Equal beating marvel

A few years back I mentioned that rank three temperaments could be given equal beating properties, involving the 3, 5 and 7 all together in a tetrad. I not only mentioned it, but composed a piece using such a tuning in the 5120/5103 temperament. However, the place to start is probably with marvel, the one which tempers out 225/224.

Anyone can use this tuning who can tune the 5-limit, as it is simply a matter of retuning 3, 5, and 7. If we take the following polynomial equations

x^4+4x^3+4x^2+32x-320 = 0
y^4-4y^3+4y^2+32y-384 = 0
z^4-44z^3+724z^2-5312z+14400 = 0

we find that there is a solution for x close to 3, for y close to 5, and for z close to 7. Moreover, x^2 * y^2 / (32 z) comes to 1, so this is a marvel tuning, and we have the relationships y = x+2, z = 10-x which gives chords simple beat relationships. Here's your chance to be the first person in the world to use synch beating marvel!

Gene wrote:

> A few years back I mentioned that rank three temperaments could be given > equal beating properties,
> involving the 3, 5 and 7 all together in a tetrad. I not only mentioned > it, but composed a piece
> using such a tuning in the 5120/5103 temperament.

Is it possible to hear this somewhere?

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Gene wrote:
>
> > A few years back I mentioned that rank three temperaments could be given
> > equal beating properties,
> > involving the 3, 5 and 7 all together in a tetrad. I not only mentioned
> > it, but composed a piece
> > using such a tuning in the 5120/5103 temperament.
>
> Is it possible to hear this somewhere?

http://www.archive.org/search.php?query=Hemifamity

5120/5103 equates 64/63 and 81/80, right? So that the syntonic comma
and septimal commas both just become "a comma"?

If this doesn't have a name - I was screwing with this earlier in the
year, saying it might be good as some kind of "educational"
temperament. My reasoning was it would be good, conceptually, for
noobs at this stuff to just add the generic interval of a "comma" to
their vocabulary, and equate all such similarly-sized commas for this
purpose. It would also be nice if we could get a single "diesis" as
well, which would be 2 commas, or something like that. Not sure what a
"kleisma" would be? Maybe a kleisma would be 2 commas, and a diesis 3
commas?

I also thought that it might be best for this purpose if the most
common very small intervals, specifically the 5-limit schisma, were
tempered out. So this could possibly be worked in as 7-limit extension
of schismatic temperament as well.

Perhaps we could use some similarly-themed name here? So the rank-1
temperament equating 5120/5103 and 32805/32768 could be like the
"7-limit pedagogical temperament" or something like that.

-Mike

On Tue, May 11, 2010 at 3:04 AM, genewardsmith
<genewardsmith@...> wrote:
>
>
>
> A few years back I mentioned that rank three temperaments could be given equal beating properties, involving the 3, 5 and 7 all together in a tetrad. I not only mentioned it, but composed a piece using such a tuning in the 5120/5103 temperament. However, the place to start is probably with marvel, the one which tempers out 225/224.
>
> Anyone can use this tuning who can tune the 5-limit, as it is simply a matter of retuning 3, 5, and 7. If we take the following polynomial equations
>
> x^4+4x^3+4x^2+32x-320 = 0
> y^4-4y^3+4y^2+32y-384 = 0
> z^4-44z^3+724z^2-5312z+14400 = 0
>
> we find that there is a solution for x close to 3, for y close to 5, and for z close to 7. Moreover, x^2 * y^2 / (32 z) comes to 1, so this is a marvel tuning, and we have the relationships y = x+2, z = 10-x which gives chords simple beat relationships. Here's your chance to be the first person in the world to use synch beating marvel!

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@> wrote:
> >
> > Gene wrote:
> >
> > > A few years back I mentioned that rank three temperaments could be given
> > > equal beating properties,
> > > involving the 3, 5 and 7 all together in a tetrad. I not only mentioned
> > > it, but composed a piece
> > > using such a tuning in the 5120/5103 temperament.
> >
> > Is it possible to hear this somewhere?
>
> http://www.archive.org/search.php?query=Hemifamity
>

Velocity CC, man, velocity! And snap to grid off, and master timeline variation. These simple MIDI things would make a huge- night and day- difference, the piece would be quite nice. Otherwise your audience is only going to be those rare people who can "hear through" the physical rendition.

Gene wrote:

> http://www.archive.org/search.php?query=Hemifamity

Wow, this sounds so "unified" ... In fact, what I find most pleasant there is the fact that the intervals are so close to pure. :-D
On the same website, I managed to find some other pieces of yours ... I think this one may well be among my high favorites together with another one which I've already heard a few years ago -- the "Ostinato on a difference set".

Petr

Actually, upon reading that again, 2 commas would have to be a diesis,
so that 81/80 * 64/63 = 36/35. So it would be comma + comma = diesis.

It would be nice if the 36/35 diesis could be equated to the 128/125
diesis as well. Actually, after writing this, I just realized that
that's basically what marvel temperament is. So have I just
rediscovered marvel or something like that?

So this 7-limit pedagogical nonsense would be the equal temperament
where marvel and schismatic collide, and hence where 32805/32768,
5120/5103, and 225/224 are tempered out. I have no idea how to solve
this problem mathematically at all yet (I really need to finish Paul's
aMP), but after screwing around I see that this corresponds to
53-equal. Sweet.

Actually, after screwing around more I see that it also seems to work
for 41-equal, so the above set of 3 intervals must in some way be
redundant.

Damn, this stuff is confusing.

-Mike

On Tue, May 11, 2010 at 4:46 AM, Mike Battaglia <battaglia01@...> wrote:
> 5120/5103 equates 64/63 and 81/80, right? So that the syntonic comma
> and septimal commas both just become "a comma"?
>
> If this doesn't have a name - I was screwing with this earlier in the
> year, saying it might be good as some kind of "educational"
> temperament. My reasoning was it would be good, conceptually, for
> noobs at this stuff to just add the generic interval of a "comma" to
> their vocabulary, and equate all such similarly-sized commas for this
> purpose. It would also be nice if we could get a single "diesis" as
> well, which would be 2 commas, or something like that. Not sure what a
> "kleisma" would be? Maybe a kleisma would be 2 commas, and a diesis 3
> commas?
>
> I also thought that it might be best for this purpose if the most
> common very small intervals, specifically the 5-limit schisma, were
> tempered out. So this could possibly be worked in as 7-limit extension
> of schismatic temperament as well.
>
> Perhaps we could use some similarly-themed name here? So the rank-1
> temperament equating 5120/5103 and 32805/32768 could be like the
> "7-limit pedagogical temperament" or something like that.
>
> -Mike
>
>
> On Tue, May 11, 2010 at 3:04 AM, genewardsmith
> <genewardsmith@...> wrote:
>>
>>
>>
>> A few years back I mentioned that rank three temperaments could be given equal beating properties, involving the 3, 5 and 7 all together in a tetrad. I not only mentioned it, but composed a piece using such a tuning in the 5120/5103 temperament. However, the place to start is probably with marvel, the one which tempers out 225/224.
>>
>> Anyone can use this tuning who can tune the 5-limit, as it is simply a matter of retuning 3, 5, and 7. If we take the following polynomial equations
>>
>> x^4+4x^3+4x^2+32x-320 = 0
>> y^4-4y^3+4y^2+32y-384 = 0
>> z^4-44z^3+724z^2-5312z+14400 = 0
>>
>> we find that there is a solution for x close to 3, for y close to 5, and for z close to 7. Moreover, x^2 * y^2 / (32 z) comes to 1, so this is a marvel tuning, and we have the relationships y = x+2, z = 10-x which gives chords simple beat relationships. Here's your chance to be the first person in the world to use synch beating marvel!
>

Mike wrote:

> Perhaps we could use some similarly-themed name here? So the rank-1
> temperament equating 5120/5103 and 32805/32768 could be like the
> "7-limit pedagogical temperament" or something like that.

Rather rank 2 than 1.
If you start with 1:2:3:5:7, you have a 4D system. If you temper out one interval, you get a 3D system. If you temper out two, you get a 2D system. That means: If you temper out 5120/5103, you get hemifamity, which is what Gene talked about. If you temper out 32805/32768 together with that, you get 7-limit schismatic where 5/4 is approximated by a diminished fourth and 7/4 is approximated with a double-diminished octave.

You can try tempering out various commas on this webpage: http://x31eq.com/temper/uv.html

Petr

I wrote:

> That means: If you temper out 5120/5103, you get hemifamity, which is what
> Gene talked about. If you temper out 32805/32768 together with that, you > get
> 7-limit schismatic where 5/4 is approximated by a diminished fourth and > 7/4
> is approximated with a double-diminished octave.

In contrast, if you start with 1:2:3:5:7 and you *only* temper out 225/224, you get marvel. If you temper out 5120/5103 together with that, this also tempers out all the multiples and divisions of these two, including 32805/32768
(which is just one divided by the other), which means you get 7-limit schismatic again.

Petr

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> 5120/5103 equates 64/63 and 81/80, right? So that the syntonic comma
> and septimal commas both just become "a comma"?

Right. Another property is that three 9/8 major whole tones come to a 10/7, and the result is that this temperament (hemifamity, I called it) wants to stretch out in the 3 direction.

> Perhaps we could use some similarly-themed name here? So the rank-1
> temperament equating 5120/5103 and 32805/32768 could be like the
> "7-limit pedagogical temperament" or something like that.

Well, we've been calling it "garibaldi" on the basis of priority.

Gene,

I'd like to try it except I have no idea how to make a scala file out
of the polynominals.
Can you help?

Chris

On Tue, May 11, 2010 at 3:04 AM, genewardsmith
<genewardsmith@...> wrote:

>
> Anyone can use this tuning who can tune the 5-limit, as it is simply a matter of retuning 3, 5, and 7. If we take the following polynomial equations
>
> x^4+4x^3+4x^2+32x-320 = 0
> y^4-4y^3+4y^2+32y-384 = 0
> z^4-44z^3+724z^2-5312z+14400 = 0
>
> we find that there is a solution for x close to 3, for y close to 5, and for z close to 7. Moreover, x^2 * y^2 / (32 z) comes to 1, so this is a marvel tuning, and we have the relationships y = x+2, z = 10-x which gives chords simple beat relationships. Here's your chance to be the first person in the world to use synch beating marvel!

People very often argue about what makes good intervals for scales, namely things like low-limit fractions.

However an interval may be very low-limit, but 2/1 over that interval may not...and that second 2 over base interval, of course, always occurs in octave-based scales. Thus it seems obvious to me that evaluation intervals based on the criteria of how well they mirror about the octave is essential in any octave-based scale.

And, coincidentally, that "check" of reflecting an interval around the octave relates to the concept I discussed before of mirroring intervals...and Gene brought it up in a fairly recent post (I believe, about one of John's scales).

In creating my own scales I have began using a system which ONLY allows me to use intervals that reflect well about the octave. I figure there must be a formal name for this concept, is there?

Interval_____Mirror (AKA 2 / Interval)

1.06666______1.875
1.0909_______1.83333333
1.1111_______1.8
1.142________1.75
1.2__________1.6666
1.222________1.63636
1.25_________1.6
1.271________1.5714
1.32_________1.515
1.333________1.5 (easy example: 2/1 over 1.3333 = 1.5)
1.363________1.466666666
1.38461______1.444444444

Some COUNTER examples (IE of why some intervals simply don't fit into the structure)...

1.375 (11/8) doesn't work as 2 over 1.375 = 16/11...and 16/11 IMVHO sounds horrible.
1.35 (27/20) doesn't work as 2 over 1.35 = 40/27...and "......"
1.625 (13/8) doesn't work as 2 over 1.625 = 16/13...""

The easiest way to use such a structure is simply try to limit your scales to having intervals that fit in it (on either the interval or mirror side of the table). You can use just the interval or just the mirror...there's no "rule" forcing you to use both unless you are strategically trying to make a mirrored scale.

I also noticed that some fractions that don't work are very near ones which do (IE 13/8 AKA 1.625 is near 8/5 AKA 1.6). Perhaps this could help explain why is some cases even a few cents can make a world of difference and, in other cases, even a 13 cent error really doesn't sound all that bad.

-Michael

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Gene,
>
> I'd like to try it except I have no idea how to make a scala file out
> of the polynominals.
> Can you help?

How many notes, more or less, would you like in a scale?

lets say 17

On Tue, May 11, 2010 at 12:04 PM, genewardsmith <genewardsmith@...
> wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > Gene,
> >
> > I'd like to try it except I have no idea how to make a scala file out
> > of the polynominals.
> > Can you help?
>
> How many notes, more or less, would you like in a scale?
>
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> lets say 17

This ought to do it:

! dwarf17marveq.scl
Semimarvelous dwarf: equal beating dwarf(<17 27 40|)
17
!
70.247930173690388400
115.13195688812420070
185.37988706181458910
269.90670087373119520
314.79072758816500750
385.03865776185539590
500.17061464997959660
570.41854482366998500
615.30257153810379730
699.82938535002040340
770.07731552371079180
814.96134223814460410
885.20927241183499250
955.45720258552538090
1000.3412292999591932
1084.8680431118757993
1200.0000000000000000
! eight tetrads/pentads, representible by [[0, -1, 0], [0, -1, 1],
! [1, -1, 1], [1, -1, 2], [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]]

Thanks!

Do the selected frequencies represent the solution line for the polynominal
used to generate the tuning set?

Thanks,

Chris

On Tue, May 11, 2010 at 12:38 PM, genewardsmith <genewardsmith@...
> wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > lets say 17
>
> This ought to do it:
>
> ! dwarf17marveq.scl
> Semimarvelous dwarf: equal beating dwarf(<17 27 40|)
> 17
> !
> 70.247930173690388400
> 115.13195688812420070
> 185.37988706181458910
> 269.90670087373119520
> 314.79072758816500750
> 385.03865776185539590
> 500.17061464997959660
> 570.41854482366998500
> 615.30257153810379730
> 699.82938535002040340
> 770.07731552371079180
> 814.96134223814460410
> 885.20927241183499250
> 955.45720258552538090
> 1000.3412292999591932
> 1084.8680431118757993
> 1200.0000000000000000
> ! eight tetrads/pentads, representible by [[0, -1, 0], [0, -1, 1],
> ! [1, -1, 1], [1, -1, 2], [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]]
>
>
>

Michael,

here's another scale I developed about two weeks ago which uses mirroring. I call it the clever scale (or clever temperament after it has been tempered).

1/1
16/15
9/8
6/5
5/4
4/3
sqrt2
3/2
8/5
5/3
16/9
15/8
2/1

When tempering this scale if, for example, the third note *up* from 1/1 is raised by 'x' cents then the third note *down* from 2/1 should be lowered by 'x' cents, and so on. This preserves the symmetry. The sqrt2 should not be tempered.

For me the maximum deviation when tempering is 6.775 cents (256/255). I tested the 3/2 interval and found that if it as little as 8 cents out of tune I can detect some slight unpleasant beating. Within 6.775 cents it sounds fine.

My own NPT scale uses a different kind of mirroring (not around the octave). The 12 notes going up from 1/1 are the inverse of the 12 notes going down from 3/2 (two tonics here: 1/1 and 3/2). Check out chapter 7 of my book where this is explained.

BTW, did you check out my M scale?

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> People very often argue about what makes good intervals for scales, namely things like low-limit fractions.
>
> However an interval may be very low-limit, but 2/1 over that interval may not...and that second 2 over base interval, of course, always occurs in octave-based scales. Thus it seems obvious to me that evaluation intervals based on the criteria of how well they mirror about the octave is essential in any octave-based scale.
>
>
> And, coincidentally, that "check" of reflecting an interval around the octave relates to the concept I discussed before of mirroring intervals...and Gene brought it up in a fairly recent post (I believe, about one of John's scales).
>
>
>
>
> In creating my own scales I have began using a system which ONLY allows me to use intervals that reflect well about the octave. I figure there must be a formal name for this concept, is there?
>
> Interval_____Mirror (AKA 2 / Interval)
>
> 1.06666______1.875
> 1.0909_______1.83333333
> 1.1111_______1.8
> 1.142________1.75
> 1.2__________1.6666
> 1.222________1.63636
> 1.25_________1.6
> 1.271________1.5714
> 1.32_________1.515
> 1.333________1.5 (easy example: 2/1 over 1.3333 = 1.5)
> 1.363________1.466666666
> 1.38461______1.444444444
>
> Some COUNTER examples (IE of why some intervals simply don't fit into the structure)...
>
> 1.375 (11/8) doesn't work as 2 over 1.375 = 16/11...and 16/11 IMVHO sounds horrible.
> 1.35 (27/20) doesn't work as 2 over 1.35 = 40/27...and "......"
> 1.625 (13/8) doesn't work as 2 over 1.625 = 16/13...""
>
> The easiest way to use such a structure is simply try to limit your scales to having intervals that fit in it (on either the interval or mirror side of the table). You can use just the interval or just the mirror...there's no "rule" forcing you to use both unless you are strategically trying to make a mirrored scale.
>
>
> I also noticed that some fractions that don't work are very near ones which do (IE 13/8 AKA 1.625 is near 8/5 AKA 1.6). Perhaps this could help explain why is some cases even a few cents can make a world of difference and, in other cases, even a 13 cent error really doesn't sound all that bad.
>
> -Michael
>

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

> Do the selected frequencies represent the solution line for the polynominal
> used to generate the tuning set?

Marvel is rank three, like the five limit, so there aren't any lines. This is an equal beating tuning of a five-limit "dwarf" scale which turns out to be the 3^3 5^3 Euler genus with an added note. Start with 1, and go through the odd 5-limit integers in order, and test if
27 times the exponent of 3 plus 40 times the exponent of 5, reduced mod 17, has appeared already or not. If it hasn't, add it to your list, and stop when you have all mod 17 possibilities and can't add any more integers. Reduce to an octave, and transpose as desired, and you have Dwarf(<17 27 40}), which has lots of 5-limit harmony, and lots of things which temper well via marvel to 7 and 9 odd limit harmony.

Hi John,

John>"here's another scale I developed about two weeks ago which uses
mirroring. I call it the clever scale (or clever temperament after it
has been tempered).

1/1
16/15
9/8
6/5
5/4
4/3
sqrt2
3/2
8/5
5/3
16/9
15/8
2/1"

Indeed...everything in that scale reflects/mirrors around sqrt(2): it truly is a mirrored scale.

And a few of the resulting ratios you get here are great IMVHO IE 16/9 over 4/3 = 4/3 and conversely 3/2 over 9/8 = 4/3 (the mirrored side of that comparison). Same goes for 5/3 over 4/3 = 5/4 and 3/2 over 6/5 = 5/4...plus 8/5 over 4/3 = 6/5 and 3/2 over 5/4 = 6/5.

******************
One thing I think of for optimization is the fact the 16/9 over 3/2 = 13/11 (1.185185)...not exactly an ideal value for a "substitute minor third" type of tone. Another thing is the 5th from 9/8 is 9/8 * 3/2 = 27/16...not exactly a match with the nearest tone of 5/3.

Here's a way to solve both issues... :-)
If you took 3/2 * 6/5 = 9/5 (to replace the 16/9), then mirrored it to get 10/9 (which just happens to be a perfect 5th away from the 5/3 already in your scale), you'd get:

1/1
16/15
10/9
6/5
5/4
4/3
sqrt2
3/2
8/5
5/3
9/5
15/8
2/1"

This is pretty cool though...and is really making me think if the square root of 2 could be an excellent "mirror point" for future scales.

-Michael

Michael,

I had the idea of mirroring years ago but dropped it when I realised I would have to use sqrt2 as the middle note. As I said, my NPT scale uses a different type of mirroring where the 12 notes going up from 1/1 are the inverse of the 12 notes going down from 3/2 (two tonics: 1/1 and 3/2).

Your substitution of 10/9 for 9/8 looks good. 9/8 goes better with 1/1 than 10/9 but 9/5 goes better with 1/1 than 16/9 which is not a good interval.

Again, did you check out the M scale?

Also I know you like the 12/11 interval and I don't. My calculator gives it -1.96. I set the zero point according to my own taste but the hierarchy of intervals is still the same. If you want to give the calculator (v7.0, you might have an older faulty version) another go then just add 1.97 to each result so that the 12/11 is good.

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Hi John,
>
> John>"here's another scale I developed about two weeks ago which uses
> mirroring. I call it the clever scale (or clever temperament after it
> has been tempered).
>
> 1/1
> 16/15
> 9/8
> 6/5
> 5/4
> 4/3
> sqrt2
> 3/2
> 8/5
> 5/3
> 16/9
> 15/8
> 2/1"
>
> Indeed...everything in that scale reflects/mirrors around sqrt(2): it truly is a mirrored scale.
>
> And a few of the resulting ratios you get here are great IMVHO IE 16/9 over 4/3 = 4/3 and conversely 3/2 over 9/8 = 4/3 (the mirrored side of that comparison). Same goes for 5/3 over 4/3 = 5/4 and 3/2 over 6/5 = 5/4...plus 8/5 over 4/3 = 6/5 and 3/2 over 5/4 = 6/5.
>
> ******************
> One thing I think of for optimization is the fact the 16/9 over 3/2 = 13/11 (1.185185)...not exactly an ideal value for a "substitute minor third" type of tone. Another thing is the 5th from 9/8 is 9/8 * 3/2 = 27/16...not exactly a match with the nearest tone of 5/3.
>
> Here's a way to solve both issues... :-)
> If you took 3/2 * 6/5 = 9/5 (to replace the 16/9), then mirrored it to get 10/9 (which just happens to be a perfect 5th away from the 5/3 already in your scale), you'd get:
>
> 1/1
> 16/15
> 10/9
> 6/5
> 5/4
> 4/3
> sqrt2
> 3/2
> 8/5
> 5/3
> 9/5
> 15/8
> 2/1"
>
> This is pretty cool though...and is really making me think if the square root of 2 could be an excellent "mirror point" for future scales.
>
> -Michael
>

>"As I said, my NPT scale uses a different type of mirroring where the 12
notes going up from 1/1 are the inverse of the 12 notes going down from
3/2 (two tonics: 1/1 and 3/2)."

Bizarre, so you are mirroring around 2/1 AKA 1/1 to get the reverse...and then "translating" by 3/2 to get the mirrored part of the scale...if I understand it correctly. As a rule of thumb though...just about every scale I have found a mirror for within 13-cents or so of the original tones sounds better in the mirrored version...again I think it's just that it helps the brain keep track of the notes. Interestingly enough the chord 3:4:5:6 is also apparently mirrored in an odd way (5/3 over 4/3 = 6/4 over 6/5)...and so is 4:5:6:7 IE 6/4 over 5/4 (1.2) = 7/5 over 7/6.

John>"Your substitution of 10/9 for 9/8 looks good. 9/8 goes better with 1/1
than 10/9 but 9/5 goes better with 1/1 than 16/9 which is not a good
interval.
Thanks! :-) I hope that sort of thinking can prove useful to others in the future IE forgiving yourself slightly from using "only perfect" intervals IE 9/8 or more consonant can often make many other dyads near-pure. Often...many gains for only a slight loss.

>"Again, did you check out the M scale?"
As in the "Michael" scale? Tried it, but never got the files.
I still have to admit, there are a whole lot of very sour dyads in that scale...and most of the sweet dyads occur from the root tone. I had a tricky time composing anything with it...though I'd still like to hear what you composed with it

Hence why a lot of my scale touch-ups occur with my own computer program which compares all the possible dyads in a scale and warns me when certain dyads (regardless) of the root are in dissonant combinations. Since it's something near or over 100 comparisons per 7-tone scale to find the dyads...I'd highly recommend either making your own program to do this or using code from mine.

It's not hard...all I do is fill, say, an array of length 14 with the scale ratios and the scale ratios * 2 and then do (pseudo-code)
For index = 7 To 14
print("Note is " + notes(index))
print("2nd before this note is " + notes(index) / notes(index - 1))
print("3rd before this note is " + notes(index) / notes(index - 2))
print("4th before this note is " + notes(index) / notes(index - 3))
...............
next index

-Michael

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> 5120/5103, and 225/224 are tempered out.
> I have no idea how to solve
> this problem mathematically at all yet
> (I really need to finish Paul's
> aMP), but after screwing around I see that this corresponds to
> 53-equal.
>
Hi Mike,
I solved the the 5120/5103 transition of retruning back
from 7-limit towards returning 5-limit by the superparticlular
bisection of that interval over the prime-number 71

5120/5103 := [568/567]*[640/639] := [(71*8)/(7*81)]*[(5*128)/(71*9)]

in order to compensate the Ragisma, that had introduced 7-limit.

see also: the 5ths step numbers 44...45 in my "Ragismatic-53":

/tuning/topicId_88725.html#89111?var=0&l=1
"
!43:C& [567/512] |-9,4,0,1> (71 142..568>) 567.0 := 7 * 3^4
!44:G& [213/128] |-7,1>*71 213.0 :=71*3 = 639/3
!45:D& [5/4] |-2,0,1> 5.0 10 ... 320 640 (>639)
"

By that way:
The quotient inbetween (64/63)/(81/80) becomes subdivided into
5120/5103 := [568/567]*[640/639]

or that's in Cent units:

1200C * ln(5120/5103) / ln(2) = ~5.7578022....C
1200C * ln(568 / 567) / ln(2) = ~3.05063347...C
1200C * ln(640 / 639) / ln(2) = ~2.70716873...C

because an ~6 Cents to wide 5th appeared me as bit to harsh,
hence I decided to aplly the above epimoric decomposition.

bye
A.S.

What I composed with the M scale is just a run up and down the notes, each note played simulataneously with the tonic (all these pairs are good). I made the scale as different from 12TET as you can get. The file is called Mscale.mp3 in the JohnOSullivan folder in the "Files" section if you're still interested. The quality isn't great.

I intend to write a tempering program some time in the future but at the moment I'm working on jazzing up the calculator and rewriting my book using all the new stuff I've learned since I joined this group, a lot of it from your good self.

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"As I said, my NPT scale uses a different type of mirroring where the 12
> notes going up from 1/1 are the inverse of the 12 notes going down from
> 3/2 (two tonics: 1/1 and 3/2)."
>
> Bizarre, so you are mirroring around 2/1 AKA 1/1 to get the reverse...and then "translating" by 3/2 to get the mirrored part of the scale...if I understand it correctly. As a rule of thumb though...just about every scale I have found a mirror for within 13-cents or so of the original tones sounds better in the mirrored version...again I think it's just that it helps the brain keep track of the notes. Interestingly enough the chord 3:4:5:6 is also apparently mirrored in an odd way (5/3 over 4/3 = 6/4 over 6/5)...and so is 4:5:6:7 IE 6/4 over 5/4 (1.2) = 7/5 over 7/6.
>
> John>"Your substitution of 10/9 for 9/8 looks good. 9/8 goes better with 1/1
> than 10/9 but 9/5 goes better with 1/1 than 16/9 which is not a good
> interval.
> Thanks! :-) I hope that sort of thinking can prove useful to others in the future IE forgiving yourself slightly from using "only perfect" intervals IE 9/8 or more consonant can often make many other dyads near-pure. Often...many gains for only a slight loss.
>
> >"Again, did you check out the M scale?"
> As in the "Michael" scale? Tried it, but never got the files.
> I still have to admit, there are a whole lot of very sour dyads in that scale...and most of the sweet dyads occur from the root tone. I had a tricky time composing anything with it...though I'd still like to hear what you composed with it
>
>
>
> Hence why a lot of my scale touch-ups occur with my own computer program which compares all the possible dyads in a scale and warns me when certain dyads (regardless) of the root are in dissonant combinations. Since it's something near or over 100 comparisons per 7-tone scale to find the dyads...I'd highly recommend either making your own program to do this or using code from mine.
>
> It's not hard...all I do is fill, say, an array of length 14 with the scale ratios and the scale ratios * 2 and then do (pseudo-code)
> For index = 7 To 14
> print("Note is " + notes(index))
> print("2nd before this note is " + notes(index) / notes(index - 1))
> print("3rd before this note is " + notes(index) / notes(index - 2))
> print("4th before this note is " + notes(index) / notes(index - 3))
> ...............
> next index
>
> -Michael
>

Funny...even with just the dyads the sense of stability just isn't there...as if it keeps switching keys. Again I think the 11/8, 15/14, 8/7 and 7/6 (they fight with each other a bit over who has that tonal space) keep the scale from sounding consistent, even when just melodically ascending through triads.

I know I'm being a nudge...but it's kind of odd having a scale named after me that doesn't very well capture the sort of intervals I'm going for in a lot of ways. If you want to build a "Michael scale" that has MANY non-12TET interval I highly recommend using a selection from the following intervals:

1
1.090909
1.111111
1.22222
1.271
1.3636363
1.38461
1.4444444
1.4666666666
1.63636363
1.75
1.833333333333333

...and then worry about trying to get the other possible dyads to fit near the above intervals (described in my "octave over interval thread")...and mirroring...if you can manage it
'*********************************
1.06666______1.875
1.0909_______1.83333333
1.1111_______1.8
1.142________1.75
1.2__________1.6666
1.222________1.63636
1.25_________1.6
1.271________1.5714
1.32_________1.515
1.333________1.5 (easy example: 2/1 over 1.3333 = 1.5)
1.363________1.466666666
1.38461______1.444444444

-Michael

On Tue, May 11, 2010 at 5:23 AM, Petr Parízek <p.parizek@...> wrote:
>
> > Perhaps we could use some similarly-themed name here? So the rank-1
> > temperament equating 5120/5103 and 32805/32768 could be like the
> > "7-limit pedagogical temperament" or something like that.
>
> Rather rank 2 than 1.
> If you start with 1:2:3:5:7, you have a 4D system. If you temper out one
> interval, you get a 3D system. If you temper out two, you get a 2D system.
> That means: If you temper out 5120/5103, you get hemifamity, which is what
> Gene talked about. If you temper out 32805/32768 together with that, you get
> 7-limit schismatic where 5/4 is approximated by a diminished fourth and 7/4
> is approximated with a double-diminished octave.

Sorry, yes, rank-2 was what I meant. It was nearly 5 AM when I wrote
that, I was a little bit brain dead. I also meant the temperament
"eliminating" 5120/5103 and 32805/32768, not "equating" them (unless
we're equating them to 1/1).

> You can try tempering out various commas on this webpage:
> http://x31eq.com/temper/uv.html

I've never actually used that before. So putting in my commas yields
Garibaldi, with a complexity of 2.341, an error in cents of 2.037, and
a mysterious entry denoting "41 & 12". How is Garibaldi 41 + 12,
exactly...?

-Mike

On Tue, May 11, 2010 at 5:47 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > 5120/5103 equates 64/63 and 81/80, right? So that the syntonic comma
> > and septimal commas both just become "a comma"?
>
> Right. Another property is that three 9/8 major whole tones come to a 10/7, and the result is that this temperament (hemifamity, I called it) wants to stretch out in the 3 direction.

By "stretch out," you mean that other intervals are generally obtained
by moving in the 3-direction?

> > Perhaps we could use some similarly-themed name here? So the rank-1
> > temperament equating 5120/5103 and 32805/32768 could be like the
> > "7-limit pedagogical temperament" or something like that.
>
> Well, we've been calling it "garibaldi" on the basis of priority.

Ah, I wasn't aware of that. I thought that Garibaldi was just another
name for schismatic, not that it was specifically 7-limit schismatic.

-Mike

Hi Michael and John.

One doesn't have to use sqrt2 to use mirroring.
All the scales I create are infact mirrored scales.

Mirroring doesn't have to be at the octave. It can be at any harmonic
overtone of choice.

Mirroring in the 2nd harmonic:
1/1 2/1

Mirroring in the 3rd harmonic:
1/1 3/2 2/1 3/1

Mirroring in the 4th harmonic:
1/1 4/3 3/2 2/1 8/3 3/1 4/1

Mirroring in the 5th harmonic:
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

Mirroring in the 6th harmonic:
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
(1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 when reduced to one octave)

Mirroring in the 7th harmonic:
1/1 7/6 6/5 5/4 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
21/10 35/16 9/4 7/3 12/5 5/2 21/8 8/3 14/5 35/12 3/1 28/9 16/5 10/3 7/2 18/5
56/15 15/4 35/9 4/1 21/5 35/8 9/2 14/3 24/5 5/1 21/4 28/5 35/6 6/1 7/1

Mirroring in the 8th harmonic:
1/1 8/7 7/6 6/5 5/4 4/3 48/35 7/5 10/7 35/24 3/2 32/21 14/9 8/5 5/3 12/7
7/4 16/9 9/5 64/35 28/15 15/8 40/21 35/18 2/1 72/35 21/10 32/15 15/7 35/16
20/9 9/4 16/7 7/3 12/5 5/2 18/7 21/8 8/3 96/35 14/5 20/7 35/12 3/1 64/21
28/9 16/5 10/3 24/7 7/2 32/9 18/5 128/35 56/15 15/4 80/21 35/9 4/1 144/35
21/5 64/15 30/7 35/8 40/9 9/2 32/7 14/3 24/5 5/1 36/7 21/4 16/3 192/35 28/5
40/7 35/6 6/1 32/5 20/3 48/7 7/1 8/1

Notice btw, that each mirroring has all the mirroring of all the previous
harmonics still inside and doubled/transposed.

Marcel

When I presented my maths Gene you said things like "mathematical proofs need more than drawing pictures". Yet here you present a set of polynomial equations, seemingly plucked from out of nowhere, that have solutions which 'almost' equal a few whole numbers, all justified by some vague idea about "equal beating properties". Well where's your motivation and proof? Even assuming that it is mathematically viable, what's so musically important about these "equal beating properties"? For eg, some time ago I calculated that PHI is best represented in a 36-tET as 2^(25/36) and its inverse 2^(11/36). Since ratios = differences then we obtain the best equal beating possible. Yet at the end of the day these PHI intervals sounded awful. If I learned anything from this it was that intellectually interesting ideas do not necessarily translate into music. At any rate I'm not buying your maths here.

-Rick

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> A few years back I mentioned that rank three temperaments could be given equal beating properties, involving the 3, 5 and 7 all together in a tetrad. I not only mentioned it, but composed a piece using such a tuning in the 5120/5103 temperament. However, the place to start is probably with marvel, the one which tempers out 225/224.
>
> Anyone can use this tuning who can tune the 5-limit, as it is simply a matter of retuning 3, 5, and 7. If we take the following polynomial equations
>
> x^4+4x^3+4x^2+32x-320 = 0
> y^4-4y^3+4y^2+32y-384 = 0
> z^4-44z^3+724z^2-5312z+14400 = 0
>
> we find that there is a solution for x close to 3, for y close to 5, and for z close to 7. Moreover, x^2 * y^2 / (32 z) comes to 1, so this is a marvel tuning, and we have the relationships y = x+2, z = 10-x which gives chords simple beat relationships. Here's your chance to be the first person in the world to use synch beating marvel!
>

Rick>"For eg, some time ago I calculated that PHI is best represented in a
36-tET as 2^(25/36) and its inverse 2^(11/36). Since ratios =
differences then we obtain the best equal beating possible. Yet at the
end of the day these PHI intervals sounded awful."
I messed around with PHI a whole lot to get this sort of thing. The things is...I rounded PHI to the slightly-less-evil 1.625 or 13/8 interval and kept more of the equal beating around without the anti-periodic nature of an exact 1.618. But still...a lot of the resulting scale really clashed with JI and this alone hurt the scale much more than equal-beating could ever help it.
----------
Howver, I found another way to split scales to ratios = differences using the Silver Ratio, whose inverse IE 1/2.414 is 0.414 by taking (1/2.414)^x + 1 AND 1 / ((1/2.414)^x + 1) to create a "Silver Sections Scale" which just happens to have its period at sqrt(2) = 1.414 (ALA 12TET) and it's period squared at the octave. This way I supposedly got equal beating without completely disobeying the heck out of JI.
I must say, equal beating does help...although one must be very careful not to disobey JI to the point where periodicity problems far outweigh any benefits of equal beating. I say if you can get equal-beating without compromising JI-compliance much, more power to you!

On 12 May 2010 03:41, Mike Battaglia <battaglia01@...> wrote:

>> You can try tempering out various commas on this webpage:
>> http://x31eq.com/temper/uv.html
>
> I've never actually used that before. So putting in my commas yields
> Garibaldi, with a complexity of 2.341, an error in cents of 2.037, and
> a mysterious entry denoting "41 & 12". How is Garibaldi 41 + 12,
> exactly...?

It means that 41- and 12-equal are examples of the temperament class.
There are also links you can click, like the name.

Graham

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> When I presented my maths Gene you said things like "mathematical proofs need more than drawing pictures". Yet here you present a set of polynomial equations, seemingly plucked from out of nowhere, that have solutions which 'almost' equal a few whole numbers, all justified by some vague idea about "equal beating properties".

It's not a vague idea, it's a perfectly clear idea, it's simply that you haven't been following the conversation.

> Well where's your motivation and proof?

This isn't a math class, but since you demand mathematical proofs you are now stuck with them. If x is the root 2.9963... of the polynomial

x^4+4*x^3+4*x^2+32*x-320

then substituting y-2 for x into the polynomial shows that x+2 will be
a root of y^4-4*y^3+4*y^2+32*y-384, and substituting 10-z for x will prove that z^4-44*z^3+724*z^2-5312*z+14400 has a root equal to 10-x.

Now consider 225/224, which factors as 2^(-5) 3^2 5^2 7^(-1). Substituting x, x+2 and 10-x in place of 3, 5, and 7, and subtracting 1 yields (x^4+4*x^3+4*x^2+32*x-320)/(32 (10-x)), which by the polynomial satisfied by x equals zero. Hence, the tuning tempers out 225/224, and is therefore a marvel tuning. This is all in theory high school algebra, please note, but I doubt any of your teachers were ever evil enough to assign such problems.

Now consider beat ratios. The beat ratio of the fifth to the major third will be (4*(x+2)/4 - 5)/(2*(x/2)-3) = 1. The beat ratio of the fifth to the 7/4 will be (4*(10-x)/4 -7)/(2*(x/2)-3) = -1. The beat ratio of the third to the 7/4 will be (4*(10-x)/4-7)/(4*(x+2)/4-5) = -1. And so on and so forth. Inspecting the approximations to 3, 5, and 7 shows they are quite close, so this is an accurate tuning which tempers out 225/224 and has synchronized beating. QED.

>Even assuming that it is mathematically viable, what's so musically important about these "equal beating properties"?

Ask George Secor or Jacques Dudon. Or try it and see if it makes any difference to your ears.

>For eg, some time ago I calculated that PHI is best represented in a 36-tET as 2^(25/36) and its inverse 2^(11/36). Since ratios = differences then we obtain the best equal beating possible.

PROVE it. Where's your proof? Because this sure sounds like horseshit to me--"Best Possible???" And just last paragraph you were scoffing at the whole notion of equal beating!

"It's not a vague idea, it's a perfectly clear idea, it's simply that you haven't been following the conversation". That's my point Gene. What I was talking about was also something quite specific and clear. The fact that for eg you called (a + b)/(p + q) the "mediant" between a/b and p/q or that we can choose any (p, q) for a given (a, b) showed that you were assuming it must all already be 'well-known'. If you had have given it the time you'd have seen that *all* questions of beating are already implied by these approx GCD's. For eg it gives a precise method for rounding off your x and y below. (As for PHI, I simply meant that it is the only number who's inverse is equal to itself minus 1 so that the whole question of differences and ratios folds in on itself. But this mathematical beauty doesn't necessarily translate into music because it sounded awful).

--- In tuning@...m, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > When I presented my maths Gene you said things like "mathematical proofs need more than drawing pictures". Yet here you present a set of polynomial equations, seemingly plucked from out of nowhere, that have solutions which 'almost' equal a few whole numbers, all justified by some vague idea about "equal beating properties".
>
> It's not a vague idea, it's a perfectly clear idea, it's simply that you haven't been following the conversation.
>
> > Well where's your motivation and proof?
>
> This isn't a math class, but since you demand mathematical proofs you are now stuck with them. If x is the root 2.9963... of the polynomial
>
> x^4+4*x^3+4*x^2+32*x-320
>
> then substituting y-2 for x into the polynomial shows that x+2 will be
> a root of y^4-4*y^3+4*y^2+32*y-384, and substituting 10-z for x will prove that z^4-44*z^3+724*z^2-5312*z+14400 has a root equal to 10-x.
>
> Now consider 225/224, which factors as 2^(-5) 3^2 5^2 7^(-1). Substituting x, x+2 and 10-x in place of 3, 5, and 7, and subtracting 1 yields (x^4+4*x^3+4*x^2+32*x-320)/(32 (10-x)), which by the polynomial satisfied by x equals zero. Hence, the tuning tempers out 225/224, and is therefore a marvel tuning. This is all in theory high school algebra, please note, but I doubt any of your teachers were ever evil enough to assign such problems.
>
> Now consider beat ratios. The beat ratio of the fifth to the major third will be (4*(x+2)/4 - 5)/(2*(x/2)-3) = 1. The beat ratio of the fifth to the 7/4 will be (4*(10-x)/4 -7)/(2*(x/2)-3) = -1. The beat ratio of the third to the 7/4 will be (4*(10-x)/4-7)/(4*(x+2)/4-5) = -1. And so on and so forth. Inspecting the approximations to 3, 5, and 7 shows they are quite close, so this is an accurate tuning which tempers out 225/224 and has synchronized beating. QED.
>
> >Even assuming that it is mathematically viable, what's so musically important about these "equal beating properties"?
>
> Ask George Secor or Jacques Dudon. Or try it and see if it makes any difference to your ears.
>
> >For eg, some time ago I calculated that PHI is best represented in a 36-tET as 2^(25/36) and its inverse 2^(11/36). Since ratios = differences then we obtain the best equal beating possible.
>
> PROVE it. Where's your proof? Because this sure sounds like horseshit to me--"Best Possible???" And just last paragraph you were scoffing at the whole notion of equal beating!
>

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> "It's not a vague idea, it's a perfectly clear idea, it's simply that you haven't been following the conversation". That's my point Gene. What I was talking about was also something quite specific and clear.

You may have thought it was clear, but getting a precise statement out of you has proven to be seemingly impossible.

> The fact that for eg you called (a + b)/(p + q) the "mediant" between a/b and p/q or that we can choose any (p, q) for a given (a, b) showed that you were assuming it must all already be 'well-known'.

No, it showed I was trying to help you out. Probably I should quit.

>If you had have given it the time you'd have seen that *all* questions of beating are already implied by these approx GCD's.

There's an example of a term you haven't defined: "approximate GCD". Unless you can define terms like these you are really just babbling when your words reach someone else, even if you have a clear idea in your head as to what you mean.

>For eg it gives a precise method for rounding off your x and y below. (As for PHI, I simply meant that it is the only number who's inverse is equal to itself minus 1 so that the whole question of differences and ratios folds in on itself.

Firstly, it isn't. It's the only *positive* number with this property. Second, language like "the whole question of differences and ratios folds in on itself" is going to convey essentially nothing to someone who isn't a mind reader.

>But this mathematical beauty doesn't necessarily translate into music because it sounded awful).

Which has what, exactly, to do with synch beating?

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

Do the selected frequencies represent the solution line for the polynominal
used to generate the tuning set?

Gene:

Marvel is rank three, like the five limit, so there aren't any lines.
This is an equal beating tuning of a five-limit "dwarf" scale which
turns out to be the 3^3 5^3 Euler genus with an added note. Start with
1, and go through the odd 5-limit integers in order, and test if
27 times the exponent of 3 plus 40 times the exponent of 5, reduced
mod 17, has appeared already or not. If it hasn't, add it to your
list, and stop when you have all mod 17 possibilities and can't add
any more integers. Reduce to an octave, and transpose as desired, and
you have Dwarf(<17 27 40}), which has lots of 5-limit harmony, and
lots of things which temper well via marvel to 7 and 9 odd limit
harmony.

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

Hi Gene,

Are we talking about these equations still?

x^4+4x^3+4x^2+32x-320 = 0
y^4-4y^3+4y^2+32y-384 = 0
z^4-44z^3+724z^2-5312z+14400 = 0

we find that there is a solution for x close to 3, for y close to 5,
and for z close to 7. Moreover, x^2 * y^2 / (32 z) comes to 1, so this
is a marvel tuning, and we have the relationships y = x+2, z = 10-x
which gives chords simple beat relationships. Here's your chance to be
the first person in the world to use synch beating marvel!

Thanks,

Chris

Rick wrote:

> Even assuming that it is mathematically viable, what's so musically > important about these
> "equal beating properties"?

I'm surprised that you, someone who was so interested in all those overtone correlations and periodicity and other stuff, can say something like this. It seems you've probably never experimented with equal-beating chords, otherwise I would assume you'd be interested in both topics.
Even if it wasn't significantly meaningful for one particular piece of music, it is undoubtedly an useful method when tuning by ear since you just do the "tempering" (whatever you want to call it) in such a way that "this fifth beats as fast as that third and twice as fast as that third" and so on. And if the overtones are close enough to regular harmonics and if they are loud enough, this is indeed audible -- I've tried it on a harpsichord at school.
BTW: Some years ago, I've actually made some quasi-meantone equal-beating temperaments and if you have an efficient way to check your calculations while doing this, the process can be quite straight-forward -- if you have Manuel's scale archive, you can look for files starting with "parizek_qmeb"-something if you want further details.

> For eg, some time ago I calculated that PHI is best represented in a > 36-tET as 2^(25/36)
> and its inverse 2^(11/36). Since ratios = differences then we obtain the > best equal beating possible.
> Yet at the end of the day these PHI intervals sounded awful.

Mixing phi and octaves probably can't bring any particularly pleasant intervals because neither phi*2 nor phi/2 makes much sense acoustically. However, a chain of consecutive phis really can sound interesting because they essentially represent both difference and summation tones of each other. I've tried it some years ago and I have to admit that it more or less works --- with the sole exception that a piece of music in such a "phi tuning" can occur only once -- because trying to make one more would sound very similar.
And anyway, I'm still more attracted by periodicity rather than phi, so I probably won't be the one who makes the significant breakthrough. :-D

Petr

Not to worry Gene, come to think of it you did enter into the discussion later on and I probably assumed you'd been following so I didn't reiterate. My apologies.

If we add two sine waves like 64:80, then the wave oscillates at the GCD frequency 16. The component freq's equal the 4th and 5th harmonics to this fundamental. But if we take instead 64:81, although their GCD is now 1, something close to this 16 still remains in the wave. For this example it is equal to (64 + 81)/(4 + 5) = 16.111..., hence the term "approximate GCD".

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > "It's not a vague idea, it's a perfectly clear idea, it's simply that you haven't been following the conversation". That's my point Gene. What I was talking about was also something quite specific and clear.
>
> You may have thought it was clear, but getting a precise statement out of you has proven to be seemingly impossible.
>
> > The fact that for eg you called (a + b)/(p + q) the "mediant" between a/b and p/q or that we can choose any (p, q) for a given (a, b) showed that you were assuming it must all already be 'well-known'.
>
> No, it showed I was trying to help you out. Probably I should quit.
>
> >If you had have given it the time you'd have seen that *all* questions of beating are already implied by these approx GCD's.
>
> There's an example of a term you haven't defined: "approximate GCD". Unless you can define terms like these you are really just babbling when your words reach someone else, even if you have a clear idea in your head as to what you mean.
>
> >For eg it gives a precise method for rounding off your x and y below. (As for PHI, I simply meant that it is the only number who's inverse is equal to itself minus 1 so that the whole question of differences and ratios folds in on itself.
>
> Firstly, it isn't. It's the only *positive* number with this property. Second, language like "the whole question of differences and ratios folds in on itself" is going to convey essentially nothing to someone who isn't a mind reader.
>
> >But this mathematical beauty doesn't necessarily translate into music because it sounded awful).
>
> Which has what, exactly, to do with synch beating?
>