Yahoo Tuning Groups Ultimate Backup tuning-math RE: [tuning-math] Digest Number 1235

Gene,

OK, so the difference betwen a tuning and a temperament
is clear to you. Not to me!

I understand that a temperament is the set of adjusted
frequency ratios that result when you choose to depart
from strict JI (in any limit), by choosing one or more
particular interval ratios, eg a comma or two, and equating
them to some other ratios, typically 1/1 in the case of a
comma. Or you might choose to equate two different
flavours of thirds, eg 5/4 and 15/11, which of course
comes down to the same thing, as it eliminates the (comma)
ratio between them, 12/11.

Please correct me if my description above misses anything
important, or includes any non-essential.

Then a tuning, it seems, is any way of assigning values to
intervals. Your choice might be strict JI (where nothing
is tempered) or it might be any temperament at all. In a
sense, that means that it is any temperament, including
the (mathematically, not musically) trivial temperament JI.
I suppose you could argue that "adaptive tuning" is a horse
of a different colour to any temperament ...

Could you please clarify what a "tuning" means to you?

Regards,
Yahya

-----Original Message-----
________________________________________________________________________
Date: Wed, 16 Mar 2005 01:01:30 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: Commas and generators

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >It won't be if it is a 5-limit comma, but in general, what you get is
> >a tempering of a subgroup of a p-limit, not the whole p-limit. The
> >subgroup is extracted from the comma, and used to temper.
>
> Sounds like TOP...

Not at all; TOP is a tuning. I'm talking about rank two temperaments
defined for subgroups of the p-limit. Of course, TOP could be used as
the tuning.

________________________________________________________________________

[Yahya] To which Carl replied -
________________________________________________________________________
...

Er...

-Carl
________________________________________________________________________

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🔗Carl Lumma <ekin@lumma.org>

>Gene,
>
>OK, so the difference betwen a tuning and a temperament
>is clear to you. Not to me!

It isn't to me either. I think I understand the difference
between temperaments and scales, but I have no idea what
a "tuning" is. Gene, is this defined on xenharmony.org
somewhere?

>I understand that a temperament is the set of adjusted
>frequency ratios that result when you choose to depart
>from strict JI (in any limit),

I don't think Gene quite defines it this way, though
if you allow for the fact that sets can be infinite
this might be equivalent.

>Please correct me if my description above misses anything
>important, or includes any non-essential.

You might try

http://tonalsoft.com/enc/temperament.htm

and

http://66.98.148.43/~xenharmo/regular.html

-Carl

🔗Herman Miller <hmiller@IO.COM>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Carl Lumma wrote:
> >>Gene,
> >>
> >>OK, so the difference betwen a tuning and a temperament
> >>is clear to you. Not to me!
> >
> >
> > It isn't to me either. I think I understand the difference
> > between temperaments and scales, but I have no idea what
> > a "tuning" is. Gene, is this defined on xenharmony.org
> > somewhere?
>
> I think "tuning" is one of those generic terms that doesn't have a
> precise meaning. One of the things "tuning" is useful for is a
> particular instance of a temperament, with the intervals tuned to a
> specific ratio; meantone would be a temperament, but quarter-comma
> meantone would be a specific tuning of meantone temperament, and
TOP
> meantone would be a different one. You could also use "tuning" to
refer
> to tuning systems that aren't close enough to just intervals to be
> considered as temperaments, like 11-EDO or Erv Wilson's golden MOS
> scales. I've used the word in this sense in cases like "superpelog
> tuning" (which could be considered a temperament, but not a very
good
> one) and "16-tone equal tuning".

The word "tuning", as I understand it, refers to a set of tones of a
specific *number* separated by specific *intervals* (expressed either
as exact ratios or a logarithmic measure, such as cents). The only
thing not necessarily specific about a tuning is the absolute pitch
or frequency of its starting tone. The definition of a tuning must
therefore be specific enough to enable one to create a .scl file for
it.

A tuning may happen to be a JI set, a temperament, a scale, an EDO,
or a subset of one or more of these, or it may happen to be a random
set of tones that is none of these.

As long as you're concerned with definitions, it would also be
helpful to distinguish between a *temperament* and a *scale*.

As I understand it, a *temperament* is an organization of tones
defined in terms of specific commas, generator(s), and period, not
generally restricted to some particular number of tones within a
period.

A *scale* is a set of tones of a specific *number* within a period
(usually an octave), in which the *intervals* between the tones fall
into (more or less) specific *size categories*. Scales are generally
temperaments (or, alternatively, JI lattices) carried out to a
specific number of tones, e.g., a major scale may be a subset of a
meantone temperament (or, alternatively, a subset of a 5-limit JI
lattice).

From the foregoing, it should be evident that EDO's and well-
temperaments would be most accurately described as *tunings*!

I encourage you to refine these thoughts into formal definitions.

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >Gene,
> > >
> > >OK, so the difference betwen a tuning and a temperament
> > >is clear to you. Not to me!
> >
> > It isn't to me either. I think I understand the difference
> > between temperaments and scales, but I have no idea what
> > a "tuning" is. Gene, is this defined on xenharmony.org
> > somewhere?
>
> In what is probably not an easily digestible form:
>
> http://66.98.148.43/~xenharmo/regular.html
>
> Meantone is a temperament. 1/4-comma meantone, 2/7-comma meantone,
> 7/26-comma meantone or Lucy meantone would be tunings of the
> temperament. So would 12, 19, 31 or 50 equal, but they collapse it
> down to rank 1.

It appears that you have a broader idea of what constitutes a tuning
than what I suggested:
/tuning-math/message/11821

I think we need another term to cover everything. Here are some
questions to think about:

1) Would you consider some random or heterogeneous set of tones
a "tuning"?

How would your definition of "tuning" cover that?

2) Are 1/4-comma meantone carried to 7 vs. 12 tones/octave different
tunings?

If not, then what term do we use to distinguish these, and how do we
define a "tuning"?

3) Are 1/4-comma meantone and 12-EDO different temperaments?

If we say "no" at this point (on the grounds that they're varieties
of "meantone" temperament), I think we're flying in the face of long-
established usage outside this group, which can only breed confusion
and/or disdain. (Do you remember Peter Sault's outrage at your
calling Pythagorean a "meantone temperament"?) Better to consider
them separate *temperaments* in the Meantone *temperament family*, or
simply *family*. I have long been accustomed to classifying EDO's
such as (12, 19, 26, 31, 43, 55, etc.) or (31, 41, 72) into what I
thought of *families*, which we now call Meantone and Miracle.

Besides, aren't 1/4-comma meantone and 12-EDO different temperaments
because more commas vanish in the latter?

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> 1) Would you consider some random or heterogeneous set of tones
> a "tuning"?

A tuning of a regular temperament simply assigns particular values to
its intervals, which can be done by assigning particular values to its
generators. Irregular temperaments and scales are not really distinct
concepts, so I think keeping the distinction focused on regular
temperaments makes sense.

> 2) Are 1/4-comma meantone carried to 7 vs. 12 tones/octave different
> tunings?

Different scales using the same tuning.

> If not, then what term do we use to distinguish these, and how do we
> define a "tuning"?

I'd call them scales, or in particular MOS/DE of meantone.

> 3) Are 1/4-comma meantone and 12-EDO different temperaments?

12-edo isn't yet a temperament until you give a mapping; then it is
different from meantone, in that it is rank one. You can factor the
mapping through meantone, so it supports meantone and can be used as a
tuning for it.

In the 5-limit, for instance, we have

JI --> meantone --> 12-et --> pitches

Here JI maps to meantone as an abstract group by sending 2^a 3^b 5^c
to 2^u g^v by 2 --> 2, 3 --> 2g, 5 --> g^4. Then 2^u g^v maps to 12-et
as an abstract group s^n by sending 2 --> s^12, g --> s^7. Finally
this rank one group can be mapped to actual pitches by for instance
sending s^n --> 440 2^(n/12) Hz. We can of course write the groups in
the various stages of this additively as well as multiplicatively
without changing the definition, only the notation. A "tuning" you
might consider to be still another stage; sending s --> 2^(1/12), so
you can map through the more abstract tuning to get to the actual
pitches in Hz.

🔗Ozan Yarman <ozanyarman@superonline.com>

Can tuning be defined as `determination in a pitch continuum of any number of meaningful frequencies`?

----- Original Message -----
From: George D. Secor
To: tuning-math@yahoogroups.com
Sent: 17 Mart 2005 Perşembe 19:21
Subject: [tuning-math] Tuning/temperament/scale (was: Digest Number 1235)

It appears that you have a broader idea of what constitutes a tuning
than what I suggested:
/tuning-math/message/11821

I think we need another term to cover everything. Here are some
questions to think about:

1) Would you consider some random or heterogeneous set of tones
a "tuning"?

How would your definition of "tuning" cover that?

2) Are 1/4-comma meantone carried to 7 vs. 12 tones/octave different
tunings?

If not, then what term do we use to distinguish these, and how do we
define a "tuning"?

3) Are 1/4-comma meantone and 12-EDO different temperaments?

Besides, aren't 1/4-comma meantone and 12-EDO different temperaments
because more commas vanish in the latter?

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

I think your definition is refined enough dear George! I appreciate your input very much. That clears my head alright.

Sincerely,
Ozan
----- Original Message -----
From: George D. Secor
To: tuning-math@yahoogroups.com
Sent: 17 Mart 2005 Perşembe 18:16
Subject: [tuning-math] Tuning/temperament/scale (was: Digest Number 1235)

A tuning may happen to be a JI set, a temperament, a scale, an EDO,
or a subset of one or more of these, or it may happen to be a random
set of tones that is none of these.

As long as you're concerned with definitions, it would also be
helpful to distinguish between a *temperament* and a *scale*.

From the foregoing, it should be evident that EDO's and well-
temperaments would be most accurately described as *tunings*!

I encourage you to refine these thoughts into formal definitions.

--George

🔗George D. Secor <gdsecor@yahoo.com>

> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > 1) Would you consider some random or heterogeneous set of tones
> > a "tuning"?
>
> A tuning of a regular temperament simply assigns particular values
to
> its intervals, which can be done by assigning particular values to
its
> generators. Irregular temperaments and scales are not really
distinct
> concepts, so I think keeping the distinction focused on regular
> temperaments makes sense.

So is that a "no"? Are you saying that I can't call something a
tuning if it isn't a regular temperament? Now you're really going to
confuse others outside this group. In _Tuning and Temperament_
Barbour distinguished a tuning from a temperament in that the former
could be described solely with rational intervals, i.e., *not*
tempered. Since that time the term "tuning" has been used to apply
to any set of tones, tempered or not. Now you seem to be saying that
we won't call something a tuning unless it's a temperament (and a
regular temperament, at that). (Okay, I went a little overboard:
you'll accept Pythagorean as a tuning, but only because it's regular.)

No, wait, on rereading your statement, I think I've got it! You'll
accept an irregular version of some regular temperament as a variant
tuning, but you find no need to consider it separately. So that's
a "yes"!

Anyway, I was going to continue by observing that,
likewise, "meantone" temperament originally meant a regular
temperament in which the major 2nd was a logarithmic mean between 1/1
and 5/4, or (5/4^(1/2). Barbour used a "m/n-comma" modifier before
the word "meantone" to specify a fraction of the 5-comma by which the
perfect 5th could be tempered (narrow) to arrive at variants of the
original (1/4-comma) meantone temperament that approximated various
EDO's, but he observed the convention that, if no fraction-modifier
is used, then the term "meantone temperament" is understood to mean
1/4-comma meantone.

Your desire to use the term "meantone temperament" to designate an
entire continuum of regular temperaments is, in my opinion, a big
mistake, because it conflicts with long-established usage outside
this group. Why not call the temperament family "Syntonic" and
thereby dispense with at least half the confusion?

Likewise, the term "equal temperament", without any additional
qualification, has always meant 12-EDO, and that term has most often
been used in distinction to "meantone temperament" and "well-
temperament" as examples of *different temperaments* (based on the
*amount* various intervals are tempered). But with your terminology
you would now call these different "tunings" of the same temperament
(Meantone) and would distinguish "temperaments" on a very different
basis (commas, generators, and periods). You're going to have enough
trouble trying to get others to follow what you're doing, much less
introducing a change in the meaning of existing terminology that's
bound to be confusing to others.

If you're talking about distinct "tunings" that just happen to be
tempered, then why don't you call them distinct "temperaments" (in
accordance with the way the word has traditionally been used) and use
the word *families* or *temperament families* (or something else, if
you please) for what you're wanting to call "temperaments"?

> > 2) Are 1/4-comma meantone carried to 7 vs. 12 tones/octave
different
> > tunings?
>
> Different scales using the same tuning.

Why not different scales using the same temperament or the same
generator(s)? If you have to tune 5 more notes, then you end up with
new intervals not found in the original 7, something many would
regard as a different tuning. Is a composition using a heptatonic
meantone-fifth sequence in the same tuning as one using that sequence
taken to 31 tones? What about a composition using a pentatonic
Pythagorean sequence vs. one taken to 53 tones? As far as the
composer or listener is concerned, it sounds like a totally different
tuning.

In my proposed definition of "tuning", two tunings must be identical
(excepting a difference in absolute pitch) in order to be the same
tuning. If you wish to use the word for something different, then
please suggest some other word that I can use for the purpose I
intend.

> > If not, then what term do we use to distinguish these, and how do
we
> > define a "tuning"?
>
> I'd call them scales, or in particular MOS/DE of meantone.

Yes, they're different scales, but a particular "scale" isn't
generally restricted to a particular tuning.

> > 3) Are 1/4-comma meantone and 12-EDO different temperaments?
>
> 12-edo isn't yet a temperament until you give a mapping;

Of course it's a temperament, because it's already the result of a
historical process that has given it an obvious mapping.

Gene, I'm not objecting to your concepts, only that you've applied
existing terms to those concepts in a way that is at variance with
previous usage, which is bound to lead to confusion.

--George

🔗George D. Secor <gdsecor@yahoo.com>

Hey, this is funny! 8-)

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> [message #11821] ...
> As I understand it, a *temperament* is an organization of tones
> defined in terms of specific commas, generator(s), and period, not
> generally restricted to some particular number of tones within a
> period.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> [message #11826, to Gene] ...
>
> Likewise, the term "equal temperament", without any additional
> qualification, has always meant 12-EDO, and that term has most
often
> been used in distinction to "meantone temperament" and "well-
> temperament" as examples of *different temperaments* (based on the
> *amount* various intervals are tempered). But with your
terminology
> you would now call these different "tunings" of the same
temperament
> (Meantone) and would distinguish "temperaments" on a very different
> basis (commas, generators, and periods). You're going to have
enough
> trouble trying to get others to follow what you're doing, much less
> introducing a change in the meaning of existing terminology that's
> bound to be confusing to others.

Oops! Looks like my definition of temperament went through a quick
change somewhere in the spin cycle. ;-)

Seriously, I would strongly advise finding a term other
than "temperament" to describe the establishment of a system of
tonality defined by commas, generators, and periods and using the
word "temperament" to refer to the specific fraction of a comma
applied to alter a generator in the distribution of that comma within
the scale. (In the case of irregular or well-temperaments, these
comma-fractions would vary from one key to another.)

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> So is that a "no"?

Yes, it's a no.

Are you saying that I can't call something a
> tuning if it isn't a regular temperament?

If you want to call it a tuning, don't you first need to explain what
it is a tuning *of*?

Now you're really going to
> confuse others outside this group. In _Tuning and Temperament_
> Barbour distinguished a tuning from a temperament in that the former
> could be described solely with rational intervals, i.e., *not*
> tempered.

I would call that a scale, unless it is something like the entire 3 or
5 limit, in which case I'd call it a JI group, giving the generators
or the prime limit.

Since that time the term "tuning" has been used to apply
> to any set of tones, tempered or not.

I don't recall seeing this definition ever, including in Barbour.

Now you seem to be saying that
> we won't call something a tuning unless it's a temperament (and a
> regular temperament, at that).

I would reserve "tuning" for the case where you have something or
other you propose to tune.

(Okay, I went a little overboard:
> you'll accept Pythagorean as a tuning, but only because it's regular.)
>
> No, wait, on rereading your statement, I think I've got it! You'll
> accept an irregular version of some regular temperament as a variant
> tuning, but you find no need to consider it separately. So that's
> a "yes"!

Yes, with the caveat that there is no sharp distinction which can be
drawn between a well-temperament and a scale.

> Your desire to use the term "meantone temperament" to designate an
> entire continuum of regular temperaments is, in my opinion, a big
> mistake, because it conflicts with long-established usage outside
> this group. Why not call the temperament family "Syntonic" and
> thereby dispense with at least half the confusion?

Because that conflicts with long-standing usage everywhere, here and
the world at large, as the Barbour usage you cite shows. "2/7-comma
meantone" *is* a species of meantone, so I call it a tuning of meantone.

> Likewise, the term "equal temperament", without any additional
> qualification, has always meant 12-EDO, and that term has most often
> been used in distinction to "meantone temperament" and "well-
> temperament" as examples of *different temperaments* (based on the
> *amount* various intervals are tempered). But with your terminology
> you would now call these different "tunings" of the same temperament
> (Meantone) and would distinguish "temperaments" on a very different
> basis (commas, generators, and periods).

I wouldn't call 12-EDO a temperament at all, because there isn't a
mapping. I'd call it an equal division. Mathematicians have a saying
that it isn't the objects, it's the morphisms; in this case, it isn't
12-EDO which makes a temperament, but how you use it, which usually
ends up meaning using it like meantone. If you send octaves to 12
steps and fifths to five steps and use it like meantone, then it
becomes a tuning of meantone. It also acquires the mapping <12 19 28|,
which define its properties vis a vis 5-limit JI.

You're going to have enough
> trouble trying to get others to follow what you're doing, much less
> introducing a change in the meaning of existing terminology that's
> bound to be confusing to others.

No matter what I do it seems to confuse people, but I think you are
equally guilty of wholesale slaughter of standard usages when you
claim that 2/7-comma meantone and 1/4-comma meantone aren't both just
"meantone".

> If you're talking about distinct "tunings" that just happen to be
> tempered, then why don't you call them distinct "temperaments" (in
> accordance with the way the word has traditionally been used) and use
> the word *families* or *temperament families* (or something else, if
> you please) for what you're wanting to call "temperaments"?

Because that suggests one thing is actually and uncountable infinity
of different things. Besides, we *already* have families of related
temperaments we want to discuss.

Is a composition using a heptatonic
> meantone-fifth sequence in the same tuning as one using that sequence
> taken to 31 tones? What about a composition using a pentatonic
> Pythagorean sequence vs. one taken to 53 tones? As far as the
> composer or listener is concerned, it sounds like a totally different
> tuning.

Not really, since the chords are the same.

RE: [tuning-math] Digest Number 1235

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Carl Lumma <ekin@lumma.org>

🔗Herman Miller <hmiller@IO.COM>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗George D. Secor <gdsecor@yahoo.com>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Gene Ward Smith <gwsmith@svpal.org>