean. Because so many averages (means) have been used in tuning through the ages, it is best to specify. And of course best to pick the mean that you really want.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Rick>"Yeah I think you are right about tonal = rational, atonal = irrational."
> ..."GCD's -> tonic (so) it follows that irrationals cannot have a GCD and are therefore 'atonal'. "
>
> I would agree with this statement minus the fact that anything (including irrationals) within 7 cents or so of a rational interval is rounded to it by the brain and thus "made rational". Thus a "near GCD" can act as a GCD.
>
> >"If we look at the wave of say the irrational minor third as 2^0.25 and
> solve for the first significant near period it gives ~GCD = (1 +
> 2^0.25)/(1 + (6/5)). IOW the formula seems to work for irrationals as
> well."
> Cool, so it does allow for "temperament/de-tune/near-GCD" error and accounts for the brain's "7 cent" rounding process.
>
> >"Even so, this doesn't really change what I said about equal roots of
> the 8ve, what I call symmetries, being atonal."
> ...........
> >"It is here where the
> rationals can never reach but only ever approximate. For 12-tET, the
> diminished 7 or root 4 is essential to much western music as a
> modulation chord."
> So let me get this right...you are implying that a-tonal essentially in this case means "between tonalities" and that the dim 7 chord essentially falls between two keys?
>
> If so, makes sense but, again, my question is why does this kind of "in between GCD/tonality" partern have to be in only 12TET? Take the x/12 and x/18 harmonic series (two of my favorites because of in how many places they overlap).
>
> If you, for example, treat the two series as two different keys you'll see they overlap exactly in many places and, with some de-tuning/temperament between two tones thrown in (noted as "averages at"), near intersect by about 12 cents error of each note or less (about the same error range as 12TET) on different notes:
> x/12__________________x/18 (intersection or near intersection with x/12)
> 13/12 = 1.0833333
> 14/12 = 1.1666666______21/18 (exact intersection)
> 15/12 = 1.25___________23/18 = 1.2777777 average at 1.2636
> 16/12 = 1.333333333____24/18 (exact intersection)
> 17/12 = 1.41666666
> 18/12 = 1.5____________27/18(exact intersection)
> 19/12 = 1.583333333____28/18 = 1.555555 average at 1.57
> 20/12 = 1.666666666666 _30/18 (exact intersection)
> 21/12 = 1.75
> 22/12 = 1.833333333333_33/18 (exact intersection)
> 23/12 = 1.916666666666_34/18 = 1.8888888 average at 1.9
> Also note, both 12 and 18 share the common factors of 2,3, and 6 which are well within the kind of low-denominator results your GCD algorithm shoots for (if I have it right).
>
> While it is true that you're going to need to use different chords than the usual 12TET ones IE the diminished 7th to modulate in the same way, my point is the chords are still there and there are many ways to find intersections or near-intersections between two different GCD series that don't involve equal temperaments (and, again, certainly aren't limited to 12TET).
>
> -Michael
>

>"Musical frequencies are percieved as rates (they are rates- measured in
cylcles per second). So the proper average is not the arithmetic mean
but the harmonic mean."

Cameron,
For sure, I am not using the arithmetic mean.
For the "mean", I am taking one frequency over the other, taking the square root of that, and then multiplying by the lower frequency. This way the same interval that separates the 1st and average frequency also separates the average frequency and 2nd (higher) frequency.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Musical frequencies are percieved as rates (they are rates- measured in
> cylcles per second). So the proper average is not the arithmetic mean
> but the harmonic mean."
>
> Cameron,
> For sure, I am not using the arithmetic mean.
> For the "mean", I am taking one frequency over the other, taking the square root of that, and then multiplying by the lower frequency. This way the same interval that separates the 1st and average frequency also separates the average frequency and 2nd (higher) frequency.
>

That works out to be the geometric mean.

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

> That works out to be the geometric mean.

...which is the arithmetic mean, logarithmically, of the intervals of course. forgot to mention.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Rick>"Yeah I think you are right about tonal = rational, atonal = irrational."
> ..."GCD's -> tonic (so) it follows that irrationals cannot have a GCD and are therefore 'atonal'. "
>
Mike>I would agree with this statement minus the fact that anything (including irrationals) within 7 cents or so of a rational interval is rounded to it by the brain and thus "made rational". Thus a "near GCD" can act as a GCD.

Yep, what I said was that I no longer believe tonal = rational, atonal = irrational. But it's more than just the fact that the interval differences lie within the 7 cents range. It's because this ~ GCD is actually a frequency within the composite wave.
>
Rick >"If we look at the wave of say the irrational minor third as 2^0.25 and
> solve for the first significant near period it gives ~GCD = (1 +
> 2^0.25)/(1 + (6/5)). IOW the formula seems to work for irrationals as
> well."
Mike> Cool, so it does allow for "temperament/de-tune/near-GCD" error and accounts for the brain's "7 cent" rounding process.

If we compare 80/64 = 5/4 with GCD = 16 and 81/64 with ~ GCD = (81 + 64)/(5 + 4) = (145/9) = 16.111...this implies that there's a 'give' in the harmonic series itself: 64/16.111...= 3.97... ~= 4 and 81/16.111...= 5.02...~= 5. Note that 4 - 4/145 = 3.97...and 5 + 4/145 = 5.02...This difference of 4/145, together with all the other possible major thirds, might be the key to understanding how much we can detune from the 'pure' harmonics. IOW the 7 cents or so difference just might be explained by wave theory and not as some type of 'fault' in human hearing.
>
> >"Even so, this doesn't really change what I said about equal roots of
> the 8ve, what I call symmetries, being atonal."
> ...........
> >"It is here where the
> rationals can never reach but only ever approximate. For 12-tET, the
> diminished 7 or root 4 is essential to much western music as a
> modulation chord."
> So let me get this right...you are implying that a-tonal essentially in this case means "between tonalities" and that the dim 7 chord essentially falls between two keys?

Yes it's a modulation chord that doesn't have a tonic. But it doesn't have a tonic precisely because every note bears the *same relation* to every other which is why they're called symmetric. In 12 ET its standard to symbolise the n in 2^(n/12) by 0,1,2,...11. (i.e. Since adding exponents is equivalent to multiplying intervals; 2^(n/12)*2^(m/12) = 2^((n+m)/12)). Now, the inverse of n is 12 - n. But since 12 = 0 (mod 12) then the inverse of n is -n. The dim 7 is 0:3:6:9. If we call the 3 the tonic 0 then it becomes -3:0:3:6 = 9:0:3:6. Next, 6 as tonic is -6:-3:0:3 = 6:9:0:3. Finally 9 = 0 gives 9:3:6:0. Since every note has equal claim to be the tonic then none of them are. If we put them back into exponential form then it becomes clear that this equalisation is *only exact for the irrational numbers*. Dim7 is 2^(0/12) = 1:2^(3/12) = 2^(1/4)etc... In 12-ET the possibilities are chromatic (12 semitones, 4ths or 5ths), whole-tone (6 tones), dim7 (4 minor thirds or major sixths), augmented (3 maj thirds or min 6ths), and the tritone, the only symmetric interval. Of course if we take any other ET then the same thing applies. 18-ET would be 0,1,2,...17 and 18 = 0.
>
> If so, makes sense but, again, my question is why does this kind of "in between GCD/tonality" pattern have to be in only 12TET? Take the x/12 and x/18 harmonic series (two of my favorites because of in how many places they overlap).

I hope I just answered that. It applies to any tuning 2^(x/y) where y is some whole number and x = 0,1,2,...(y - 1). Any number that divides into y an integral amount of times will be a symmetry. 18 for eg divides 1, 2, 3, 6 and 9 plus their inverses 17, 16, 15 and 12. Adding these numbers to itself will arrive back at 18 or a multiple thereof. But don't confuse this symmetry with the approx GCD's. These are trying to explain *tonalities* between intervals. 81 and 64 are close to the 5th and 4th harmonics of tonic 16.111...for eg.
>
> If you, for example, treat the two series as two different keys you'll see they overlap exactly in many places and, with some de-tuning/temperament between two tones thrown in (noted as "averages at"), near intersect by about 12 cents error of each note or less (about the same error range as 12TET) on different notes:
> x/12__________________x/18 (intersection or near intersection with x/12)
> 13/12 = 1.0833333
> 14/12 = 1.1666666______21/18 (exact intersection)
> 15/12 = 1.25___________23/18 = 1.2777777 average at 1.2636
> 16/12 = 1.333333333____24/18 (exact intersection)
> 17/12 = 1.41666666
> 18/12 = 1.5____________27/18(exact intersection)
> 19/12 = 1.583333333____28/18 = 1.555555 average at 1.57
> 20/12 = 1.666666666666 _30/18 (exact intersection)
> 21/12 = 1.75
> 22/12 = 1.833333333333_33/18 (exact intersection)
> 23/12 = 1.916666666666_34/18 = 1.8888888 average at 1.9
> Also note, both 12 and 18 share the common factors of 2,3, and 6 which are well within the kind of low-denominator results your GCD algorithm shoots for (if I have it right).

This doesn't seem like an unreasonable 12 note scale in its own right. But it's important not to confuse the fact that this has 12 notes with a 12-ET. The 14/12 = 7/6 flattened min 3rd for eg would give a very bad dim 7: (7/6)^2 = 49/36 = 1.36111... doesn't equate with your b5 as 17/12, (7/6)^3 = 343/216 = 1.5879..is fairly close to your 19/12, but (7/6)^4 = 2401/1296 = 1.8526...is way below the 8ve of 2.
>
> While it is true that you're going to need to use different chords than the usual 12TET ones IE the diminished 7th to modulate in the same way, my point is the chords are still there and there are many ways to find intersections or near-intersections between two different GCD series that don't involve equal temperaments (and, again, certainly aren't limited to 12TET).

Not at all, while the symmetries of dim7 are not unique to 12-ET they are essential to it. As I said, they will belong to any ET 2^(x/y) where y is divisible by 4.

-Rick
>
> -Michael
>

Rick>"Yep, what I said was that I no longer believe tonal = rational, atonal = irrational. But it's more than just the fact that the interval
differences lie within the 7 cents range. It's because this ~ GCD is
actually a frequency within the composite wave."
Ah...ok....makes much more sense now...now you are open about that concept of "periodic error tolerance".

>"IOW the 7 cents or so difference just might be explained by wave theory
and not as some type of 'fault' in human hearing. "
I can believe it...and best luck on proving it mathematically.

me>"
> x/12________ _________ _x/18 (intersection or near intersection
with x/12)
> 13/12 = 1.0833333
> 14/12 = 1.1666666___ ___21/18 (exact intersection)
> 15/12 = 1.25________ ___23/18 = 1.2777777 average at 1.2636
> 16/12 = 1.333333333_ ___24/18 (exact intersection)"

>"This doesn't seem like an unreasonable 12 note scale in its own right.
But it's important not to confuse the fact that this has 12 notes with a 12-ET. The 14/12 = 7/6 flattened min 3rd for eg would give a very bad
dim 7: (7/6)^2 = 49/36 = 1.36111... doesn't equate with your b5 as
17/12, (7/6)^3 = 343/216 = 1.5879..is fairly close to your 19/12, but
(7/6)^4 = 2401/1296 = 1.8526...is way below the 8ve of 2. "

You seem to be missing the point. I'm not trying to mimic ET-type intervals and I am not concerned how well I, say, mimic a diminished chord. Easy example, what is the 13/12 interval called under 12TET?...nothing, it (or any near equivalent) is not there.

I'm simply trying to show the idea of no single note having a claim to the tonic...just like it seems you have in your example only without an ET tuning. Between 18 and 12 we gets the factors of 18, 9,6,3,2 (for 18) and 12,6,4,3,2 (for 12)...and it's very easy to get lots of chords that point to many different tonics this way. Another way to say it: the scale includes many different separate/disparate harmonic series all pointing to different roots. You probably won't get "as" smooth modulations as you do under an ET-type tuning, but you can get close.

I'm looking at the rest of your proof now...may take me a while to fully digest it all.

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Is this tuning suitable for accidentals, modulations?

Depends on how brave you are--it has two fifths flat by a comma, and one sharp by a diaschisma.

Hi Mike,

Sorry I do seem to be missing the point. Are you taking ratios between 18 or 12 and these factors? But this would of course give the lower note as tonic. Explain it to me again with an example pleeeease.

Rick

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Rick>"Yep, what I said was that I no longer believe tonal = rational, atonal = irrational. But it's more than just the fact that the interval
> differences lie within the 7 cents range. It's because this ~ GCD is
> actually a frequency within the composite wave."
> Ah...ok....makes much more sense now...now you are open about that concept of "periodic error tolerance".
>
> >"IOW the 7 cents or so difference just might be explained by wave theory
> and not as some type of 'fault' in human hearing. "
> I can believe it...and best luck on proving it mathematically.
>
> me>"
> > x/12________ _________ _x/18 (intersection or near intersection
> with x/12)
> > 13/12 = 1.0833333
> > 14/12 = 1.1666666___ ___21/18 (exact intersection)
> > 15/12 = 1.25________ ___23/18 = 1.2777777 average at 1.2636
> > 16/12 = 1.333333333_ ___24/18 (exact intersection)"
>
>
> >"This doesn't seem like an unreasonable 12 note scale in its own right.
> But it's important not to confuse the fact that this has 12 notes with a 12-ET. The 14/12 = 7/6 flattened min 3rd for eg would give a very bad
> dim 7: (7/6)^2 = 49/36 = 1.36111... doesn't equate with your b5 as
> 17/12, (7/6)^3 = 343/216 = 1.5879..is fairly close to your 19/12, but
> (7/6)^4 = 2401/1296 = 1.8526...is way below the 8ve of 2. "
>
> You seem to be missing the point. I'm not trying to mimic ET-type intervals and I am not concerned how well I, say, mimic a diminished chord. Easy example, what is the 13/12 interval called under 12TET?...nothing, it (or any near equivalent) is not there.
>
> I'm simply trying to show the idea of no single note having a claim to the tonic...just like it seems you have in your example only without an ET tuning. Between 18 and 12 we gets the factors of 18, 9,6,3,2 (for 18) and 12,6,4,3,2 (for 12)...and it's very easy to get lots of chords that point to many different tonics this way. Another way to say it: the scale includes many different separate/disparate harmonic series all pointing to different roots. You probably won't get "as" smooth modulations as you do under an ET-type tuning, but you can get close.
>
>
> I'm looking at the rest of your proof now...may take me a while to fully digest it all.
>

Ok (trying to clarify the point after you/Rick noted some confusion), back to the scales...

x/12__________________x/18 (intersection or near intersection with x/12)
********************************************
12/12 18/18 (exact intersection)
13/12 = 1.0833333 ---other x/18 notes----
14/12 = 1.1666666______21/18 (exact intersection)
15/12 = 1.25___________23/18 = 1.2777777 average at
1.2636
16/12 = 1.333333333____24/18 (exact intersection)
17/12 =
1.41666666 ---other x/18 notes----
18/12 = 1.5____________27/18 (exact intersection)
19/12 = 1.583333333____28/18 = 1.555555 average at 1.57
20/12 =
1.666666666666 _30/18 (exact intersection)
21/12 =
1.75 ---other x/18 notes----
22/12 = 1.833333333333_33/18 (exact
intersection)
23/12 = 1.916666666666_34/18 = 1.8888888 average at 1.9

Now look at how many chords and tones are in common between both scales and can be used as transition points.

The 14:16:18:20 (AKA 7:8:9:10) chord in the x/12 scale matches exactly with the 21:24:27:30 chord in the x/18 scale.
Now if you "transpose" it to start at 16/12 you get 16:18:20:22 in x/12 and the exact match 24:27:30:33 in x/18.
And if again "transposed" to 22/12 you get a similar pattern. And, of course, same goes for starting at any of the 6-tones 14,16,18,20,22 in x/12...you can make chords starting at that point match. Those tetrachord-like 4-note chords may not be "diminished chords" or have exactly the same intervals in each instance of the chord, but that's not the point...the point is just how many chords you can match between different scales even without "perfect" TET style transposition.

And yes, I'm using 6 notes (not the full 7 in "common music") in this example, but if you also use the near matches IE the temperate averages near 23/18, 28/18, and 34/18 (barely any more tempered than 12TET is from JI)...you can get about 9 notes that all be used to form chords to use as transition points between the x/12 and x/18 scales.

If you look a step closer, you'll notice the intervals in the above chords share common LCD factors. Easy example: 16/12 (4/3) and 21/18 (7/6) share the common denominator of 6, an obvious common factor between 12 and 18. Another example: 16/12 (4/3) which shares its LCD with 20/12 (5/3).

In fact when you drill down you notice any combination of the notes that match have a LCD of either 6 or one of its factors IE the x/6 harmonic series (which contains the other common factors between 12 and 18, 3 and 2 IE x/3 and x/2). So anything in x/12 and x/18 that also fits x/6 can be used as a transition chord for modulation.

Wow, you disappeared from these parts for several years and made your grand re-entrance with that! I love it. Welcome back, Gene.

(You've probably already noticed that i'm scarce around here these days ... kid number 2 arrived a couple of months ago ... i've been busy ...)

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "jonszanto" <jszanto@> wrote:
>
> >Of course, I cut my teeth on some of Partch's guitars
>
> Did it hurt?
>

I get it now, thanks. The 12 note scale is b/w 12/12 and 24/12. Does the 18 note scale have 18 notes b/w 18/18 and 36/18?

Of course the point of a true ET would mean that these transpositions would be the same from every note, which is not what you're saying right? From a math'l point of view this would need 2^(1/2), 2^(1/3), 2^(1/4)...2^(1/n) which don't have a solution in rational numbers a/b [e.g. Pythagoras' proof that 2^(1/2) = a/b is a reductio ad absurdum].

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Ok (trying to clarify the point after you/Rick noted some confusion), back to the scales...
>
> x/12__________________x/18 (intersection or near intersection with x/12)
> ********************************************
> 12/12 18/18 (exact intersection)
> 13/12 = 1.0833333 ---other x/18 notes----
> 14/12 = 1.1666666______21/18 (exact intersection)
> 15/12 = 1.25___________23/18 = 1.2777777 average at
> 1.2636
> 16/12 = 1.333333333____24/18 (exact intersection)
> 17/12 =
> 1.41666666 ---other x/18 notes----
> 18/12 = 1.5____________27/18 (exact intersection)
> 19/12 = 1.583333333____28/18 = 1.555555 average at 1.57
> 20/12 =
> 1.666666666666 _30/18 (exact intersection)
> 21/12 =
> 1.75 ---other x/18 notes----
> 22/12 = 1.833333333333_33/18 (exact
> intersection)
> 23/12 = 1.916666666666_34/18 = 1.8888888 average at 1.9
>
>
> Now look at how many chords and tones are in common between both scales and can be used as transition points.
>
> The 14:16:18:20 (AKA 7:8:9:10) chord in the x/12 scale matches exactly with the 21:24:27:30 chord in the x/18 scale.
> Now if you "transpose" it to start at 16/12 you get 16:18:20:22 in x/12 and the exact match 24:27:30:33 in x/18.
> And if again "transposed" to 22/12 you get a similar pattern. And, of course, same goes for starting at any of the 6-tones 14,16,18,20,22 in x/12...you can make chords starting at that point match. Those tetrachord-like 4-note chords may not be "diminished chords" or have exactly the same intervals in each instance of the chord, but that's not the point...the point is just how many chords you can match between different scales even without "perfect" TET style transposition.
>
> And yes, I'm using 6 notes (not the full 7 in "common music") in this example, but if you also use the near matches IE the temperate averages near 23/18, 28/18, and 34/18 (barely any more tempered than 12TET is from JI)...you can get about 9 notes that all be used to form chords to use as transition points between the x/12 and x/18 scales.
>
> If you look a step closer, you'll notice the intervals in the above chords share common LCD factors. Easy example: 16/12 (4/3) and 21/18 (7/6) share the common denominator of 6, an obvious common factor between 12 and 18. Another example: 16/12 (4/3) which shares its LCD with 20/12 (5/3).
>
> In fact when you drill down you notice any combination of the notes that match have a LCD of either 6 or one of its factors IE the x/6 harmonic series (which contains the other common factors between 12 and 18, 3 and 2 IE x/3 and x/2). So anything in x/12 and x/18 that also fits x/6 can be used as a transition chord for modulation.
>

>"Does the 18 note scale have 18 notes b/w 18/18 and 36/18?"
Yes...I just skipped the ones that didn't have near matches with the 12/12-24/12 scale because the idea was to find possible points of modulation between the scales based on those two series.

>"Of course the point of a true ET would mean that these transpositions
would be the same from every note, which is not what you're saying
right?"
Right, but I'm focusing on modulation, not transposition...on doing things like swapping keys in the middle of a verse where x/12 counts as one key and x/24 as another. You gave the diminished 7th as a good modulation chord, as I understand it, so I gave an example listing lots of modulation chords in non-12TET scales with the idea "they aren't the same chords you'd use to do that in 12TET...but they work just as well for the purpose of modulation".

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> Wow, you disappeared from these parts for several years and made your grand re-entrance with that! I love it. Welcome back, Gene.
>
> (You've probably already noticed that i'm scarce around here these days ... kid number 2 arrived a couple of months ago ... i've been busy ...)

Congrats!

ok thanks. Of course in a true ET questions of modulation and transposition become the same thing. Since all notes are equal, each has a right to be called 0.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Does the 18 note scale have 18 notes b/w 18/18 and 36/18?"
> Yes...I just skipped the ones that didn't have near matches with the 12/12-24/12 scale because the idea was to find possible points of modulation between the scales based on those two series.
>
> >"Of course the point of a true ET would mean that these transpositions
> would be the same from every note, which is not what you're saying
> right?"
> Right, but I'm focusing on modulation, not transposition...on doing things like swapping keys in the middle of a verse where x/12 counts as one key and x/24 as another. You gave the diminished 7th as a good modulation chord, as I understand it, so I gave an example listing lots of modulation chords in non-12TET scales with the idea "they aren't the same chords you'd use to do that in 12TET...but they work just as well for the purpose of modulation".
>