Is 19-et j.i.?

HI Kraig,

I've been thinking that this actually ties in with my
doctoral work on maths.

I was working on an aproach to maths that treats
infinite precision intervals as non standard finite
ratios - i.e. you have a ratio just like any other
one but the numbers on top and bottom are
non standard finite.

So in a way I was treating all intervals as
rational intervals.

The non standard finite numbers work just like the
ordinary finite ones, but are larger. It takes
some care to describe how the two types of numbers
work with each other.

However this is no more tricky really
than explaining how you can do calculations
using numbers with infinitely many decimal
places, which is what conventional maths
is based on. Both alas are not quite
intuitive.

If you could go back a few
centuries and try to explain the modern idea
of an inifinite precision decimal I think
you'd be met with soem scepticism too.
and if one were asked a few questions
by a sceptical mathematician of the time one
might be quite hard pressed to explain how
they can work, if one hadn't researched into
it. Easy to explain how to work with finite
decimals but not so easy to explain how infinite
decimals work and what you can do with them.
We don't usually need to do that kind of thing
in daily life - unless you are a mathematican
perhaps.

The way it works is that instead of building
an infinite decimal expansion outwards,
you group together the non standard ratios
into equivalence classes, by saying that the ones that
are within an "infinitesimal" of each other
are identical.

Anyway to take a more practical example,
I discovered while programaming FTS that
one degree of 19-et (i.e. the nineteenth
root of two) is very close to
a rational number.

2^(1/19) = 1.0371550444461919
419914/404871 = 1.0371550444463

I wonder if this is the closest one
can get to an equal temperament
note with, say, a ratio involving
only twelve digits in total,
as we have it here?

One thing I've wondered is if you
can get this close to an n-et note
using a ratio that is factorisable completely
into primes less than, say, 50.
If it can be done, it is certainly
fairly rare.

It would be quite easy to program
such a search actually - go through all
the n-ets up to say 100 or something
(most interesting for small ones)
and for each one check to find the
closest 50 limit ratios for all the
scale degrees - not a particularly
efficient way of doing it but a modern
computer would be fast enough to do that
so there wouldn't be that much incentive
to speed up the algorithm. Some time when
I have some leisure I plan to do it for fun.

All purely of theoretical interest
- the difference between that
19-et note and the rational
approximation is about one
beat every 6904473296317137
seconds at 440 herz,
which I figure out as
one beat every 218,939,411.98367 years
(probably the last few places can
be ignored here)

Robert

Robert,

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> All purely of theoretical interest
> - the difference between that
> 19-et note and the rational
> approximation is about one
> beat every 6904473296317137
> seconds at 440 herz,
> which I figure out as
> one beat every 218,939,411.98367 years
> (probably the last few places can
> be ignored here)

That last bit (between the parens) has to be one of the most unitentionally funny things I've read in a long time! "probably"?!?

Cheers,
Jon

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> I was working on an aproach to maths that treats
> infinite precision intervals as non standard finite
> ratios - i.e. you have a ratio just like any other
> one but the numbers on top and bottom are
> non standard finite.

I've long thought taking a non-standard model of the rationals and then moding out the ring of finite (according to some valuation) rationals by the maximal ideal defined by the corresponding infinitesimals is a neat way of constructing the reals and p-adics all in one fell swoop, but I'd be surprised to see a use for non-standard arithmetic in music theory, much as I'd like it if there was one. Perhaps this and the arithmetical questions you ask below could be moved off the main list to one of the tuning-math lists.

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:
> Robert,

> > one beat every 218,939,411.98367 years
> > (probably the last few places can
> > be ignored here)

> That last bit (between the parens) has to be one of the most unitentionally funny things I've read in a long time! "probably"?!?

Unintentionally? :)

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Unintentionally? :)

Educated guess, as Robert knows how to use emoticons...

<grin>,
Jon

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:

/tuning/topicId_38060.html#38064

> Robert,
>
> --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> > All purely of theoretical interest
> > - the difference between that
> > 19-et note and the rational
> > approximation is about one
> > beat every 6904473296317137
> > seconds at 440 herz,
> > which I figure out as
> > one beat every 218,939,411.98367 years
> > (probably the last few places can
> > be ignored here)
>
> That last bit (between the parens) has to be one of the most
unitentionally funny things I've read in a long time! "probably"?!?
>
> Cheers,
> Jon

***The thing is, though, Jon, you'd want to be there at that very
second, perhaps milisecond, or you might miss the beat after waiting
all that time!

Joe

Robert Walker wrote:

> HI Kraig,
>
> I've been thinking that this actually ties in with my
> doctoral work on maths.
>
> I was working on an aproach to maths that treats
> infinite precision intervals as non standard finite
> ratios - i.e. you have a ratio just like any other
> one but the numbers on top and bottom are
> non standard finite.
>
> So in a way I was treating all intervals as
> rational intervals.

It appears we have all type of infinite continuums ranging for JI generators to ET generators as
well as irrational generators. There is no reason to not map it out in an way one can think of. If
fact the more approaches beside stand ET and JI i think the better.

>
> The way it works is that instead of building
> an infinite decimal expansion outwards,
> you group together the non standard ratios
> into equivalence classes, by saying that the ones that
> are within an "infinitesimal" of each other
> are identical.

>
>
> Anyway to take a more practical example,
> I discovered while programaming FTS that
> one degree of 19-et (i.e. the nineteenth
> root of two) is very close to
> a rational number.
>
> 2^(1/19) = 1.0371550444461919
> 419914/404871 = 1.0371550444463

if you are asking how i would see this , I would say (being of a gestalt leaning psychology) I
would say you have a 19 ET scale if you were superimposing this interval. My understanding is the
mind will attempt to place any perception into its simplest form or combinations there of

>
>
> I wonder if this is the closest one
> can get to an equal temperament
> note with, say, a ratio involving
> only twelve digits in total,
> as we have it here?

One interesting feature i have found going the other direction with an ET being associated with an
irrational (phi). If one forms an 13 MOS of this interval you end up with a close interval with
the 12 ET semitone. Iy has made me wonder if all that circus music might be drawing upon this
interpretation of this interval. What could be more turning our world upside down that giving a
different meaning to what is already here.

>
>
> One thing I've wondered is if you
> can get this close to an n-et note
> using a ratio that is factorisable completely
> into primes less than, say, 50.
> If it can be done, it is certainly
> fairly rare.
>
> It would be quite easy to program
> such a search actually - go through all
> the n-ets up to say 100 or something
> (most interesting for small ones)
> and for each one check to find the
> closest 50 limit ratios for all the
> scale degrees - not a particularly
> efficient way of doing it but a modern
> computer would be fast enough to do that
> so there wouldn't be that much incentive
> to speed up the algorithm. Some time when
> I have some leisure I plan to do it for fun.
>
> All purely of theoretical interest
> - the difference between that
> 19-et note and the rational
> approximation is about one
> beat every 6904473296317137
> seconds at 440 herz,
> which I figure out as
> one beat every 218,939,411.98367 years
> (probably the last few places can
> be ignored here)

I think this type of difference might be of interest to Mayan Astronomers. In fact isn't this
close to one revolution of the Milky way? or am i about 400 million short?

One thing that has not been mentioned in there are assumption to JI that have not been touched
upon and that is that the ratios will be used in certain ways and that lattices and this has been
the practice so far. So while it is possible for some JI user to use 419914/404871 such an
interval would be reached by the simpler intervals in between.
Lou Harrison mentioned that JI has only examined half of it universe in that it has been scale
oriented as opposed to interval oriented. In the latter we would have a scale of intervals at hand
regardless of weather it forms a cohesive whole or not. This for me is out of my interest in that
i find expression in such structures that the intervals outline out.
It seems possible to have an array of different ET scales or intervals or a matrix of any
Irrationals from which to proceed through the continuum

>
>
> Robert
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
>
> ... Anyway to take a more practical example,
> I discovered while programaming FTS that
> one degree of 19-et (i.e. the nineteenth
> root of two) is very close to
> a rational number.
>
> 2^(1/19) = 1.0371550444461919
> 419914/404871 = 1.0371550444463

To answer the question in the title of this posting:

<< Is 19-et j.i.? >>

Absolutely not! JI includes only systems or sets of tones related by
intervals that are defined as small-number (i.e., rational) ratios.
This involves two concepts:

1) Small-number

How small the numbers must be in order for the ratios to be
considered "small" is open to debate, but that is not generally an
issue, because the specification of a harmonic limit (e.g., 11-limit
JI) automatically defines the consonances.

There is some point, however, at which a harmonic limit can be so
high as to be meaningless. I think that I can safely conclude that
you have exceeded it.

2) Ratio

Any tuning in which a rational interval (no matter how small)
vanishes is by definition a temperament, not a rational tuning.

Yes, there are temperaments that are so accurate that they are, for
all practical purposes, indistinguishable from JI (e.g., the 1/8-
schisma Helmholtzian temperament or its even more precise 1/9-schisma
variant), but they are still temperaments. And, yes, such a
temperament will be closer to exact JI than the very best efforts of
singers or musicians to perform a composition accurately in JI. And
who can argue but that the *intent* of the temperament and of the
performer is exactly the same: to produce accurate *representations*
(i.e., approximations) of rational intervals.

However, we could save ourselves a lot of grief by calling these low-
error temperaments "near-just" tunings or temperaments and refraining
from misrepresenting them as "just intonation", because, by
definition, they aren't JI. And using six-digit rational ratios to
represent an interval in some ET isn't going to make that a JI
interval -- it's not a consonance that falls within anyone's intended
harmonic limit.

May I suggest that (taking 72-ET as a point of departure) only
temperaments having an error less than 4 cents for all of the
consonances being represented within the specified harmonic limit
(and less than 2 cents for 2:3 and 3:4) be designated "near-just"?

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> >
> > ... Anyway to take a more practical example,
> > I discovered while programaming FTS that
> > one degree of 19-et (i.e. the nineteenth
> > root of two) is very close to
> > a rational number.
> >
> > 2^(1/19) = 1.0371550444461919
> > 419914/404871 = 1.0371550444463
>
> To answer the question in the title of this posting:
>
> << Is 19-et j.i.? >>
>
> Absolutely not! JI includes only systems or sets of tones related
by
> intervals that are defined as small-number (i.e., rational)
ratios.
> This involves two concepts:
>
> 1) Small-number
>
> How small the numbers must be in order for the ratios to be
> considered "small" is open to debate, but that is not generally an
> issue, because the specification of a harmonic limit (e.g., 11-
limit
> JI) automatically defines the consonances.
>
> There is some point, however, at which a harmonic limit can be so
> high as to be meaningless. I think that I can safely conclude that
> you have exceeded it.
>
> 2) Ratio
>
> Any tuning in which a rational interval (no matter how small)
> vanishes is by definition a temperament, not a rational tuning.
>
> Yes, there are temperaments that are so accurate that they are, for
> all practical purposes, indistinguishable from JI (e.g., the 1/8-
> schisma Helmholtzian temperament or its even more precise 1/9-
schisma
> variant),

used by eduardo sabat-garibaldi on his dinarra (guitar with 53 frets
per octave).

> but they are still temperaments. And, yes, such a
> temperament will be closer to exact JI than the very best efforts
of
> singers or musicians to perform a composition accurately in JI.
And
> who can argue but that the *intent* of the temperament and of the
> performer is exactly the same: to produce accurate
*representations*
> (i.e., approximations) of rational intervals.
>
> However, we could save ourselves a lot of grief by calling these
low-
> error temperaments "near-just" tunings or temperaments and
refraining
> from misrepresenting them as "just intonation", because, by
> definition, they aren't JI. And using six-digit rational ratios to
> represent an interval in some ET isn't going to make that a JI
> interval -- it's not a consonance that falls within anyone's
intended
> harmonic limit.

tell that to jacky! ;)

> May I suggest that (taking 72-ET as a point of departure) only
> temperaments having an error less than 4 cents for all of the
> consonances being represented within the specified harmonic limit
> (and less than 2 cents for 2:3 and 3:4) be designated "near-just"?
>
> --George

a chain of extremely near-just 5:3s is identical to 19-equal. so
maybe 19-equal is "juster" than you thought?

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> > To answer the question in the title of this posting:
> >
> > << Is 19-et j.i.? >>
> >
> > Absolutely not! JI includes only systems or sets of tones
related by
> > intervals that are defined as small-number (i.e., rational)
ratios.
>
> a chain of extremely near-just 5:3s is identical to 19-equal. so
> maybe 19-equal is "juster" than you thought?

Just a minute here: I already knew that, so your last sentence is
just not so -- 19-equal is just near-just, but not just.

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> Just a minute here: I already knew that, so your last sentence is
> just not so -- 19-equal is just near-just, but not just.

Just thought you should know that 2^(-14) 3^(-19) 5^19 is just a little comma of 2.8 cents, and we might justify calling a chain of
19 justly-tuned 5/3's a form of 19-equal, I suppose.

355/113 gets PI to within one part in over 11 million (almost 12
million). Any irrational number can be approximated to any desired
level of accuracy with a rational number. It means nothing.

Respectfully,

Bob

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> >
> > ... Anyway to take a more practical example,
> > I discovered while programaming FTS that
> > one degree of 19-et (i.e. the nineteenth
> > root of two) is very close to
> > a rational number.
> >
> > 2^(1/19) = 1.0371550444461919
> > 419914/404871 = 1.0371550444463
>
> To answer the question in the title of this posting:
>
> << Is 19-et j.i.? >>
>
> Absolutely not! JI includes only systems or sets of tones related
by
> intervals that are defined as small-number (i.e., rational)
ratios.
> This involves two concepts:
>
> 1) Small-number
>
> How small the numbers must be in order for the ratios to be
> considered "small" is open to debate, but that is not generally an
> issue, because the specification of a harmonic limit (e.g., 11-
limit
> JI) automatically defines the consonances.
>
> There is some point, however, at which a harmonic limit can be so
> high as to be meaningless. I think that I can safely conclude that
> you have exceeded it.
>
> 2) Ratio
>
> Any tuning in which a rational interval (no matter how small)
> vanishes is by definition a temperament, not a rational tuning.
>
> Yes, there are temperaments that are so accurate that they are, for
> all practical purposes, indistinguishable from JI (e.g., the 1/8-
> schisma Helmholtzian temperament or its even more precise 1/9-
schisma
> variant), but they are still temperaments. And, yes, such a
> temperament will be closer to exact JI than the very best efforts
of
> singers or musicians to perform a composition accurately in JI.
And
> who can argue but that the *intent* of the temperament and of the
> performer is exactly the same: to produce accurate
*representations*
> (i.e., approximations) of rational intervals.
>
> However, we could save ourselves a lot of grief by calling these
low-
> error temperaments "near-just" tunings or temperaments and
refraining
> from misrepresenting them as "just intonation", because, by
> definition, they aren't JI. And using six-digit rational ratios to
> represent an interval in some ET isn't going to make that a JI
> interval -- it's not a consonance that falls within anyone's
intended
> harmonic limit.
>
> May I suggest that (taking 72-ET as a point of departure) only
> temperaments having an error less than 4 cents for all of the
> consonances being represented within the specified harmonic limit
> (and less than 2 cents for 2:3 and 3:4) be designated "near-just"?
>
> --George

As most of us here know, 19- and 31-EDO are essentially equivalent to
1/3-comma and 1/4-comma meantone temperaments respectively.
These temperaments are defined by their just minor thirds and major
thirds respectively. To turn around and call either them or their ET
equivalents just seems to be begging the question, does it not?

Mildly perplexed,

Bob

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> > --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> > > To answer the question in the title of this posting:
> > >
> > > << Is 19-et j.i.? >>
> > >
> > > Absolutely not! JI includes only systems or sets of tones
> related by
> > > intervals that are defined as small-number (i.e., rational)
> ratios.
> >
> > a chain of extremely near-just 5:3s is identical to 19-equal. so
> > maybe 19-equal is "juster" than you thought?
>
> Just a minute here: I already knew that, so your last sentence is
> just not so -- 19-equal is just near-just, but not just.
>
> --George

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

> 355/113 gets PI to within one part in over 11 million (almost 12
> million).

More to the point around here, 835/827 as an approximation to 2^(1/72)
is off by only 1.45 * 10^(-8), which is more than sufficiently precise.

Any irrational number can be approximated to any desired
> level of accuracy with a rational number. It means nothing.

Can be useful--didn't the Hammond organ use rational approximations?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
> > 355/113 gets PI to within one part in over 11 million (almost 12
> > million).
>
> More to the point around here, 835/827 as an approximation to 2^
(1/72)
> is off by only 1.45 * 10^(-8), which is more than sufficiently
precise.
>
> > Any irrational number can be approximated to any desired
> > level of accuracy with a rational number. It means nothing.
>
> Can be useful--didn't the Hammond organ use rational approximations?

that's right. from a central 320 Hz gear, a full octave of 12-equal
(in A-440) is obtained using gear ratios (hence frequency ratios)
with at most two digits in the numerator and at most two digits in
the denominator. the maximum error is about 1 cent. i posted this a
while back. (source: barbour)

> "gdsecor" <gdsecor@yahoo.com>
>
> --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
>>
>> ... Anyway to take a more practical example,
>> I discovered while programaming FTS that
>> one degree of 19-et (i.e. the nineteenth
>> root of two) is very close to
>> a rational number.
>>
>> 2^(1/19) = 1.0371550444461919
>> 419914/404871 = 1.0371550444463
>
> To answer the question in the title of this posting:
>
> << Is 19-et j.i.? >>
>
> Absolutely not! JI includes only systems or sets of tones related by
> intervals that are defined as small-number (i.e., rational) ratios.

By "19-et", may I presume you (Robert) mean 19EDO (Equal
Divisions of the "Octave") ? If so, the it's up to you whether
it's "just" or not :-) But, 419914/404871 *is* within the
harmonic series.

I'm no math expert by any stretch; I just use it when I have to.
But, that is not a correct definition of a rational number, i.e.
the size of the number is of no consequence.

"rational: 5. Math. (of a quantity or ratio) expressible as a
ratio of whole numbers." (Oxford Concise Dictionary)

"...rational numbers, i.e. any number of the form p/q where p
and q are integers (positive or negative) and q ≠ (is not equal
to) 0." (Additional Mathematics, Patrick Murphy

"integer: 1 a whole number." (Oxford Concise Dictionary)

"whole number: a number without fractions; an integer." (Oxford
Concise Dictionary)

> This involves two concepts:
>
> 1) Small-number
>
> How small the numbers must be in order for the ratios to be
> considered "small" is open to debate, but that is not generally an
> issue, because the specification of a harmonic limit (e.g., 11-limit
> JI) automatically defines the consonances.
>

"Consonance" is a subjective impression, much like "Atonality".
The OED, for example, defines consonance as "a harmonious
combination of notes; a harmonious interval." One musicians
consonance is another's "eew @#%!". How many here have played
microtonal music to an unsuspecting lab rat, uh I mean friend,
to have her/him say it's "out of tune" ?

> There is some point, however, at which a harmonic limit can be so
> high as to be meaningless. I think that I can safely conclude that
> you have exceeded it.
>

But, one may be thinking simply in terms of the harmonics series
with no sense or need for a "limit".

> 2) Ratio
>
> Any tuning in which a rational interval (no matter how small)
> vanishes is by definition a temperament, not a rational tuning.
>

This is partly correct. First it is scales we're concerned in
this instance, not tunings. One tunes *to* a scale. Second, a
scale consisting of non-rational intervals is *not* "by
definition" a temperament. It may just be someone's (perfectly
valid) notion of a nice sounding bunch of notes. Plus, the
meaning of temperament in a musical context is (mostly) clearly
defined, i.e., as stated by the OED, "an adjustment of intervals
in tuning a piano etc. so as to fit the scale for use in all
keys".

The human brain, marvel that it is, finds inharmonic intervals
interesting & appealing, so non-rational intervals are
completely validated ("justified" ?!) in their existence. Please
note, I'm just stating a few things here not in direct response
to you, George, but to expand on thoughts. Anyway, the point is,
my initial idea is that scales not consisting solely of rational
(harmonic) intervals may be labelled as "inharmonic"
(irrational). My instinct is not to use "rational scale" or
"irrational scale" to avoid suggestions of ideology, however
vague, similar to the labelling of intervals and scales as
"Just". Plus, I think "harmonic" and "inharmonic" would more apt
in musical use.

Before anyone misconstructs this in a hurry, let me clarify:

I am not confusing "harmonic" and "inharmonic" with "consonance"
and "dissonance". I'm working with ref. again to the OED:

"harmonic: adj. 2 Mus. b (of a tone) produced by vibration of a
string etc. in an exact fraction of its length."

"inharmonic: adj. esp. Mus. not harmonic."

> Yes, there are temperaments that are so accurate that they are, for
> all practical purposes, indistinguishable from JI (e.g., the 1/8-
> schisma Helmholtzian temperament or its even more precise 1/9-schisma
> variant), but they are still temperaments. And, yes, such a
> temperament will be closer to exact JI than the very best efforts of
> singers or musicians to perform a composition accurately in JI. And
> who can argue but that the *intent* of the temperament and of the
> performer is exactly the same: to produce accurate *representations*
> (i.e., approximations) of rational intervals.

> However, we could save ourselves a lot of grief by calling these low-
> error temperaments "near-just" tunings or temperaments and refraining
> from misrepresenting them as "just intonation", because, by
> definition, they aren't JI.

"Low-error" (for their intended purpose) temperaments are (or
should be), by design, just that. However, there is something to
what you propose, but it's not quite so simple.

> And using six-digit rational ratios to
> represent an interval in some ET isn't going to make that a JI
> interval -- it's not a consonance that falls within anyone's intended
> harmonic limit.

This is subjective. As long as it's a rational interval, we're
still within the realm of the harmonic series, the sequence of
pitches being integer multiples of a fundamental (base)
frequency (the definition paraphrased from the article "Pianos
and Continued Fractions" by Edward G. Dunne). See above for my
thoughts on "consonance".

> May I suggest that (taking 72-ET as a point of departure) only
> temperaments having an error less than 4 cents for all of the
> consonances being represented within the specified harmonic limit
> (and less than 2 cents for 2:3 and 3:4) be designated "near-just"?
>

Interesting, but personally don't see it as a good idea if used
with a broad stroke, especially not for equal-intervalled
scales. Someone working with a scale with a Werntz-like
"xenharmonic" approach may not always be interested in
approximations to rational intervals. But for something like
meantone, well-temperaments, yes.

An additional stray thought - it may be perceived by some that,
"Just Intonation" is "practised", as it were, by a group of
people, as a school of thought. If so, it's practitioners may be
entitled to place a small number limit on the ratios that they
see as valid. That would allow for others to use and discuss the
musical use of the harmonic series as they please, leaving out
the term "Just", without upsetting anyone. OK, make that a very
stray thought...

> --George

- Joel

---
PS. The above are evolving personal opinions & views formed in a
continuing strive to clarify & simply my own thinking and work,
such that it is. Mostly just me thinking out loud.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
> > 355/113 gets PI to within one part in over 11 million (almost 12
> > million).
>
> More to the point around here, 835/827 as an approximation to 2^
(1/72)
> is off by only 1.45 * 10^(-8), which is more than sufficiently
precise.
>
> Any irrational number can be approximated to any desired
> > level of accuracy with a rational number. It means nothing.
>
> Can be useful--didn't the Hammond organ use rational approximations?

Bob:
Never said it wasn't useful. Rational approximations can be
wonderfully useful. My statement that "in means nothing" is to be
taken only in the context in which it was embedded, namely that a
rational approximation of an irrational number is absolutely
meaningless in terms of establishing the justness of an interval.

I have used rational approximations for the tangents of angles to
draw very precise angles on graph paper simply by using the results
to count horizontal and vertical squares. The errors due to
approximation were always much less than the mechanical errors
inherent in doing it any other way, certainly much less than the
width of a pencil line. No one taught me this. I just came up with it
to solve a problem. So I know they're useful, having used them many
other ways since.

--- In tuning@y..., Joel Rodrigues <joelrodrigues@m...> wrote:

> "rational: 5. Math. (of a quantity or ratio) expressible as a
> ratio of whole numbers." (Oxford Concise Dictionary)
>
> "...rational numbers, i.e. any number of the form p/q where p
> and q are integers (positive or negative) and q ≠ (is not equal
> to) 0." (Additional Mathematics, Patrick Murphy
>
> "integer: 1 a whole number." (Oxford Concise Dictionary)
>
> "whole number: a number without fractions; an integer." (Oxford
> Concise Dictionary)

This isn't going to help much if we allow non-standard integers. :)

> "Consonance" is a subjective impression, much like "Atonality".

It means something a little different in discussions of "odd limit" harmoni=
es, which was Robert's point. As for atonality, why is that a subjective imp=
ression? It seems to me you could define it statistically if you wanted to, =
as a score which had no tonal center for lengths of time beyond some small a=
mount. It certainly would be easy enough to produce triadic harmony with no =
tonal center, and whatever subjective impression it gave, it would be atonal=
.

> This is subjective. As long as it's a rational interval, we're
> still within the realm of the harmonic series, the sequence of
> pitches being integer multiples of a fundamental (base)
> frequency (the definition paraphrased from the article "Pianos
> and Continued Fractions" by Edward G. Dunne). See above for my
> thoughts on "consonance".

You are mixing up two different things here--the rational interval in your =
head, and the actual physical sound. Rational intervals in your head can be =
anything which fits in your head, but the series of partial tones does not g=
o on to infinity and consists of integer multiples of a fundamental only wit=
hin some margin.

Sorry if it took several days to reply to this, but I just got this
in digest form in my e-mail this morning. (Call it e-snail-mail.)

--- In tuning@y..., Joel Rodrigues <joelrodrigues@m...> wrote:
> > "gdsecor" <gdsecor@y...>
> >
> > --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> >>
> >> ... Anyway to take a more practical example,
> >> I discovered while programaming FTS that
> >> one degree of 19-et (i.e. the nineteenth
> >> root of two) is very close to
> >> a rational number.
> >>
> >> 2^(1/19) = 1.0371550444461919
> >> 419914/404871 = 1.0371550444463
> >
> > To answer the question in the title of this posting:
> >
> > << Is 19-et j.i.? >>
> >
> > Absolutely not! JI includes only systems or sets of tones
related by
> > intervals that are defined as small-number (i.e., rational)
ratios.
>
> By "19-et", may I presume you (Robert) mean 19EDO (Equal
> Divisions of the "Octave") ? If so, the it's up to you whether
> it's "just" or not :-) But, 419914/404871 *is* within the
> harmonic series.

Theoretically, yes; for all practical purposes, no. You can't hear a
harmonic that high in any musical tone because:

1) It wouldn't have sufficient amplitude;

2) Either it or its fundamental (or both) would be out of our range
of hearing.

> I'm no math expert by any stretch; I just use it when I have to.
> But, that is not a correct definition of a rational number, i.e.
> the size of the number is of no consequence.

The size of the number is very important when you are dealing with
musical acoustics. What makes just intervals or chords "just"? It's
the elimination of roughness, i.e., perceived beating between off-
tuned harmonics or combination tones. Once the numbers reach a
certain size, intervals begin to lose their identities, at which
point they begin to sound like approximations of lower-numbered
ratios. Then they can no longer be considered distinct consonances.
So there is a practical limit to which one can take a harmonic limit.

Taking a string of tones separated by 5:6's and calling the interval
between the tones at opposite ends a "just" interval does not do
justice to the term "just". It's a rational interval, but it's not
a "just" consonance, and using it in a chord is not going to make the
chord sound "just".

This is why the Hammond organ could have so-called "rational"
intervals, yet could still be considered (for all intents and
purposes) to be in 12-ET -- it was approximating an ET, which is in
turn an approximation.

> "rational: 5. Math. (of a quantity or ratio) expressible as a
> ratio of whole numbers." (Oxford Concise Dictionary)
>
> "...rational numbers, i.e. any number of the form p/q where p
> and q are integers (positive or negative) and q ≠(is not equal
> to) 0." (Additional Mathematics, Patrick Murphy
>
> "integer: 1 a whole number." (Oxford Concise Dictionary)
>
> "whole number: a number without fractions; an integer." (Oxford
> Concise Dictionary)
>
> > This involves two concepts:
> >
> > 1) Small-number
> >
> > How small the numbers must be in order for the ratios to be
> > considered "small" is open to debate, but that is not generally an
> > issue, because the specification of a harmonic limit (e.g., 11-
limit
> > JI) automatically defines the consonances.
>
> "Consonance" is a subjective impression, much like "Atonality".
> The OED, for example, defines consonance as "a harmonious
> combination of notes; a harmonious interval." One musicians
> consonance is another's "eew @#%!". How many here have played
> microtonal music to an unsuspecting lab rat, uh I mean friend,
> to have her/him say it's "out of tune" ?

In rational intervals consonance is closely correlated with the size
of the numbers in the ratio -- the larger the numbers, the less
consonant the interval -- so long as the interval doesn't lose its
identity and start sounding like an approximation of some lower-
numbered ratio. There is some subjectivity, but it involves:

1) At what point (or p-harmonic limit) one no longer recognizes
intervals as consonant; and

2) At what point one no longer recognizes intervals as having
distinct identities.

Regarding one's unsuspecting friends: one of the challenges of
writing a microtonal composition is to use new intervals without
making the result sound as if it is out of tune. The problem is
greatest when using temperaments in which the intervals have large
errors, which is a good rationale for the employment of JI or near-
just temperaments.

> > There is some point, however, at which a harmonic limit can be so
> > high as to be meaningless. I think that I can safely conclude
that
> > you have exceeded it.
>
> But, one may be thinking simply in terms of the harmonics series
> with no sense or need for a "limit".

Then one has failed to recognize that the extent to which a harmonic
limit may be successfully employed is not unlimited.

> > 2) Ratio
> >
> > Any tuning in which a rational interval (no matter how small)
> > vanishes is by definition a temperament, not a rational tuning.
> >
>
> This is partly correct. First it is scales we're concerned in
> this instance, not tunings. One tunes *to* a scale. Second, a
> scale consisting of non-rational intervals is *not* "by
> definition" a temperament. It may just be someone's (perfectly
> valid) notion of a nice sounding bunch of notes. Plus, the
> meaning of temperament in a musical context is (mostly) clearly
> defined, i.e., as stated by the OED, "an adjustment of intervals
> in tuning a piano etc. so as to fit the scale for use in all
> keys".

Where was it that we were talking about scales?

In stating that temperaments are non-rational tunings, one would not
logically conclude that all non-rational tunings are temperments.
The essence of a temperament is to represent two rational intervals
by a single (tempered) interval that serves as an approximation for
either or both; the difference between the two rational intervals is
called a comma, which vanishes in that temperament. This is not in
conflict with the OED definition you gave, although it is a bit
broader.

> The human brain, marvel that it is, finds inharmonic intervals
> interesting & appealing, so non-rational intervals are
> completely validated ("justified" ?!) in their existence. Please
> note, I'm just stating a few things here not in direct response
> to you, George, but to expand on thoughts. Anyway, the point is,
> my initial idea is that scales not consisting solely of rational
> (harmonic) intervals may be labelled as "inharmonic"
> (irrational). My instinct is not to use "rational scale" or
> "irrational scale" to avoid suggestions of ideology, however
> vague, similar to the labelling of intervals and scales as
> "Just". Plus, I think "harmonic" and "inharmonic" would more apt
> in musical use.
>
> Before anyone misconstructs this in a hurry, let me clarify:
>
> I am not confusing "harmonic" and "inharmonic" with "consonance"
> and "dissonance". I'm working with ref. again to the OED:
>
> "harmonic: adj. 2 Mus. b (of a tone) produced by vibration of a
> string etc. in an exact fraction of its length."
>
> "inharmonic: adj. esp. Mus. not harmonic."

The term "inharmonic" is generally applied to timbres having non-
harmonic partials, not to scales or tunings.

> > Yes, there are temperaments that are so accurate that they are,
for
> > all practical purposes, indistinguishable from JI (e.g., the 1/8-
> > schisma Helmholtzian temperament or its even more precise 1/9-
schisma
> > variant), but they are still temperaments. And, yes, such a
> > temperament will be closer to exact JI than the very best efforts
of
> > singers or musicians to perform a composition accurately in JI.
And
> > who can argue but that the *intent* of the temperament and of the
> > performer is exactly the same: to produce accurate
*representations*
> > (i.e., approximations) of rational intervals.
>
> > However, we could save ourselves a lot of grief by calling these
low-
> > error temperaments "near-just" tunings or temperaments and
refraining
> > from misrepresenting them as "just intonation", because, by
> > definition, they aren't JI.
>
> "Low-error" (for their intended purpose) temperaments are (or
> should be), by design, just that. However, there is something to
> what you propose, but it's not quite so simple.
>
> > And using six-digit rational ratios to
> > represent an interval in some ET isn't going to make that a JI
> > interval -- it's not a consonance that falls within anyone's
intended
> > harmonic limit.
>
> This is subjective. As long as it's a rational interval, we're
> still within the realm of the harmonic series, the sequence of
> pitches being integer multiples of a fundamental (base)
> frequency (the definition paraphrased from the article "Pianos
> and Continued Fractions" by Edward G. Dunne). See above for my
> thoughts on "consonance".

I responded to this above under "small-number".
>
> > May I suggest that (taking 72-ET as a point of departure) only
> > temperaments having an error less than 4 cents for all of the
> > consonances being represented within the specified harmonic limit
> > (and less than 2 cents for 2:3 and 3:4) be designated "near-just"?
>
> Interesting, but personally don't see it as a good idea if used
> with a broad stroke, especially not for equal-intervalled
> scales. Someone working with a scale with a Werntz-like
> "xenharmonic" approach may not always be interested in
> approximations to rational intervals. But for something like
> meantone, well-temperaments, yes.

As long as the ET fits the definition, why shouldn't it be included?
(So what if someone wants to use it differently?)

Meantone temperament and well-temperaments have errors large enough
that I wouldn't call them near-just. The requirement is that the
consonances be slow-beating, relatively speaking.

> An additional stray thought - it may be perceived by some that,
> "Just Intonation" is "practised", as it were, by a group of
> people, as a school of thought. If so, it's practitioners may be
> entitled to place a small number limit on the ratios that they
> see as valid. That would allow for others to use and discuss the
> musical use of the harmonic series as they please, leaving out
> the term "Just", without upsetting anyone. OK, make that a very
> stray thought...

The term "just intonation" is widely accepted, and that is not likely
to change for the foreseeable future. But I don't think that it
should apply to near-just temperaments or large-numbered rational
intervals that don't sound like JI.

> ---
> PS. The above are evolving personal opinions & views formed in a
> continuing strive to clarify & simply my own thinking and work,
> such that it is. Mostly just me thinking out loud.

It's always good to know that someone out there is thinking!

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

/tuning/topicId_38060.html#38221

of no consequence.
>
> The size of the number is very important when you are dealing with
> musical acoustics. What makes just intervals or chords "just"?
It's
> the elimination of roughness, i.e., perceived beating between off-
> tuned harmonics or combination tones. Once the numbers reach a
> certain size, intervals begin to lose their identities, at which
> point they begin to sound like approximations of lower-numbered
> ratios. Then they can no longer be considered distinct
consonances.

***This is the idea of Paul Erlich's "Harmonic Entropy..." Right,
Paul?? Paul, did the term "Harmonic Entropy" *originate* with *you*
or did somebody ever use it before??

Thanks!

Joseph

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> /tuning/topicId_38060.html#38221
>
> of no consequence.
> >
> > The size of the number is very important when you are dealing
with
> > musical acoustics. What makes just intervals or chords "just"?
> It's
> > the elimination of roughness, i.e., perceived beating between off-
> > tuned harmonics or combination tones. Once the numbers reach a
> > certain size, intervals begin to lose their identities, at which
> > point they begin to sound like approximations of lower-numbered
> > ratios. Then they can no longer be considered distinct
> > consonances.
>
> ***This is the idea of Paul Erlich's "Harmonic Entropy..." Right,
> Paul??

well, not really:

(1) harmonic entropy is indended to model a different component of
discordance, distinct from the roughness, beating, or combinational
tones to which george refers above. to get a better idea of what this
component of discordance is and its relevance to music, see, for
example, _Harmony: A Psychoacoustical Approach_ by Richard Parncutt.
however, for simple dyads, the function is not too different from a
beating/roughness-based model, such as those of helmholtz,
kameoka&kutiygawa, and sethares.

(2) what you may be thinking about is this: many of the mathematical
measures of concordance/discordance that have been proposed, from
euler to barlow, tenney, and wilson, have been functions of the
numbers in the ratio. the problem is that this produces a highly
*discontinuous* function -- the curve is extremely jumpy, no matter
how finely you zoom in, as a function of interval size. and none of
these measures can cope with *irrational* intervals, which are
actually infinitely more common than rational intervals. so not only
is the curve extremely jumpy, but it has an infinite number of
*holes* in it, and the holes actually take up infinitely more of the
intervallic continuum. now, harmonic entropy is a mathematical
measure of concordance/discordance which is *continuous* -- there are
no jumps or holes in the curve. this certainly accords with how
almost everyone hears intervallic discordance: change the size of the
interval by a very small amount, and the discordance will change by a
very small amount -- it won't suddenly jump a huge amount or fail to
be defined at all.

> Paul, did the term "Harmonic Entropy" *originate* with *you*
> or did somebody ever use it before??

i made it up.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> > Once the numbers reach a
> > certain size, intervals begin to lose their identities, at which
> > point they begin to sound like approximations of lower-numbered
> > ratios. Then they can no longer be considered distinct
> > consonances.
>
> ***This is the idea of Paul Erlich's "Harmonic Entropy..." Right,
> Paul??

let me try this again, joseph. i was feeling uncharitable for a
moment there.

yes, the quote above is very closely related to what goes on with
harmonic entropy. any interval is going to evoke the sound of the
simple-integer ratios that it is close in size to. for example, 440
cents can sound like it has 5:4 qualities and 4:3 qualities at the
same time -- not just in that it resembles them in size, but that it
actually evokes, to an incomplete degree, harmonic "roots" that would
correspond to both interpretations. harmonic entropy measures
the "confusion" of this sensation for any given interval. it turns
out that for the simplest ratios, harmonic entropy predicts a
dissonance function that agrees almost perfectly with tenney's
harmonic distance function, which is log(n*d) for a ratio n/d.
however, for ratios more complex than a certain "cutoff", the
probability that the ratio is heard in its own right (that is,
evoking the actual fundamental that would be implied by such high
members of a harmonic series) becomes quite small, and the simpler
ratios that it approximates become more important. for example,
30001:20001 is virtually the same as 3:2, and is heard primarily as
such, not as the 30001st and 20001st harmonics of a (verrrrrrry
loooow) fundamental.

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_38060.html#38229

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > /tuning/topicId_38060.html#38221
> >
> > of no consequence.
> > >
> > > The size of the number is very important when you are dealing
> with
> > > musical acoustics. What makes just intervals or
chords "just"?
> > It's
> > > the elimination of roughness, i.e., perceived beating between
off-
> > > tuned harmonics or combination tones. Once the numbers reach a
> > > certain size, intervals begin to lose their identities, at
which
> > > point they begin to sound like approximations of lower-numbered
> > > ratios. Then they can no longer be considered distinct
> > > consonances.
> >
> > ***This is the idea of Paul Erlich's "Harmonic Entropy..."
Right,
> > Paul??
>
> well, not really:
>
> (1) harmonic entropy is indended to model a different component of
> discordance, distinct from the roughness, beating, or combinational
> tones to which george refers above. to get a better idea of what
this
> component of discordance is and its relevance to music, see, for
> example, _Harmony: A Psychoacoustical Approach_ by Richard
Parncutt.
> however, for simple dyads, the function is not too different from a
> beating/roughness-based model, such as those of helmholtz,
> kameoka&kutiygawa, and sethares.
>
> (2) what you may be thinking about is this: many of the
mathematical
> measures of concordance/discordance that have been proposed, from
> euler to barlow, tenney, and wilson, have been functions of the
> numbers in the ratio. the problem is that this produces a highly
> *discontinuous* function -- the curve is extremely jumpy, no matter
> how finely you zoom in, as a function of interval size. and none of
> these measures can cope with *irrational* intervals, which are
> actually infinitely more common than rational intervals. so not
only
> is the curve extremely jumpy, but it has an infinite number of
> *holes* in it, and the holes actually take up infinitely more of
the
> intervallic continuum. now, harmonic entropy is a mathematical
> measure of concordance/discordance which is *continuous* -- there
are
> no jumps or holes in the curve. this certainly accords with how
> almost everyone hears intervallic discordance: change the size of
the
> interval by a very small amount, and the discordance will change by
a
> very small amount -- it won't suddenly jump a huge amount or fail
to
> be defined at all.

***Thanks, Paul, for "refining" these definitions...

>
> > Paul, did the term "Harmonic Entropy" *originate* with *you*
> > or did somebody ever use it before??
>
> i made it up.

***Great! It seems to have "stuck..."

Joseph

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_38060.html#38231

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > > Once the numbers reach a
> > > certain size, intervals begin to lose their identities, at
which
> > > point they begin to sound like approximations of lower-numbered
> > > ratios. Then they can no longer be considered distinct
> > > consonances.
> >
> > ***This is the idea of Paul Erlich's "Harmonic Entropy..."
Right,
> > Paul??
>
> let me try this again, joseph. i was feeling uncharitable for a
> moment there.

:) Naah... You wouldn't do that! :)

Besides, it's always easier to say *no* than *yes!* No has only 2
letters in it, yes, 3. The "math" is clear... :)

That's funny... :)

Thanks for the additional info!

Joseph

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_38060.html#38231

>
> yes, the quote above is very closely related to what goes on with
> harmonic entropy. any interval is going to evoke the sound of the
> simple-integer ratios that it is close in size to. for example, 440
> cents can sound like it has 5:4 qualities and 4:3 qualities at the
> same time -- not just in that it resembles them in size, but that
it actually evokes, to an incomplete degree, harmonic "roots" that
would correspond to both interpretations.

***Oh... I forgot to ask about this example:

Is this the schismatic situation that Margo talks about that happened
in the Renaissance which Lindley talks about??

And is it the 11th partial that would be evoked as contrasted with
the 5th partial??

Or am I "all wet..." here... (It's Summer after all, time to go
swimming...)

Joseph

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> /tuning/topicId_38060.html#38231
>
> >
> > yes, the quote above is very closely related to what goes on with
> > harmonic entropy. any interval is going to evoke the sound of the
> > simple-integer ratios that it is close in size to. for example,
440
> > cents can sound like it has 5:4 qualities and 4:3 qualities at
the
> > same time -- not just in that it resembles them in size, but that
> it actually evokes, to an incomplete degree, harmonic "roots" that
> would correspond to both interpretations.
>
> ***Oh... I forgot to ask about this example:
>
> Is this the schismatic situation that Margo talks about that
>happened
> in the Renaissance which Lindley talks about??

it's always easier to type "no" . . .

> And is it the 11th partial that would be evoked as contrasted with
> the 5th partial??

at this point, i have "no" idea what you're thinking. it would help
if you would elaborate, slowly and clearly.

> Or am I "all wet..." here... (It's Summer after all, time to go
> swimming...)

that *might* be a better use of your time :)

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_38060.html#38238

>
> > And is it the 11th partial that would be evoked as contrasted
with
> > the 5th partial??
>
> at this point, i have "no" idea what you're thinking. it would help
> if you would elaborate, slowly and clearly.
>

***Well, wasn't the schismatic "situation" in the Renaissance the use
of a pitch somewhere around E# or so that was actually created from
Pythagorean fifths and fourths??

Does that have anything to do with your 440 cents example??

I guess probably *not* since, apparently you were just making a
simple case for an interval *between* a simple major third and
perfect fourth. But then, why wasn't it 450, rather than 440??

I guess it has *nothing* to do with the 11th partial... that would be
around *550* cents, yes? Sorry...

Thanks!

Joseph

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> /tuning/topicId_38060.html#38238
>
> >
> > > And is it the 11th partial that would be evoked as contrasted
> with
> > > the 5th partial??
> >
> > at this point, i have "no" idea what you're thinking. it would
help
> > if you would elaborate, slowly and clearly.
> >
>
> ***Well, wasn't the schismatic "situation" in the Renaissance the
use
> of a pitch somewhere around E# or so that was actually created from
> Pythagorean fifths and fourths??

you must be thinking of Fb, which turns out to be 384 cents above C,
2 cents off a just 5:4? in practice, it was Gb, Db, and Ab which
served as "just major thirds" for roots on D, A, and E, respectively.

> Does that have anything to do with your 440 cents example??
>
> I guess probably *not* since, apparently you were just making a
> simple case for an interval *between* a simple major third and
> perfect fourth. But then, why wasn't it 450, rather than 440??

because a major third is 386. B-)

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> Regarding one's unsuspecting friends: one of the challenges of
> writing a microtonal composition is to use new intervals without
> making the result sound as if it is out of tune. The problem is
> greatest when using temperaments in which the intervals have large
> errors, which is a good rationale for the employment of JI or near-
> just temperaments.

In my experience many, perhaps most, people will find 11-limit JI out of tune, no matter how well in tune it is. However, the 7-limit seems to go down smoothly, despite the fact that few people have had a chance to hear it. In any event, I find some people love 11-limit harmonies, and others hate them, but I've never heard quacking about the 7-limit, though people can tell there's something a little unusual going on. Things like 9/7 start to get on some people's nerves, but 7/6 seems to be easily assimilated. I suppose 10/7 might be more of a problem but for the fact that people are used to tritones.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > Regarding one's unsuspecting friends: one of the challenges of
> > writing a microtonal composition is to use new intervals without
> > making the result sound as if it is out of tune. The problem is
> > greatest when using temperaments in which the intervals have
large
> > errors, which is a good rationale for the employment of JI or
near-
> > just temperaments.
>
> In my experience many, perhaps most, people will find 11-limit JI
>out of tune, no matter how well in tune it is. However, the 7-limit
>seems to go down smoothly, despite the fact that few people have had
>a chance to hear it.

including 8/7? you used to claim it was more dissonant than 9/7 . . .

>In any event, I find some people love 11-limit >harmonies, and
>others hate them, but I've never heard quacking about >the 7-limit,
>though people can tell there's something a little >unusual going on.
>Things like 9/7 start to get on some people's >nerves, but 7/6 seems
>to be easily assimilated. I suppose 10/7 might >be more of a problem
>but for the fact that people are used to >tritones.

this corresponds i think to how an average person hears dyadic
discordance in an average register and average timbre:

/tuning/files/perlich/tenney/te01_13p2877
.jpg

10/7 and 11/7 are pretty neutral, and it looks like they're at
inflection points rather than local maxima or minima. 11:8, 11:9,
11:10, 12:11 -- all are looking pretty discordant here.

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_38060.html#38241

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> >
> > /tuning/topicId_38060.html#38238
> >
> >
> > ***Well, wasn't the schismatic "situation" in the Renaissance the
> use
> > of a pitch somewhere around E# or so that was actually created
from
> > Pythagorean fifths and fourths??
>
> you must be thinking of Fb, which turns out to be 384 cents above
C,
> 2 cents off a just 5:4? in practice, it was Gb, Db, and Ab which
> served as "just major thirds" for roots on D, A, and E,
respectively.
>

***Thanks, Paul. That's exactly what I was forgetting about. In
this case, surely the nomenclature would make a difference...

Joseph

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> /tuning/topicId_38060.html#38241
>
> > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > > --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> > >
> > > /tuning/topicId_38060.html#38238
> > >
> > >
> > > ***Well, wasn't the schismatic "situation" in the Renaissance
the
> > use
> > > of a pitch somewhere around E# or so that was actually created
> from
> > > Pythagorean fifths and fourths??
> >
> > you must be thinking of Fb, which turns out to be 384 cents above
> C,
> > 2 cents off a just 5:4? in practice, it was Gb, Db, and Ab which
> > served as "just major thirds" for roots on D, A, and E,
> respectively.
> >
>
> ***Thanks, Paul. That's exactly what I was forgetting about. In
> this case, surely the nomenclature would make a difference...
>
> Joseph

also, this was before the period usually called the renaissance in
music -- late medieval is more like it.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_38060.html#38250

> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > Regarding one's unsuspecting friends: one of the challenges of
> > writing a microtonal composition is to use new intervals without
> > making the result sound as if it is out of tune. The problem is
> > greatest when using temperaments in which the intervals have
large
> > errors, which is a good rationale for the employment of JI or
near-
> > just temperaments.
>
> In my experience many, perhaps most, people will find 11-limit JI
out of tune, no matter how well in tune it is. However, the 7-limit
seems to go down smoothly, despite the fact that few people have had
a chance to hear it. In any event, I find some people love 11-limit
harmonies, and others hate them, but I've never heard quacking about
the 7-limit, though people can tell there's something a little
unusual going on. Things like 9/7 start to get on some people's
nerves, but 7/6 seems to be easily assimilated. I suppose 10/7 might
be more of a problem but for the fact that people are used to
tritones.

***Actually, this is very similar to the experience I was having with
Blackjack as well...

Joseph