72 tet, standardization

You can play in:
>
>N=8
>N=9
>N=12
>N=18
>N=24
>N=36
>

OK, I've been on this list for a year and 1/2, admittedly skimming at
times, but this list of et's available in 72 seems to miss all of the ones
I've seen discussed: 5, 7, 11, 13, 17, 19, 22, 29, 31, 51, 53, 144.

Aside from an occasional usage by Dan or Margo, or an appearance in
[whoever's] "I can make music in every ET"-Suit, I haven't seen any of
these being widely used.

so is this really so advantageous? (not to belittle 72tet's other
advantages.)

***From: Christopher Bailey******************

http://music.columbia.edu/~chris

**********************************************

--- In tuning@y..., Christopher Bailey <cb202@c...> wrote:

/tuning/topicId_24270.html#24270

> > You can play in:
> >
> > N=8
> > N=9
> > N=12
> > N=18
> > N=24
> > N=36
> >
>
> OK, I've been on this list for a year and 1/2, admittedly
> skimming at times, but this list of et's available in 72
> seems to miss all of the ones I've seen discussed:
> 5, 7, 11, 13, 17, 19, 22, 29, 31, 51, 53, 144.
>
> Aside from an occasional usage by Dan or Margo, or an
> appearance in [whoever's] "I can make music in every ET"-Suit,
> I haven't seen any of these being widely used.
>
> so is this really so advantageous? (not to belittle
> 72tet's other advantages.)

Hi Chris,

You post prompted me to create some interval matrices for
some of these EDOs and compare them with 72-EDO.
I've done 5-, 7-, 8-, 9-, 10-, and 11-EDO.

I'm not arguing one way or the other for 72-EDO's ability
to approximate these other EDOs... just presenting the
mathematical facts about one way of seeing it:

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

-monz
http://www.monz.org
"All roads lead to n^0"

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

/tuning/topicId_24270.html#24301

> Hi Chris,
>
>
> You post prompted me to create some interval matrices for
> some of these EDOs and compare them with 72-EDO.
> I've done 5-, 7-, 8-, 9-, 10-, and 11-EDO.
>
> I'm not arguing one way or the other for 72-EDO's ability
> to approximate these other EDOs... just presenting the
> mathematical facts about one way of seeing it:
>
> http://www.ixpres.com/interval/dict/intervalmatrix.htm
>
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"

Monz... I have a question about your chart. I can understand why you
would want to put the 12-tET equivalences to EDO's in one chart.
HOWEVER, why do you want to show the deviations from 12-tET in the 72-
tET comparison??

Wouldn't it be MORE interesting to show the fractions of 72-tET in a
chart and how close the 72-tET notes are cominng to the EDO using 72-
tET as THE STANDARD, without getting 12-tET involved in the mix??

Or am I just misunderstanding something??

_________ ________ ______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_24270.html#24311

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > ...interval matrices for
> > some of these EDOs and compare them with 72-EDO.
> > I've done 5-, 7-, 8-, 9-, 10-, and 11-EDO.
> >
> > http://www.ixpres.com/interval/dict/intervalmatrix.htm
>
>
> Monz... I have a question about your chart. I can
> understand why you would want to put the 12-tET equivalences
> to EDO's in one chart. HOWEVER, why do you want to show
> the deviations from 12-tET in the 72-tET comparison??
>
> Wouldn't it be MORE interesting to show the fractions of
> 72-tET in a chart and how close the 72-tET notes are cominng
> to the EDO using 72-tET as THE STANDARD, without getting
> 12-tET involved in the mix??
>
> Or am I just misunderstanding something??

Well... I suppose perhaps there's a tiny bit of misunderstanding,
in that I really wasn't *intentionally* showing "deviations
from 12-tET"... I was simply using 12-EDO as my standard
measurement of Semitones as I've always done.

But, Joe, you're absolutely right... if we're going to try
to instate 72-EDO as a tuning standard, might as well start
right now by *using* it as a standard!

So I've added new tables to all of these interval matrices,
which give all the small-number EDOs (5, 7, 8, 9, 10, 11,
and 13) as 72-EDO degrees ("morias") and fractions thereof.

I used actual fractions instead of decimals because I think
it's much clearer in this case, since an EDO will always
divide cents equally according to some fractional unit.

(Thanks to Dan Stearns for - somewhat unintentionally,
I think - enlightening me to this way of perceiving EDOs.)

From the 72-EDO degrees (and their fractional parts), you
should easily be able to convert to the nearest 72-EDO pitch
of your chosen notation. (whichever one that is...)
But watch out for the inconsistencies!

I guess one of the saddest things is that now we have
two definitions for "moria". Because of the historical
precedence of defining moria as (4/3)^(1/30), we really
should have just invented a new term for 1 degree of
72-EDO... perhaps we still can. Suggestions?

How about "habatu", for _Hába T_uning _U_nit?, since
he seems to have been the first person to have used
72-EDO as a division of the pitch-continuum.

-monz
http://www.monz.org
"All roads lead to n^0"

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

/tuning/topicId_24270.html#24320

> But, Joe, you're absolutely right... if we're going to try
> to instate 72-EDO as a tuning standard, might as well start
> right now by *using* it as a standard!
>
> So I've added new tables to all of these interval matrices,
> which give all the small-number EDOs (5, 7, 8, 9, 10, 11,
> and 13) as 72-EDO degrees ("morias") and fractions thereof.
>
> I used actual fractions instead of decimals because I think
> it's much clearer in this case, since an EDO will always
> divide cents equally according to some fractional unit.
>

Thank you very much, Monz, for this revision and I would have written
you off list again about it (I'm trying to reduce my number of
posts!!) but I do have a couple of questions that might be of value
for some others.

1) Where does the term "moria" come from?? And, regrettably, I'm
not understanding the math you posted for it.

2) Why do you use fractional divisions of 72-tET that are the same
number of divisions of the tiny unit as the number of divisions of
the octave of the basic scale you are investigating?

I should be "getting" this, but I'm not. Maybe it's related to point
one. If you could run this down a bit more with some examples I
would GREATLY appreciate it!

And how does this all relate to "cents" again??

Sorry to be so mystified... and this isn't even the MATH list, which
is not heartening...

__________ _________ ______
Joseph Pehrson

Hello
"Moria" is the Greek word for molecules. The term is used in traditional
Greek music to mean a few cents change in the 12tone scale note , to give an
oriental effect , such as the blue notes for blues.
In Greece both the terms moria and quarter tones are used for the same
thing.

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_24270.html#24324

> ... I do have a couple of questions that might be of value
> for some others.
>
> 1) Where does the term "moria" come from?? And, regrettably,
> I'm not understanding the math you posted for it.

Did you look at my new defintion of "moria"?
http://www.ixpres.com/interval/dict/moria.htm

It explains all the math and why there are two different
values for this small interval.

Even tho I've read Cleonides, who originated the term
(according to Manuel at
<http://www.xs4all.nl/~huygensf/doc/measures.html>),
I actually don't remember ever having seen it before,
and I got it from Manuel's webpage.

I figured that since you asked me to use 72-EDO as the
tuning unit of measurement for these interval matrix tables,
I'd might as well use the term that goes with it.

:)

Here's the bottom line on the math of "moria":

There are two different sizes which are audibly identical.

The older one comes from Cleonides, analyzing the work of
Aristoxenos. Cleonides divides the "perfect 4th" into
30 presumably equal parts. (There is a question about
whether they were intended to be equal.) So the formula
is: 1 moria = (4/3)^(1/30), just a fly-hair wider than
16 & 3/5 (16.6) cents.

The newer definition (does it originate with you, Manuel?)
equates the moria with 1 degree of 72-EDO [= 2^(1/72) ],
which is exactly 16 & 2/3 (16.66...) cents.

>
> 2) Why do you use fractional divisions of 72-tET that
> are the same number of divisions of the tiny unit as
> the number of divisions of the octave of the basic scale
> you are investigating?
>
> I should be "getting" this, but I'm not. Maybe it's
> related to point one. If you could run this down a bit
> more with some examples I would GREATLY appreciate it!

I didn't plan those fractional divisions, Joe... that's
how the math works out. Any time you divide the 2:1 into
an equal number of parts, it is going to divide into cents
in even fractions without remainders, because the cents
are also an equal division.

As I said, I never realized this on my own and it was
brought to my attention in an email Dan sent me. He
didn't say anything about it, but it struck me how the
cents values for 13-EDO divided into 1/13ths of a cent:

> [private email from Dan Stearns:]
>
> 13-tET becomes:
>
> [cents] [144-EDO notation]
>
> 0 C
> 92 & 4/13 +~Db...(+~C#)
> 184 & 8/13 +D
> 276 & 12/13 >~Eb...(>~Eb)
> 369 & 3/13 >E...(>Fb)
> 461 & 7/13 #~F...(#~E#)
> 553 & 11/13 #F#
> 646 & 2/13 bGb
> 738 & 6/13 b~G
> 830 & 10/13 <Ab...(<G#)
> 923 & 1/13 <~A
> 1015 & 5/13 -Bb...(-A#)
> 1107 & 9/13 -~B...(-~Cb)
> 1200 C/B#
>

I had always seen cents with decimal places rather than
fractions, and upon encountering this for the first time,
I noticed right away that something very interesting was
going on here...

So there's no "magic" to it. If you divide 1200 cents
by 13 (for example), each part is going to carry a multiple
of 1/13th of a cent until when you get to the last note,
you have 13/13ths which adds a whole extra cent to bring
you to the "octave".

>
> And how does this all relate to "cents" again??
>
> Sorry to be so mystified... and this isn't even the MATH
> list, which is not heartening...

I repeated above both cents-values for the two types of moria.

If you think that your maximum 72-EDO error of (16 & 2/3) / 2
= 8 & 1/3 cents is small enough to disregard, then you can
simply forget about the fractions in my tables and round to
the nearest 72-EDO degree. If 8+ cents won't bother you
(which it really probably won't since we're approximating
other EDOs here, and not easier-to-hear-out-of-tune JI),
you can ignore the fractions and round up or down.

I only included the fractions to show where the inconsistencies
occur. By keeping track of the amount of error as you go
along, you might be able to find advantageous places to
hide the commatic shifts. And to my eyes it's much easier
to keep track of the error when that error is expressed
in a simple system of fractions that can be easily added.

One reason why 72 is such an important number thru-out
history (kabalah, numerology, gematria, etc. ... see the
spiritual_tuning list for recent posts on this stuff)
is because, like 60 and 12, it's very easy to divide a
number of different ways.

This easy divisibility is a big part of the reason why
the 72-EDO notation is so compact. Lower levels of
cardinality are easily expressed by a single symbol.
Part of the problem with notating such "good" EDO divisions
as 29, 31, 41, and 53 is that they are all prime numbers
and therefore not amenable to such subdivision.

-monz
http://www.monz.org
"All roads lead to n^0"

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

/tuning/topicId_24270.html#24326

Thank you very much, Monz, for your explanation of the "moria..."

It does seem to be a "convenient" description for one degree of 72.
It's interesting that it has such a close historical approximation as
well. It's great that Manuel thought of it! I hope everybody will
continue using it that way!
>

And on the second "arithmetic" item...

>
> So there's no "magic" to it. If you divide 1200 cents
> by 13 (for example), each part is going to carry a multiple
> of 1/13th of a cent until when you get to the last note,
> you have 13/13ths which adds a whole extra cent to bring
> you to the "octave".
>

But, of course, on your page, we are taking about division of 72-
equal, not cents, and it works the same way.

This is probably pretty rudimentary, and should be on the remedial-
tuning-arithmetic@yahoogroups.com group, but it looks as though ANY
number when divided by another number that gives answers which have
remainders, gives fractional remainders that have a denominator equal
to the original divisor... or at least are a simple reduction of that
divisor...

Is that correct?? Unfortunately, I never thought about that before...

_________ _______ _______
Joseph Pehrson

Alexandros,

what does "epimoria" mean, in ancient and current usage?

Some friends of mine have been wondering what the plural for epimore is.

Thanks,

GZ

Alexandros Papadopoulos wrote:

> Hello
> "Moria" is the Greek word for molecules. The term is used in traditional
> Greek music to mean a few cents change in the 12tone scale note , to give an
> oriental effect , such as the blue notes for blues.
> In Greece both the terms moria and quarter tones are used for the same
> thing.
>
>
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on 6/5/01 12:38 AM, George Zelenz at ploo@mindspring.com wrote:

> Alexandros,
>
> what does "epimoria" mean, in ancient and current usage?
>
> Some friends of mine have been wondering what the plural for epimore is.
>
> Thanks,
>
> GZ

I haven't heard the word "epimoria" before . Is it Greek? I haven't even
found it on the dictionary.
Sorry

--- In tuning@y..., Alexandros Papadopoulos <Alexmoog@o...> wrote:
> on 6/5/01 12:38 AM, George Zelenz at ploo@m... wrote:
>
> > Alexandros,
> >
> > what does "epimoria" mean, in ancient and current usage?
> >
> > Some friends of mine have been wondering what the plural for
epimore is.
> >
> > Thanks,
> >
> > GZ
>
> I haven't heard the word "epimoria" before . Is it Greek? I haven't
even
> found it on the dictionary.
> Sorry

"Epimoric ratios" are also called "superparticular ratios" and are
ratios of the form (N+1)/N. So 2/1, 3/2, 4/3, 5/4, 6/5, 7/6 . . .
16/15 . . . 32/31 . . . 81/80 . . . 2401/2400 . . . Many ancient
Greek and Roman theorists thought that the intervals between adjacent
scale degrees always had to be epimoric ratios . . . Erv Wilson and
perhaps Pierre Lamothe seem to also put some stock in this theory,
though I don't know why . . .

>Erv Wilson and perhaps Pierre Lamothe seem to also put some
>stock in this theory, though I don't know why . . .

David Canright offers...

Among all just intervals, superparticular intervals have been
preferred by such notables as Ptolemy and Lou Harrison. The
reasons for preferring superparticular ratios are not entirely
clear, but one special property concerns difference tones.
When two tones related by a superparticular interval sound
together, the primary difference tone is the fundamental of the
harmonic series to which both tones belong. (For other just
intervals, the primary difference tone is another tone in the
harmonic series, not the fundamental.)

-Carl

Sure, but why must all scale steps be epimoric? It would seem more
important to make as many _intervals_ as possible epimoric, rather
than as many _scale steps_ as possible epimoric . . . besides, even
Partch admits that 25:24 can't be tuned by ear, or recognized as part
of a harmonic series . . . somewhere between 12:11 and 25:24, he
states, the ear gives up.

Paul,

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Sure, but why must all scale steps be epimoric? It would seem more
> important to make as many _intervals_ as possible epimoric, rather
> than as many _scale steps_ as possible epimoric

Could this be considered an aesthetic issue? Choosing the materials
one would like to work with? It seems clear that it doesn't *have* to
be this way, but maybe it was only an issue of taste or preference,
as opposed to any other. Only a hunch on my part...

> . . . besides, even Partch admits that 25:24 can't be tuned by ear,
> or recognized as part of a harmonic series . . . somewhere between
> 12:11 and 25:24, he states, the ear gives up.

Hell, I would have given up a few steps earier, but what the heck! :)

Cheers,
Jon

In message 24355 Paul Erlich wrote:

<< Many ancient Greek and Roman theorists thought that the
intervals between adjacent scale degrees always had to be
epimoric ratios . . . Erv Wilson and perhaps Pierre Lamothe
seem to also put some stock in this theory, though I don't
know why . . . >>

There is no a priori about epimoric ratios in gammoid theory. Ordering all
coherent (*) sets of steps, non-epimoric steps appear very soon. It is
simply a fact, rather than a theoretic choice, that most of combinatory
possibilities in first ranks give epimoric ratios.

With the first 50 gammiers (**) we have in order :

rank set non-epimoric

(1) 9/8-10/9-6/5
(2) 9/8-8/7-7/6
(3) 12/11-9/8-11/9 11/9
(4) 12/11-11/10-10/9-9/8
(5) 13/12-9/8-16/13 16/13
(6) 14/13-13/12-9/8-8/7

(16) 16/15-13/12-15/13-6/5 15/13
also in
(18)(19)(21)(23) 15/13

(27) 18/17-17/16-12/11-11/9 11/9
(29) 18/17-17/16-13/12-16/13 16/13
(31) 17/16-16/15-20/17-6/5 20/17
(33) 18/17-10/9-17/15-5/4 17/15
also in
(36)(41) 17/15
(37) 20/17
(38) 15/13
(39) 17/15 15/13

(43) 19/18-9/8-24/19 24/19
(45) 19/18-12/11-9/8-22/19 22/19
(47) 20/19-16/15-19/16-6/5 19/16
also in
(50) 19/16

-----

(*) Coherent means here obeying the congruity condition
what is equivalent to (CS) constant structure.

(**) Gammier obeys the supplemental fertility condition
meaning simply "more degrees than elements in the
minimal harmonic generator".

Pierre

--- In tuning@y..., George Zelenz <ploo@m...> wrote:

> Alexandros,
>
> what does "epimoria" mean, in ancient and current usage?
>
> Some friends of mine have been wondering what the plural for
> epimore is.

Hi George, don't mind my stepping in here...

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

-monz
http://www.monz.org
"All roads lead to n^0"

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

> "Epimoric ratios" are also called "superparticular ratios"
> and are ratios of the form (N+1)/N. So 2/1, 3/2, 4/3, 5/4,
> 6/5, 7/6 . . . 16/15 . . . 32/31 . . . 81/80 . . . 2401/2400
> . . . Many ancient Greek and Roman theorists thought that
> the intervals between adjacent scale degrees always had to
> be epimoric ratios . . . Erv Wilson and perhaps Pierre Lamothe
> seem to also put some stock in this theory, though I don't
> know why . . .

This is interesting to me.

I think that a preference for superparticular (or epimoric)
ratios may perhaps be due to a perceived ease of recognition.
In other words, the two ratio terms are easy to compare
because they only differ by one, or to look at it another
way, they are successive pitches in a pair of partials
sebsetted from the overtone series.

I'm willing to grant that there's something to this.
I say "perceived ease of recognition" because I'd bet
that many times the actual tuning is *not* superparticular
and a listener perceives it as such. Certainly many
ancient Greek theorists placed much emphasis on it.

-monz
http://www.monz.org
"All roads lead to n^0"

Joe wrote:
>The newer definition (does it originate with you, Manuel?)
>equates the moria with 1 degree of 72-EDO [= 2^(1/72) ],

No, from John Chalmers' Divisions.

Manuel

Paul Erlich schrieb:
>
> Sure, but why must all scale steps be epimoric? It would seem more
> important to make as many _intervals_ as possible epimoric, rather
> than as many _scale steps_ as possible epimoric . . . besides, even
>

Because one entails the other when you divide "melodically" (as I am
likely to do on my trombone - at least I think so). It goes like
this:

You take an intervallic frame (let's say a fifth and a fourth) and
want to build a scale around it (let's take pentatonic). Can be done
by dividing the fifth 3 times, the fourth twice, in a harmonic
division.

3/2=9/6: 7/6*8/7*9/8;
4/3=8/6: 7/6*8/7

The resulting scale is 7/6, 4/3, 3/2, 7/4, 2/1. Step combinations
within one of the framing intervals always reduce to other epimoric
ratios, combinations across the division of the octave do so
sometimes (can this be optimized? I doubt it).

This does not take vertical harmony into account and there is no
symmetry in the scale, but the method ("equal" [=harmonic] division
of other intervals) seems very intuitive to me from a melodic point
of view. Doesn't seem to be mentioned in the psycho-acoustic
literature, though. The harmony mafia... (Please tell me if I'm
wrong here.)

Nice to see that some of the Ancients were as wise as I!

Klaus

klaus schmirler does not want to be too inexact if he can help it:
>
> You take an intervallic frame (let's say a fifth and a fourth) and
> want to build a scale around it (let's take pentatonic). Can be done
> by dividing the fifth 3 times, the fourth twice, in a harmonic
> division.
>
> 3/2=9/6: 7/6*8/7*9/8;
> 4/3=8/6: 7/6*8/7
>
> The resulting scale is 7/6, 4/3, 3/2, 7/4, 2/1. Step combinations
> within one of the framing intervals always

No, often. Sometimes you trip over prime factors.

> reduce to other epimoric
> ratios, combinations across the division of the octave do so
> sometimes (can this be optimized? I doubt it).
>

Sorry I only started reading this thread rather late. It's the
language police here. :-)

I suggest that we drop the use of the words

epimore (n)
epimoria (n pl)
epimoric (adj)

and related, on the list, when referring to ratios of the form
(n+1)/n.

These words just confuse people. They are not English words. They will
not be found in any English dictionary.

There is no need to bastardise ancient Greek for this purpose when
there have been perfectly good words available for this concept, in
English, since at least the year 1557. Namely

superparticular (n)
superparticulars (n pl)
superparticular (adj)

And even if the "epimor-" are actual Greek words (not bastardised), I
shouldn't have to point out that (for better or worse) the language of
this list is English.

Regards,
-- Dave Keenan

>
>
> And even if the "epimor-" are actual Greek words (not bastardised), I
> shouldn't have to point out that (for better or worse) the language of
> this list is English.
>
> Regards,
> -- Dave Keenan
>

Ach doo lieben!

Pinche' pendejo.

Jorge Zelenzkahovitch

Dave Keenan wrote:

> I suggest that we drop the use of the words
>
> epimore (n)
> epimoria (n pl)
> epimoric (adj)
>
> and related, on the list, when referring to ratios of the form
> (n+1)/n.
>
> These words just confuse people. They are not English words. They will
> not be found in any English dictionary.
>
> There is no need to bastardise ancient Greek for this purpose when
> there have been perfectly good words available for this concept, in
> English, since at least the year 1557. Namely
>
> superparticular (n)
> superparticulars (n pl)
> superparticular (adj)
>
> And even if the "epimor-" are actual Greek words (not bastardised), I
> shouldn't have to point out that (for better or worse) the language of
> this list is English.

"Superparticular" is Latin - at least to the extent that "epimoric" is
Greek. Why try to prevent the use of one or the other? There are plenty of
valuable English-language texts that use "epimoric" over "superparticular"
- look at Barker's translations in "Greek Musical Writings". I suspect I'm
not the only one to have encountered "epimoric" before "superparticular".
And re: English dictionaries, there are plenty of terms used frequently on
this list that you won't find in the OED. And how confusing can it be,
anyway, if someone does ask, to say "epimoric" means "of the form (n+1)/n"
- you'll probably have to explain "superparticular" almost as often. I
also find that using an originally Greek word reminds us where these
concepts came from...

Best --Jon

> ----- Original Message -----
> From: jon wild <wild@fas.harvard.edu>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, June 06, 2001 9:30 AM
> Subject: [tuning] Re: less is "moria"
>

> I also find that using an originally Greek word reminds
> us where these concepts came from...

I'm pretty sure that the concept of epimoric/superparticular
(n+1)/n did not originate with the Greeks, and that the oldest
texts containing the idea are Babylonian, which in my opinion
probably propagate knowledge of the earlier Sumerians, thus
making it a few millennia older than ancient Greece.

It's been almost a year since I buried myself in this stuff...
I'd have to search for evidence. But I'm quite certain that
most of the Greek music-theory was tranmitted *to* them
from Babylon.

-monz
http://www.monz.org
"All roads lead to n^0"

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