Fun to see good friends here from the tuning group! Much of the

language here is new to me, so I've been using Monz' online

dictionary extensively. However, I think it would be productive for

me to actually study some concise presentation of these concepts to

get myself up to speed a bit, since frankly much of the discussion

here in this group is still going around my ears.

I do have a good technical background and no math phobias, since

before I switched to music, I was a physics and math student for two

years and loved it. Music was just an even more powerful draw.

Nonetheless, I spent a good part of my life after graduating working

in electronics, and since I've also done recording engineering,

acoustics and psychoacoustics are not alien to me either.

So I would love to know of any recommendations you might offer as a

good first primer on lattices, unison vectors, periodicity blocks,

linear temperaments, etc. and their relationships to each other.

Thanks tons in advance!

Sincerely,

Bob

--- In tuning-math@y..., BobWendell@t... wrote:

> So I would love to know of any recommendations you might offer as a

> good first primer on lattices, unison vectors, periodicity blocks,

> linear temperaments, etc. and their relationships to each other.

>

> Thanks tons in advance!

>

> Sincerely,

>

> Bob

As far as I know, we here are on the cutting edge of this stuff, and

it really isn't dealt with anywhere else. My advice would be to go

through the archives of this list, and of the tuning list before this

list split off from it. Ask us lots of questions.

Thanks so much, Paul! I feel very fortunate indeed to have run across

you all here. I was hoping for some kind of more coherent

presentation format, but I guess that doesn't exist yet. I will take

your generous advice and hope that I don't pester you too much. You

all seem to have a high tolerance for my naivete in such matters and

I deeply appreciate that!

Sincerely,

Bob

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

> --- In tuning-math@y..., BobWendell@t... wrote:

>

> > So I would love to know of any recommendations you might offer as

a

> > good first primer on lattices, unison vectors, periodicity

blocks,

> > linear temperaments, etc. and their relationships to each other.

> >

> > Thanks tons in advance!

> >

> > Sincerely,

> >

> > Bob

>

> As far as I know, we here are on the cutting edge of this stuff,

and

> it really isn't dealt with anywhere else. My advice would be to go

> through the archives of this list, and of the tuning list before

this

> list split off from it. Ask us lots of questions.

--- In tuning-math@y..., BobWendell@t... wrote:

> Thanks so much, Paul! I feel very fortunate indeed to have run

across

> you all here. I was hoping for some kind of more coherent

> presentation format, but I guess that doesn't exist yet. I will

take

> your generous advice and hope that I don't pester you too much. You

> all seem to have a high tolerance for my naivete in such matters

and

> I deeply appreciate that!

>

> Sincerely,

>

> Bob

Bob, I feel particularly close to you since we seem to share a very

similar set of opinions and ideals, in both of our cases acquired

through genuine musical experience.

I'm curious whether my last response to you on the tuning list made

sense to you, and if you have any outstanding questions.

-Paul

--- In tuning-math@y..., BobWendell@t... wrote:

> Thanks so much, Paul! I feel very fortunate indeed to have run

across

> you all here. I was hoping for some kind of more coherent

> presentation format

I trust you've read the _Gentle Introduction to Fokker Periodicity

Blocks_ (http://www.ixpres.com/interval/td/erlich/intropblock1.htm)?

Paul:

I trust you've read the _Gentle Introduction to Fokker Periodicity

> Blocks_

Bob:

Yes, but I need to peruse it with time and the depth of contemplation

that affords. I apologize for not having responded yet to your last

replies. I do have replies floating around in my head, but no time to

get them down and to you yet. Will do hopefully soon. Thanks for your

solicitous follow-up, Paul.

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

> --- In tuning-math@y..., BobWendell@t... wrote:

> > Thanks so much, Paul! I feel very fortunate indeed to have run

> across

> > you all here. I was hoping for some kind of more coherent

> > presentation format

>

> I trust you've read the _Gentle Introduction to Fokker Periodicity

> Blocks_ (http://www.ixpres.com/interval/td/erlich/intropblock1.htm)?

The feeling is mutual, Paul. Thank you!

Most sincerely,

Bob

P.S. I sometimes feel a bit isolated here, not so much because of the

location in rural Iowa, but the University of Iowa only a bit over an

hour away and other schools around here have ignored my overtures in

seeking mentoring and cooperation in my endeavors with Cantus

Angelicus. The philosophies around here, although I've never really

had the opportunity to discuss them explicitly, seem to be in

opposition to my approach. We are on the Web, as you must know, and

quite visible to them. I have seen a significant number of hits from

U of I, so I know they have poked around the site.

Or perhaps it has to do with my having only a BA in Mus Ed from a

small state university in Tennessee. I've notice that many academics

don't like to have their tea parties crashed by uncredentialed souls

like me, no matter what the quality of our musical product,

especially when our Website proclaims a unique approach to choral

training and we boast a small chamber choir of virtual choral

neophytes in most cases that sounds better than many of their choirs

richly stocked with music majors. It takes us longer to get our music

to market, so to speak, but it's usually great once we get it there

as long as the recent turnover rate has been reasonable.

At any rate, whatever the motives may actually be, it's given me a

significant antidote to my feelings of marginalization to interact

with you and others in the tuning group on themes so dear to my heart

for so long. I feel like I've been wondering in the wilderness for

all these long years and finally found a warm and commodious place to

settle. Thank you all again.

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

> --- In tuning-math@y..., BobWendell@t... wrote:

> > Thanks so much, Paul! I feel very fortunate indeed to have run

> across

> > you all here. I was hoping for some kind of more coherent

> > presentation format, but I guess that doesn't exist yet. I will

> take

> > your generous advice and hope that I don't pester you too much.

You

> > all seem to have a high tolerance for my naivete in such matters

> and

> > I deeply appreciate that!

> >

> > Sincerely,

> >

> > Bob

>

> Bob, I feel particularly close to you since we seem to share a very

> similar set of opinions and ideals, in both of our cases acquired

> through genuine musical experience.

>

> I'm curious whether my last response to you on the tuning list made

> sense to you, and if you have any outstanding questions.

>

> -Paul

> From: <BobWendell@technet-inc.com>

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

> Sent: Friday, August 17, 2001 2:53 PM

> Subject: [tuning-math] Re: Hi! Seeking advice

>

>

> Or perhaps it has to do with my having only a BA in Mus Ed from a

> small state university in Tennessee. I've notice that many academics

> don't like to have their tea parties crashed by uncredentialed souls

> like me, no matter what the quality of our musical product,

> especially when our Website proclaims a unique approach to choral

> training and we boast a small chamber choir of virtual choral

> neophytes in most cases that sounds better than many of their choirs

> richly stocked with music majors. It takes us longer to get our music

> to market, so to speak, but it's usually great once we get it there

> as long as the recent turnover rate has been reasonable.

Hi Bob,

Don't be concerned about being "uncredentialed". I get plenty of

respect from lots of people on these different tuning lists, and it's

based entirely on my webpages and on what I've posted to those lists;

and I have no degrees... nothing beyond a high school diploma.

love / peace / harmony ...

-monz

http://www.monz.org

"All roads lead to n^0"

_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

Hi, Monz! You have an impressive level of knowledge. I'm not

personally concerned about credentials, and certainly not here! No

indication ever that I should be. For me the bottom line is the

ability to come up with the goods.

I was simply referring to the lack of response from local university

types to my work with the choir. We just came back from Vienna and

Sopron, Hungary in early July where we delivered stellar performances

at an international choral festival. We have received local

recognition in terms of corporate sponsorship and private donations

to help get us there. We have also been scheduled for the 2002

Community Concert Series.

Yet no recognition from academia so far. Zero! I think it's utterly

stupid and don't really know what the motives are. Just guessing

about the credentials, etc. Oh well, it's their problem, right? But

it does have an impact on us in the practical realm, making it more

difficult to become well known in the larger community around us, for

example, without their support.

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

> > From: <BobWendell@t...>

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

> > Sent: Friday, August 17, 2001 2:53 PM

> > Subject: [tuning-math] Re: Hi! Seeking advice

> >

> >

> > Or perhaps it has to do with my having only a BA in Mus Ed from a

> > small state university in Tennessee. I've notice that many

academics

> > don't like to have their tea parties crashed by uncredentialed

souls

> > like me, no matter what the quality of our musical product,

> > especially when our Website proclaims a unique approach to choral

> > training and we boast a small chamber choir of virtual choral

> > neophytes in most cases that sounds better than many of their

choirs

> > richly stocked with music majors. It takes us longer to get our

music

> > to market, so to speak, but it's usually great once we get it

there

> > as long as the recent turnover rate has been reasonable.

>

>

> Hi Bob,

>

>

> Don't be concerned about being "uncredentialed". I get plenty of

> respect from lots of people on these different tuning lists, and

it's

> based entirely on my webpages and on what I've posted to those

lists;

> and I have no degrees... nothing beyond a high school diploma.

>

>

>

> love / peace / harmony ...

>

> -monz

> http://www.monz.org

> "All roads lead to n^0"

>

>

>

>

>

> _________________________________________________________

> Do You Yahoo!?

> Get your free @yahoo.com address at http://mail.yahoo.com

Bob, I feel particularly close to you since we seem to share a very

> similar set of opinions and ideals, in both of our cases acquired

> through genuine musical experience.>

> I'm curious whether my last response to you on the tuning list made

> sense to you, and if you have any outstanding questions.> > -Paul

Hi again, Paul! Still not gotten back to you since I haven't yet

gotten a satisfactory perusal of your "Gentle Intro..."(to PBs,

etc.). I've been tied up with a little project that has obsessed me

for the last two days or so. I have long been interested in coming up

with an EDO that would meet the following requirements:

1) P5s and M3s less than 3 cents from JI

2) all primes through 19 less than 5 cents from JI

3) The smallest number of scale degrees possible while meeting

requirements 1 and 2

The purpose is to provide myself and anyone else interested with a

simple, quick and but rather clean tool for dealing conceptually with

JI and tuning systems in general. 1200 is a huge number. The goal is

not to find a scale division so low as to be terribly practical

physically, but to serve as a finite analytical tool that reflects

the realities of intonation as accurately as possible with the least

encumberance, requiring only simple arithmetical tools for its

practical implementation in anaylzing or composing.

I just finished some research that took as a conceptual starting

point 12-EDO for the P5 as derived from the 7/12 lowest rational

approximation of a JI P5 in terms of its pitchwise portion of the

octave. Using the same principle of rational approximation of JI

pitch ratios (NOT their frequency ratios), I found the lowest

rational approximation of P5 to M3 (exact pitch ratio = 1.8171) to be

9/5. 9 X 12 = 108-EDO. I'm sure this must be a well-known EDO

division, since it has so many compelling characteristics, but I

don't know about it from any source other than my own research of the

last two days.

It does satisfy the above conditions, except for #3 if someone finds

a lesser division that meets conditions 1 & 2. The greatest error is

5.0 cents for 17/16. The P5 is -2.0 cents as in 12-EDO, and the M3 is

+2.6 cents. The lowest deviation is the 16/15 M2 at -0.6 cents. The

M2+ (8/7) is 2.2 cents, so the greatest error for 7-limit is 2.6

cents for the M3.

Do you or does anyone else have some feedback? Thanks in advance,

Paul et al.

- Bob

--- In tuning-math@y..., BobWendell@t... wrote:

This isn't what you asked for, but 311 represents all primes up

through 41 with an error of less than a cent; it's a remarkable et in

some ways. Your question could be answered easily with a brute-force

search, however; and I might do it if you define P3 and M5 for me.

> This isn't what you asked for, but 311 represents all primes up

> through 41 with an error of less than a cent; it's a remarkable et

> in some ways. Your question could be answered easily with a brute-

> force search, however; and I might do it if you define P3 and M5

> for me.

As I'm sure others will be quick to point out, if one intends to

use anything more than dyadic harmony, he must also check the errors

of these larger chords in the given et. This gave rise to the

notion of consistentcy. See:

http://library.wustl.edu/~manynote/music.html

Also, aside from looking at max errors, Paul Erlich has suggested

considering an et as an an approximation. See:

http://www-math.cudenver.edu/~jstarret/Question1.html

As far as replacing cents with the units of some et, it isn't

likely to happen. Ellis' cents enjoy wide use, provide good

accuracy regardless of wether the target interval is just, make

the octave highly divisable without fractional results, and

provide a handy reference to 12-tone equal temperament.

-Carl

--- In tuning-math@y..., BobWendell@t... wrote:

> I have long been interested in coming up

> with an EDO that would meet the following requirements:

>

> 1) P5s and M3s less than 3 cents from JI

What about m3s and M6ths?

>

> 2) all primes through 19 less than 5 cents from JI

Why just primes?

> 3) The smallest number of scale degrees possible while meeting

If you insist on _consistency_ through the 19-limit (which I would),

and maximum error less than 5 cents in the 19-limit, then 121-tET is

for you. It has maximum errors of

2.18 cents in the 3-limit

2.18 cents in the 5-limit

3.07 cents in the 7-limit

4.35 cents in the 9-limit

4.35 cents in the 11-limit

4.35 cents in the 13-limit

4.35 cents in the 15-limit

4.35 cents in the 17-limit

4.35 cents in the 19-limit

111-tET would probably satisfy you, though, since its maximum errors

are

0.75 cents in the 3-limit

2.88 cents in the 5-limit

4.15 cents in the 7-limit

4.15 cents in the 9-limit

4.15 cents in the 11-limit

4.15 cents in the 13-limit

4.15 cents in the 15-limit

4.15 cents in the 17-limit

5.19 cents in the 19-limit

>

> The purpose is to provide myself and anyone else interested with a

> simple, quick and but rather clean tool for dealing conceptually

with

> JI and tuning systems in general.

These systems (121 and 111) will not deal with adaptive JI very well.

> 1200 is a huge number. The goal is

> not to find a scale division so low as to be terribly practical

> physically, but to serve as a finite analytical tool that reflects

> the realities of intonation as accurately as possible with the

least

> encumberance, requiring only simple arithmetical tools for its

> practical implementation in anaylzing or composing.

See above.

>

> I just finished some research that took as a conceptual starting

> point 12-EDO for the P5 as derived from the 7/12 lowest rational

> approximation of a JI P5 in terms of its pitchwise portion of the

> octave. Using the same principle of rational approximation of JI

> pitch ratios (NOT their frequency ratios), I found the lowest

> rational approximation of P5 to M3 (exact pitch ratio = 1.8171) to

be

> 9/5. 9 X 12 = 108-EDO. I'm sure this must be a well-known EDO

> division, since it has so many compelling characteristics, but I

> don't know about it from any source other than my own research of

the

> last two days.

It isn't. As for multiples of 12, 72-tET is superior in all respects.

I'm not sure what your derivation seeks to accomplish.

>

> It does satisfy the above conditions, except for #3

But it's not even consistent beyond the 7-limit.

if someone finds

> a lesser division that meets conditions 1 & 2. The greatest error

is

> 5.0 cents for 17/16. The P5 is -2.0 cents as in 12-EDO, and the M3

is

> +2.6 cents. The lowest deviation is the 16/15 M2 at -0.6 cents. The

> M2+ (8/7) is 2.2 cents, so the greatest error for 7-limit is 2.6

> cents for the M3.

108-tET has a maximum error of 4.53 cents in the 5-limit and 4.73

cents in the 7-limit.

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

> --- In tuning-math@y..., BobWendell@t... wrote:

>

> This isn't what you asked for, but 311 represents all primes up

> through 41 with an error of less than a cent

Not only that, it's actually _consistent_ through the 41-limit. We

noticed this amazing fact a few years ago -- it's a huge spike in the

consistency-limit as a function of increasing ET cardinality. Gene,

you still haven't brought up anything about the zetafunction. Could

it help explain why 311 is so special?

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

> Not only that, it's actually _consistent_ through the 41-limit.

That's right--this is what really made my eyes open wide when I

noticed it some time ago.

We

> noticed this amazing fact a few years ago -- it's a huge spike in

the

> consistency-limit as a function of increasing ET cardinality. Gene,

> you still haven't brought up anything about the zetafunction. Could

> it help explain why 311 is so special?

It can express it, but it can't really explain it any better than the

charts I've been looking at by you and Paul Hahn (which was like

meeting old friends!) I promise to explain it sometime, but in fact

there are lots of things still locked away in my head unpublished

(because there didn't seem to be any place to publish.) I'm still

trying to figure out your hypothesis. :(

--- In tuning-math@y..., "Carl" <carl@l...> wrote:

> As far as replacing cents with the units of some et, it isn't

> likely to happen. Ellis' cents enjoy wide use, provide good

> accuracy regardless of wether the target interval is just, make

> the octave highly divisable without fractional results, and

> provide a handy reference to 12-tone equal temperament.

I've often used the 612 system for my own use, but I don't expect

anyone else to adopt it. I like it partly because I have some of it

memorized.

P5 is a Perfect fifth, frequency ration = 3/2

M3 is Major third = 5/4

Thanks! Brute force is fine with me, Gene (chuckle).

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

> --- In tuning-math@y..., BobWendell@t... wrote:

>

> This isn't what you asked for, but 311 represents all primes up

> through 41 with an error of less than a cent; it's a remarkable et

in

> some ways. Your question could be answered easily with a brute-

force

> search, however; and I might do it if you define P3 and M5 for me.

Thanks, Carl! Not trying to replace cents, but simply trying to offer

myself and others a simplified option to complement the use of cents.

--- In tuning-math@y..., "Carl" <carl@l...> wrote:

> > This isn't what you asked for, but 311 represents all primes up

> > through 41 with an error of less than a cent; it's a remarkable et

> > in some ways. Your question could be answered easily with a brute-

> > force search, however; and I might do it if you define P3 and M5

> > for me.

>

> As I'm sure others will be quick to point out, if one intends to

> use anything more than dyadic harmony, he must also check the errors

> of these larger chords in the given et. This gave rise to the

> notion of consistentcy. See:

>

> http://library.wustl.edu/~manynote/music.html

>

> Also, aside from looking at max errors, Paul Erlich has suggested

> considering an et as an an approximation. See:

>

> http://www-math.cudenver.edu/~jstarret/Question1.html

>

> As far as replacing cents with the units of some et, it isn't

> likely to happen. Ellis' cents enjoy wide use, provide good

> accuracy regardless of wether the target interval is just, make

> the octave highly divisable without fractional results, and

> provide a handy reference to 12-tone equal temperament.

>

> -Carl

> Thanks, Carl! Not trying to replace cents, but simply trying to

> offer myself and others a simplified option to complement the use

> of cents.

Sure. And these large ets are interesting in their own right.

-Carl

--- In tuning-math@y..., BobWendell@t... wrote:

> I have long been interested in coming up

> with an EDO that would meet the following requirements:

>

> 1) P5s and M3s less than 3 cents from JI

>

> 2) all primes through 19 less than 5 cents from JI

>

> 3) The smallest number of scale degrees possible while meeting

> requirements 1 and 2

Bob,

These and many such questions are easily answered thanks to Paul Hahn.

All tuning-math-ers should have copies of the 3 files that I just put

up at

</tuning-math/files/EDO%20consistency%20a

nd%20accuracy/>

Are these already available somewhere else on the web?

Paul Hahn didn't even put his name in the files so I can't see him

worrying about copyright. I'll take that risk. But does anyone know

what year he created them so I can include it in the acknowledgement?

Regards,

-- Dave Keenan

Thanks, Dave! However, the link you gave below doesn't seem to work

for me. It says no connection with the server could be established.

/tuning-math/files/EDO%20consistency%

20and%20accuracy/

Cheers,

Bob

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> --- In tuning-math@y..., BobWendell@t... wrote:

> > I have long been interested in coming up

> > with an EDO that would meet the following requirements:

> >

> > 1) P5s and M3s less than 3 cents from JI

> >

> > 2) all primes through 19 less than 5 cents from JI

> >

> > 3) The smallest number of scale degrees possible while meeting

> > requirements 1 and 2

>

> Bob,

>

> These and many such questions are easily answered thanks to Paul

Hahn.

>

> All tuning-math-ers should have copies of the 3 files that I just

put

> up at

> </tuning-math/files/EDO%20consistency%

20a

> nd%20accuracy/>

>

> Are these already available somewhere else on the web?

>

> Paul Hahn didn't even put his name in the files so I can't see him

> worrying about copyright. I'll take that risk. But does anyone know

> what year he created them so I can include it in the

acknowledgement?

>

> Regards,

> -- Dave Keenan

Never mind, Dave! Just got there through the menu (duh!). there are

no column labels. I inferred from the structure that the consisitency

tables showed levels of consistency through successive odd limits?

Apparently the accuracy table only shows the accuracy for the

intervals that are consistent in that temperament?

Also what does the number for consistency levels mean? I don't

understand the meaning of level in this context. From the definition

of consistency in Monz' dictionary I saw nothing defining levels of

consistency.

Thanks,

Bob

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> --- In tuning-math@y..., BobWendell@t... wrote:

> > I have long been interested in coming up

> > with an EDO that would meet the following requirements:

> >

> > 1) P5s and M3s less than 3 cents from JI

> >

> > 2) all primes through 19 less than 5 cents from JI

> >

> > 3) The smallest number of scale degrees possible while meeting

> > requirements 1 and 2

>

> Bob,

>

> These and many such questions are easily answered thanks to Paul

Hahn.

>

> All tuning-math-ers should have copies of the 3 files that I just

put

> up at

> </tuning-math/files/EDO%20consistency%

20a

> nd%20accuracy/>

>

> Are these already available somewhere else on the web?

>

> Paul Hahn didn't even put his name in the files so I can't see him

> worrying about copyright. I'll take that risk. But does anyone know

> what year he created them so I can include it in the

acknowledgement?

>

> Regards,

> -- Dave Keenan

--- In tuning-math@y..., BobWendell@t... wrote:

> Never mind, Dave! Just got there through the menu (duh!). there are

> no column labels. I inferred from the structure that the

consisitency

> tables showed levels of consistency through successive odd limits?

Right.

> Apparently the accuracy table only shows the accuracy for the

> intervals that are consistent in that temperament?

Right.

>

> Also what does the number for consistency levels mean?

A consistency "level" of N, for N > 1.5, simply means that the

largest error is 1/2N steps of the ET. Paul Hahn had something else

in mind but this is the way I think of it, since it's mathematically

equivalent.

I wrote,

> A consistency "level" of N, for N > 1.5, simply means that the

> largest error is 1/2N steps of the ET.

Should be "the largest error is _less than_ 1/2N steps of the ET".

So 111-EDO (3*37 divisions) looks like it wins hands down on both

consistency and accuracy?

By the way, 72-EDO does indeed look compelling, but 13/8 (50 steps)

is flat by 7.2 cents, so it doesn't meet the criteria I specified.

Considering its other merits and the high order prime 13, I suppose

7.2 cents ain't half bad!

Thanks!

Bob

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

> I wrote,

>

> > A consistency "level" of N, for N > 1.5, simply means that the

> > largest error is 1/2N steps of the ET.

>

> Should be "the largest error is _less than_ 1/2N steps of the ET".

--- In tuning-math@y..., BobWendell@t... wrote:

> So 111-EDO (3*37 divisions) looks like it wins hands down on both

> consistency and accuracy?

Well sort of . . . but consider that 120-tET _can't_ have more than a

5 cent error for _anything_ (since it's increments of 10 cents). So

one would have hoped for a "special" tuning with much fewer than 120

notes to satisfy your criteria. Unfortunately there isn't one.

>

> By the way, 72-EDO does indeed look compelling, but 13/8 (50 steps)

> is flat by 7.2 cents, so it doesn't meet the criteria I specified.

> Considering its other merits and the high order prime 13, I suppose

> 7.2 cents ain't half bad!

Right . . . but I think the whole idea of 72 or 111 or 121 as a least-

common-denominator way of describing ideal musical practice kind of

falters when _adaptive JI_ comes into the picture . . . doesn't it?

See bottom:

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

> --- In tuning-math@y..., BobWendell@t... wrote:

>

> > So 111-EDO (3*37 divisions) looks like it wins hands down on both

> > consistency and accuracy?

>

> Well sort of . . . but consider that 120-tET _can't_ have more than

a

> 5 cent error for _anything_ (since it's increments of 10 cents). So

> one would have hoped for a "special" tuning with much fewer than

120

> notes to satisfy your criteria. Unfortunately there isn't one.

> >

> > By the way, 72-EDO does indeed look compelling, but 13/8 (50

steps)

> > is flat by 7.2 cents, so it doesn't meet the criteria I

specified.

> > Considering its other merits and the high order prime 13, I

suppose

> > 7.2 cents ain't half bad!

>

> Right . . . but I think the whole idea of 72 or 111 or 121 as a

least-

> common-denominator way of describing ideal musical practice kind of

> falters when _adaptive JI_ comes into the picture . . . doesn't it?

I don't have the experience with these tunings in practice to address

that, Paul. Not by a long shot (how about zero?) As far as a daptive

JI, you may be right. (How would I know?) But I was looking at it as

more of a potential compositional tool for microtonal polyphony.

Thanks for all your reponses. It's very stimulating to interact with

everyone here. Makes you think through things rather deeply.

--- In tuning-math@y..., BobWendell@t... wrote:

>

> I don't have the experience with these tunings in practice to

address

> that, Paul. Not by a long shot (how about zero?) As far as a

daptive

> JI, you may be right. (How would I know?)

Well clearly you understand the Vicentino adaptive JI scheme, based

on two 1/4-comma meantone chains. Since 53-tET is a "scale of

commas", you would need 53*4=212-tET to implement this scheme. See?

> But I was looking at it as

> more of a potential compositional tool for microtonal polyphony.

If you don't care about melody or "necessarily tempered chords" (such

as CEGAD) and only care about just-style harmony, then sure, that's

great thinking. Looking forward to hearing some harmonically-oriented

111-tET music!

Well, thanks to you, I'm leaning toward 72-tET now, since it has the

obvious advantage of greater simplicity and is apparently consistent

through 7-limit.

Gratefully,

Bob

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

> --- In tuning-math@y..., BobWendell@t... wrote:

> >

> > I don't have the experience with these tunings in practice to

> address

> > that, Paul. Not by a long shot (how about zero?) As far as a

> daptive

> > JI, you may be right. (How would I know?)

>

> Well clearly you understand the Vicentino adaptive JI scheme, based

> on two 1/4-comma meantone chains. Since 53-tET is a "scale of

> commas", you would need 53*4=212-tET to implement this scheme. See?

>

> > But I was looking at it as

> > more of a potential compositional tool for microtonal polyphony.

>

> If you don't care about melody or "necessarily tempered chords"

(such

> as CEGAD) and only care about just-style harmony, then sure, that's

> great thinking. Looking forward to hearing some harmonically-

oriented

> 111-tET music!

--- In tuning-math@y..., BobWendell@t... wrote:

> Well, thanks to you, I'm leaning toward 72-tET now, since it has

the

> obvious advantage of greater simplicity and is apparently

consistent

> through 7-limit.

>

> Gratefully,

> Bob

It's consistent through 17-limit!

--- In tuning-math@y..., BobWendell@t... wrote:

> Well, thanks to you, I'm leaning toward 72-tET now

But it's useless for adaptive JI. You'll have to deal with drifts or

shifts of almost a full comma in typical diatonic triadic

progressions.

Well, rats! LOL....Maybe you have some other recommendation that

would satisfy the stated goals and still be useful for adaptvie JI?

Or maybe for compositional purposes, I don't need to be concerned

with that?

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

> --- In tuning-math@y..., BobWendell@t... wrote:

> > Well, thanks to you, I'm leaning toward 72-tET now

>

> But it's useless for adaptive JI. You'll have to deal with drifts

or

> shifts of almost a full comma in typical diatonic triadic

> progressions.

--- In tuning-math@y..., BobWendell@t... wrote:

> Well, rats! LOL....Maybe you have some other recommendation that

> would satisfy the stated goals and still be useful for adaptvie JI?

Are cents really so bad?

> Or maybe for compositional purposes, I don't need to be concerned

> with that?

If you're not planning on worrying about abrupt commatic shifts in

your melodies (that is, if they don't bother you at all), then maybe

111-tET would be fine for you.

Personally, I see 152-tET as my "Universal tuning". But I'm not going

past 11-limit, and I care very much about certain melodic systems.

Hi, Paul! Thanks...

I've been pondering this a bit, and I'm wondering:

If I'm only interested in choosing some n-tET as a tool strictly to

to serve as a kind of simplified compositional calculus, do I really

need to be concerned about more than accurate approximation of just

intervals and consistency?

Why ask? Each tuning strategy has its own characteristics and it

occurs to me that a composer needn't and perhaps even shouldn't think

in terms of how it interfaces with the character of others.

Comma drifts, if writing original compositions as if one had never

been exposed to any other system, might actually be viewed as

microtonal modulations, and one can write in such a way as to return

home just as we do in 12-tET. We come back home if we chose to

because we compose with the intention of doing so within the

constraints the tuning offers us.

If we freely exploit the character of a high-order EDO, why not

conceive things in terms of its indigenous character and modulate

away from and back to a pitch center as that system's character

constrains our freedom to choose as any system does. Why shouldn't

our art reflect the intonational medium in which we are working

rather than superimposed ideas from media alien to it?

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

> --- In tuning-math@y..., BobWendell@t... wrote:

>

> > Well, rats! LOL....Maybe you have some other recommendation that

> > would satisfy the stated goals and still be useful for adaptvie

JI?

>

> Are cents really so bad?

>

> > Or maybe for compositional purposes, I don't need to be concerned

> > with that?

>

> If you're not planning on worrying about abrupt commatic shifts in

> your melodies (that is, if they don't bother you at all), then

maybe

> 111-tET would be fine for you.

>

> Personally, I see 152-tET as my "Universal tuning". But I'm not

going

> past 11-limit, and I care very much about certain melodic systems.

--- In tuning-math@y..., BobWendell@t... wrote:

> Comma drifts, if writing original compositions as if one had never

> been exposed to any other system, might actually be viewed as

> microtonal modulations, and one can write in such a way as to

return

> home just as we do in 12-tET. We come back home if we chose to

> because we compose with the intention of doing so within the

> constraints the tuning offers us.

A "comma drift" is simply another word for a microtonal modulation,

and the idea that avoiding the use of just intonatation will spare us

from comma drifts is incorrect. It will set any microtonal modulation

which belongs to the kernel to unison, but not any that do not--in

fact, these may become exaggerated from what they would have been had

just intonation been employed.

> If we freely exploit the character of a high-order EDO, why not

> conceive things in terms of its indigenous character and modulate

> away from and back to a pitch center as that system's character

> constrains our freedom to choose as any system does. Why shouldn't

> our art reflect the intonational medium in which we are working

> rather than superimposed ideas from media alien to it?

I think microtonalists should use whatever scale suits them; however

it would be well if they understood the structure of the system they

intend to use.

--- In tuning-math@y..., BobWendell@t... wrote:

> Hi, Paul! Thanks...

>

> I've been pondering this a bit, and I'm wondering:

>

> If I'm only interested in choosing some n-tET as a tool strictly to

> to serve as a kind of simplified compositional calculus, do I

really

> need to be concerned about more than accurate approximation of just

> intervals and consistency?

Well, sure -- each ET will imply a different set of progressions that

drift, and a different set that don't.

>

> Why ask? Each tuning strategy has its own characteristics and it

> occurs to me that a composer needn't and perhaps even shouldn't

think

> in terms of how it interfaces with the character of others.

Sure! But you need to be aware of what those characteristics are!

>

> Comma drifts, if writing original compositions as if one had never

> been exposed to any other system, might actually be viewed as

> microtonal modulations, and one can write in such a way as to

return

> home just as we do in 12-tET. We come back home if we chose to

> because we compose with the intention of doing so within the

> constraints the tuning offers us.

>

> If we freely exploit the character of a high-order EDO, why not

> conceive things in terms of its indigenous character and modulate

> away from and back to a pitch center as that system's character

> constrains our freedom to choose as any system does. Why shouldn't

> our art reflect the intonational medium in which we are working

> rather than superimposed ideas from media alien to it?

I agree completely -- but these characteristics exert a powerful

effect on one's compositional possibilities. Hence, it seems

premature to settle for a given ET as one's "tool" without being sure

that one wants to be constrained by its particular behaviors.

Paul said:

Hence, it seems

> premature to settle for a given ET as one's "tool" without being

sure

> that one wants to be constrained by its particular behaviors.

Bob answers:

I see your point, Paul. I suppose one has to start somewhere though,

since I would think that it would require significant experience with

the alternatives under consideration before the decision becomes

adequately informed.

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

> --- In tuning-math@y..., BobWendell@t... wrote:

> > Hi, Paul! Thanks...

> >

> > I've been pondering this a bit, and I'm wondering:

> >

> > If I'm only interested in choosing some n-tET as a tool strictly

to

> > to serve as a kind of simplified compositional calculus, do I

> really

> > need to be concerned about more than accurate approximation of

just

> > intervals and consistency?

>

> Well, sure -- each ET will imply a different set of progressions

that

> drift, and a different set that don't.

> >

> > Why ask? Each tuning strategy has its own characteristics and it

> > occurs to me that a composer needn't and perhaps even shouldn't

> think

> > in terms of how it interfaces with the character of others.

>

> Sure! But you need to be aware of what those characteristics are!

> >

> > Comma drifts, if writing original compositions as if one had

never

> > been exposed to any other system, might actually be viewed as

> > microtonal modulations, and one can write in such a way as to

> return

> > home just as we do in 12-tET. We come back home if we chose to

> > because we compose with the intention of doing so within the

> > constraints the tuning offers us.

> >

> > If we freely exploit the character of a high-order EDO, why not

> > conceive things in terms of its indigenous character and modulate

> > away from and back to a pitch center as that system's character

> > constrains our freedom to choose as any system does. Why

shouldn't

> > our art reflect the intonational medium in which we are working

> > rather than superimposed ideas from media alien to it?

>

> I agree completely -- but these characteristics exert a powerful

> effect on one's compositional possibilities. Hence, it seems

> premature to settle for a given ET as one's "tool" without being

sure

> that one wants to be constrained by its particular behaviors.

Gene said:

> I think microtonalists should use whatever scale suits them;

however

> it would be well if they understood the structure of the system

they

> intend to use.

Bob answers:

Thanks, Gene, for your complete comments and not just this quote from

it. Regarding the above quote from you, see my response in post #880

to Paul Erlich and Paul's comments just below that response.

I'm interested in 72-tET because it is reasonably accurate in its

approximation of just intervals, is strong in consistency, and

reduces the set of pitches available for composition to a finite

number that is conceptually manageable with simple arithmetic you can

do quickly in your head.

Given time and experience with it, you could learn to think

comprehensively in terms of it and its implications while in the act

of composing. These criteria comprise the essential motivations for

my quest.

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

> --- In tuning-math@y..., BobWendell@t... wrote:

>

> > Comma drifts, if writing original compositions as if one had

never

> > been exposed to any other system, might actually be viewed as

> > microtonal modulations, and one can write in such a way as to

> return

> > home just as we do in 12-tET. We come back home if we chose to

> > because we compose with the intention of doing so within the

> > constraints the tuning offers us.

>

> A "comma drift" is simply another word for a microtonal modulation,

> and the idea that avoiding the use of just intonatation will spare

us

> from comma drifts is incorrect. It will set any microtonal

modulation

> which belongs to the kernel to unison, but not any that do not--in

> fact, these may become exaggerated from what they would have been

had

> just intonation been employed.

>

> > If we freely exploit the character of a high-order EDO, why not

> > conceive things in terms of its indigenous character and modulate

> > away from and back to a pitch center as that system's character

> > constrains our freedom to choose as any system does. Why

shouldn't

> > our art reflect the intonational medium in which we are working

> > rather than superimposed ideas from media alien to it?

>

> I think microtonalists should use whatever scale suits them;

however

> it would be well if they understood the structure of the system

they

> intend to use.

--- In tuning-math@y..., BobWendell@t... wrote:

> Gene said:

> > I think microtonalists should use whatever scale suits them;

> however

> > it would be well if they understood the structure of the system

> they

> > intend to use.

>

> Bob answers:

> Thanks, Gene, for your complete comments and not just this quote

from

> it. Regarding the above quote from you, see my response in post

#880

> to Paul Erlich and Paul's comments just below that response.

>

> I'm interested in 72-tET because it is reasonably accurate in its

> approximation of just intervals, is strong in consistency, and

> reduces the set of pitches available for composition to a finite

> number that is conceptually manageable with simple arithmetic you

can

> do quickly in your head.

>

> Given time and experience with it, you could learn to think

> comprehensively in terms of it and its implications while in the

act

> of composing. These criteria comprise the essential motivations for

> my quest.

Sounds great! If you look back a few months in the tuning list

archives, you will see a great deal of discussion on 72-tET, and

you'll see me advocating it as a certain sort of "standard" . . . I'm

sure you'll find some useful information in all that.

Terrific, Paul! Thanks again!

- Bob

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

> --- In tuning-math@y..., BobWendell@t... wrote:

> > Gene said:

> > > I think microtonalists should use whatever scale suits them;

> > however

> > > it would be well if they understood the structure of the system

> > they

> > > intend to use.

> >

> > Bob answers:

> > Thanks, Gene, for your complete comments and not just this quote

> from

> > it. Regarding the above quote from you, see my response in post

> #880

> > to Paul Erlich and Paul's comments just below that response.

> >

> > I'm interested in 72-tET because it is reasonably accurate in its

> > approximation of just intervals, is strong in consistency, and

> > reduces the set of pitches available for composition to a finite

> > number that is conceptually manageable with simple arithmetic you

> can

> > do quickly in your head.

> >

> > Given time and experience with it, you could learn to think

> > comprehensively in terms of it and its implications while in the

> act

> > of composing. These criteria comprise the essential motivations

for

> > my quest.

>

> Sounds great! If you look back a few months in the tuning list

> archives, you will see a great deal of discussion on 72-tET, and

> you'll see me advocating it as a certain sort of "standard" . . .

I'm

> sure you'll find some useful information in all that.