Yahoo Tuning Groups Ultimate Backup tuning [tuning] What We've called Relative Cents

It was suggested maybe "elastic" would be a better term.
We've used "relative" because it's used
to express 100ths of a non-12 temperament.
Also there was a VERY BAD mathematical typo in that post.

Having never given a second thought to it,
after naming it Relative 10 years ago or so,
(to the Lab it's a very old concept...)
Possibly the most accurate word to use,
would be "proportional" cents.

Well here's the idea anyway, mathematically:
(if you can think of a suitable term, let us know)

Taking 12, 19 and 31 as an example:

Referencing the 3:2 interval:
in 12 the nearest note is 1.955/1200 of an octave flat.
in 19 the nearest note is 7.219/1200 of an octave flat.
in 31 the nearest note is 5.182/1200 of an octave flat.

The three results have no other use than
seeing which is more or less than the other.

However: (this is what we discovered)

Again, looking at the 3:2 -
in 12 the nearest note is 1.955/1200 of an octave flat.
in 19 the nearest note is 11.430/1900 of an octave flat.
in 31 the nearest note is 13.385/3100 of an octave flat.

...and 11.430 + 1.955 = 13.385!

The inaccuracy of a reference tone in the sum of two temperaments
in terms of "[YOUR ADJECTIVE HERE] Cents",
is equal to the sum of the inaccuracies
of the reference tone in each of the two temperaments.

Asbolute and Relative were the first thing that ever came to mind.
Static and Dynamic don't really fit either.
Elastic is a little more analog sounding...
Traditional and Proportional?
Looky, though,
I wouldn't expect a lot of people to see the use of this right away.
Took about a year or so to sink in. Found it by accident.

See - there are a few theorems and otherwise useful operations
that don't necessarily come from any isolateable principle,
but that definitely include the above example.
It pretty much drains back to modular arithmetic.

Evidently, if you were to pick some 8- or 9- digit number out of a hat,
that temperament would be able to approximate any interval
to *well* within a fraction of what you call a cent.
Any temperament higher than 1201
is automatically less than a cent away from everything.

After studying the resources of the Brun algorithm, though,
besides useful lists of temperaments in a vague sense,
and besides the hint of discreet web convergence, (q.v. a day or so ago)
there arises a sort of hint of magnetism,
(which for some reason we nicknamed it "polarity"... ha.)
wherein if you look at the [proportional] accuracies
of the driving intervals as the algorithm progresses,
it's not so much that obviously when you hit 78005
that everything's getting a lot closer just by increase in number,
but there's almost an inverse square effect,
when you see the [proportional] accuracies
are being systematically diminished.
This makes enough sense if you imagine the entire web convergence
of any one temperament, in an algortihm, for specific a set of intervals,
is going to be *more* accurate than the previous temperament,
this places more of a burden on the intervals to *be* more accurate.
(qv qv qv)

Again there's no almighty principle any of this teeters on,
it all just sort of floats back to modular arithmetic and such things.

Here's the makings of another theorlet:

7/12 is 1.955/1200 of an octave flat from 3:2.

if you cycle the 7/12, the next step as you know is 2/12...

2/12 is 3.911/1200 of an octave flat from 9:8.

But also, since:
7/12 is 1.955/1200 of an octave flat from 3:2, or
14/24 is 1.955/1200 of an octave flat from 3:2, then
14/24 is 3.911/2400 of an octave flat from 3:2.

2/12 is 3.911/1200 of an octave flat from 9:8, see, and
14/24 is 3.911/2400 of an octave flat from 3:2.

9/12 is 5.867/1200 of an octave flat from 27:16,
21/36 is 5.867/3600 of an octave flat from 3:2.

9/12 is 7.822/1200 of an octave flat from 81:64,
28/48 is 7.822/4800 of an octave flat from 3:2.

So, the number of times you can cycle an interval in a temperament
is proportional to what multiple of that temperament
still uses the same note to represent that interval. :)
The turning point in the example is a cycle of 25 fifths,
the nearest note to the 26th fifth is slightly closer;
In 300, the same note as 7/12 (175/300) is the nearest note to 3:2,
while in 312, the nearest note to 3:2 is 183, not 182.
At 25x, the numerator (1.955, 3.911 etc) is 48.89,
and at 26x, it's 50.84.
In more words, it's more than 50/1200 flat,
that is, it's more than a quartertone flat,
so the next note higher is actually the more accurate.

Take something like 19,
if you know that 38, 57, and 76 have the same nearest fifth,
but you know that 95 has a closer one (56/95, not 55)
then you can deduce that the fifth in 19
is only accurate after 4 turns of the cycle
and the fifth one will be off!
(that is, the 18/19 is actually closer to 243:128 than 17/19.)

The 5:4 in 146 is extremely accurate, at the 47th note.
But since it's an odd note,
this leaves you with both candidates for the 5:4 in 73
being both almost completely 50/7300 away.
That either candidate is only 8 cents (1200ths) away,
gives you a sort of arbitrary coin toss as to which to use.
In 118, where both 3:2 and 5:4 are scorchingly accurate,
it's easy enough to visualize 118 as a great 5th limit system.
But in terms of trying to separate behavior by harmonic tendencies,
since the 5:4 is almost 100% ambiguous in 73,
that kind of study might consider 73 more appropriate
for what would be a very 17tET-esque 3rd limit.

Orph-on-and-off

🔗paul@stretch-music.com

--- In tuning@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
>
> However: (this is what we discovered)
>
> Again, looking at the 3:2 -
> in 12 the nearest note is 1.955/1200 of an octave flat.
> in 19 the nearest note is 11.430/1900 of an octave flat.
> in 31 the nearest note is 13.385/3100 of an octave flat.
>
> ...and 11.430 + 1.955 = 13.385!

We've discussed that on the list. At least Paul Hahn and I have. When it comes up, it's usually
expressed in degrees.
So:
in 12 the nearest note is .01955 of a degree flat.
in 19 the nearest note is .11430 of a degree flat.
in 31 the nearest note is .13385 of a degree flat.

. . . and .01955 + .11430 = .13385.

>
> Any temperament higher than 1201
> is automatically less than a cent away from everything.

So is any temperament from 601-1200-tET.
>
> but there's almost an inverse square effect,
> when you see the [proportional] accuracies
> are being systematically diminished.

I've studied this and, by correcting for it, came up with some lists of "most special" ETs, back
when this list was on the Mills server.

> So, the number of times you can cycle an interval in a temperament
> is proportional to what multiple of that temperament
> still uses the same note to represent that interval. :)
> The turning point in the example is a cycle of 25 fifths,
> the nearest note to the 26th fifth is slightly closer;

Yes, that's because 53-tET is essentially just fifths, and 53/2 = 26.5.
>
> then you can deduce that the fifth in 19
> is only accurate after 4 turns of the cycle
> and the fifth one will be off!
> (that is, the 18/19 is actually closer to 243:128 than 17/19.)

The way I see it, we never hear 243:128 directly, but only as a construction by fifths and
possibly 9:8s . . . so functionally, 243:128 is always 17 degrees of 19-tET. But that's a very
interesting point when directly comparing the melodic distances of Pythagorean and 19-tET.

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

On 5/21/01 12:24 AM, "paul@stretch-music.com" <paul@stretch-music.com>
wrote:

> --- In tuning@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
>>
>> However: (this is what we discovered)
>>
>> Again, looking at the 3:2 -
>> in 12 the nearest note is 1.955/1200 of an octave flat.
>> in 19 the nearest note is 11.430/1900 of an octave flat.
>> in 31 the nearest note is 13.385/3100 of an octave flat.
>>
>> ...and 11.430 + 1.955 = 13.385!
>
> We've discussed that on the list. At least Paul Hahn and I have. When it comes
> up, it's usually
> expressed in degrees.
> So:
> in 12 the nearest note is .01955 of a degree flat.
> in 19 the nearest note is .11430 of a degree flat.
> in 31 the nearest note is .13385 of a degree flat.
>
> . . . and .01955 + .11430 = .13385.

Degree? I think I *have* heard that before.

>> Any temperament higher than 1201
>> is automatically less than a cent away from everything.
>
> So is any temperament from 601-1200-tET.

Bravo! I stand improved upon.
That's actually what I meant.
Math typos get that way...

>> but there's almost an inverse square effect,
>> when you see the [proportional] accuracies
>> are being systematically diminished.
>
> I've studied this and, by correcting for it, came up with some lists of "most
> special" ETs, back
> when this list was on the Mills server.

Have any reference? I mean post number or whatever,
are they still available?

>> So, the number of times you can cycle an interval in a temperament
>> is proportional to what multiple of that temperament
>> still uses the same note to represent that interval. :)

^-- *this* was the point though. Just a theorem.
Since you've talked about this using the word "degree",
it's an interesting way of using the information.
It's at the very least a mnemonic device
that can cut what you have to retain about temperaments in half.

>> The turning point in the example is a cycle of 25 fifths,
>> the nearest note to the 26th fifth is slightly closer;
>
> Yes, that's because 53-tET is essentially just fifths, and 53/2 = 26.5.

That kind of caught my eye the same way, speaking of mnemonics...
I think because of the sequential numbers and one of them being 26.
I'm glad you saw that the way you did, that's pretty interesting.

But...
No, actually it's because the 1.955% deviation in 12
is almost 1/51 of 100; and 51/2 = 25.5!!!
Which means two things,
that the fifth will wrap 25 times without being more than a quartertone off,
and the fifth will wrap 51 times withough being more than a semitone off.
Which by the above theorem, though, also implies two *more* things:
that the closest fifth in 25*12 (300) will be 25*7,
and the closest fifth in 26*12 (312) will *not* be 26*7. :)

Or actually it's because 12*26 is the first *multiple* of 12
that has a more accurate fifth than 12.
"Which is... oh yeah... 612." :)
I'd forgotten. And yes, 306 is up there.

So make it 5 things? No actually.

Because "actually", there isn't any specific "because" here,
like I was saying, it doesn't all go back to any one math principle.
It's just sort of an implied set of ways of looking at something,
a group of facts that are all corollaries of each other.
that float around commutativity like a molecule.
Big digression, I know, I just don't know if there's a name for it.

Anyway.

This is what I mean about numerology can't be monolithic!
I hate when numbers get close like this.
In my earlier days, when I first wondered what 31 + 34 was,
I came up with 67 and it messed up my work for 8 months.

But since you brought up ol' 53...!
Since the fifth in 53 is .003039 of a "degree" flat,
.5 / .003039 = 164.527805
So:
the fifth in 53 can wrap around 164 times before being misrepresented,
the closest fifth in 329*53 is 329*31,
the closest fifth in 330*53 is not 330*31.
10231/17490 is a little more accurate than 10230/17490.

>> then you can deduce that the fifth in 19
>> is only accurate after 4 turns of the cycle
>> and the fifth one will be off!
>> (that is, the 18/19 is actually closer to 243:128 than 17/19.)
>
> The way I see it, we never hear 243:128 directly, but only as a construction
> by fifths and
> possibly 9:8s . . . so functionally, 243:128 is always 17 degrees of 19-tET.
> But that's a very
> interesting point when directly comparing the melodic distances of Pythagorean
> and 19-tET.

Since 19 isn't as accurate,
I was using this to illustrate that the way 4 turns falls too flat,
with 4 as the threshold,
that while 19, 38, 57, and 76 share the same closest 5th,
95 has one that's closer.

...that the number of possible turns of an interval in a temperament
is the same of the multiple of that temperament that can handle *one* turn.

Why do I feel like a new list is going to start soon...

🔗paul@stretch-music.com

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

On 5/21/01 1:51 PM, "paul@stretch-music.com" <paul@stretch-music.com> wrote:

>> Since the fifth in 53 is .003039 of a "degree" flat,
>> .5 / .003039 = 164.527805
>> So:
>> the fifth in 53 can wrap around 164 times before being
> misrepresented,
>
> Isn't it 165? This is what Paul Hahn means when he says that 53-tET
> is level-165 consistent in the 3-limit. See, for example,
> http://library.wustl.edu/~manynote/consist.txt.

I don't really know his terminology,
sometimes you can't tell if someone's using
a 0-base or 1-base counting system,
but it looks like he might have rounded instead of truncated.
You know what I mean,
in seeing the way you have to basically divide,
to produce a chart like that you'd run some kind of program,
and I think I would have done just that,
rounded instead of truncated, since truncation is rare.

In other words, if he's counting the root, it's 165.
If you count the fifth as one,
the major second as two, etc, then it's 164.
Or you could say it's the 165th one that becomes inaccurate.

Cool chart though.
Worst case is some of the numbers might be one less.
The general idea is still there. I'll look into it.

I tend to only notice subtle discrepancies,
if it seems like I'm nitpicking.
In text I can't really tell.
It's a part of my mental imbalance.
I always catch obscure details but I miss main points.
There's a lot of *major* ideas I don't really grasp.
The 0-base/1-base problem is a serious issue, though.
I've found bugs in Quark XPress that relate to this;
as far as the way things line up on the screen vs the printer.