--- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <perlich@a...>

wrote:

/makemicromusic/topicId_6820.html#6852

> Hi George,

>

> For quite a few years on these lists, the term "middle path" has

been

> used by Margo Schulter, Dave Keenan, and myself, in discussions

with

> Jacky Ligon and others, to refer to a tuning concept or philosophy

> that is between those of JI and the various ETs.

Back in the 70's I found myself sorely in need of a term that would

encompass what I thought of as a "third world" of tuning: all tunings

other than strict JI and ET's (which I understood at the time to

include only equal divisions of the octave), which are generally

thought to be at opposite ends of the tuning philosophical spectrum.

Your term "middle path" seemed to have filled that void very nicely.

However, there are two reasons that I think non-octave ET's more

properly belong in a "third-world" or "middle-path" category:

1) This category includes tunings that are rather esoteric and, from

a practical standpoint, in a category rather distinct from EDO's;

2) Theoretically, non-octave ET's are open systems.

> Some posts mentioning the concept include:

>

> /tuning/topicId_29348.html#29348

> /tuning/topicId_28984.html#29212

>

> The idea is that, while ETs (not EDOs) take JI and simplify it

> by "regularly" (or uniformly) tempering out a sufficient number of

> independent commas, the "middle path" tunings regularly temper out

a

> smaller number of independent commas. The most famous example is

> meantone temperament, whose various varieties are solutions to the

> problem of best tempering out the syntonic comma (81:80) without

> regard to any other commas vanishing. Schismic (or schismatic)

> temperament is another classic example (q.v. Helmholtz, Groven, and

> Sabat-Garibaldi), while recently we have seen the discovery and

> rediscovery of quite a few other such schemes, such as Miracle,

> Hanson, and the "Pajara" system which forms the underpinning for my

> paper on the decatonic scales.

Since "middle path" doesn't sound particularly technical, and since

all of the times I have seen it used it seemed to apply to "third

world" tunings (not a good name, of course, because of its political

and economic connotations), I assumed that this is how you were using

it, and I didn't see anything in the above examples that would have

given me a clue that I was mistaken. My apologies.

> Like JI and ETs, middle path tunings are "regular" -- a given

> consonance is represented the same way no matter where it appears

in

> the tuning system. It seems to me that if a property such

> as "regularity" is possessed by both JI and the ETs, then it ought

to

> be possessed by any "middle path" between them.

Yes, that seems logical. However, I could also argue that, since

well-temperament has historical importance in the transition from

meantone temperament to 12-ET, it therefore occupies a middle-ground

between the two and could thereby be included in the broader middle

ground (or middle path) between 5-limit JI and 12-ET.

> My XH18 paper is a complete survey of 5- and 7-limit, two-

dimensional

> (like those mentioned above) "middle path" systems within a certain

> boundary on error and complexity. In fact, the title of the paper

is

> _The Middle Path. Part 1: Fifty Floragrams_.

>

> It seems that you are now putting forth a different definition

> of "middle path", similar to the "well-temperament" concept, which

> refers to closed systems (while mine refers to open systems) and

> irregular systems (while mine refers to closed [s/b regular]

systems).

About my well-temperaments you are absolutely correct. However, my

high-tolerance temperament sets are basically multiple chains of

uniformly tempered fifths, hence are not really irregular, at least

from a philosophical standpoint, and they are closed sets only as a

matter of convenience, for the same sense that Blackjack or Canasta

might be considered closed sets. You could conceive of the 29-tone

high-tolerance temperament as starting with chains of fifths

including the tones 1/1, 5/4, 7/4, 11/8, and 13/8, then tempering the

fifths so that 7/4, 11/8, and 13/8 are all in the same chain. Fifths

perceived to be "irregular" are actually intervals between tones in

different chains, and otonal ogdoads in 6 keys (B-flat through A)

require 28 tones. Yes, there is one exception: an additional tone (E-

flat) is required to fill a hole in the 29-tone mapping, but even

that could be interpreted as 19/16 of C in a one-tone chain (hey, cut

me a little slack here, huh?).

In the other HTT sets there are two more chains of fifths that

include the tones 65/64 and 91/64, giving 9-limit otonalities on

(approximated) ratios of 11 and 13. Ratios of 11 and 13 may be

approximated on those roots using tones from the 1/1 chain, which is

tantamount to substituting tempered 22:27:32 intervals for 9:11:13.

While the preceding 2 paragraphs may seem to be a bit of a

digression, I was hoping to make 2 points:

1) Not all of my tunings that you were excluding from the "middle

path" category are well-temperaments;

2) Like JI vs. temperament or rational vs. irrational, a practical

distinction between regular vs. irregular or open vs. closed may not

always be as clear as we expected.

> I can

> certainly see how a JI periodicity block might be created, then

> tempered first near the edges, and finally tempered uniformly

> everywhere to yield an ET, such that the middle step would be a

well-

> temperament. However, I feel this unfairly omits the whole universe

> of possibilities where fewer of the periodicity blocks' unison

> vectors are tempered out, and which are thus also in the "middle"

> between JI and ETs.

As I said above, my impression was that the term "middle path" would

not include any tuning not covered by the two opposite extremes, JI

and ET (or as I would prefer, EDO).

> While I certainly respect your thinking process, and the musical

> value of well-tempered systems is unquestionable, I wonder if we're

> heading toward a terminological nightmare here.

Well, "middle path" is your term, so I think I'm obligated to abide

by whatever definition you choose for it. My problem is that you

have used the term to exclude certain tunings that, given a similar

number of tones, offer better intonation than EDO's and more

modulation than JI, thereby offering a practical middle-ground

between the two.

Hmmm, maybe I should call this broader category the "middle-ground",

to include tunings both on and off the "middle path". (I always

fancied most of my tunings as being off the beaten path, anyway. ;-)

Question: would all regular middle-ground tunings then be middle-path?

--George

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

> --- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <perlich@a...>

> wrote:

>

> /makemicromusic/topicId_6820.html#6852

>

> > Hi George,

> >

> > For quite a few years on these lists, the term "middle path" has

> been

> > used by Margo Schulter, Dave Keenan, and myself, in discussions

> with

> > Jacky Ligon and others, to refer to a tuning concept or

philosophy

> > that is between those of JI and the various ETs.

>

> Back in the 70's I found myself sorely in need of a term that would

> encompass what I thought of as a "third world" of tuning: all

tunings

> other than strict JI and ET's (which I understood at the time to

> include only equal divisions of the octave), which are generally

> thought to be at opposite ends of the tuning philosophical

spectrum.

> Your term "middle path" seemed to have filled that void very nicely.

>

> However, there are two reasons that I think non-octave ET's more

> properly belong in a "third-world" or "middle-path" category:

> 1) This category includes tunings that are rather esoteric and,

from

> a practical standpoint, in a category rather distinct from EDO's;

Where do you draw the line? Pianos are tuned to a stretched 12-equal -

- strictly speaking, a non-octave ET.

> 2) Theoretically, non-octave ET's are open systems.

Huh? I don't get it. ETs, whether octave-based or not, have a

discrete set of pitches, each one separated from the others by a

given interval, which would seem to make them closed rather than open

systems.

> > Some posts mentioning the concept include:

> >

> > /tuning/topicId_29348.html#29348

> > /tuning/topicId_28984.html#29212

> >

> > The idea is that, while ETs (not EDOs) take JI and simplify it

> > by "regularly" (or uniformly) tempering out a sufficient number

of

> > independent commas, the "middle path" tunings regularly temper

out

> a

> > smaller number of independent commas. The most famous example is

> > meantone temperament, whose various varieties are solutions to

the

> > problem of best tempering out the syntonic comma (81:80) without

> > regard to any other commas vanishing. Schismic (or schismatic)

> > temperament is another classic example (q.v. Helmholtz, Groven,

and

> > Sabat-Garibaldi), while recently we have seen the discovery and

> > rediscovery of quite a few other such schemes, such as Miracle,

> > Hanson, and the "Pajara" system which forms the underpinning for

my

> > paper on the decatonic scales.

>

> Since "middle path" doesn't sound particularly technical, and since

> all of the times I have seen it used it seemed to apply to "third

> world" tunings

> (not a good name, of course, because of its political

> and economic connotations), I assumed that this is how you were

using

> it, and I didn't see anything in the above examples that would have

> given me a clue that I was mistaken.

If you start with the above examples and click on "up thread" enough

times, you'll see more than a clue.

> > My XH18 paper is a complete survey of 5- and 7-limit, two-

> dimensional

> > (like those mentioned above) "middle path" systems within a

certain

> > boundary on error and complexity. In fact, the title of the paper

> is

> > _The Middle Path. Part 1: Fifty Floragrams_.

> >

> > It seems that you are now putting forth a different definition

> > of "middle path", similar to the "well-temperament" concept,

which

> > refers to closed systems (while mine refers to open systems) and

> > irregular systems (while mine refers to closed [s/b regular]

> systems).

I'm pretty sure you've distorted this quote. I wouldn't have

said "mine refers to open systems", and then said "mine refers to

closed systems"! Or did I really goof that badly?

> 2) Like JI vs. temperament or rational vs. irrational, a practical

> distinction between regular vs. irregular or open vs. closed may

not

> always be as clear as we expected.

But the theoretical distinction is clear -- in open temperaments,

including more and more of the lattice means adding more and more

tones between the existing ones, with no limit to how far this

process may proceed. In closed temperaments, the entire infinite

lattice is already mapped to a finite set of tones.

> > I can

> > certainly see how a JI periodicity block might be created, then

> > tempered first near the edges, and finally tempered uniformly

> > everywhere to yield an ET, such that the middle step would be a

> well-

> > temperament. However, I feel this unfairly omits the whole

universe

> > of possibilities where fewer of the periodicity blocks' unison

> > vectors are tempered out, and which are thus also in the "middle"

> > between JI and ETs.

>

> As I said above, my impression was that the term "middle path"

would

> not include any tuning not covered by the two opposite extremes, JI

> and ET

Exactly. Is there anything in the above paragraph that would suggest

otherwise?

> (or as I would prefer, EDO).

I would prefer EDO to refer to tuning systems conceived as equal

divisions of the octave and not as temperaments. ET should refer to

temperament, a tempering of JI, in particular that of a sort which

leads to a finite number of equally-spaced tones. 24-equal, for

example, would be awkward to derive as an ET, unless you began with

the restriction that only primes 3, 11, and perhaps 13 would figure

into your initial JI system. 24-equal, at least in the West, is

usually best considered an EDO. Not so for 12-equal.

> > While I certainly respect your thinking process, and the musical

> > value of well-tempered systems is unquestionable, I wonder if

we're

> > heading toward a terminological nightmare here.

>

> Well, "middle path" is your term, so I think I'm obligated to abide

> by whatever definition you choose for it. My problem is that you

> have used the term to exclude certain tunings that, given a similar

> number of tones, offer better intonation than EDO's and more

> modulation than JI, thereby offering a practical middle-ground

> between the two.

>

> Hmmm, maybe I should call this broader category the "middle-

ground",

> to include tunings both on and off the "middle path". (I always

> fancied most of my tunings as being off the beaten path, anyway. ;-

)

Sounds good to me.

> Question: would all regular middle-ground tunings then be middle-

>path?

Shoot me an example to test.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

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

wrote:

> >

> > Back in the 70's I found myself sorely in need of a term that

would

> > encompass what I thought of as a "third world" of tuning: all

tunings

> > other than strict JI and ET's (which I understood at the time to

> > include only equal divisions of the octave), which are generally

> > thought to be at opposite ends of the tuning philosophical

spectrum.

> > Your term "middle path" seemed to have filled that void very

nicely.

> >

> > However, there are two reasons that I think non-octave ET's more

> > properly belong in a "third-world" or "middle-path" category:

> > 1) This category includes tunings that are rather esoteric and,

from

> > a practical standpoint, in a category rather distinct from EDO's;

>

> Where do you draw the line? Pianos are tuned to a stretched 12-

equal -

> - strictly speaking, a non-octave ET.

Strictly speaking that's not an ET, since the octaves are not

stretched equally over the entire range of the instrument. The

amount of stretch varies according to the specific dimensions of each

string.

And even if the amount of stretch were equal over all octaves, this

example would still not be a tuning *theoretically* distinct from 12-

ET, since the amount of stretch would vary with the implementation of

the tuning on one particular instrument vs. another. The intention

is to tune the instrument so that the octaves sound exact (or as

nearly exact as possible), in order to make the result (what the ear

hears in *practice*) most closely match the *theoretical* tuning.

Similarly, are we to insist that any implementation of JI on acoustic

instruments is a temperament, since it's impossible to achieve exact

phase-locked ratios? Of course not; this is a *practical* problem

rather than another (theoretical) tuning.

So I would draw the line at the point where the *theoretical tuning*

is not mathematically an EDO, where the octaves are intentionally

tempered by a specified number of cents, usually for the purpose of

approximating other consonances more accurately.

> > 2) Theoretically, non-octave ET's are open systems.

>

> Huh? I don't get it. ETs, whether octave-based or not, have a

> discrete set of pitches, each one separated from the others by a

> given interval, which would seem to make them closed rather than

open

> systems.

But the set can be extended indefinitely up or down without any pitch

ever repeating any previous pitch at a different octave (taking the

term "octave" in the strict sense of the word). The principle of

octave equivalence exists whether or not an ET is octave-based.

(Please read further before replying to this.)

> > ... I didn't see anything in the above examples that would have

> > given me a clue that I was mistaken.

>

> If you start with the above examples and click on "up thread"

enough

> times, you'll see more than a clue.

I didn't look that far, but I'll take your word for it.

> > > My XH18 paper is a complete survey of 5- and 7-limit, two-

dimensional

> > > (like those mentioned above) "middle path" systems within a

certain

> > > boundary on error and complexity. In fact, the title of the

paper is

> > > _The Middle Path. Part 1: Fifty Floragrams_.

> > >

> > > It seems that you are now putting forth a different definition

> > > of "middle path", similar to the "well-temperament" concept,

which

> > > refers to closed systems (while mine refers to open systems)

and

> > > irregular systems (while mine refers to closed [s/b regular]

systems).

>

> I'm pretty sure you've distorted this quote. I wouldn't have

> said "mine refers to open systems", and then said "mine refers to

> closed systems"! Or did I really goof that badly?

See for yourself (in the next-to-last paragraph):

/makemicromusic/topicId_6820.html#6852

Hey, this can happen to the best of us, especially when we start

getting on in years. Once you've reached 22 or so, it's all downhill

from there. ;-)

> > 2) Like JI vs. temperament or rational vs. irrational, a

practical

> > distinction between regular vs. irregular or open vs. closed may

not

> > always be as clear as we expected.

>

> But the theoretical distinction is clear --

Well, I'm glad that we seem to agree that restricting distinctions to

what is theoretically intended can clarify the issues at hand.

> in open temperaments,

> including more and more of the lattice means adding more and more

> tones between the existing ones, with no limit to how far this

> process may proceed. In closed temperaments, the entire infinite

> lattice is already mapped to a finite set of tones.

Ah, now I see that you're counting the tones to determine how many

actual pitches occur in the range of a given octave, whereas I'm

would count them according to how many actual pitches there are when

all of them are reduced to a single octave in order to determine how

many distinct pitches there might be when they are evaluated

harmonically. I realize that there's a practical limit to how many

pitches fall within the range of audibility, but think for a moment:

couldn't you have an organ tuned in a non-octave temperament with

ranks of pipes pitched an octave or two apart, or two instruments

(such as flute and piccolo) in that same temperament pitched an

octave apart? Are there any rules for avoiding such things with non-

octave temperaments, and if so, who's going to enforce them?

> > > I can

> > > certainly see how a JI periodicity block might be created, then

> > > tempered first near the edges, and finally tempered uniformly

> > > everywhere to yield an ET, such that the middle step would be a

well-

> > > temperament. However, I feel this unfairly omits the whole

universe

> > > of possibilities where fewer of the periodicity blocks' unison

> > > vectors are tempered out, and which are thus also in

the "middle"

> > > between JI and ETs.

> >

> > As I said above, my impression was that the term "middle path"

would

> > not include any tuning not covered by the two opposite extremes,

JI

> > and ET

>

> Exactly. Is there anything in the above paragraph that would

suggest

> otherwise?

Oops, I think my brain flew out of gear at that point. I meant to

say, my impression was that the term "middle path" would not exclude

any tuning covered by the two opposite extremes, JI and ET. But in

light of your response (below), I would now substitute "middle

ground" for "middle path" in that statement (and I rather think that

I would prefer EDO to ET for one of those extremes).

> > (or as I would prefer, EDO).

>

> I would prefer EDO to refer to tuning systems conceived as equal

> divisions of the octave and not as temperaments. ET should refer to

> temperament, a tempering of JI, in particular that of a sort which

> leads to a finite number of equally-spaced tones. 24-equal, for

> example, would be awkward to derive as an ET, unless you began with

> the restriction that only primes 3, 11, and perhaps 13 would figure

> into your initial JI system. 24-equal, at least in the West, is

> usually best considered an EDO. Not so for 12-equal.

Then should we call a non-octave equal spacing of tones that does not

approximate JI particularly well an ET, and, if not, then what name

should we use and where do we put the boundary between the two? ;-)

I would much prefer to make the two philosophical extremes JI and

EDO's rather than JI and ET's, because the former is unambiguous.

> > > While I certainly respect your thinking process, and the

musical

> > > value of well-tempered systems is unquestionable, I wonder if

we're

> > > heading toward a terminological nightmare here.

> >

> > Well, "middle path" is your term, so I think I'm obligated to

abide

> > by whatever definition you choose for it. My problem is that you

> > have used the term to exclude certain tunings that, given a

similar

> > number of tones, offer better intonation than EDO's and more

> > modulation than JI, thereby offering a practical middle-ground

> > between the two.

> >

> > Hmmm, maybe I should call this broader category the "middle-

ground",

> > to include tunings both on and off the "middle path". (I always

> > fancied most of my tunings as being off the beaten path,

anyway. ;-)

>

> Sounds good to me.

Agreed, then -- nightmare averted!

> > Question: would all regular middle-ground tunings then be middle-

> >path?

>

> Shoot me an example to test.

I didn't have anything particular in mind. I just wondered if you

could think of an example for which that would not hold true.

What about a linear tuning that doesn't approximate JI well? Or if I

considered JI and EDO's to be the two extremes between which there

exists a middle ground (while you keep JI vs. ET's): then non-octave

ET's would be regular middle-ground tunings that are not middle-

path. (Hmmm, that could get confusing.)

--George

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

> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>

wrote:

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

wrote:

> > > ...

> > > As I said above, my impression was that the term "middle path"

would

> > > not include any tuning not covered by the two opposite

extremes, JI

> > > and ET

> >

> > Exactly. Is there anything in the above paragraph that would

suggest

> > otherwise?

>

> Oops, I think my brain flew out of gear at that point. I meant to

> say, my impression was that the term "middle path" would not

exclude

> any tuning covered by the two opposite extremes, JI and ET.

Oops, again. How about: my impression was that the term "middle

path" would not exclude any tuning *not* covered by the two opposite

extremes ...

Which shows to go, you shouldn't never use no double negative.

--George

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

> And even if the amount of stretch were equal over all octaves, this

> example would still not be a tuning *theoretically* distinct from

12-

> ET, since the amount of stretch would vary with the implementation

of

> the tuning on one particular instrument vs. another.

What if it didn't?

> The intention

> is to tune the instrument so that the octaves sound exact (or as

> nearly exact as possible), in order to make the result (what the

ear

> hears in *practice*) most closely match the *theoretical* tuning.

What if the theoretical tuning *does* have tempered octaves? For

example, the 3-limit TOP tuning of 12-equal has stretched octaves,

while the 5-limit TOP tuning of 12-equal has compressed octaves.

>

> Similarly, are we to insist that any implementation of JI on

acoustic

> instruments is a temperament, since it's impossible to achieve

exact

> phase-locked ratios? Of course not; this is a *practical* problem

> rather than another (theoretical) tuning.

>

> So I would draw the line at the point where the *theoretical

tuning*

> is not mathematically an EDO, where the octaves are intentionally

> tempered by a specified number of cents, usually for the purpose of

> approximating other consonances more accurately.

I still see these as ETs. I'm not sure what you would see them as.

>

> > > 2) Theoretically, non-octave ET's are open systems.

> >

> > Huh? I don't get it. ETs, whether octave-based or not, have a

> > discrete set of pitches, each one separated from the others by a

> > given interval, which would seem to make them closed rather than

> open

> > systems.

>

> But the set can be extended indefinitely up or down without any

pitch

> ever repeating any previous pitch at a different octave (taking the

> term "octave" in the strict sense of the word). The principle of

> octave equivalence exists whether or not an ET is octave-based.

> (Please read further before replying to this.)

OK.

> > in open temperaments,

> > including more and more of the lattice means adding more and more

> > tones between the existing ones, with no limit to how far this

> > process may proceed. In closed temperaments, the entire infinite

> > lattice is already mapped to a finite set of tones.

>

> Ah, now I see that you're counting the tones to determine how many

> actual pitches occur in the range of a given octave, whereas I'm

> would count them according to how many actual pitches there are

when

> all of them are reduced to a single octave in order to determine

how

> many distinct pitches there might be when they are evaluated

> harmonically.

Slightly tempered octaves are really no different than pure octaves

in terms of the harmonic "equivalence" they convey. If you start with

an EDO, and then slightly stretch or compress the tuning, the

subjective effect is not one of a sudden and great increase in the

number of pitches.

> I realize that there's a practical limit to how many

> pitches fall within the range of audibility, but think for a

moment:

> couldn't you have an organ tuned in a non-octave temperament with

> ranks of pipes pitched an octave or two apart, or two instruments

> (such as flute and piccolo) in that same temperament pitched an

> octave apart? Are there any rules for avoiding such things with

non-

> octave temperaments, and if so, who's going to enforce them?

By this reasoning, 12-equal on most instruments is an open tuning,

since the harmonics don't coincide with the fundamentals. I don't

think this is a very practical way of viewing the situation.

> Then should we call a non-octave equal spacing of tones that does

not

> approximate JI particularly well an ET,

If it's not theoretically approximating JI, it's merely an equal

tuning, not an equal temperament.

> I would much prefer to make the two philosophical extremes JI and

> EDO's rather than JI and ET's, because the former is unambiguous.

"JI" is anything but unambiguous.

> What about a linear tuning that doesn't approximate JI well?

Again, if there's no approximation to JI intended, I'd call it a 2D

tuning rather than a 2D temperament (I'm not too thrilled with the

term "linear" -- it's equal tunings are better regarded as one-

dimensional).

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

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

wrote:

> > ...

> > So I would draw the line at the point where the *theoretical

tuning*

> > is not mathematically an EDO, where the octaves are intentionally

> > tempered by a specified number of cents, usually for the purpose

of

> > approximating other consonances more accurately.

>

> I still see these as ETs. I'm not sure what you would see them as.

Okay, I see your point. I will concede that ET's with slightly

tempered octaves can be considered closed systems, just as if they

were EDO's.

> > > > 2) Theoretically, non-octave ET's are open systems.

> > >

> > > Huh? I don't get it. ETs, whether octave-based or not, have a

> > > discrete set of pitches, each one separated from the others by

a

> > > given interval, which would seem to make them closed rather

than open

> > > systems.

> >

> > But the set can be extended indefinitely up or down without any

pitch

> > ever repeating any previous pitch at a different octave (taking

the

> > term "octave" in the strict sense of the word). The principle of

> > octave equivalence exists whether or not an ET is octave-based.

> > (Please read further before replying to this.)

>

> OK.

>

> > > in open temperaments,

> > > including more and more of the lattice means adding more and

more

> > > tones between the existing ones, with no limit to how far this

> > > process may proceed. In closed temperaments, the entire

infinite

> > > lattice is already mapped to a finite set of tones.

> >

> > Ah, now I see that you're counting the tones to determine how

many

> > actual pitches occur in the range of a given octave, whereas I'm

> > would count them according to how many actual pitches there are

when

> > all of them are reduced to a single octave in order to determine

how

> > many distinct pitches there might be when they are evaluated

> > harmonically.

>

> Slightly tempered octaves are really no different than pure octaves

> in terms of the harmonic "equivalence" they convey. If you start

with

> an EDO, and then slightly stretch or compress the tuning, the

> subjective effect is not one of a sudden and great increase in the

> number of pitches.

Okay. When I wrote the above I was thinking principally of tunings

such as the Bohlen-Pierce scale, where intervals with new harmonic

significance are encountered in each successive octave and therefore

do not behave harmonically as closed systems. Since one may

construct a continuum of ET's with the closest interval to an octave

ranging all the way from a few cents difference to an entire diesis

(or more), I was thinking that there may be no clear boundary between

a harmonically open or closed system, so I was inclined to lump

together all non-EDO ET's as open systems and therefore middle-ground

tunings.

Now I'm inclined just to throw in the towel and put the ET's

(including all EDO's) and JI at opposite extremes and just let

everything else be middle-ground (of which the subset consisting of

regular temperaments is middle-path). Is this okay by you?

But I'm still not convinced that tunings such as Bohlen-Pierce should

be considered closed systems.

--George

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

> Okay. When I wrote the above I was thinking principally of tunings

> such as the Bohlen-Pierce scale, where intervals with new harmonic

> significance are encountered in each successive octave and

therefore

> do not behave harmonically as closed systems.

The concept behind the BP scale requires octave equivalence to give

way to tritave equivalence. Though I haven't experienced this

phenomenon myself, I feel it is wrong to conceptually "impose" the

implications of octave-equivalence on users of the scale.

> But I'm still not convinced that tunings such as Bohlen-Pierce

should

> be considered closed systems.

If you can consider pitch without octave-equivalence or any other

interval of equivalence applying, then you can see BP as a closed

system.