this page is great and very important (by contrast, check out joe

monzo's definition of linear temperament if you want to turn red).

graham, it seems you missed negri's system, which looks to be the

same as "bastoni" and, if so, should replace it -- search the tuning

list or this list for "negri" . . .

hi Paul,

> From: "emotionaljourney22" <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, May 10, 2002 3:25 PM

> Subject: [tuning-math] graham's linear temperament page

>

>

> this page is great and very important (by contrast, check out joe

> monzo's definition of linear temperament if you want to turn red).

how about a more constructive criticism? post something that

i can use to replace or supplement my definition to make it better.

(and also please specify whether i should be replacing or supplementing!)

thanks.

(BTW, i'm only checking in on the lists sporadically these days.

too much other "life" stuff happening...)

-monz

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

> this page is great and very important

So great and so important you couldn't take the time to include a link? Sure, we can find it from our collection of links, or do a search, etc. - I thought you were the consummate cross-referenced poster.

> (by contrast, check out joe

> monzo's definition of linear temperament if you want to turn red).

I didn't want to comment on this unless Joe had seen it, and he has. Paul, if you ever want a good example of the kind of thing that makes people really sad and crazy and distrustful, just save that above fragment. And maybe don't write about other people so uncharitably in public when you could just as easily be helpful about it.

Some people never learn...

Jon

In-Reply-To: <abmetp+mul3@eGroups.com>

jonszanto wrote:

> So great and so important you couldn't take the time to include a link?

> Sure, we can find it from our collection of links, or do a search, etc.

> - I thought you were the consummate cross-referenced poster.

Everybody does or doesn't include links according to their mood. This one

is <http://x31eq.com/catalog.htm>.

I personally would like somebody to include the definitions of the

temperaments I'm supposed to be adding. It's been quite difficult to keep

up over the past year.

> > (by contrast, check out joe

> > monzo's definition of linear temperament if you want to turn red).

After following the links, I thought the "temperament" definition was

lacking, and decided to clarify it. Then scrolled down, and saw I already

had. Although I now disagree with my own definition -- temperaments can

approximate "ideal" tunings other than just intonation.

It looks like the problem with the "linear temperament" definition is that

there isn't one! You simply say it's "another term for unequal

temperaments" which isn't the case. Not all unequal temperaments are

linear.

A linear temperament is fully described by a generator and equivalence

interval. The mapping from generators to consonances is fixed. For

example, meantones are defined by the generator approximating 3:2 and four

generators approximating 5:4. The generator can have different sizes,

giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same

temperament class. 53-equal, however, isn't a meantone because the best

5:4 is 8 fourths rather than 4 fifths.

Planar temperaments are like linear temperaments, but with two generators

and an equivalence interval. The difference between a generator and an

equivalence interval is only a matter of formalism.

Graham

--- In tuning-math@y..., graham@m... wrote:

> Everybody does or doesn't include links according to their mood.

Sure, to be expected. But if someone says "hey, here is something to note" and then expects you to hunt it down yourself, it means they think their time is more valuable than yours. Pity.

> is <http://x31eq.com/catalog.htm>.

I'll take a look, thanks Graham.

Cheers,

Jon

many thanks, Graham! so the Tuning Dictionary now has

a couple of new definitions:

linear temeperament

http://www.ixpres.com/interval/dict/lineartemp.htm

planar temperament

http://www.ixpres.com/interval/dict/planartemp.htm

as well as an important addition to an old definition:

http://www.ixpres.com/interval/dict/consonance.htm

see Paul, that's the *right* way to do it!

(and if anyone would care to explain that "matter of formalism"

which differentiates a "generator" from an "equivalence interval",

i'd sure like to read it! after much discussion of it a few months

ago, i'm still not clear about it.)

-monz

----- Original Message -----

From: <graham@microtonal.co.uk>

To: <tuning-math@yahoogroups.com>

Sent: Monday, May 13, 2002 2:57 AM

Subject: [tuning-math] Re: graham's linear temperament page

> In-Reply-To: <abmetp+mul3@eGroups.com>

> jonszanto wrote:

>

> > So great and so important you couldn't take the time to include a link?

> > Sure, we can find it from our collection of links, or do a search, etc.

> > - I thought you were the consummate cross-referenced poster.

>

> Everybody does or doesn't include links according to their mood. This one

> is <http://x31eq.com/catalog.htm>.

>

> I personally would like somebody to include the definitions of the

> temperaments I'm supposed to be adding. It's been quite difficult to keep

> up over the past year.

>

> > > (by contrast, check out joe

> > > monzo's definition of linear temperament if you want to turn red).

>

> After following the links, I thought the "temperament" definition was

> lacking, and decided to clarify it. Then scrolled down, and saw I already

> had. Although I now disagree with my own definition -- temperaments can

> approximate "ideal" tunings other than just intonation.

>

> It looks like the problem with the "linear temperament" definition is that

> there isn't one! You simply say it's "another term for unequal

> temperaments" which isn't the case. Not all unequal temperaments are

> linear.

>

> A linear temperament is fully described by a generator and equivalence

> interval. The mapping from generators to consonances is fixed. For

> example, meantones are defined by the generator approximating 3:2 and four

> generators approximating 5:4. The generator can have different sizes,

> giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same

> temperament class. 53-equal, however, isn't a meantone because the best

> 5:4 is 8 fourths rather than 4 fifths.

>

> Planar temperaments are like linear temperaments, but with two generators

> and an equivalence interval. The difference between a generator and an

> equivalence interval is only a matter of formalism.

>

>

> Graham

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> (and if anyone would care to explain that "matter of formalism"

> which differentiates a "generator" from an "equivalence interval",

> i'd sure like to read it! after much discussion of it a few months

> ago, i'm still not clear about it.)

Unfortunately I have a slightly different take on this to Graham's

recent statement. I'm wondering if Graham really meant to say the

following.

A linear temperament is generated by two "generating intervals" (or

generators for short), one of which is distinguished as the "interval

of periodicity" (or "period" for short) and the other is simply

called "the generator". The "period" is distinguished from the other

generator by the fact that the "interval of equivalence" is a whole

number multiple of the period. The interval of equivalence is usually

the octave. A particular scale within a linear temperament (e.g. a

diatonic scale within meantone) will have a fixed number of

(non period) generators but an unspecified number of intervals of

equivalence.

Hi Graham,

Interesting. This definition--"A linear temperament is fully described

by a generator and equivalence interval. The mapping from generators

to consonances is fixed."--is basically the same one I tried to use to

generalize meantone. (The way I looked at the Kleismic in this sense

was as a 3:5 in 1D and a 3:4:5 in 2D.)

So true Meantones according to your definition would be found in ETs

5, 7, 12, 19, 26, 31, 43, 45, 50, 55, etc. and true Kleismics in ETs

4, 15, 19, 23, 34, 49, 53, etc., right?

Hmm, I like it, but there must be a zillion tunings like this!

The 6-tone scale in 11-tet that Margo brought up recently would be one

example. I'd see it as a 4:7 in 1D and a 8:11:14 in 2D with true

whatever-you-want-to-call-thems found in ETs 5, 6, 11, 12, 16, 17, 23,

27, etc.

Then again maybe that's too coarse (11 and then 12)? It seems like it

might be tough to say (exactly) when a simple JI interpretation would

be the 'correct' one?

take care,

--Dan Stearns

----- Original Message -----

From: <graham@microtonal.co.uk>

To: <tuning-math@yahoogroups.com>

Sent: Monday, May 13, 2002 2:57 AM

Subject: [tuning-math] Re: graham's linear temperament page

> In-Reply-To: <abmetp+mul3@eGroups.com>

> jonszanto wrote:

>

> > So great and so important you couldn't take the time to include a

link?

> > Sure, we can find it from our collection of links, or do a search,

etc.

> > - I thought you were the consummate cross-referenced poster.

>

> Everybody does or doesn't include links according to their mood.

This one

> is <http://x31eq.com/catalog.htm>.

>

> I personally would like somebody to include the definitions of the

> temperaments I'm supposed to be adding. It's been quite difficult

to keep

> up over the past year.

>

> > > (by contrast, check out joe

> > > monzo's definition of linear temperament if you want to turn

red).

>

> After following the links, I thought the "temperament" definition

was

> lacking, and decided to clarify it. Then scrolled down, and saw I

already

> had. Although I now disagree with my own definition -- temperaments

can

> approximate "ideal" tunings other than just intonation.

>

> It looks like the problem with the "linear temperament" definition

is that

> there isn't one! You simply say it's "another term for unequal

> temperaments" which isn't the case. Not all unequal temperaments

are

> linear.

>

> A linear temperament is fully described by a generator and

equivalence

> interval. The mapping from generators to consonances is fixed. For

> example, meantones are defined by the generator approximating 3:2

and four

> generators approximating 5:4. The generator can have different

sizes,

> giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same

> temperament class. 53-equal, however, isn't a meantone because the

best

> 5:4 is 8 fourths rather than 4 fifths.

>

> Planar temperaments are like linear temperaments, but with two

generators

> and an equivalence interval. The difference between a generator and

an

> equivalence interval is only a matter of formalism.

>

>

> Graham

>

>

> ------------------------ Yahoo! Groups

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> Your use of Yahoo! Groups is subject to

http://docs.yahoo.com/info/terms/

>

>

--- In tuning-math@y..., graham@m... wrote:

> A linear temperament is fully described by a generator and

equivalence

> interval.

I disagree. I'd be a lot closer to agreeing if you had said "a

generator and a periodicity interval" (which can be a fraction of the

equivalence interval), but I think you want to know the equivalence

interval as well as the period.

But what if there isn't an equivalence interval but there are still

two generators; does it still make sense to call it a "linear"

temperament. That's a tough one.

>The mapping from generators to consonances is fixed. For

> example, meantones are defined by the generator approximating 3:2

and four

> generators approximating 5:4. The generator can have different

sizes,

> giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same

> temperament class. 53-equal, however, isn't a meantone because the

best

> 5:4 is 8 fourths rather than 4 fifths.

So the mapping defines a "temperament class" while specific generators

define a "temperament (instance)". I like that.

But what do we call it when we have different numbers of notes per

equivalence interval (typically MOS) for the same temperament. I guess

we just call them things like "12 note meantone" or "miracle-21".

> Planar temperaments are like linear temperaments, but with two

generators

> and an equivalence interval. The difference between a generator and

an

> equivalence interval is only a matter of formalism.

Again, substitute "period" for "equivalence interval" and "only a

matter of formalism" for the definitions in my preceeding post.

--- In tuning-math@y..., graham@m... wrote:

> I personally would like somebody to include the definitions of the

> temperaments I'm supposed to be adding. It's been quite difficult to keep

> up over the past year.

I think it would be nice to include links to music in the various temperaments discussed. I have examples for Orwell and Wonder, and am working on something in Magic. I also have 126/125 and 64/63 if you want to add planar temperaments. Porcupine, Miracle and of course Meantone are out there--what else?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> A linear temperament is generated by two "generating intervals" (or

> generators for short), one of which is distinguished as the "interval

> of periodicity" (or "period" for short) and the other is simply

> called "the generator".

One of them may or may not be an interval of peridoicity--there's no requirement to have one, it's simply always possible to express them in that way. The key fact is that a linear temperament is rank two--ie, it has two generators.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> But what if there isn't an equivalence interval but there are still

> two generators; does it still make sense to call it a "linear"

> temperament. That's a tough one.

That's more or less my definition of a linear temperament, except I would add that the generators are mapped to from some subgroup of the positive rationals. Making an interval of equivalence part of the definition gives us one meantone for 2, one for 3/2, one for 3, one for 5/2, another for 5/3 and so forth. I don't like it, and don't plan to use it.

In-Reply-To: <abq076+gjf0@eGroups.com>

genewardsmith wrote:

> I think it would be nice to include links to music in the various

> temperaments discussed. I have examples for Orwell and Wonder, and am

> working on something in Magic. I also have 126/125 and 64/63 if you

> want to add planar temperaments. Porcupine, Miracle and of course

> Meantone are out there--what else?

I'd prefer to link to a "home page" for each temperament. I already have

these in place for meantone, miracle, schismic and diaschismic. If you

can supply your own pages for these other temperaments, I can link to them

and you can handle the links to examples. I should do something for magic

and I plan to tune up the multiple-29 soon as well. The 29&19 temperament

that we're apparently calling Negri does look promising on its own terms

and as a way of slicing up multiple-29, maybe even giving a planar

temperament.

Graham

In-Reply-To: <abpt98+61vd@eGroups.com>

dkeenanuqnetau wrote:

> I disagree. I'd be a lot closer to agreeing if you had said "a

> generator and a periodicity interval" (which can be a fraction of the

> equivalence interval), but I think you want to know the equivalence

> interval as well as the period.

I should have said period. It can be fully described by the period and

generator, although it can also be convenient to think in terms of an

equivalence interval. If you put aside contorsion, the number of periods

to an equivalence is the GCD of the generator mapping, but that's not

something the definition should concern itself with. Unless we want to

pronounce on whether temperaments with contorsion really count as

temperaments. But to cover them you need to state the full mapping, in

terms of both period and generator, anyway. In which case we still have

to specify that the mapping should only involve integers.

> But what if there isn't an equivalence interval but there are still

> two generators; does it still make sense to call it a "linear"

> temperament. That's a tough one.

You can always call one of the generators an equivalence interval.

There's really little difference as far as the mathematics is concerned.

But I expect musicians will find it easier to think if the period as a

generalisation of an octave and the generator as the generalisation of a

fifth. They'll find it obvious that a fifth and an octave function

differently in meantone. So that's how the simple definition should be

framed, and mathematical caveats added later.

The word "linear" does strongly suggest a single generator, and music

theory usually assumes octave equivalence, sometimes implicitly. If we

define linear temperaments as having 2 generators, people will get

confused by the off-by-one error.

> But what do we call it when we have different numbers of notes per

> equivalence interval (typically MOS) for the same temperament. I guess

> we just call them things like "12 note meantone" or "miracle-21".

I think those are called "scales". These ones in particular being MOS or

Well Formed.

> Again, substitute "period" for "equivalence interval" and "only a

> matter of formalism" for the definitions in my preceeding post.

Those definitions look fine to me.

Oh, and Joe, when I said "consonances" I meant something much closer to

your definition of "consonance" than "concordance". Musical context is

important for this.

Graham

> From: <graham@microtonal.co.uk>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, May 14, 2002 3:57 AM

> Subject: [tuning-math] Re: graham's linear temperament page

>

>

> Oh, and Joe, when I said "consonances" I meant something much closer to

> your definition of "consonance" than "concordance". Musical context is

> important for this.

oops ... OK, it's been fixed.

http://www.ixpres.com/interval/dict/lineartemp.htm

and what's "contorsion"? that's a new one for me!

-monz

monz wrote:

> and what's "contorsion"? that's a new one for me!

If you define 5-limit meantone using neutral thirds instead of fifths, you

find a fifth is two generators and a major third is two generators.

Because all consonances involve an even number of steps it has a

contorsion of 2. Whether it counts as a new temperament or an odd way of

mapping meantone isn't currently addressed by the definition.

Graham

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

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> > A linear temperament is generated by two "generating intervals"

(or

> > generators for short), one of which is distinguished as the

"interval

> > of periodicity" (or "period" for short) and the other is simply

> > called "the generator".

>

> One of them may or may not be an interval of peridoicity--there's no

requirement to have one, it's simply always possible to express them

in that way. The key fact is that a linear temperament is rank

two--ie, it has two generators.

I agree, but point out that this is a fairly recent way of looking at

them, or at least a particularly mathematical way. To a musician it is

rank 1 where the arithmetic is modulo the interval of equivalence. If

it is "always possible to express them in that way" then they will

always want you to.

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

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>

> > But what if there isn't an equivalence interval but there are

still

> > two generators; does it still make sense to call it a "linear"

> > temperament. That's a tough one.

>

> That's more or less my definition of a linear temperament, except I

would add that the generators are mapped to from some subgroup of the

positive rationals. Making an interval of equivalence part of the

definition gives us one meantone for 2, one for 3/2, one for 3, one

for 5/2, another for 5/3 and so forth. I don't like it, and don't plan

to use it.

OK. I agree with Gene and Graham. Although it doesn't make any _sense_

to call it linear when there is no IoE, we will anyway. If only

because it _can_ be treated the same mathematically (rank two) whether

there is an IoE or not. But I wouldn't want a definition to be based

on that since having no IoE is so rare.

I'd prefer to ignore the question of whether the IoE or the other

consonances _must_ be rational ratios, and thereby avoid a whole can

of worms associated with temperaments for inharmonic timbres.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I'd prefer to ignore the question of whether the IoE or the other

> consonances _must_ be rational ratios, and thereby avoid a whole can

> of worms associated with temperaments for inharmonic timbres.

We could leave off the mapping, though that would make it murky in some respects. Another region of murk are degenerate cases--what if we start with related generators, and find that that the actual rank is less than the number of generators?

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

> --- In tuning-math@y..., graham@m... wrote:

>

> > I personally would like somebody to include the definitions of

the

> > temperaments I'm supposed to be adding. It's been quite

difficult to keep

> > up over the past year.

>

> I think it would be nice to include links to music in the various

>temperaments discussed. I have examples for Orwell and Wonder, and

>am working on something in Magic. I also have 126/125 and 64/63 if

>you want to add planar temperaments. Porcupine, Miracle and of

>course Meantone are out there--what else?

pajara -- 'decatonic swing'

the piece 'glassic' is <2,3,5[,11]> porcupine in the first section,

then goes to the 64:63 temperament in <2,3,7>.

then again, you may decide that 15-equal or 22-equal or 72-equal

should count as examples of equal temperament (even if linear-based

modes are used), not linear temperament . . .

anyway, ara is working out his system bugs over on makemicromusic --

once he's ready, we'll begin making pajara and other-tempered music

in earnest.

by the way, i'd like to send a cd to anyone who'd like to listen to

some enjoyable (i hope) and professional (finally) music. please send

me your snail-mail address if interested. there may be a slight delay

as i'm awaiting receipt of the cd from mastering in NYC.

--- In tuning-math@y..., "monz" <monz@a...> wrote:

>

> > From: <graham@m...>

> > To: <tuning-math@y...>

> > Sent: Tuesday, May 14, 2002 3:57 AM

> > Subject: [tuning-math] Re: graham's linear temperament page

> >

> >

> > Oh, and Joe, when I said "consonances" I meant something much

closer to

> > your definition of "consonance" than "concordance". Musical

context is

> > important for this.

>

>

> oops ... OK, it's been fixed.

> http://www.ixpres.com/interval/dict/lineartemp.htm

hi joe, sorry if i seemed mean before . . . hope you can forgive me!

it's just that i've done a lot of work trying to help you correct

your et definition and the pages it links to, and you, quite

understandably, have had little time to consider my many e-mails,

posts, and IM comments to you on these issues. hope you've at least

saved my e-mails, and made note of the relevant posts, for later

consideration -- that's all i can ask. maybe in 10 years we can get

back to this -- i'd be very happy with that.

anyway, on the current topic, it seems you missed an important fix

that everyone here agrees on -- please replace "interval of

equivalence" with "period" in your definitions.

--- In tuning-math@y..., graham@m... wrote:

> monz wrote:

>

> > and what's "contorsion"? that's a new one for me!

>

> If you define 5-limit meantone using neutral thirds instead of

fifths, you

> find a fifth is two generators and a major third is two

generators.

> Because all consonances involve an even number of steps it has a

> contorsion of 2. Whether it counts as a new temperament or an odd

way of

> mapping meantone isn't currently addressed by the definition.

>

>

> Graham

i thought that, a few months ago, gene and i argued, and dave finally

concurred, that contorsion is disallowed in the definition of linear

temperament -- we seemed able to convince dave that "temperament"

means "temperament of JI", and that contorsion involves a far more

radical departure from JI (thought of as a single, infinitely-

extending lattice of notes, connected by justly-intoned concordant

intervals).

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I agree, but point out that this is a fairly recent way of looking

at

> them, or at least a particularly mathematical way. To a musician it

is

> rank 1 where the arithmetic is modulo the interval of equivalence.

If

> it is "always possible to express them in that way" then they will

> always want you to.

i suspect you can go through the tuning list archives and find two or

three or more musicians who thought otherwise; i.e., in register-

specific terms. i see no reason to discriminate against such

musicians/thinkers -- especially if gene's current mathematical

system handles such a view in a nice, natural way . . .

--- In tuning-math@y..., graham@m... wrote:

> In-Reply-To: <abpt98+61vd@e...>

> dkeenanuqnetau wrote:

>

> > I disagree. I'd be a lot closer to agreeing if you had said "a

> > generator and a periodicity interval" (which can be a fraction of

the

> > equivalence interval), but I think you want to know the

equivalence

> > interval as well as the period.

>

> I should have said period.

monz, i hope you'll take note of this.

hugs,

paul

> From: "emotionaljourney22" <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, May 15, 2002 2:36 PM

> Subject: [tuning-math] Re: graham's linear temperament page

>

>

> > http://www.ixpres.com/interval/dict/lineartemp.htm

>

> hi joe, sorry if i seemed mean before . . . hope you can forgive me!

no sweat, paul -- i'm used to your style by now. :)

consider it water under the bridge.

the main thing is that i want the Dictionary to be as correct

as possible.

> it's just that i've done a lot of work trying to help you correct

> your et definition and the pages it links to, and you, quite

> understandably, have had little time to consider my many e-mails,

> posts, and IM comments to you on these issues. hope you've at least

> saved my e-mails, and made note of the relevant posts, for later

> consideration -- that's all i can ask. maybe in 10 years we can get

> back to this -- i'd be very happy with that.

well, i hope it doesn't take 10 years! but right now i am

kind of tied up with other projects, and have pretty successfully

weaned myself off the tuning list(s) addiction(s). i'm just

checking in these days and fighting the urge to reply to everything.

>

> anyway, on the current topic, it seems you missed an important fix

> that everyone here agrees on -- please replace "interval of

> equivalence" with "period" in your definitions.

ok, look at the linear temperament definition now.

NOW -- regarding "equivalence interval" and "periodicity interval",

you'll all see that i provided links to definitions for those two

terms, but still don't have any content in them. that's because

i'm *still* not clear on how to define them. can someone(s)

*please* post simply a definition for each of them which i can

put into those Dictionary entries. if others disagree, then

i'll try to keep updating the webpages the way i've done with

"linear temperament" -- but please give me something to start with.

-monz

In-Reply-To: <000f01c1fca0$7d449ca0$af48620c@dsl.att.net>

monz wrote:

> ok, look at the linear temperament definition now.

Fancy that!

> NOW -- regarding "equivalence interval" and "periodicity interval",

> you'll all see that i provided links to definitions for those two

> terms, but still don't have any content in them. that's because

> i'm *still* not clear on how to define them. can someone(s)

> *please* post simply a definition for each of them which i can

> put into those Dictionary entries. if others disagree, then

> i'll try to keep updating the webpages the way i've done with

> "linear temperament" -- but please give me something to start with.

Dave gave some, and I didn't notice anybody disagreeing. The only quibble

I have is that the period definition uses the terminology differently to

the linear temperament one. And that it's circular, but that's probably

unavoidable.

"""

"interval of equivalence" = "equivalence interval" = "formal octave"

is that interval (much larger than a unison) which, when it occurs

between two pitches, we consider them to be, in some sense, (formally

if not perceptibly) the same note. For most scales this is the octave

1:2, and when it is not the octave it is usually some other highly

consonant interval such as the "tritave" 1:3. But the essential

feature of the interval of equivalence in relation to definitions of

scales and types of scales is that when we describe a scale we

describe only the pitches that fall within a single interval of

equivalence, and we leave it up to the instrument builder to decide

the range of the instrument and therefore how many times (including

fractions) the interval of equivalence should be repeated.

"interval of periodicity" = "periodic interval" = "period"

is that generator of a regular temperament (whether linear, planar, or

n-dimensional) which generates the interval of equivalence all by

itself. This means that the period is either equal to the interval of

equivalence or fits into the interval of equivalence a whole number of

times.

"""

Graham

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> > anyway, on the current topic, it seems you missed an important

fix

> > that everyone here agrees on -- please replace "interval of

> > equivalence" with "period" in your definitions.

>

>

> ok, look at the linear temperament definition now.

the line

"The difference between a generator and an equivalence interval is

only a matter of formalism."

should read

"The difference between a generator and a period is only a matter of

formalism."

if you're going to say "interval of periodicity" for "period", i'd

suggest that "interval of repetition" is even better -- it's more to

the point, and doesn't cause confusion with periodicity blocks.