Yahoo Tuning Groups Ultimate Backup tuning about range of 12 degrees in EDOs

Hi all

I have a question :

Can anyone help me to know the ranges of degrees like 'major 2nd''minor
tone' & etc in cent or ratio,

Or the range of any interval in the scale ... has it any role? What is
the maximum and minimum value of each degree?

As i read in mail from monz:

during the meantone era, the size of the "major-2nd"
(diatonic "whole-tone") in actual practice could range
anywhere from ~182.4037121 cents (exactly the JI
"small tone" with ratio 10:9) in 1/2-comma meantone,
to 200 cents in 12-et.

but in theoretical treatises, nearly every theorist
continued to define the "tone" as 9:8, as had been
done already for millennia.

Shahin mohajeri

tombak player and researcher , composer

www.geocities.com/acousticsoftombak

my tombak musics : www.rhythmweb.com/gdg

🔗monz <monz@tonalsoft.com>

hi Mohajeri,

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...> wrote:

> Hi all
>
> I have a question :
>
> Can anyone help me to know the ranges of degrees
> like 'major 2nd''minor tone' & etc in cent or ratio,
>
> Or the range of any interval in the scale ... has
> it any role? What is the maximum and minimum value
> of each degree?
>
>
>
> As i read in mail from monz:
>
>
>
> during the meantone era, the size of the "major-2nd"
> (diatonic "whole-tone") in actual practice could range
> anywhere from ~182.4037121 cents (exactly the JI
> "small tone" with ratio 10:9) in 1/2-comma meantone,
> to 200 cents in 12-et.
>
> but in theoretical treatises, nearly every theorist
> continued to define the "tone" as 9:8, as had been
> done already for millennia.

the interval names "major-2nd" etc. really have no
meaning outside of a diatonic type of scale. so they
only apply in cases where a composer used a tuning
which provides diatonic scales

... by "diatonic", i mean a scale with these properties:

- an equivalence-interval with a ratio of 2:1 (or an
approximation to it);

- two different step-sizes in the scale, the "tone"
and "semitone" (or 3 step-sizes in the case of the
"harmonic minor" scale, which also has an "augmented-2nd");

- the equivalence-interval is spanned by 8 scale steps.

for a scale which has a different number of steps per
equivalence-interval, it does not make sense to use
these traditional interval names, because they are
based on the third property listed above.

the two main tuning families for which ranges for these
interval names may be provided are, in historical
chronological order, pythagorean and meantone.

here is a table showing the cents values of all intervals
commonly found in diatonic music.

on the left is the number of generators which make up
that interval (8ve-reduced), assuming a generator which
is a "5th" or a tempered "5th".

the first main column is pythagorean, next comes
1/4-comma meantone, and last is 1/2-comma meantone.

the last column gives an abbreviation of the interval name.

the pythagorean and 1/2-comma meantone show the extreme
ranges for each interval. 1/4-comma meantone was
included only for purposes of comparison.

it's possible that the ranges may be still be extended
further on either side ... this is just my opinion of
what is applicable to the standard Western intervals.
comments from others are welcome.

generators --------------- cents --------------------- interval
.......... pythagorean .... 1/4-comma ..... 1/2-comma

... 11 ... 521.5050095 ... 462.3627131 ... 403.2204167 ... aug3
... 10 .. 1019.550009 .... 965.7842847 ... 912.0185607 ... aug6
.... 9 ... 317.5950078 ... 269.2058562 ... 220.8167046 ... aug2
.... 8 ... 815.6400069 ... 772.6274277 ... 729.6148485 ... aug5
.... 7 ... 113.6850061 .... 76.04899926 ... 38.41299247 .. aug1
.... 6 ... 611.7300052 ... 579.4705708 ... 547.2111364 ... aug4
.... 5 .. 1109.775004 ... 1082.892142 ... 1056.00928 ..... maj7
.... 4 ... 407.8200035 ... 386.3137139 ... 364.8074243 ... maj3
.... 3 ... 905.8650026 ... 889.7352854 ... 873.6055682 ... maj6
.... 2 ... 203.9100017 ... 193.1568569 ... 182.4037121 ... maj2
.... 1 ... 701.9550009 ... 696.5784285 ... 691.2018561 ... p5
.... 0 ..... 0 ............. 0 ............. 0 ........... prime
... -1 ... 498.0449991 ... 503.4215715 ... 508.7981439 ... p4
... -2 ... 996.0899983 .. 1006.843143 ... 1017.596288 .... min7
... -3 ... 294.1349974 ... 310.2647146 ... 326.3944318 ... min3
... -4 ... 792.1799965 ... 813.6862861 ... 835.1925757 ... min6
... -5 .... 90.22499567 .. 117.1078577 ... 143.9907197 ... min2
... -6 ... 588.2699948 ... 620.5294292 ... 652.7888636 ... dim5
... -7 .. 1086.314994 ... 1123.951001 ... 1161.587008 .... dim1
... -8 ... 384.3599931 ... 427.3725723 ... 470.3851515 ... dim4
... -9 ... 882.4049922 ... 930.7941438 ... 979.1832954 ... dim7
.. -10 ... 180.4499913 ... 234.2157153 ... 287.9814393 ... dim3
.. -11 ... 678.4949905 ... 737.6372869 ... 796.7795833 ... dim6
.. -12 .. 1176.53999 ...... 41.05885841 .. 105.5777272 ... dim2

the only EDOs which could be plugged into this table are
those which approximate a pythagorean or meantone tuning.
i'll do that table for you later. 26-edo is the one which
approaches 1/2-comma meantone, and 53-edo is nearly identical
to pythagorean.

some approximate equivalences between EDOs and meantones:

EDO ... comma tempering of "5th"

26 ..... -1/2
19 ..... -1/3
88 ..... (LucyTuning)
50 ..... -2/7, -7/26, -5/18
81 ..... (golden meantone)
31 ..... -1/4
43 ..... -1/5
55 ..... -1/6
12 ..... -1/11

41 also resembles pythagorean, but less closely than 53.

-monz

🔗Graham Breed <graham@microtonal.co.uk>

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@tonalsoft.com>

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@tonalsoft.com>

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

> >> monz schrieb...
> >>
> >> >for a scale which has a different number of steps per
> >> >equivalence-interval, it does not make sense to use
> >> >these traditional interval names,
> >>
> >> Er, many theorists have used these terms on scales with
> >> other numbers of tones.
> >
> >yes, i suppose i should have stated that explicitly.
> >but, IMO, it still doesn't make sense to use those
> >names for non-diatonic scales.
>
> So what do you call scale intervals in something like
> Paul's decatonics?
>
> -Carl

i've never used those scales, nor written about them
in any significant manner -- so i've never had to call
them anything, myself.

but considering that the decatonic scales have 10 notes
per "8ve", i think the interval names should reflect
that fact. it doesn't make any sense to base the
decatonic interval names on the same 8-based arithmetic
as the diatonic scales.

i regret that the regular diatonic intervals have these
names in the first place, because the "8-based" arithmetic
is really modulo-7, and so the intervals should be named
from zero thru 7. but the names were invented at a time
when people didn't recognize "zero" as a number, which
is why musical math can so stupidly say that 3+3=5 and
be correct.

-monz

🔗Carl Lumma <ekin@lumma.org>

>> >> >for a scale which has a different number of steps per
>> >> >equivalence-interval, it does not make sense to use
>> >> >these traditional interval names,
>> >>
>> >> Er, many theorists have used these terms on scales with
>> >> other numbers of tones.
>> >
>> >yes, i suppose i should have stated that explicitly.
>> >but, IMO, it still doesn't make sense to use those
>> >names for non-diatonic scales.
>>
>> So what do you call scale intervals in something like
>> Paul's decatonics?
>
>i've never used those scales, nor written about them
>in any significant manner -- so i've never had to call
>them anything, myself.
>
>but considering that the decatonic scales have 10 notes
>per "8ve", i think the interval names should reflect
>that fact. it doesn't make any sense to base the
>decatonic interval names on the same 8-based arithmetic
>as the diatonic scales.

Well, they "go to 11". But of course you will still have
2nds and 3rds. I don't see any way around that.

>i regret that the regular diatonic intervals have these
>names in the first place, because the "8-based" arithmetic
>is really modulo-7, and so the intervals should be named
>from zero thru 7. but the names were invented at a time
>when people didn't recognize "zero" as a number, which
>is why musical math can so stupidly say that 3+3=5 and
>be correct.

I tend to doubt they antedate the use of zero.

-Carl

🔗monz <monz@tonalsoft.com>

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

> >i regret that the regular diatonic intervals have these
> >names in the first place, because the "8-based" arithmetic
> >is really modulo-7, and so the intervals should be named
> >from zero thru 7. but the names were invented at a time
> >when people didn't recognize "zero" as a number, which
> >is why musical math can so stupidly say that 3+3=5 and
> >be correct.
>
> I tend to doubt they antedate the use of zero.

guess again.

the interval we call "perfect-4th" was referred to
by the ancient Greeks as "diatessaron", which means ...
"thru 4 [strings]" -- at least as early as Philolaus,
in the 400s BC.

similarly, they used "diapente" ("thru 5") to mean
the "perfect-5th".

zero didn't come on the scene until much later:

http://india.coolatlanta.com/GreatPages/sudheer/maths.html

>> "The ancient India astronomer Brahmagupta is credited
>> with having put forth the concept of zero for the
>> first time: Brahmagupta is said to have been born
>> the year 598 A.D. at Bhillamala (today's Bhinmal)
>> in Gujarat, Western India".

that's more than 1,000 years after the Greek theory
i referenced above.

-monz

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Carl Lumma <ekin@lumma.org>

>> >i regret that the regular diatonic intervals have these
>> >names in the first place, because the "8-based" arithmetic
>> >is really modulo-7, and so the intervals should be named
>> >from zero thru 7. but the names were invented at a time
>> >when people didn't recognize "zero" as a number, which
>> >is why musical math can so stupidly say that 3+3=5 and
>> >be correct.
>>
>> I tend to doubt they antedate the use of zero.
>
>guess again.
>
>the interval we call "perfect-4th" was referred to
>by the ancient Greeks as "diatessaron", which means ...
>"thru 4 [strings]" -- at least as early as Philolaus,
>in the 400s BC.
>
>similarly, they used "diapente" ("thru 5") to mean
>the "perfect-5th".
>
>zero didn't come on the scene until much later:
>
>http://india.coolatlanta.com/GreatPages/sudheer/maths.html
>
>>> "The ancient India astronomer Brahmagupta is credited
>>> with having put forth the concept of zero for the
>>> first time: Brahmagupta is said to have been born
>>> the year 598 A.D. at Bhillamala (today's Bhinmal)
>>> in Gujarat, Western India".
>
>that's more than 1,000 years after the Greek theory
>i referenced above.

Wow.

-Carl

🔗Mohajeri Shahin <shahinm@kayson-ir.com>

Dear keenan

Thanks a lot for your sheet of calculating ratios. It is very useful for
me ....

I'm working on intervals and scales in Iranian traditional and folklore
music so need to standards for naming intervals.i know that there are
different theories but want to know which is useful and practical.

The structure of one you told me to look is systematically good but
what about for example 155 cent in :

....

Cent intvl.name note from C ratio

150 N2 Dv 11:12

167 WN2 Dt 10:1

....

Is it n2 or wn2?

________________________________

From: Dave Keenan [mailto:d.keenan@bigpond.net.au]
Sent: Wednesday, August 25, 2004 11:34 AM
To: tuning@yahoogroups.com
Subject: [tuning] Re: about range of 12 degrees in EDOs

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...>
wrote:
> Hi all
>
> I have a question :
>
> Can anyone help me to know the ranges of degrees like 'major
2nd''minor
> tone' & etc in cent or ratio,
>
> Or the range of any interval in the scale ... has it any role?
What is
> the maximum and minimum value of each degree?

Dear Mahajeri,

These ranges are definitely somewhat fuzzy, but this
http://dkeenan.com/Music/Miracle/MiracleIntervalNam
ing.txt
can be read as suggesting that the ranges can be taken as
approximately +-8.3 cents from the cents given by their nearest
approximations in 72-tone equal temperament. This suits some
purposes but not others. Perhaps if we knew what use you wish to
make of such ranges, we could help you better.

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

Yahoo! Groups Sponsor

ADVERTISEMENT
click here
<http://us.ard.yahoo.com/SIG=129880tik/M=306910.5263657.6380312.2248467/
D=grplch/S=1705032401:HM/EXP=1093503887/A=2290377/R=0/SIG=12so36p36/*htt
p:/ad.doubleclick.net/jump/N3390.yahoocom/B1408521.15;sz=300x250;ord=109
3417487516870?>

<http://us.adserver.yahoo.com/l?M=306910.5263657.6380312.2248467/D=grplc
h/S=:HM/A=2290377/rand=395942411>

________________________________

Yahoo! Groups Links

* To visit your group on the web, go to:
/tuning/

* To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com
<mailto:tuning-unsubscribe@yahoogroups.com?subject=Unsubscribe>

* Your use of Yahoo! Groups is subject to the Yahoo! Terms of
Service <http://docs.yahoo.com/info/terms/> .

🔗Brad Lehman <bpl@umich.edu>

> > here is a table showing the cents values of all intervals
> > commonly found in diatonic music.
>
> i've added the table and a graph of it, at the bottom
> of my "interval" page:
> http://tonalsoft.com/enc/index2.htm?interval.htm

A good resource. But, how about a second and third table below it
showing the same values represented as fractions of comma (one table
for syntonic comma, the other for Pythagorean), instead of cents?
For example, in regular 1/4 syntonic comma, all the correctly-spelled
minor thirds are exactly 1/4 comma out of tune from pure (too narrow)
and the major sixths are consequently 1/4 comma too wide.
Representations as "310.264..." and "889.735..." for those really
give no suggestion of the way the scales are organized as a result of
the choice of fifth-size.

From the perspective of tonal music, cents values are meaningless in
the attempt to represent any neat little portions like that, and the
stepwise organization of the errors through the different degrees of
the scale. And, for those of us who do all our tuning work by ear
with zero aid from electronics, cents are (again) absolutely
worthless. Not to mention the deep bias the measurement system
itself has toward 12edo, as to judging quality of the results
(Barbour's whole book....)!

The *only* thing, conceptually, that I find cent-values useful for is
the convenient measurement and comparisons of semitones and tones.

It makes very little sense to me, theoretically, to pick a system
based on irrational numbers (i.e. the spacing of notes in 12edo) and
use that as a standard of measurement against which everything else
is measured, by deviation! It's a totally obfuscatory way of
representing the interval relationships that are really happening in
there...in my opinion, of course.

Far clearer, to me, is that intervals either hit one of the easy JI
points dead-on, or they are some easily understandable portion of one
comma or the other away from that point (preserving their function
and derivation from the cycles of fifths or major thirds). "Cents"
measurement muddies that picture. 9/8 is approx "203.91" and 10/9 is
approx "182.40", but those cent numbers make it appear that the sound
of them is somehow impure or otherwise inferior, else they would be
nice integers of some sort that's easier to conceptualize.

When playing, for example, in regular 1/6 syntonic comma, it's FAR
more interesting to know that any tritones arising in the music are
going to sound suddenly dead-on spooky in tune [hitting 45/32 exactly]
, and that they're therefore static by melodic/harmonic tendency (not
wanting to resolve anywhere), than that they measure the meaningless
value of "590.22" cents. They're dead-on spooky and beatless because
we've dropped exactly one syntonic comma in the six intervening
fifths; i.e. a pure 9/8 plus a pure 5/4 to build it (although the 9/8
and 5/4 themselves are not in that particular scale). Sort of like
suddenly encountering the standing-wave nodes in a room.

And, playing in 1/3 syntonic comma (aka 19edo, give or take a
smidge), our diminished triads and diminished-seventh chords are all
dead-on, because all the component minor thirds are pure. But "315.
64" says nothing about the way that sound is so startling. A better
measurement system should make that immediately obvious in its
numbers.

Brad Lehman

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> A good resource. But, how about a second and third table below it
> showing the same values represented as fractions of comma (one table
> for syntonic comma, the other for Pythagorean), instead of cents?

Monz, here is a fine opportunity for you to make use of your
fractional monzo computations. :)

> The *only* thing, conceptually, that I find cent-values useful for is
> the convenient measurement and comparisons of semitones and tones.

Logarithms are useful; there's not much point claiming otherwise.

> It makes very little sense to me, theoretically, to pick a system
> based on irrational numbers (i.e. the spacing of notes in 12edo) and
> use that as a standard of measurement against which everything else
> is measured, by deviation!

(81/80)^(1/4) is just as irrational as 2^(1/12). I could demonstrate
how to use Kirnberger atoms to do everything using 5-limit rational
numbers, but what would be the point?

> When playing, for example, in regular 1/6 syntonic comma, it's FAR
> more interesting to know that any tritones arising in the music are
> going to sound suddenly dead-on spooky in tune [hitting 45/32 exactly]
> , and that they're therefore static by melodic/harmonic tendency (not
> wanting to resolve anywhere), than that they measure the meaningless
> value of "590.22" cents. They're dead-on spooky and beatless because
> we've dropped exactly one syntonic comma in the six intervening
> fifths; i.e. a pure 9/8 plus a pure 5/4 to build it (although the 9/8
> and 5/4 themselves are not in that particular scale). Sort of like
> suddenly encountering the standing-wave nodes in a room.

This is an extremely interesting comment, since it suggests that I am
on an interesting track with my ideas such as "ratwolf". Ratwolf
tuning is almost identical to 2/7 comma, but instead the wolf fifth is
given the *exact* value of 20/13. It seems to me that if it is still
possible for 45/32 to sound spookily-dead-on, it must be possible for
20/13, tuned very precisely, to have the same effect. Moreover if
45/32 can somehow manage to be beatless, a pure 9/7 ought to be able
to also. A meantone with a fifth of the size exactly (224/9)^(1/8)
will have two interesting properties; first it will have supermajor
triads which are spookily dead on, and second it will have nearly
synchronized beating for a major triad in root position in case that
makes a difference to how you hear it. This fifth, of size about 695.6
cents, is about 5/17-comma, even flatter than 2/7-comma, and is close
to the 69-equal fifth.

> And, playing in 1/3 syntonic comma (aka 19edo, give or take a
> smidge), our diminished triads and diminished-seventh chords are all
> dead-on, because all the component minor thirds are pure.

Not really, because stacking four 6/5s is not a reasonable way to make
a diminished 7th chord. A tempered (6/5)^3 * (7/6), assuming you are
in a temperament which (like meantone) tempers out 126/125, is a
better idea. In 1/3-comma the 7/6 is so flat (13.795 cents) it seems
likely it doesn't work like one unless as a part of a chord giving it
that interpretation. Taken by itself, it's more a 15/13 or even, if
you believe in the 19-limit, a 22/19. (The commas involved in those
interpretations of this chord would be 325/324 and 2376/2375.)

🔗Brad Lehman <bpl@umich.edu>

> > A good resource. But, how about a second and third table below
it
> > showing the same values represented as fractions of comma (one
table
> > for syntonic comma, the other for Pythagorean), instead of cents?
>
> Monz, here is a fine opportunity for you to make use of your
> fractional monzo computations. :)
>
> > The *only* thing, conceptually, that I find cent-values useful
for
is
> > the convenient measurement and comparisons of semitones and
tones.

>
> Logarithms are useful; there's not much point claiming otherwise.

Of course they are. My objection is to the arbitrary assignment of
merely 1200 pieces to the octave, and then slashing those up further
into little bits of thousandths and millionths.

There are much better arbitrary assignments available, to make better
use of logarithms. Logarithmic base can be assigned any way one
chooses. The one I like best gives 720 pieces to the Pythagorean
comma, and therefore 660 pieces to the syntonic comma, 60 to the
schisma, and 1260 to the lesser diesis. Then, all these can be
portioned up nicely into the most commonly used amounts, all with
integer arithmetic. Much of the tuning math is reduced to very easy
paperwork. Compared with this, the "cents" system is far too coarse
and its resulting numbers look almost randomly organized.

>
> > It makes very little sense to me, theoretically, to pick a system
> > based on irrational numbers (i.e. the spacing of notes in 12edo)
and
> > use that as a standard of measurement against which everything
else
> > is measured, by deviation!
>
> (81/80)^(1/4) is just as irrational as 2^(1/12). I could demonstrate
> how to use Kirnberger atoms to do everything using 5-limit rational
> numbers, but what would be the point?

I fear you've unfortunately missed the main part of the point I was
trying to make; I'll try to say it again better. The cents system
assigns its nice even 100s to an interval that is itself irrational.
What a silly place to put a baseline: on an irrational interval! I
suggest that the baselines should be the JI ones of 3/2, 4/3, 5/4,
6/5, maybe also 7/6, and then measure deviations from those
blissfully
easy and consonant points. We can still measure the *deviations* in
whatever units we want (I like fractions or percentages of commas),
but the baseline points are easy places that make acoustical sense.

> > And, playing in 1/3 syntonic comma (aka 19edo, give or take a
> > smidge), our diminished triads and diminished-seventh chords are
all
> > dead-on, because all the component minor thirds are pure.
>
> Not really, because stacking four 6/5s is not a reasonable way to
make
> a diminished 7th chord. A tempered (6/5)^3 * (7/6), assuming you are
> in a temperament which (like meantone) tempers out 126/125, is a
> better idea. In 1/3-comma the 7/6 is so flat (13.795 cents) it seems
> likely it doesn't work like one unless as a part of a chord giving
it
> that interpretation.

A diminished-seventh chord is built of THREE stacked 6/5s, not four.
G#-B-D-F...that's three intervals.

And it works wonderfully in practice. Check out Edward Parmentier's
"17th Century French Harpsichord Music" CD (Wildboar 8502) which is
*mostly* French, and is done throughout in 1/3 syntonic comma
(19edo).
Soon into the first track, a toccata by Froberger, he lands on a
pure
diminished 7th chord and it really jumps out.

Brad Lehman

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> > Logarithms are useful; there's not much point claiming otherwise.

> Of course they are. My objection is to the arbitrary assignment of
> merely 1200 pieces to the octave, and then slashing those up further
> into little bits of thousandths and millionths.

You've got to have logs base something.

> There are much better arbitrary assignments available, to make better
> use of logarithms.

I agree; for years for my own use I divided the octave into 612 parts,
which allowed me to get good accuracy while rounding off to the
nearest integer. But if everyone uses cents you pretty well have to go
with the flow.

Logarithmic base can be assigned any way one
> chooses. The one I like best gives 720 pieces to the Pythagorean
> comma, and therefore 660 pieces to the syntonic comma, 60 to the
> schisma, and 1260 to the lesser diesis.

This leads to a set of linear equations which has a one-parameter
family of solutions; the one making the most sense being 36828 equal.
I'm not clear why you chose this, if in fact you did, but it does have
the following property--it is divisible among other things by 12, 22,
27, 31, and 99, and in fact is the least common multiple of these, and
so those divisions are all expressible in terms of integers. By other
measures it is an undistinguished division, and is consistent only to
the 9-limit.

Then, all these can be
> portioned up nicely into the most commonly used amounts, all with
> integer arithmetic. Much of the tuning math is reduced to very easy
> paperwork. Compared with this, the "cents" system is far too coarse
> and its resulting numbers look almost randomly organized.

If we ignore the business about divisors we can certainly do much
better than 36828, but perhaps that is a key issue for you. What would
you regard as the salient virtues of a musical logarithm system?

> >
> > > It makes very little sense to me, theoretically, to pick a system
> > > based on irrational numbers (i.e. the spacing of notes in 12edo)
> and
> > > use that as a standard of measurement against which everything
> else
> > > is measured, by deviation!
> >
> > (81/80)^(1/4) is just as irrational as 2^(1/12). I could demonstrate
> > how to use Kirnberger atoms to do everything using 5-limit rational
> > numbers, but what would be the point?
>
> I fear you've unfortunately missed the main part of the point I was
> trying to make; I'll try to say it again better. The cents system
> assigns its nice even 100s to an interval that is itself irrational.
> What a silly place to put a baseline: on an irrational interval!

2, clearly a rational number, is set to 1200 cents.

I
> suggest that the baselines should be the JI ones of 3/2, 4/3, 5/4,
> 6/5, maybe also 7/6, and then measure deviations from those
> blissfully
> easy and consonant points.

If any of these is set exactly to an integer an octave will not be an
integer. However, if 3/2, 4/3, 5/4, 6/5 and 7/6 are all approximately
integers, then we are in the 7-limit, since any 7-limit interval can
be expressed as a product of intervals in the above set. Hence, what
we would want to do is to find good 7-limit micro equal temperaments.
We might, for instance, pick on 18355, 84814 or Paul Erlich's 103169;
if we want divisibility by 12 there is 11664, 21480 or 33144.

> A diminished-seventh chord is built of THREE stacked 6/5s, not four.
> G#-B-D-F...that's three intervals.

That's my point; then you get F-G#, which is something else. You can
call it an augmented second if you like, but what it actually sounds
like is dependent on tuning.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> > There are much better arbitrary assignments available, to make
better
> > use of logarithms.
>
> I agree; for years for my own use I divided the octave into 612
parts,
> which allowed me to get good accuracy while rounding off to the
> nearest integer. But if everyone uses cents you pretty well have
to go
> with the flow.

I have to agree with Gene here, so many people have come up with so
many different log bases for working with tuning. They all can tell
you why their's is the best. (I even had one myself for a while but
I won't bore you with it.) About the only thing we agree on is that
cents are far from perfect.

It's fine to use your own system for your own work, but if you want
others to follow what you are saying, you'd better use cents. But by
all means describe intervals as JI ratios plus or minus cents.
That's quite common on this list. e.g. the 12-equal major third can
be described as 4:5 +14c.

It's rarely necessary to use decimal places of cents since few
people claim to be able to hear +-0.5 cent. But sometimes it is
necessary, such as when giving an optimal generator for some
temperament, where this generator will be stacked many times to
approximate various just intervals, since the errors would
accumulate.

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>> >It seems to me that if it is still possible for 45/32 to sound
>> >spookily-dead-on, it must be possible for 20/13, tuned very
>> >precisely, to have the same effect. Moreover if 45/32 can
>> >somehow manage to be beatless, a pure 9/7 ought to be able to
>> >also.
>>
>> Why?
>
>It's much higher on the consonance totem-pole, obviously.

That's not obvious, or even necessarily true. Tenney height
stops working as the numbers get large.

>> >> And, playing in 1/3 syntonic comma (aka 19edo, give or take a
>> >> smidge), our diminished triads and diminished-seventh chords
>> >> are all dead-on, because all the component minor thirds are pure.
>> >
>> >Not really, because stacking four 6/5s is not a reasonable way to make
>> >a diminished 7th chord.
>>
>> Why?
>
>Because (6/5)^4 = 2 (648/625) is 63 cents sharper than an octave, and
>that's a dissonance.

So you don't play the octave.

>Aren't you asking me questions you already know the answer to?

No.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@tonalsoft.com>

hi Brad,

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> > [Gene:]
> > Logarithms are useful; there's not much point claiming otherwise.
>
>
> Of course they are. My objection is to the arbitrary
> assignment of merely 1200 pieces to the octave, and then
> slashing those up further into little bits of thousandths
> and millionths.

yes, well ... in America you're still free to use cubits to
measure length if you think it makes more sense ... but
the rest of the country will still be using inches, feet,
and miles, and the rest of the world will use meters.

the point i'm making is that cents are by far the most
standard form of small-interval measurement. i'm not
crazy about using them either.

> There are much better arbitrary assignments available, to
> make better use of logarithms. Logarithmic base can be
> assigned any way one chooses. The one I like best gives
> 720 pieces to the Pythagorean comma, and therefore 660 pieces
> to the syntonic comma, 60 to the schisma, and 1260 to the
> lesser diesis. Then, all these can be portioned up nicely
> into the most commonly used amounts, all with integer
> arithmetic. Much of the tuning math is reduced to very easy
> paperwork. Compared with this, the "cents" system is far
> too coarse and its resulting numbers look almost randomly
> organized.

the unit you're talking about here is the "tuning unit"
advocated by John Brombaugh:

http://tonalsoft.com/enc/index2.htm?tu.htm

it divides the 8ve into ~36828.6282 equal sections.

depending on what type of JI tuning you're trying to
emulate, there are many different logarithmic measures
that may be appropriate.

for example, to study 5-limit JI, 612-edo is much more
logical than cents, because it gives nearly-integer
values for every JI ratio for a very large section of
the lattice. and for even more accuracy in 5-limit,
4296-edo can't be beat.

but if you're going to go all the way to ~36829, why not
just go all the way to the finest MIDI tuning resolution?
196608-edo.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >It seems to me that if it is still possible for 45/32 to sound
> >> >> >spookily-dead-on, it must be possible for 20/13, tuned very
> >> >> >precisely, to have the same effect. Moreover if 45/32 can
> >> >> >somehow manage to be beatless, a pure 9/7 ought to be able to
> >> >> >also.
> >> >>
> >> >> Why?
> >> >
> >> >It's much higher on the consonance totem-pole, obviously.
> >>
> >> That's not obvious, or even necessarily true. Tenney height
> >> stops working as the numbers get large.
> >
> >9/7 isn't large, but obviously it has to stop working, because the
> >rational numbers are dense. However, the claim was that 45/32 was
> >spookily dead-on; this can hardly be the case if 9/7 isn't.
>
> There are a lot of things that could happen here. 9/7 can be
> in the field of attraction of 5/4, and when it is it usually
> sounds terrible. 45/32 is in a realm that enjoys a lot of
> flexibility, like the minor third realm.

45/32 is only 225/224 away from 7/5, whereas 9/7 is a quartertone
(36/35) away from 5/4, and that's a huge difference. If you stick it
together with 3/2 and form the supermajor triad, I don't think that
helps move it into the field of attraction for 5/4; it sounds to me to
be an interval with its own field of attraction. The more dubious
claim is that 45/32 has a life independent of 7/5.

🔗Brad Lehman <bpl@umich.edu>

> > There are much better arbitrary assignments available, to
> > make better use of logarithms. Logarithmic base can be
> > assigned any way one chooses. The one I like best gives
> > 720 pieces to the Pythagorean comma, and therefore 660 pieces
> > to the syntonic comma, 60 to the schisma, and 1260 to the
> > lesser diesis. Then, all these can be portioned up nicely
> > into the most commonly used amounts, all with integer
> > arithmetic. Much of the tuning math is reduced to very easy
> > paperwork. Compared with this, the "cents" system is far
> > too coarse and its resulting numbers look almost randomly
> > organized.
>
> the unit you're talking about here is the "tuning unit"
> advocated by John Brombaugh:
> http://tonalsoft.com/enc/index2.htm?tu.htm
> it divides the 8ve into ~36828.6282 equal sections.

Brombaugh actually calls them "Temperament Units" and so do I, these
days. I got them from him back in 1997 in private correspondence,
and wrote them up (accidentally mislabelled as "tuning units") on my
page
http://how.to/tune
--which appears to be the uncredited source for the explanation on
http://tonalsoft.com/enc/index2.htm?tu.htm

I re-established conversation with Brombaugh earlier this year, and
we've been exchanging later incarnations of our spreadsheets; we both
use "Temperament Units" and I simply haven't bothered to update my
web page yet. Ever since I learned them from him seven years ago,
they've been adequate for all my practical and theoretical needs. I
feel they're a brilliantly clear way of thinking about the issues
relevant to historical temperaments.

> but if you're going to go all the way to ~36829, why not
> just go all the way to the finest MIDI tuning resolution?
> 196608-edo.

Because I care almost zero about MIDI.

It matters not how many ~36828.6282 equal sections the Temperaments
Units scheme divides the octave. Division of the octave is
irrelevant, for historical tuning issues (except that 19edo, 31edo,
and 55edo are convenient ways to round off regular comma-fraction
systems, and some of the 17th and 18th century theorists did so as
their own shortcuts to explain tuning to common musicians).

Instead of logarithms base 2, which (if we stick with integers) locks
us into N-edo systems, it's logarithms base PythagoreanComma, to 720
units. If a given pitch is 360TU away from making a pure JI interval
with some other pitch, I can see immediately that it's 1/2
Pythagorean comma out. If another pitch is 165TU away from making
another JI interval, it's immediately obvious that this is 1/4
syntonic comma because a syntonic comma is 660TU. Since almost
everything in the literature deals with commas whacked into 2, 3, 4,
5, 6, or 12 pieces, it's nice to have integers for both commas that
are easily divisible by all those divisors, yielding other integers.

That is, Brombaugh's "Temperament Units" system offers Diophantine
clarity. It's quite a bit easier to keep track of the lesser diesis
at 1260TU (two syntonic commas less a schisma) than to tote a bunch
of fractions around; or worse, micro-portions of cents. Logarithms
base PC actually do something useful.

Brad Lehman

🔗monz <monz@tonalsoft.com>

hi Gene, Carl, Brad,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > There are a lot of things that could happen here.
> > 9/7 can be in the field of attraction of 5/4, and when
> > it is it usually sounds terrible. 45/32 is in a realm
> > that enjoys a lot of flexibility, like the minor third
> > realm.
>
> 45/32 is only 225/224 away from 7/5, whereas 9/7 is a
> quartertone (36/35) away from 5/4, and that's a huge
> difference. If you stick it together with 3/2 and form
> the supermajor triad, I don't think that helps move it
> into the field of attraction for 5/4; it sounds to me to
> be an interval with its own field of attraction. The more
> dubious claim is that 45/32 has a life independent of 7/5.

pairing 9/7 and 3/2 (with 1/1) into the supermajor triad
gives you a chord with the proportions 14:18:21.

but if you look at it from the utonal point of view,
it's much simpler: 1/(9:7:6) .

but anyway, Carl's point about 9/7 being within the
field-of-attraction of 5/4 is a good one. if a listener
is expecting a 5/4 and gets a 9/7 instead, the effect
should be jarring.

but i also do agree with Gene ... from the very beginning
when Brad wrote about the 45/32 being "spookily dead-on",
i too was doubting if that interval has "a life independent
of 7/5" -- of course a compositional context could make
it so, but i doubt if that's been done very often, if
ever at all.

-monz

🔗monz <monz@tonalsoft.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> It matters not how many ~36828.6282 equal sections the Temperaments
> Units scheme divides the octave.

If you want a system which can actually deal with the 5-limit as a
whole, yes it does matter.

Division of the octave is
> irrelevant, for historical tuning issues

Considering that historical tuning theory refers itself to pure
octaves, this is hardly the case.

> Instead of logarithms base 2, which (if we stick with integers) locks
> us into N-edo systems, it's logarithms base PythagoreanComma, to 720
> units.

Which "locks you in" to Pythagorean commas; following your reasoning
all we can now talk about are powers of the Pythagorean comma, which
is pretty useless by itself. In fact, however, you want to be able to
discuss the Pythagorean comma, but also 81/80, 5/4, 3/2 and 2--and all
in terms of the same system.

A system designed specifically to discuss temperament of 12-note
scales in terms of the 5-limit would be well advised to make use of
atomic methods, by which I mean tempering out the atom of Kirnberger;
and in fact 36828 does exactly this, so I think it is lurking in this
TU system. 36828 is a "contorted" 5-limit system; 36828 = 3*12276;
12276 is an atomic 5-limit system, but actually also a pretty decent
atomic 11-limit system. In an atomic system, the Pythagorean comma is
12 schismas, 81/80 is 11 schismas, and 12-equal is one schisma shy of
a pure fifth. Dividing the Pythagorean comma into 720 parts and the
Ddidymus comma into 660 parts seems to be done with an eye to getting
a lot of useful divisors (of 720 and 660.) You could do something
similar, only with a good deal more tuning accuracy, by dividing the
Pythagorean comma into 576 parts and the Didymus comma into 528 parts.
In any case it seems perverse to ignore the fact that the natural
interpretation of all of this is that 5/4 gets 11856 parts, 3/2 gets
21543 parts, and the octave 36828 parts. If you are tempering a fifth,
you should know how big a fifth is. If you are tempering to
approximate 5/4, you should know how big 5/4 is. And if you are
dividing the octave into parts, you should know how big an octave is.

> That is, Brombaugh's "Temperament Units" system offers Diophantine
> clarity.

I think your clarity would become more Diophanine if you realize you
implicitly are dividing an octave, whether tuned justly or not, into
36828 parts, that this is in effect a 3*12276 system, so that integer
numbers of TUs, unless resulting from a temperament, always seem to
turn out to be multiples of 3, and that atomic temperament is
implicitly in use. All of this is below the surface as you present the
matter, but it is there, even so.

I don't see any cosmic significance attached to tuning the Pythagorean
comma pure. TOP tuning would make the octave 0.00023 cents flat and
the Pythagorean comma 0.00040 cents sharp, which gives you an idea of
the small size of the ajustments involved. To get the Pythagorean
comma pure you need to tune flatly far out of the optimal range for
5-limit consonances, which seems fairly pointless. I would suggest
simply using the integer values and ignoring the tuning.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> This branch of this thread appears to have very little to do with
> the subject heading. And why are you guys talking about "fields of
> attraction" for individual pitches?
>
> What's that you say? You're actually talking about dyads! Well then,
> how about using the correct notation, so readers don't get confused,
> e.g. 7:9 and 4:5.

I am not going to use a confusing notation for something I don't even
really mean anyway. When I say 9/7, I normally mean neither a dyad nor
a pitch, but a directed interval--that is, to me 9/7 and 7/9 are not
the same, and intervals have a group structure. If 9/7 does refer in
some sense to a pitch, there must be a reference 1 around somewhere,
in which case you get a directed interval anyway. The antique a:b
notation, dating to the days people did not possess the field of
rational numbers, is inherently confusing, since it fails to
distinguish a:b from b:a, and inherently inferior, since it fails to
denote a number or form a group.

Rational numbers are your friend. Learn them, use them. I would be
very happy most of the time (it *is* useful in some cases) if people
tossed this old-fashioned rubbish a:b out the window, but if you do
use it it hardly makes sense to assume someone who does not secretly
meant to. I did not.

🔗monz <monz@tonalsoft.com>

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > You might want to include the connection with
> > 12276 equal and atomic temperament on this page.
>
> Some other information: the poptimal range for the
> atomic temperament schisma

doesn't atomic also have a "schismina"? if so, then
what is its TU value?

> in TU terms is 60.00389 to 60.00416; hence if we took
> for instance the schisma to be exactly 60.004 TUs, we
> would get a Pythagorean comma of 720.048 TUs and a
> Didymus comma of 660.044 TUs. The whole point of this
> sort of thing, of course, is that you plan to round off
> to the nearest integer, but if you do not this would
> be a good set of tuning values for the TU. It would give
> a 5/4 as 4*3069-7*60.004 = 11855.972 TUs, 3/2 as
> 7*3069+60.004 = 21543.004 TUs, and of course the octave
> is exacty 12*3069=36828 TUs. You can see how closely
> these lie to the rounded values.

i've simply included your post at the bottom of my
"tuning unit" page.

can you please fill out one of those temperament templates
for atomic? thanks.

-monz

🔗monz <monz@tonalsoft.com>

hi Gene and Brad,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> Dividing the Pythagorean comma into 720 parts and the
> Ddidymus comma into 660 parts seems to be done with an
> eye to getting a lot of useful divisors (of 720 and 660.)
> You could do something similar, only with a good deal
> more tuning accuracy, by dividing the Pythagorean comma
> into 576 parts and the Didymus comma into 528 parts.

i know you had an easy way to figure that out! ;-)

please elaborate over on tuning-math.

> I don't see any cosmic significance attached to tuning
> the Pythagorean comma pure. TOP tuning would make the
> octave 0.00023 cents flat and the Pythagorean comma
> 0.00040 cents sharp, which gives you an idea of the
> small size of the ajustments involved. To get the
> Pythagorean comma pure you need to tune flatly far out
> of the optimal range for 5-limit consonances, which seems
> fairly pointless. I would suggest simply using the integer
> values and ignoring the tuning.

Gene, (presuming to speaking for Brad) from what i know
of his work, which is only what i've seen here and on his
website, i think Brad is really only interested in knowing
deviations from JI in terms of fractions of the pythagorean
and syntonic commas, and not specifically in "tuning the
Pythagorean comma pure".

if that's obvious to you, then i guess i'm not clear on
what you guys are really arguing here. maybe you can
elaborate.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > This branch of this thread appears to have very little to do
with
> > the subject heading. And why are you guys talking about "fields
of
> > attraction" for individual pitches?
> >
> > What's that you say? You're actually talking about dyads! Well
then,
> > how about using the correct notation, so readers don't get
confused,
> > e.g. 7:9 and 4:5.
>
> I am not going to use a confusing notation for something I don't
even
> really mean anyway. When I say 9/7, I normally mean neither a dyad
nor
> a pitch, but a directed interval--

That's fine. Whether you want to think of it as a dyad or an
interval it's helpful to distinguish it from a pitch. This is
analogous to the distinction between a vector and a point in
geometry. You just as often 9/7 to represent a pitch when you list
the notes of a scale.

> that is, to me 9/7 and 7/9 are not
> the same,

I totally agree.

> and intervals have a group structure. If 9/7 does refer in
> some sense to a pitch, there must be a reference 1 around
somewhere,

No. Look at Erv Wilson's CPS scales.

> in which case you get a directed interval anyway.

You might well get one, but we'd still like to be certain whether
you are talking about the interval or the pitch.

> The antique a:b
> notation, dating to the days people did not possess the field of
> rational numbers, is inherently confusing, since it fails to
> distinguish a:b from b:a, and inherently inferior, since it fails
to
> denote a number or form a group.

Then don't think about it in that way. Just think of the form a:b as
an alternative representation of the rational number
max(a,b)/min(a,b). So this form is incapable of representing
rationals less than 1, but when representing intervals we don't need
to represent rationals less than one.

> Rational numbers are your friend. Learn them, use them.

I hope you understand now, that I _am_ using rational numbers.

> I would be
> very happy most of the time (it *is* useful in some cases) if
people
> tossed this old-fashioned rubbish a:b out the window,

Sigh. I think most people find 4:5:6 quite meaningful for the JI
major chord. I can't see your alternative 1-5/4-3/2 (did I get that
right?) catching on any time soon. Most people are used to thinking
of a "-" between numbers as a minus sign for one thing.

It seems obvious to me, to want to say that a 4:5:6 consists of a
4:5 and a 5:6 stacked.

> but if you do
> use it it hardly makes sense to assume someone who does not
secretly
> meant to. I did not.

I think I eventually managed to parse the above correctly, as
indicated by the commas below:
"but if you do use it, it hardly makes sense to assume someone who
does not, secretly meant to."

I assumed no such thing. I did however assume you were talking about
intervals rather than pitches. But I wasn't sure whether Brad was
talking about intervals or pitches when he said that 45/32 was dead-
on. Since a 45/32 pitch can clearly be heard as dead-on relative to
a 9/8 pitch or a 15/8 pitch, but it is seriously doubtful whether
this is the case for a lone interval of 32:45.

By the way, what is the difference between a dyad and an interval,
musically speaking?

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> That's fine. Whether you want to think of it as a dyad or an
> interval it's helpful to distinguish it from a pitch. This is
> analogous to the distinction between a vector and a point in
> geometry. You just as often 9/7 to represent a pitch when you list
> the notes of a scale.

9/7 is not a pitch even then. It becomes a pitch if you give the scale
in Hertz, but this is rarely done.

> > that is, to me 9/7 and 7/9 are not
> > the same,
>
> I totally agree.
>
> > and intervals have a group structure. If 9/7 does refer in
> > some sense to a pitch, there must be a reference 1 around
> somewhere,
>
> No. Look at Erv Wilson's CPS scales.

The reference 1 is around, it simply isn't a member of the scale.
Since we are not talking about cycles per second but combination
product sets, it *has* to be around.

> > in which case you get a directed interval anyway.
>
> You might well get one, but we'd still like to be certain whether
> you are talking about the interval or the pitch.

If I am talking about a pitch I use Hz.

> > The antique a:b
> > notation, dating to the days people did not possess the field of
> > rational numbers, is inherently confusing, since it fails to
> > distinguish a:b from b:a, and inherently inferior, since it fails
> to
> > denote a number or form a group.
>
> Then don't think about it in that way. Just think of the form a:b as
> an alternative representation of the rational number
> max(a,b)/min(a,b). So this form is incapable of representing
> rationals less than 1, but when representing intervals we don't need
> to represent rationals less than one.

Somehow making it not to be a group does not seem to me to be an
improvement.

> > Rational numbers are your friend. Learn them, use them.
>
> I hope you understand now, that I _am_ using rational numbers.

The monoid of rational numbers >= 1, under multiplication. You can use
it, but why would you want to make your life difficult?

> > I would be
> > very happy most of the time (it *is* useful in some cases) if
> people
> > tossed this old-fashioned rubbish a:b out the window,
>
> Sigh. I think most people find 4:5:6 quite meaningful for the JI
> major chord.

I did say it is useful in some cases; this is one of them.

> I assumed no such thing. I did however assume you were talking about
> intervals rather than pitches.

Of course I was not talking about pitches--did you see a "Hz"?

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > That's fine. Whether you want to think of it as a dyad or an
> > interval it's helpful to distinguish it from a pitch. This is
> > analogous to the distinction between a vector and a point in
> > geometry. You just as often 9/7 to represent a pitch when you
list
> > the notes of a scale.
>
> 9/7 is not a pitch even then.

So what does each number in a Scala .scl file (other than the
cardinality and those in comments) represent for you. Manuel sure
thinks they represent pitches (see the Scala help), and so does
everyone else as far as I know.

If you want to call them "relative pitches" that's fine by me. They
are still quite different beasties to "intervals", at least to a
musician.

> > > If 9/7 does refer in
> > > some sense to a pitch, there must be a reference 1 around
> > somewhere,
> >
> > No. Look at Erv Wilson's CPS scales.
>
> The reference 1 is around, it simply isn't a member of the scale.

This is clearly a different meaning of "around somewhere", from the
one I am used to. In your sense, any number you like to think of
is "around somewhere".

> Since we are not talking about cycles per second but combination
> product sets, it *has* to be around.

1 need not be one of the factors being combined in a Wilson CPS. But
of course you can always rescale so that some arbitrarily choosen
pitch in the scale becomes 1/1. But it remains true that a scale can
be described without including a 1/1 and therefore it may contain a
pitch called 9/7 without containing any 7:9 interval.

> > Then don't think about it in that way. Just think of the form
a:b as
> > an alternative representation of the rational number
> > max(a,b)/min(a,b). So this form is incapable of representing
> > rationals less than 1, but when representing intervals we don't
need
> > to represent rationals less than one.
>
> Somehow making it not to be a group does not seem to me to be an
> improvement.
>
> > > Rational numbers are your friend. Learn them, use them.
> >
> > I hope you understand now, that I _am_ using rational numbers.
>
> The monoid of rational numbers >= 1, under multiplication. You can
use
> it, but why would you want to make your life difficult?

I don't recall ever seeing you use a rational less than 1 to
describe a musical interval. So what is lost?

Anyway, I'm not restricting myself to this monoid in performing
calculations relating to intervals. It's just that any inputs or
outputs that represent intervals (as opposed to representing
relative pitches) always seem to be >=1. So I don't see that
anything is lost by using a representation for these that cannot
represent <1 but allows me to distinguish intervals from relative
pitches for the sake of the musicians on this list who don't have a
PhD in pure math.

> > > I would be
> > > very happy most of the time (it *is* useful in some cases) if
> > people
> > > tossed this old-fashioned rubbish a:b out the window,
> >
> > Sigh. I think most people find 4:5:6 quite meaningful for the JI
> > major chord.
>
> I did say it is useful in some cases; this is one of them.

I'm very glad of that. So doesn't it make sense to use 4:5 and 5:6
to represent "chords of 2 notes" or dyads? You're still free to read
them as the rationals 5/4 and 6/5 when it comes time to make
calculations with them.

I can understand you being upset if you thought we would then have
to read 5:4 as 4/5. I hope I've set your mind at rest, that this is
not the case, with my explanation that
a:b = max(a,b)/(min(a,b)

You still haven't explained the difference between a dyad and an
interval.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> So what does each number in a Scala .scl file (other than the
> cardinality and those in comments) represent for you.

They represent directed intervals, taken in the given order, from the
unison interval. When using a Scala .scl file for tuning purposes, the
reference frequency (in Hertz) must always be supplied. A tun file is
different--that *does* consist of pitches.

Manuel sure
> thinks they represent pitches (see the Scala help), and so does
> everyone else as far as I know.

A strange belief. However telling me everyone believes something which
is plainly false does not make it true; it doesn't even make it true
that everyone believes it. I'd be interested to hear from anyone else
who cares to cast their ballot--is 3/2 a pitch, and is so, what pitch
is it?

> If you want to call them "relative pitches" that's fine by me. They
> are still quite different beasties to "intervals", at least to a
> musician.

And the difference is?
But it remains true that a scale can
> be described without including a 1/1 and therefore it may contain a
> pitch called 9/7 without containing any 7:9 interval.

A 1 need not be part of the set, though in fact Scala assumes it
always is, a fact you had better keep in mind if you use it.

> I don't recall ever seeing you use a rational less than 1 to
> describe a musical interval. So what is lost?

I have, however. What's lost is the math which makes tuning theory a
lot easier. If you want to go Pierre Lamothe's route be my guest, but
I see no point in obscuring issues or making subjects more difficult
than they need to be.

> Anyway, I'm not restricting myself to this monoid in performing
> calculations relating to intervals. It's just that any inputs or
> outputs that represent intervals (as opposed to representing
> relative pitches) always seem to be >=1. So I don't see that
> anything is lost by using a representation for these that cannot
> represent <1 but allows me to distinguish intervals from relative
> pitches for the sake of the musicians on this list who don't have a
> PhD in pure math.

Why do you care about undirected intervals? What use are they? I
understand why you might want to talk about dyads, which is two notes
playing at once, but what do you gain by not allowing 3/4 into your
universe of intervals?

> > > > I would be
> > > > very happy most of the time (it *is* useful in some cases) if
> > > people
> > > > tossed this old-fashioned rubbish a:b out the window,
> > >
> > > Sigh. I think most people find 4:5:6 quite meaningful for the JI
> > > major chord.
> >
> > I did say it is useful in some cases; this is one of them.
>
> I'm very glad of that. So doesn't it make sense to use 4:5 and 5:6
> to represent "chords of 2 notes" or dyads?

You can if you like. I see nothing gained by it; I prefer actual numbers.

You're still free to read
> them as the rationals 5/4 and 6/5 when it comes time to make
> calculations with them.

Eh, but they aren't defined as numbers, or not exactly. That is to
say, some people think it is an actual number, but then they are happy
to write it as a/b and not a:b. Just to confuse the issue, when you
are doing odds a:b means a probability of b/(a+b); if it is 10:1
against, it has a probability of 1/11. You put down 1, and either you
get to keep the 1 and get 10 back if you win, or you lose the 1. If
the winning probabilty is 1/11, there is no house take and it is a
zero-sum game.

> I can understand you being upset if you thought we would then have
> to read 5:4 as 4/5. I hope I've set your mind at rest, that this is
> not the case, with my explanation that
> a:b = max(a,b)/(min(a,b)

Some of the time. :)

But...is it a number, or isn't it?

> You still haven't explained the difference between a dyad and an
> interval.

I regard a dyad as a chord.

🔗monz <monz@tonalsoft.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > If you want to call them "relative pitches" that's fine by me.
They
> > are still quite different beasties to "intervals", at least to a
> > musician.
>
> And the difference is?

Are you serious? If someone is playing a 3/2 pitch, they are playing
a single note. If they are playing a 2:3 interval they are playing
_two_ notes.

> I see no point in obscuring issues or making subjects more
difficult
> than they need to be.

That's the funniest thing I've heard from you in quite a while. :-)

> Why do you care about undirected intervals? What use are they?

You can play them and hear them. How does a 3/4 interval sound
different to a 4/3 interval?

> I
> understand why you might want to talk about dyads, which is two
notes
> playing at once, but what do you gain by not allowing 3/4 into your
> universe of intervals?

Like I said, I gain the use of a notation that lets me distinguish
intervals or dyads (two note things) from pitches (one note things).

I still don't understand what math it is exactly that you are
claiming I lose the use of.

> Just to confuse the issue, when you
> are doing odds a:b means a probability of b/(a+b); if it is 10:1
> against, it has a probability of 1/11. You put down 1, and either
you
> get to keep the 1 and get 10 back if you win, or you lose the 1. If
> the winning probabilty is 1/11, there is no house take and it is a
> zero-sum game.

This certainly _is_ confusing the issue. I can't remember the last
time someone on this list got confused because they though 2:3 was
the odds in a horse race. :-)

> But...is it a number, or isn't it?

In the context of tuning, a:b is the number max(a,b)/(min(a,b)
whenever you need it to be a number. Surely it's not too hard to
accept that it can also be read as a two element version of however
you read a:b:c, a:b:c:d, etc.

> > You still haven't explained the difference between a dyad and an
> > interval.
>
> I regard a dyad as a chord.

And an interval is ...?

I agree there is sometimes a difference in emphasis between the
terms "interval" and "dyad", but this is a rather subtle difference
compared to the difference between one-note-at-a-time things like
pitches (whether relative or absolute) and two-note-at-a-time things
like dyads or intervals.

🔗monz <monz@tonalsoft.com>

hi Gene and Carl,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> 9/7 is not a pitch even then. It becomes a pitch if you
> give the scale in Hertz, but this is rarely done.

see below ...

> > > and intervals have a group structure. If 9/7 does refer in
> > > some sense to a pitch, there must be a reference 1 around
> > somewhere,
> >
> > No. Look at Erv Wilson's CPS scales.
>
> The reference 1 is around, it simply isn't a member of the scale.
> Since we are not talking about cycles per second but combination
> product sets, it *has* to be around.

Carl, i agree with Gene here.

> > > in which case you get a directed interval anyway.
> >
> > You might well get one, but we'd still like to be certain whether
> > you are talking about the interval or the pitch.
>
> If I am talking about a pitch I use Hz.

Gene, i have to argue with you here that this is
not a good idea.

a value expressed in Hz refers to a *frequency*,
not a pitch.

"pitch" is our logarithmic perception of frequency.
therefore, logarithmic values like cents or flus
(dc's? ... how about "dioclu's"?) are referring
to pitch.

-monz

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > > If you want to call them "relative pitches" that's fine by me.
> They
> > > are still quite different beasties to "intervals", at least to a
> > > musician.
> >
> > And the difference is?
>
> Are you serious? If someone is playing a 3/2 pitch, they are playing
> a single note. If they are playing a 2:3 interval they are playing
> _two_ notes.

No one is playing a "3/2 pitch", since it doesn't exist. They could be
playing a note 3/2 times the frequency of some other note, or one
notated as "3/2" in some scale, but in that case, they are not playing
an interval.

> > Why do you care about undirected intervals? What use are they?
>
> You can play them and hear them. How does a 3/4 interval sound
> different to a 4/3 interval?

One goes down and the other up; that is, 1 followed by 3/4 is
different than 1 followed by 4/3.

> I still don't understand what math it is exactly that you are
> claiming I lose the use of.

All the group theory and linear and multilinear algebra. All that
stuff about commas in the kernel, and on and on. In return, I get a
mixed-up, ugly mess. Why do you think people invented negative
integers, when clearly you don't actually have minus 7 apples on the
table? Because it makes a group, and suddenly your life is much easier.

> In the context of tuning, a:b is the number max(a,b)/(min(a,b)
> whenever you need it to be a number.

If it's a number, why not just say a/b? Then you know what the heck it
is, and what you can do with it.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > "pitch" is our logarithmic perception of frequency.
>
> The only person I know who has proposed a logarithmic
> measure of frequency is me, with my business about dollars.
> I probably am not the first to do this but obviously it
> is hardly in common use around here, and not anywhere else
> that I know of. What all the usual logs measure
> is *relative* frequency, or frequency ratios.

hmm ... the dollars idea sounds interesting. i only
vaguely recall it ... can you point to some links?
i'll put it in the Encyclopaedia.

this seems somewhat (but not too much) related to my
idea of using either C-1 Hz (a very low inaudible frequency)
or C-256 (= 2^8) Hz (middle-C) as the reference note.

my preference is for the one based on middle-C, because
then you can call middle-C n^0, and pretty much the
entire range of notes used in music is contained within
2^-4 ... 2^+4 -- a nice symmetrical arrangement centered
on the note most often learned first by beginning musicians.

on the other hand, using C-1 Hz as the reference makes
for a nice neat JI system where the monzos of all ratios
give a quick identification of the frequencies.

you can read my proposal here:
http://tonalsoft.com/index2.htm?../monzo/article/article.htm#reference

> > therefore, logarithmic values like cents or flus
> > (dc's? ... how about "dioclu's"?) are referring
> > to pitch.
>
> These are not measures of absolute pitch, but of
> *relative* pitch, and that is a completely different
> thing.

hmm ... the whole reason Dave, Carl, and Paul (and me)
like to distinguish the notation of the ratios, is to
distinguish between absolute and relative pitch.

we like to use the slash for absolute pitches, and
the colon for relative pitches. thus, 6/5 means the
note which is associated with the ratio 6/5 in some
tuning system, but 5:6 is a ratio which only represents
an interval or relative pitch distance.

-monz

🔗Carl Lumma <ekin@lumma.org>

>> > > and intervals have a group structure. If 9/7 does refer in
>> > > some sense to a pitch, there must be a reference 1 around
>> > > somewhere,
>> >
>> > No. Look at Erv Wilson's CPS scales.
>>
>> The reference 1 is around, it simply isn't a member of the scale.
>> Since we are not talking about cycles per second but combination
>> product sets, it *has* to be around.
>
>Carl, i agree with Gene here.

That twernt me.

Sigh. Another enthralling conversation on the tuning list. This
time about the difference between "intervals" and "dyads". It's
claimed that group theory is relevant to the way we write ratios.
If I were any kind of moderator I'd turn this crap off at the
faucet.

Gene, the colon notation was developed to placate the "American
Gamelan" school of JI, after Harrison and Partch, where pitches
notated with a slash DO denote pitches, with the 1/1 always
around somewhere and explicitly stated in Hz. This argument
predates your tenure here and you've apparently not read Partch
or understood his influence, so why don't you just leave it alone?
Dave, you know Gene always writes with slashes, he isn't going to
change, not now or ever, so why don't you just leave him alone?
The number of members here who were privy to the colon notation
consensus are now in the minority. Even I find borderline cases
where it isn't clear which one to use. And there are friendlier
ways to ask what someone meant than, 'Why don't you use the
correct notation so I have a chance of knowing what you meant?'.

Thanks,

-Carl

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> hi Dave and Gene,
>
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > I agree there is sometimes a difference in emphasis
> > between the terms "interval" and "dyad", but this is
> > a rather subtle difference compared to the difference
> > between one-note-at-a-time things like pitches
> > (whether relative or absolute) and two-note-at-a-time
> > things like dyads or intervals.
>
>
> but if a pitch is an absolute pitch, it's also (as
> Gene said) and interval, because the 1/1 is always
> implied.
>
> i can appreciate why you (Dave), Carl, and Paul
> like to distinguish so carefully between pitches
> and intervals. but whenever any reference is invoked,
> all pitches automatically become intervals.

Yes. But we just had an example where someone seemed to be claiming
that a 45/32 interval could be tuned precisely by ear. This is very
different from claiming that a 45/32 pitch can be tuned by ear. The
former is extremely difficult, the latter is easy if you have a 9/8
(or any of several other pitches) to tune it against.

This sort of misunderstanding has happened before. If it can be
eliminated by the substitution of a single character, in a way that
is consistent with accepted notations for chords of more than two
notes, then why not do so? I don't care so much whether you write
32:45 or 45:32, although I personally prefer the former.

Then we have Paul's notation that uses a semicolon instead of the
colon when we want to make it clear that we are talking about the
tempered size of some interval.

> Dave, could you explain exactly what is the
> "rather subtle difference" between "interval"
> and "dyad"?

It's just what you already wrote.

Sometimes there's no difference (both can mean a two note chord) and
sometimes when we use the term "interval" we are focussing more on
the distance between the two notes (or even the continuum of pitches
between them) than the two notes themselves.

On a lattice diagram, a pitch is represented by a point while an
interval or dyad is represented by a line segment of a particular
length and direction. Sometimes we're thinking more about the two
points at the ends of the line segment (dyad or interval) and
sometimes the line-segment itself (interval).

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Gene, the colon notation was developed to placate the "American
> Gamelan" school of JI, after Harrison and Partch, where pitches
> notated with a slash DO denote pitches, with the 1/1 always
> around somewhere and explicitly stated in Hz. This argument
> predates your tenure here and you've apparently not read Partch
> or understood his influence, so why don't you just leave it alone?

I've read Partch, though some while back. Obviously, I am not going to
toss standard mathematical notation out the door of a kind we all
learned in grade school because of some other school, even if it is
the American Gamelan school. To me, 3/2 is a rational number. It has
many uses; one of these is the use where you set 1 to a particular
pitch value and then 3/2 will denote a pitch with a frequency 3/2
times that pitch. That's fine, but what is NOT fine is telling me that
is what 3/2 *means*. It's not just that I know better, *you* know
better, and so does Dave. Why people want to twist my arm about this
is beyond my understanding. I am going to continue to use standard
language.

> Dave, you know Gene always writes with slashes, he isn't going to
> change, not now or ever, so why don't you just leave him alone?

Good idea. Never try to teach Grandma how to suck eggs.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> hmm ... the whole reason Dave, Carl, and Paul (and me)
> like to distinguish the notation of the ratios, is to
> distinguish between absolute and relative pitch.
>
> we like to use the slash for absolute pitches, and
> the colon for relative pitches. thus, 6/5 means the
> note which is associated with the ratio 6/5 in some
> tuning system, but 5:6 is a ratio which only represents
> an interval or relative pitch distance.

Sorry Monz, but I don't see it that way.

"6/5" (in a .scl file for instance) can be considered as specifying
only a relative pitch. It only specifies an absolute pitch when we
say what the frequency of the 1/1 is. This is what Gene is saying.

But what I'm saying is that since most people only have a crude
sense of absolute pitch we often don't bother to do this, although
there's a common default of 1/1 = middle C. For most of the purposes
of this list, relative pitch _is_ pitch.

The most important distinction I'm trying to make with slash versus
colon is whether it relates to a single note (a pitch, whether
absolute or relative) or a pair of notes (a dyad or interval). i.e.
what it sounds like when you play it.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > Are you serious? If someone is playing a 3/2 pitch, they are
playing
> > a single note. If they are playing a 2:3 interval they are
playing
> > _two_ notes.
>
> No one is playing a "3/2 pitch", since it doesn't exist. They
could be
> playing a note 3/2 times the frequency of some other note, or one
> notated as "3/2" in some scale, but in that case, they are not
playing
> an interval.

If I said to a musician "play a 3/2 pitch" I think they would know
that I meant them to play the pitch notated as 3/2 in the scale
we're using, or the one labelled 3/2 on the keyboard, or the one
whose frequency is 3/2 relative to our previously agreed 1/1 = x Hz.
In fact I wouldn't even need to include the word "pitch". I could
just say "play a 3/2".

And if they didn't already know what scale or key or 1/1 reference,
they would ask. The point is, if they played anything at all, they
would play a single pitch.

And yes, I totally agree that they would not be playing an interval.

So now I'm _really_ confused about what you are saying. I thought
from what you said previously, that you were making the apparently
ridiculous claim that even though they are only playing a single
note, they are _really_ playing an interval (because the 1/1
is "around somewhere").

In the case of an interval, you wouldn't normally just say "play a
3/2 interval" because you would normally want a particular one, so
you'd say "play a 3/2 interval above the 5/4 pitch" or some such.
Once again, you wouldn't need to bother with the terms "interval"
or "pitch" here because it's clear that "play a 3/2 above the 5/4"
means to play an interval (two pitches) and "play a 3/2" means to
play a single pitch.

If however you really _did_ want the performer to play any old 3/2
interval (leaving the decision up to them as to which particular
member of the class to play), then it would be no use just
saying "play a 3/2" because they would assume you only wanted that
single pitch.

But what would be unusual in actual performance is quite usual on
this list. We are usually talking about intervals in the abstract.
i.e. interval _classes_, not _particular_ pairs of pitches. We are
often considering the tuning of _all_ the 3/2 intervals in a scale,
not the individual intervals such as the 3/2 interval above the 1/1
pitch and the 3/2 interval above the 5/4 pitch etc.

So if we just refer to "the tuning of the 3/2 in this scale" it is
unclear whether we are talking about the single (relative) pitch
notated as 3/2, or the set of all 3/2 intervals in the scale.

In the written form, it is a simple matter to substitute a colon for
the slash to make it clear when we are talking about an interval
class.

> > > Why do you care about undirected intervals? What use are they?
> >
> > You can play them and hear them. How does a 3/4 interval sound
> > different to a 4/3 interval?
>
> One goes down and the other up; that is, 1 followed by 3/4 is
> different than 1 followed by 4/3.

So you're saying that, to you, 3/4 is that particular perfect fourth
above the pitch notated as 3/4 (or below the 1/1), while 4/3 is that
particular perfect fourth above the pitch notated as 1/1?

If so, then how would you refer (numerically) to the particular
perfect fourth above the pitch notated as say 3/2?

How would you refer (numerically) to perfect fourths in the
abstract. e.g. the class of all perfect fourths in the infinite JI
lattice, which is what we most often mean on this list when we
write "3:4" or "4/3 interval".

Maybe in your terms, what I'm saying is use slash for
specific "intervals" but use colon for the interval class.

> > I still don't understand what math it is exactly that you are
> > claiming I lose the use of.
>
> All the group theory and linear and multilinear algebra. All that
> stuff about commas in the kernel, and on and on. In return, I get a
> mixed-up, ugly mess. Why do you think people invented negative
> integers, when clearly you don't actually have minus 7 apples on
the
> table? Because it makes a group, and suddenly your life is much
easier.
>

As I said, just because interval classes happen to be representable
by numbers >= 1 doesn't mean you have to stop using numbers less
than 1 in your calculations. That would indeed be some kind of
mathematical masochism.

> > In the context of tuning, a:b is the number max(a,b)/(min(a,b)
> > whenever you need it to be a number.
>
> If it's a number, why not just say a/b? Then you know what the
heck it
> is, and what you can do with it.

Because numbers can mean many different things. That's the beauty of
them. But in an applied field such as tuning, when you've finished
your calculations with them, you have to tell us what real-world
objects they are representing. You have to put headings on your
columns of numbers, or give units, or indicate this in some other
way (such as using a colon to indicate an interval class).

It seems that you find it extremely difficult to empathise with your
readers. That's OK. We can't all posess all skills. I just fervently
wish you could understand that you have this blind spot and would
believe us when we present you with tried and tested ways to make
your meaning clearer.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Sigh. Another enthralling conversation on the tuning list. This
> time about the difference between "intervals" and "dyads". It's
> claimed that group theory is relevant to the way we write ratios.
> If I were any kind of moderator I'd turn this crap off at the
> faucet.

And you probably wouldn't insult the posters.

> Dave, you know Gene always writes with slashes, he isn't going to
> change, not now or ever, so why don't you just leave him alone?

I actually didn't know this. I actually thought there was some
chance he might understand why we do it. But I see now that I am
wasting my time.

> The number of members here who were privy to the colon notation
> consensus are now in the minority.

Maybe so, but I suspect the number who think it is a bad idea are in
a very serious minority.

> Even I find borderline cases
> where it isn't clear which one to use.

That's OK. The point is, there are many cases where it is clear.

> And there are friendlier
> ways to ask what someone meant than, 'Why don't you use the
> correct notation so I have a chance of knowing what you meant?'.

You're right. It's nice that you are concerned for Gene's feelings.

I have to tell you, it is incredibly hard for me to watch silently
while everything we've worked for over the years, in regard to
facilitating communication on this list, is buried under a deluge of
pure-math diarrhoea. If it had stayed on tuning-math it wouldn't be
so bad.

Since I don't know of any easy way to install a Gene filter on my
list reading (including blocking quotes inside other people's
posts), I have no alternative but to stop reading the tuning lists.

You're right. I have better things to do with my life than bang my
head against a wall.

-- Dave Keenan

🔗Carl Lumma <ekin@lumma.org>

>> Sigh. Another enthralling conversation on the tuning list. This
>> time about the difference between "intervals" and "dyads". It's
>> claimed that group theory is relevant to the way we write ratios.
>> If I were any kind of moderator I'd turn this crap off at the
>> faucet.
>
>And you probably wouldn't insult the posters.

Uh-huh.

>> Dave, you know Gene always writes with slashes, he isn't going to
>> change, not now or ever, so why don't you just leave him alone?
>
>I actually didn't know this. I actually thought there was some
>chance he might understand why we do it. But I see now that I am
>wasting my time.

Haven't we both tried to explain it to him in the past?

>> The number of members here who were privy to the colon notation
>> consensus are now in the minority.
>
>Maybe so, but I suspect the number who think it is a bad idea are
>in a very serious minority.

I'm beginning to change my mind about it, for reasons that aren't
worth discussing.

>> And there are friendlier
>> ways to ask what someone meant than, 'Why don't you use the
>> correct notation so I have a chance of knowing what you meant?'.
>
>You're right. It's nice that you are concerned for Gene's feelings.

In fact I reamed you both equally.

>I have to tell you, it is incredibly hard for me to watch silently
>while everything we've worked for over the years, in regard to
>facilitating communication on this list, is buried under a deluge
>of pure-math diarrhoea. If it had stayed on tuning-math it wouldn't
>be so bad.
>
>Since I don't know of any easy way to install a Gene filter on my
>list reading (including blocking quotes inside other people's
>posts), I have no alternative but to stop reading the tuning lists.

You would be the fourth person in recent memory to do so, for
those keeping score at home. Gene, would you be willing to look
at your patterns to see what could be upsetting people?

>You're right. I have better things to do with my life than bang
>my head against a wall.

You have better things to do than engage in discussion topics
that are proven wastes of time, like whether to notate ratios
with a colon or a slash. Egad, I can't believe how ludicrous
it sounds!

The problem with this list is the endless recycling of arguments
over trivialities. There are still scholarly expositions,
friendly news items, sincere questions (one of which you recently
fielded as no one else could), and links to incredible music (like
Aaron's Bull rendering), but I fear the regulars, including myself,
constantly threaten to crush it all with incessant arguing over
minutia, drama-filled walk-outs followed by sheepish re-entries...
Why not just take a step back?

This has been going on for a long time. As Daniel Wolf said,
"Far too often, I'd practically sweat blood over a message to
make sure that it was as accurate and temperate as I could
... only to find out that my efforts were never enough ... no
matter what I wrote, a Thad or an Edgar would scream in
disagreement."

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> >Since I don't know of any easy way to install a Gene filter on my
> >list reading (including blocking quotes inside other people's
> >posts), I have no alternative but to stop reading the tuning lists.
>
> You would be the fourth person in recent memory to do so, for
> those keeping score at home. Gene, would you be willing to look
> at your patterns to see what could be upsetting people?

Do you think "do things my way or I leave" is polite, helpful or
likely to facilitate communication? Why am I the bad guy here, when
Dave is resorting to this kind of thing?

> The problem with this list is the endless recycling of arguments
> over trivialities.

Which we could reduce if we would adopt a live-and-let-live policy. We
might, crazy as it sounds, allow people to use and discuss pure JI,
regular temperaments, irregular tunings, or all of the above. We might
allow people to use whatever notations they find suitable. I could
stop frowning when people say "lattice" when I don't see one. Birds
could sing, bands could play.

> Why not just take a step back?

Why indeedy?

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>>> Since I don't know of any easy way to install a Gene filter on
>>> my list reading (including blocking quotes inside other people's
>>> posts), I have no alternative but to stop reading the tuning
>>> lists.
>>
>> You would be the fourth person in recent memory to do so, for
>> those keeping score at home. Gene, would you be willing to look
>> at your patterns to see what could be upsetting people?
>
>Do you think "do things my way or I leave" is polite, helpful or
>likely to facilitate communication? Why am I the bad guy here,
>when Dave is resorting to this kind of thing?

Gene, you're not a bad guy here, drop the Christ complex already.
Four people have left the list in the last six months citing you
as a primary reason. Probably scores have left on account of me
in the past umpteen years. In fact I'm reminded how much of an
Edgar I was to Daniel Wolf. I'm going to go back and try to find
and read 15 messages containing bitter arguments I've had. Are
you not willing to do the same?

Dave, you get so convinced you're right, it means anybody who
disagrees has to be that much wrong. Why do that to people?
Can't imagine anybody disagreeing with you about ratio notation?
Try a little harder. Or try weighing the value of the best case:
In a series of 20 e-mails, you argue so convincingly for colon
notation that everyone adopts it. Was it worth it? Maybe not!
It isn't *that* important how people write ratios. Replacing
those 20 mails, possibly intimidating to the 300 lurkers here,
with one friendly answer to one question could do more for the
health of discourse here than standardized ratio notation. Or
write a position paper and be done with it -- keep the link
somewhere. Or put it on your web page so it can't be dissected.

Part of the problem may be quote-and-reply. It's like, a
disease, man.

That's it: I'm lighting a J. And next time, rather than try
to play psychotherapist, I'm heading over to Wikipedia.

-Carl

🔗Carl Lumma <ekin@lumma.org>

>> I still don't know what 3:2 is supposed to be; it seems to mean
>> various things; sometimes a dyad, sometimes an interval. I would
>> prefer for it to mean what it normally means, which is a proportion,
>
>That should be "ratio". However, I still find it astonishing to be
>treated with such disdain for not using a terminology which does not
>seem to be standard and for which contradictory explanations have been
>provided. Does anyone actually know what this "officially" means in
>tuning connections? Becuase I seriously doubt anyone does.

It's a ratio between two pitches, descriptive of either a melodic
or harmonic interval.

In the colon notation, 3/2 is a NOT an interval or a dyad but a pitch,
with concert pitch (1/1=Hz.) specified elsewhere.

This came about because some here were using 3/2 to refer to an
interval, while others would take it to be a pitch, causing massive
confusion. An extreme case was Paul, for two years, reading me as
talking about strict JI when I had always meant relative or "adaptive"
JI. It was a particularly painful waste, and it occurs to me that
you might not even be onboard with terms like "strict JI" and
"adaptive JI", that were so important to the development (but I
recognize, not necessarily the future) of discourse here.

As monz so correctly points out, there's really very little difference
*in practice* between these two interpretations, and in retrospect
the system seems a rather artificial way to communicate with strict
JI folks. But to put it in perspective, the very notion of thinking
of JI ratios as intervals may be fairly new. At least that's how it
plays in Boomsliter and Creel: as a major breakthrough. The idea of
writing 3/2 to mean a 3/2 above anything other than 1/1 was apparently
new to their readers.

-Carl

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> It's a ratio between two pitches, descriptive of either a melodic
> or harmonic interval.

I've been told it's a ratio between pitches, that it is simply a ratio
(and that, incidentally, is the usual meaning) and that it is a
two-note chord.

> In the colon notation, 3/2 is a NOT an interval or a dyad but a pitch,
> with concert pitch (1/1=Hz.) specified elsewhere.

And if I happen to want to use the usual notation for rational numbers
as the notation for rational numbers, now I can't? That's hardly a
sensible proposal.

> This came about because some here were using 3/2 to refer to an
> interval, while others would take it to be a pitch, causing massive
> confusion.

I don't suppose anyone thought about using it as a number?

An extreme case was Paul, for two years, reading me as
> talking about strict JI when I had always meant relative or "adaptive"
> JI. It was a particularly painful waste, and it occurs to me that
> you might not even be onboard with terms like "strict JI" and
> "adaptive JI", that were so important to the development (but I
> recognize, not necessarily the future) of discourse here.

"Adaptive JI" I associate with such things as deLaubenfels, and
"strict JI" as rational intonation.

> As monz so correctly points out, there's really very little difference
> *in practice* between these two interpretations, and in retrospect
> the system seems a rather artificial way to communicate with strict
> JI folks. But to put it in perspective, the very notion of thinking
> of JI ratios as intervals may be fairly new. At least that's how it
> plays in Boomsliter and Creel: as a major breakthrough. The idea of
> writing 3/2 to mean a 3/2 above anything other than 1/1 was apparently
> new to their readers.

This is so bizarre I really find it difficult to believe. Anyway, I've
seen no reason yet to toss out the standard interpretations, which is
that 3/2 is a number and 3:2 or 2:3 a ratio, to be understood in an
old-fashioned, Greek mathematics perspective. Since I regard the
distinction between 3:2 and 3/2 to be inherently confusing, I prefer
not to use 3:2 unless it happens to make things convenient somehow.
That might happen if you *want* to invoke the old-fashioned, Greek
mathematics point of view, if you want to show a series of terms such
as 3:4:5, or if you want to goof around with stuff like 2:3::4:6 for
some some obscure reason. It could even be you have some explicit
meaning, usable in musical contexts, for 2:3 or 3:2, but if so you'd
better be sure you can explain what it is without contradicting
yourself in the next paragraph, becuase you've just given a definition.

🔗Aaron K. Johnson <akjmicro@comcast.net>

🔗Carl Lumma <ekin@lumma.org>

>> It's a ratio between two pitches, descriptive of either a melodic
>> or harmonic interval.
>
>I've been told it's a ratio between pitches, that it is simply a
>ratio (and that, incidentally, is the usual meaning) and that it
>is a two-note chord.

A "harmonic interval" is a dyad/chord in music theory parlance, yo.

>> In the colon notation, 3/2 is a NOT an interval or a dyad but a
>> pitch, with concert pitch (1/1=Hz.) specified elsewhere.
>
>And if I happen to want to use the usual notation for rational numbers
>as the notation for rational numbers, now I can't? That's hardly a
>sensible proposal.

Ok. I see this point of view. There shouldn't be any problems if
authors are sensitive to the issues when building context in their
messages.

>> This came about because some here were using 3/2 to refer to an
>> interval, while others would take it to be a pitch, causing massive
>> confusion.
>
>I don't suppose anyone thought about using it as a number?

Does anyone really know what a number is? :)

> An extreme case was Paul, for two years, reading me as
>> talking about strict JI when I had always meant relative or
>> "adaptive" JI. It was a particularly painful waste, and it
>> occurs to me that you might not even be onboard with terms
>> like "strict JI" and "adaptive JI", that were so important
>> to the development (but I recognize, not necessarily the
>> future) of discourse here.
>
>"Adaptive JI" I associate with such things as deLaubenfels, and
>"strict JI" as rational intonation.

This is somewhat true. John's latest software actually does
*adaptive temperament*. Meanwhile, I'm blanking on the precise
meaning of "rational intonation". The tonalsoft entry isn't
particularly enlightening. I remember it being a term Dave
cooked up to placate those who cannot accept any amount of
approximation, or even the idea of approximation. Strict JI
simply means we tend to keep a fixed key center, or at least
think in terms of a fixed key center. As opposed to thinking
of JI as a vertical (in the score) concept, as we seem to do
most often on tuning-math, strict JI extends it to a
melodic (horizontal in the score) thing.

>> As monz so correctly points out, there's really very little difference
>> *in practice* between these two interpretations, and in retrospect
>> the system seems a rather artificial way to communicate with strict
>> JI folks. But to put it in perspective, the very notion of thinking
>> of JI ratios as intervals may be fairly new. At least that's how it
>> plays in Boomsliter and Creel: as a major breakthrough. The idea of
>> writing 3/2 to mean a 3/2 above anything other than 1/1 was apparently
>> new to their readers.
>
>This is so bizarre I really find it difficult to believe. Anyway, I've
>seen no reason yet to toss out the standard interpretations, which is
>that 3/2 is a number and 3:2 or 2:3 a ratio, to be understood in an
>old-fashioned, Greek mathematics perspective.

If we do so, does it not lead to the distinction advocated by Dave?

>Since I regard the
>distinction between 3:2 and 3/2 to be inherently confusing, I prefer
>not to use 3:2 unless it happens to make things convenient somehow.
>That might happen if you *want* to invoke the old-fashioned, Greek
>mathematics point of view, if you want to show a series of terms such
>as 3:4:5, or if you want to goof around with stuff like 2:3::4:6 for
>some some obscure reason. It could even be you have some explicit
>meaning, usable in musical contexts, for 2:3 or 3:2, but if so you'd
>better be sure you can explain what it is without contradicting
>yourself in the next paragraph, becuase you've just given a definition.

I see no defects whatever in the definitions I've given (above).
Are they clear (or not) yet?

-Carl

🔗Aaron K. Johnson <akjmicro@comcast.net>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@tonalsoft.com>

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

> ... Meanwhile, I'm blanking on the precise
> meaning of "rational intonation". The tonalsoft entry isn't
> particularly enlightening. I remember it being a term Dave
> cooked up to placate those who cannot accept any amount of
> approximation, or even the idea of approximation. Strict JI
> simply means we tend to keep a fixed key center, or at least
> think in terms of a fixed key center. As opposed to thinking
> of JI as a vertical (in the score) concept, as we seem to do
> most often on tuning-math, strict JI extends it to a
> melodic (horizontal in the score) thing.

the huge debate which culminated in the decision of
some of us to distinguish "JI" as a subset of the
larger "rational intonations", essentially boils
down to this:

- if your music *sounds* like it's beatless, "pure",
and based on small-integer ratios, then you call it "JI";

- if it uses ratios but the numbers in each term of
the ratio are large, or the prime-factors are very high,
or it *doesn't sound* beatless, "pure", etc., then
you call it "rational intonation" (RI);

- if it doesn't use ratios at all, then it's not RI or JI,
and most likely is a temperament of some sort.

the textbook example of RI is the Hammond Organ tuning,
which *is* rational (because of the gearing system in the
tone-wheels), but is meant to *sound* like 12-edo -- an
entirely irrational tuning.

you're discussing "strict-JI" which is to be opposed
to adaptive-JI. that's a whole different thing.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_55795.html#55938

> >> It's a ratio between two pitches, descriptive of either a melodic
> >> or harmonic interval.
> >
> >I've been told it's a ratio between pitches, that it is simply a
> >ratio (and that, incidentally, is the usual meaning) and that it
> >is a two-note chord.
>
> A "harmonic interval" is a dyad/chord in music theory parlance, yo.
>
> >> In the colon notation, 3/2 is a NOT an interval or a dyad but a
> >> pitch, with concert pitch (1/1=Hz.) specified elsewhere.
> >
> >And if I happen to want to use the usual notation for rational
numbers
> >as the notation for rational numbers, now I can't? That's hardly a
> >sensible proposal.
>
> Ok. I see this point of view. There shouldn't be any problems if
> authors are sensitive to the issues when building context in their
> messages.
>
> >> This came about because some here were using 3/2 to refer to an
> >> interval, while others would take it to be a pitch, causing
massive
> >> confusion.
> >
> >I don't suppose anyone thought about using it as a number?
>
> Does anyone really know what a number is? :)
>
> > An extreme case was Paul, for two years, reading me as
> >> talking about strict JI when I had always meant relative or
> >> "adaptive" JI. It was a particularly painful waste, and it
> >> occurs to me that you might not even be onboard with terms
> >> like "strict JI" and "adaptive JI", that were so important
> >> to the development (but I recognize, not necessarily the
> >> future) of discourse here.
> >
> >"Adaptive JI" I associate with such things as deLaubenfels, and
> >"strict JI" as rational intonation.
>
> This is somewhat true. John's latest software actually does
> *adaptive temperament*. Meanwhile, I'm blanking on the precise
> meaning of "rational intonation". The tonalsoft entry isn't
> particularly enlightening. I remember it being a term Dave
> cooked up to placate those who cannot accept any amount of
> approximation, or even the idea of approximation. Strict JI
> simply means we tend to keep a fixed key center, or at least
> think in terms of a fixed key center. As opposed to thinking
> of JI as a vertical (in the score) concept, as we seem to do
> most often on tuning-math, strict JI extends it to a
> melodic (horizontal in the score) thing.
>

***My understanding is that Dave Keenan invoked a distinction based
on an *audible* differentiation.... In other words, if a ratio got
*really large* it was no longer JI, but would be considered RI. I
bought this interpretation is being very reasonable, but I'm sure
there could be others to disagree with it...

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> However, I don't see why we need a notation
> for pitches other than Hertz, and moreover a/b is not, in fact, being
> used as a notation for a pitch so far as I can make out. If it was,
> someone could tell me what pitch 1/1 represents, and it doesn't seem
> anyone can.

From what I've been told, it is never used as a notation for a pitch,
but as a notation for pitches relative to 1/1, but 1/1 is itself
assignable to any pitch you want to use it to standardize. In this
sense, I've often used a/b to mean a pitch myself, which makes we
wonder why I am angrily denounced; presumably because I don't assume
it has to mean a relative pitch, since it is the standard notation for
a rational number?

We are getting into the murky distiction between a rational number,
used sometimes to denote a relative pitch, at other times a directed
interval, and in other contexts anything else a rational number can
mean (eg 1/4-comma meantone) and the notation for rational numbers
being hijacked and made unavailable for other purposes.

It's just a fact that the rational numbers act on themselves. If you
assign a base frequency to 1/1 they can be regarded as a space of
musical pitches a la David Lewin; but then they also act on
themselves, since you can multiply one rational number by another. In
math jargon you get the positive rational numbers as a "homogenous
space" or "symmetric space", the positive rational numbers
representing pitches; on this "space" of pitches the positive rational
numbers also act as transpositions; in math jargon again, the group of
positive rational numbers acts on itself as a symmetric space.
Moreover, between any ordered pair of pitches in the musical space,
you also get a rational number which is their ratio, which denotes the
interval between them, and if that isn't enough, unordered pairs (set
pairs) of pitches also have a ratio, only now it must be positive. You
cannot clarify these issues by getting mad at people and denouncing
them for heresy; this is just the way it is.

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

/tuning/topicId_55795.html#55963

***Hi Gene,

Well, I'm a little confused here, since this seems to imply that you
are endorsing a/b to describe a pitch as long as the 1/1 is defined.
I think generally it is in the Partch and other contexts. I admit,
it seemed a little peculiar when I saw these kind of ratios on
Partch's Chromelodeon and in countless other JI scores of many
composers (including Lou Harrison...)

In any case, I don't see why everybody needs to get so "huffy" about
all of this: this list is a good place to discuss these issues and
modify the terminology in current coinage as necessary.

After all, I kicked and screamed about not using the Ezra Sims
notation anymore (and for good reason since there is a current
performance tradition... although not a huge one...) but finally
decided to go with Wilson-Sagittal, simply because it was better...
Hopefully others might see that too and follow along, but it has to
start someplace, I now agree...

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> Well, I'm a little confused here, since this seems to imply that you
> are endorsing a/b to describe a pitch as long as the 1/1 is defined.

I;m endorsing a/b for any purpose for which a rational number is required.

> I think generally it is in the Partch and other contexts. I admit,
> it seemed a little peculiar when I saw these kind of ratios on
> Partch's Chromelodeon and in countless other JI scores of many
> composers (including Lou Harrison...)

I've written many scores myself where I wrote down a fraction and
intended it to be interpreted (when some squack box got hold of it) as
a pitch, but of course I knew I'd need to give a base frequency.

> After all, I kicked and screamed about not using the Ezra Sims
> notation anymore (and for good reason since there is a current
> performance tradition... although not a huge one...) but finally
> decided to go with Wilson-Sagittal, simply because it was better...
> Hopefully others might see that too and follow along, but it has to
> start someplace, I now agree...

My feeling is that a live-and-let-live philosophy about such issues is
appropriate.