large integer ratios

hello all,

it is my impression - and some will disagree - that small integer ratios are
"natural", and appropriately precise ET schemes approximate them well enough
for most purposes. this begs the question of when one would purposefully
employ other intervals (large integer ratios, i suppose). meaningful music
can be composed with, say, the first dozen small integer ratios, but some
people like to use more and more obscure ones. i have seen mention in here
of ratios in the tens (e.g., 40/27); does anyone use larger values (e.g.,
400/271)?

how to determine the practical upper limit of integers for forming useful
ratios? perhaps some citation of study on the lower limit of perceptible
differences or beating would help.

best,

aaron brick.
- | | | | | | | | | | -
- -
- -
- aaron brick -
- aa@lithic.org -
- -
- -
- | | | | | | | | | | -

Hello, there, Aaron Brick and everyone.

You've raised an often exciting question:

> it is my impression - and some will disagree - that small integer
> ratios are "natural", and appropriately precise ET schemes
> approximate them well enough for most purposes. this begs the
> question of when one would purposefully employ other intervals
> (large integer ratios, i suppose).

Actually, at least in my experience, large ratios and small ones are
both quite natural in rational tuning systems, with large ratios often
arising from combinations or differences of small ones.

This is true not only in "avant-garde" 21st-century tunings, but in
many traditions of rational intonation (RI) ranging from ancient
Greece and China to the medieval Near East and Europe, for example.

Consider, for example, a traditional tuning in a chain of pure fifths
or fourths. Very quickly we get intervals with ratios at once very
large and very useful, like regular diatonic semitones and major
sevenths at 256:243 (~90.22 cents) and 243:128 (~1109.78 cents). This
is the system often known as Pythagorean tuning, used or approximated
in a range of world musics.

There's a standing discussion or debate about just which ratios or
sonorities are "distinctive" or "primary colors," and which tend to be
more "secondary colors" heard more or less to resemble simpler ratios.

However, if one accepts this distinction, then both types of intervals
are still very useful in my opinion, and the basic patterns of
intonation often product both even if one is not specifically seeking
this goal.

> meaningful music can be composed with, say, the first dozen small
> integer ratios, but some people like to use more and more obscure
> ones. i have seen mention in here of ratios in the tens (e.g.,
> 40/27); does anyone use larger values (e.g., 400/271)? how to
> determine the practical upper limit of integers for forming useful
> ratios? perhaps some citation of study on the lower limit of
> perceptible differences or beating would help.

In my approach to rational intonation, there's no limit to the size of
a useful ratio, although one could argue that often very large ratios
draw some of their usefulness from the fact that they resemble smaller
ones.

For example, one tuning system I use has two 12-note Pythagorean
chains at three Pythagorean commas apart. Each comma has a ratio of
531441:524288, or about 23.46 cents, so that this "tricomma" is equal
to 150094635296999121:144115188075855872, or about 70.38 cents.

This interval makes a very nice cadential semitone. It's very close,
as it happens, to the simpler ratio of 25:24 (~70.67 cents).

The usual Pythagorean major third at 81:64 (~407.82 cents) less this
interval produces a very useful "semi-neutral" third at a ratio of
2251799813685248:1853020188851841 or about 337.44 cents, quite close
to 17:14 (~336.13 cents), and even closer to the somewhat larger ratio
of 243:200 (~337.15 cents).

Of course, one could argue that any of these semi-neutral or
supraminor thirds is pretty much musically equivalent, but I find it
very convenient to have such "synonyms" available.

Views on these questions vary: in saying that I find a large or
"obscure" ratio in a tuning system often a special ornament, I'm
speaking for myself, of course.

Most appreciatively,

Margo Schulter
mschulter@value.net

margo,

thanks for your reply! i only just found your mail from well over a year
ago. :/ in any case, i now rather agree with you about the silliness of
trying avoid "useless" ratios. the question which now comes to mind for me
regards interval traversal. as you observed, repeated jumps make for larger
and larger integer ratios; and ET steps (requiring an irrational component)
cannot even be written as ratios. if we want to represent the resultant
interval as a ratio, at what precision should these values be transcribed
into fractions?

my brain is spinning....

best,
aaron.

so said MSCHULTER@VALUE.NET in 4.2K bytes on Fri, Jul 26, 2002:

>
> Hello, there, Aaron Brick and everyone.
>
> You've raised an often exciting question:
>
> > it is my impression - and some will disagree - that small integer
> > ratios are "natural", and appropriately precise ET schemes
> > approximate them well enough for most purposes. this begs the
> > question of when one would purposefully employ other intervals
> > (large integer ratios, i suppose).
>
> Actually, at least in my experience, large ratios and small ones are
> both quite natural in rational tuning systems, with large ratios often
> arising from combinations or differences of small ones.
>
> This is true not only in "avant-garde" 21st-century tunings, but in
> many traditions of rational intonation (RI) ranging from ancient
> Greece and China to the medieval Near East and Europe, for example.
>
> Consider, for example, a traditional tuning in a chain of pure fifths
> or fourths. Very quickly we get intervals with ratios at once very
> large and very useful, like regular diatonic semitones and major
> sevenths at 256:243 (~90.22 cents) and 243:128 (~1109.78 cents). This
> is the system often known as Pythagorean tuning, used or approximated
> in a range of world musics.
>
> There's a standing discussion or debate about just which ratios or
> sonorities are "distinctive" or "primary colors," and which tend to be
> more "secondary colors" heard more or less to resemble simpler ratios.
>
> However, if one accepts this distinction, then both types of intervals
> are still very useful in my opinion, and the basic patterns of
> intonation often product both even if one is not specifically seeking
> this goal.
>
> > meaningful music can be composed with, say, the first dozen small
> > integer ratios, but some people like to use more and more obscure
> > ones. i have seen mention in here of ratios in the tens (e.g.,
> > 40/27); does anyone use larger values (e.g., 400/271)? how to
> > determine the practical upper limit of integers for forming useful
> > ratios? perhaps some citation of study on the lower limit of
> > perceptible differences or beating would help.
>
> In my approach to rational intonation, there's no limit to the size of
> a useful ratio, although one could argue that often very large ratios
> draw some of their usefulness from the fact that they resemble smaller
> ones.
>
> For example, one tuning system I use has two 12-note Pythagorean
> chains at three Pythagorean commas apart. Each comma has a ratio of
> 531441:524288, or about 23.46 cents, so that this "tricomma" is equal
> to 150094635296999121:144115188075855872, or about 70.38 cents.
>
> This interval makes a very nice cadential semitone. It's very close,
> as it happens, to the simpler ratio of 25:24 (~70.67 cents).
>
> The usual Pythagorean major third at 81:64 (~407.82 cents) less this
> interval produces a very useful "semi-neutral" third at a ratio of
> 2251799813685248:1853020188851841 or about 337.44 cents, quite close
> to 17:14 (~336.13 cents), and even closer to the somewhat larger ratio
> of 243:200 (~337.15 cents).
>
> Of course, one could argue that any of these semi-neutral or
> supraminor thirds is pretty much musically equivalent, but I find it
> very convenient to have such "synonyms" available.
>
> Views on these questions vary: in saying that I find a large or
> "obscure" ratio in a tuning system often a special ornament, I'm
> speaking for myself, of course.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@value.net
>
>
>
>
> You do not need web access to participate. You may subscribe through
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- | | | | | | | | | | -
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- aaron brick -
- aa@lithic.org -
- -
- -
- | | | | | | | | | | -

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:
> margo,
>
> thanks for your reply! i only just found your mail from well over a
year
> ago. :/ in any case, i now rather agree with you about the
silliness of
> trying avoid "useless" ratios. the question which now comes to mind
for me
> regards interval traversal. as you observed, repeated jumps make
for larger
> and larger integer ratios; and ET steps (requiring an irrational
component)
> cannot even be written as ratios. if we want to represent the
resultant
> interval as a ratio, at what precision should these values be
transcribed
> into fractions?
>
> my brain is spinning....
>
> best,
> aaron.

Hi Aaron,

Welcome back after all this time, and I'm pleased to see you joining
Aaron Johnson, Aaron Wolf, and Aaron Hunt among the recent
contributors to this list.

As I'm sure Monz will explain, it's often easier to just write out
the exponents of the primes making up a ratio, instead of the full
ratio itself. For example, the 'atom of Kirnberger', which is a tiny
interval of about 0.015 cent obtained from a chain of lots of simple-
ratio intervals when tuning up Kirnberger's approximation to 12-tone
equal temperament, is, as a ratio,

2923003274661805836407369665432566039311865085952
-------------------------------------------------
2922977339492680612451840826835216578535400390625

Who wants to look at that? Instead, one can write the ratio as

2^161
-----------
3^84 * 5^12

or as

2^161 * 3^-84 * 5^-12

or, as a "monzo", as [161 -84 -12].

A lot simpler, and no precision is lost!

Best,
Paul

evening paul! i've seen some other aarons in my mail log, it did kind of get
my attention. :)

i agree that reducing the ratio to the product of exponents is the best way
to represent cumulative intervals, but i don't think you can express ET
steps in this notation, on account of their irrationality. 12ET's
"half-note" is the 12th root of 2 iirc and it cannot be expressed as a
fraction. please elaborate!

best,

aaron.

> Hi Aaron,
>
> Welcome back after all this time, and I'm pleased to see you joining
> Aaron Johnson, Aaron Wolf, and Aaron Hunt among the recent
> contributors to this list.
>
> As I'm sure Monz will explain, it's often easier to just write out
> the exponents of the primes making up a ratio, instead of the full
> ratio itself. For example, the 'atom of Kirnberger', which is a tiny
> interval of about 0.015 cent obtained from a chain of lots of simple-
> ratio intervals when tuning up Kirnberger's approximation to 12-tone
> equal temperament, is, as a ratio,
>
> 2923003274661805836407369665432566039311865085952
> -------------------------------------------------
> 2922977339492680612451840826835216578535400390625
>
> Who wants to look at that? Instead, one can write the ratio as
>
> 2^161
> -----------
> 3^84 * 5^12
>
> or as
>
> 2^161 * 3^-84 * 5^-12
>
> or, as a "monzo", as [161 -84 -12].
>
> A lot simpler, and no precision is lost!
>
> Best,
> Paul
>
>
> You do not need web access to participate. You may subscribe through
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- -
- aaron brick -
- aa@lithic.org -
- -
- -
- | | | | | | | | | | -

hi Aaron,

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:

> evening paul! i've seen some other aarons in my mail log,
> it did kind of get my attention. :)

yes indeed, this list has been very "Aaron-heavy" of late.

> i agree that reducing the ratio to the product of exponents
> is the best way to represent cumulative intervals, but i
> don't think you can express ET steps in this notation,
> on account of their irrationality. 12ET's "half-note" is
> the 12th root of 2 iirc and it cannot be expressed as a
> fraction. please elaborate!

well, many others here don't see the value of using fractional
exponents in a "monzo" prime-factor notation, but i use them
all the time.

the 12ET "half-tone" (which i would call "12edo Semitone")
is simply notated as 2^(1/12).

since the Pythagorean comma is the difference between
12 "5ths" and 7 "8ves", it can be notated as
((2^-12)*(3^12)) / (2^7) = (2^-19)*(3^12).

taking the 12th root of this interval gives the amount
of tempering of each 12edo "5th", and has fractional
values in the exponents of the prime-factors:
(2^(-19/12))*(3^(12/12)).

subtracting this amount from the 3:2 ratio ("perfect-5th")
gives (2^((12-(-19))/12))*(3^((12-12)/12)) = 2^(7/12),
which is precisely the value of the 12edo "5th".

many irrational intervals can still be prime-factored
by using fractional exponents for the primes, and for
example, all "fraction-of-a-comma" meantones can be
expressed with mathematical exactitude in this manner.

for some examples, see my webpages:

http://tonalsoft.com/enc/12-eq.htm
http://tonalsoft.com/enc/19edo.htm
http://tonalsoft.com/enc/1-4cmt.htm
http://tonalsoft.com/enc/2-7cmt.htm

and two of the most interesting examples of using
prime-factor notation with strange (i.e., non-integer)
exponents:

http://tonalsoft.com/enc/lucy.htm
http://tonalsoft.com/enc/golden.htm

-monz

hello monz,

> well, many others here don't see the value of using fractional
> exponents in a "monzo" prime-factor notation, but i use them
> all the time.

ahh! this is a sensible solution. fractional exponents didn't occur to me.
nonetheless, the less similar the intervals you combine, the hairier the
equation. in this sense i am inclined to use decimals to some precision, so
that all notes can be described with the same amount of information. (i am
inclined to represent them as base-2 logs as well, so that we can see that
440 Hz (2^8.7814) is in the eighth octave, and that an added interval of "1"
means exactly one octave.)

> http://tonalsoft.com/enc/12-eq.htm

thanks for the links, i found this one illuminating.

> and two of the most interesting examples of using
> prime-factor notation with strange (i.e., non-integer)
> exponents:
>
> http://tonalsoft.com/enc/lucy.htm
> http://tonalsoft.com/enc/golden.htm

strange indeed. :) i haven't yet been convinced of any reason to use
irrational ratios other than ET stepping in Hz or precise jumping in a log
scale.

aaron.

_ _ _ _ _ _ _ _
/ \
- -
- aaron brick -
- aa@lithic.org -
- -
\ _ _ _ _ _ _ _ _ /

hi Aaron,

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:

> hello monz,
>
> > well, many others here don't see the value of using fractional
> > exponents in a "monzo" prime-factor notation, but i use them
> > all the time.
>
> ahh! this is a sensible solution. fractional exponents
> didn't occur to me. nonetheless, the less similar the
> intervals you combine, the hairier the equation. in this
> sense i am inclined to use decimals to some precision, so
> that all notes can be described with the same amount of
> information. (i am inclined to represent them as base-2
> logs as well, so that we can see that 440 Hz (2^8.7814)
> is in the eighth octave, and that an added interval of "1"
> means exactly one octave.)

there's certainly a lot of sense (pun on "cents" intended)
in describing pitches as a logarithmic value. i see no
need to replace it with prime-factor notation, only to
*supplement* it with that.

i view logarithmic pitch-height space and prime-space
as two entirely different measurements of the pitch continuum,
and probably irreconcilable. so using them *both* together
is very reasonable.

the prime-space approach is interesting in that it gives
a multidimensional spin on pitch relationships. logarithmic
pitch-height space is linear, that is, 1-dimensional, and
assuming 8ve-equivalence, so is Pythagorean (3-limit) tuning.
simple 5-limit JI is 2-dimensional, 7-limit JI is 3-D,
and 11-limit is 4-D.

at that point visualizing things already becomes quite
tricky, altho the mathematics can continue into higher
dimensions with perfect ease.

the software being released later this year by my company
can create lattice diagrams of up to 7 dimensions, but
things become very complicated with anything above 3.

> > http://tonalsoft.com/enc/12-eq.htm
>
> thanks for the links, i found this one illuminating.
>
> > and two of the most interesting examples of using
> > prime-factor notation with strange (i.e., non-integer)
> > exponents:
> >
> > http://tonalsoft.com/enc/lucy.htm
> > http://tonalsoft.com/enc/golden.htm
>
> strange indeed. :) i haven't yet been convinced of any
> reason to use irrational ratios other than ET stepping
> in Hz or precise jumping in a log scale.

different kinds of mathematics have different kinds of appeal.

someone who believes that the golden proportion can be
perceived aurally as a measure of pitch-space can find
a way to justify the validity of golden meantone, for example.

it all comes down to the fact that humans love to find
patterns in the universe. music gives us a way of
expressing patterns in sound, and the more deeply a
composer/performer can embed different layers of
recognizability in the patterns of his/her music,
the more satisfying a listener finds it to be.
at least that's my theory.

"great" music is an interesting balance between art
and science, and between manipulating emotional feelings
(right-brain) and manipulating the mathematics of
pattern recognition (left-brain). in my book,
Mahler stands supreme.

-monz
(coming down off soapbox now)

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:

> strange indeed. :) i haven't yet been convinced of any reason to use
> irrational ratios other than ET stepping in Hz or precise jumping
in a log
> scale.

Aaron, are you familiar with meantone temperament, perhaps the most
important tuning system in the history of Western music?

::steals Monz's soapbox before someone else does::

In a message dated 2004:01:03 09:55:18 PM, monzo bonzo writes:

>it all comes down to the fact that humans love to find
>patterns in the universe.

oh don't be so species-centric ;) tsk tsk, you know better, too...
esp'ly since ya have known Herman Miller longer than I have
(Ouch... eek, that doesn't sound quite right...)
ohwell... but examples from nature that HM might cite if he were so inclined
---
examples of pattern loving non-anthros: chimps, gorillas, etc., whales & even
pea-brained birds (supposedly the descendants of eek! dinosaurs...)

[weird tangent: if ya like sci-fi, David Brin's Uplift series are pretty
intriguing for non-anthro heroes and villains...]

> music gives us a way of
>expressing patterns in sound, and the more deeply a
>composer/performer can embed different layers of
>recognizability in the patterns of his/her music,
>the more satisfying a listener finds it to be.
>at least that's my theory.

Good one. But that doesn't even begin to explain the attraction to the
juicy, spicy-spikey-ness of the asymmetrical, the non-harmonic/atonal, the
accidental, etc. -
the "Dark Side" in music (as W.A. Mathieu calls it), the stuffings "off to
the side or in[to] 'outer space'..."
my personal pet theory ;) is that both order and chaos make for
intriguing musics. Us MonkeyBrained Ones love surprises, right, lotsa "highper" brain
stimulation, right? We hate "mindless monotony and repetition," right? Even
Muzak tech-geeks know this... Am I right or I am right?

>"great" music is an interesting balance between art
>and science, and between manipulating emotional feelings
>(right-brain) and manipulating the mathematics of
>pattern recognition (left-brain). in my book,
>Mahler stands supreme.

::begs to seriously differ with one of his "on-line" music teachers, but
keeps damn mouth shut as this is not the contextual time or place... proud to
know his place ;) time enuff for friendly headbanging, brainstungunning debate
later, hehe...::

>-monz
>
>(coming down off soapbox now)

::sets fire to soapbox
... to roast a few Chinese pork sausages, a seaslug,
& sugarless marshmellows with good wheatcrackers & dark chocolate ;)::

why post if ya can't have some serious FUN?
::mourns the death of witty repartee & wittier intellectually indense
entertaining conversation (wacky "edutainment"?) ::

---///// __/_//_/ __/_//_////// __/_//_/ __/_//_/

in the yera of 2004 CE, year 4702 of the Huangdi era,
Year of the Wooden Monkey, _Jia-Shen_...

_Nom de Guerre et Nom de 'Nick'_: Hanuman "Stitch/626" Zhang
AIN Resistance partisan code name: "Z23-4C"
WOGeR (Wiley Oriental Gentleman/Rogue)
BBC (BritishBornChinaman)
Avatar of Sun Wu K'ung, a.k.a. Sun Wukong, a.k.a _Ma-Lau_ ("Monkey
King")
a.k.a. "TricksterGod of the Glorious Anti-Imperialist Chinese Boxers"
¡¡¡ TricksterShapeShifterIncarnate !!! >^..^< ';'
;P~~~
"one o' dem best-est & bright-est a' de bottom o' de barrel"
<A HREF="http://www.friendster.com/user.jsp">Friendster - zHANgster</A>

=> om hung hanumatay rudratmakai hung phat <=
mantra to Hanuman the Hindu Monkey TricksterGod

"I rather be whole than good." - Carl Jung

<= thee prIs ov eXistenZ iz aetern'l warfaer 'N' kreativ playf'llnizz... =>
"Mea typo, mea typo, mea maxima typo... "

"Life is all a great joke, but only the brave ever get the point."
- Kenneth Rexroth

googolgigglabyte
goegolgiechelbijt - of - met een vette megagrijns
GoogolGekicherByte
googolrisibyte ===> el byte de la risita de googol
googolrisadinhabyte ===> o byte de risadinha de googol
googolspassoctet
guugoIllolbijt
gugolhihibajt
gugolngisibayt
okukolkikikol
egúgelegigalibaith
kiletstroknolyadgigabaiti
cimacimakekehapi
baitakhakhweifayatrauni
ufi'auayinisuguguluarkhar
pokatra oemadroabhethetre
inarevuta yhiyhayhake nawyo
AnekoMeppathmoTtilvatelmDiggulgyttahat
va'i utne tuktukt'ishushukuko`g tuk go`go`o`gwgaga
ggsngngsbd [gugulaNexebidi]

http://www.boheme-magazine.net

hi hanuman,

--- In tuning@yahoogroups.com, czhang23@a... wrote:

>
> ::steals Monz's soapbox before someone else does::
>
> In a message dated 2004:01:03 09:55:18 PM, monzo bonzo writes:
>
> > music gives us a way of
> > expressing patterns in sound, and the more deeply a
> > composer/performer can embed different layers of
> > recognizability in the patterns of his/her music,
> > the more satisfying a listener finds it to be.
> > at least that's my theory.
>
> Good one. But that doesn't even begin to explain
> the attraction to the juicy, spicy-spikey-ness of the
> asymmetrical, the non-harmonic/atonal, the accidental,
> etc. - the "Dark Side" in music (as W.A. Mathieu calls it),
> the stuffings "off to the side or in[to] 'outer space'..."
> my personal pet theory ;) is that both order and chaos
> make for intriguing musics. Us MonkeyBrained Ones love
> surprises, right, lotsa "highper" brain stimulation, right?
> We hate "mindless monotony and repetition," right? Even
> Muzak tech-geeks know this... Am I right or I am right?

good points. ... but i'd bet that even in "chaotic"
music listeners will be trying to find the patterns!

> >"great" music is an interesting balance between art
> >and science, and between manipulating emotional feelings
> >(right-brain) and manipulating the mathematics of
> >pattern recognition (left-brain). in my book,
> >Mahler stands supreme.
>
> ::begs to seriously differ with one of his "on-line"
> music teachers, but keeps damn mouth shut as this is not
> the contextual time or place... proud to know his place ;)
> time enuff for friendly headbanging, brainstungunning debate
> later, hehe...::

are you saying that you don't like Mahler? >:-(

... we'd better continue this on metatuning ...

-monz

> > strange indeed. :) i haven't yet been convinced of any reason to use
> > irrational ratios other than ET stepping in Hz or precise jumping in a
> > log scale.
>
> Aaron, are you familiar with meantone temperament, perhaps the most
> important tuning system in the history of Western music?

hello paul,

i have just read up on it. please correct me, but it doesn't sound like
mathematical innovation, instead just a set of intervals which are somewhat
more functional than JI. could you please elaborate on why it's still
relevant?

thanks,
aaron.

- | | | | | | | | | | -
- -
- -
- aaron brick -
- aa@lithic.org -
- -
- -
- | | | | | | | | | | -

hiya monz,

> i view logarithmic pitch-height space and prime-space
> as two entirely different measurements of the pitch continuum,
> and probably irreconcilable. so using them *both* together
> is very reasonable.

accepting a small amount of imprecision we can write one of the
"prime-space" values as a fraction or decimal, or a log thereof. so as far
as i can tell, they are reconcilable (though i bet it's more computationally
expensive to translate from a decimal to prime notation than vice versa).

> simple 5-limit JI is 2-dimensional, 7-limit JI is 3-D, and 11-limit is
> 4-D.

i understand now. how can you leverage these prime components - what is the
use in keeping track of them?

> different kinds of mathematics have different kinds of appeal.

you can say that again!

> it all comes down to the fact that humans love to find patterns in the
> universe. music gives us a way of expressing patterns in sound, and the
> more deeply a composer/performer can embed different layers of
> recognizability in the patterns of his/her music, the more satisfying a
> listener finds it to be. at least that's my theory.

i quite like that. i think you're largely on target there.

> "great" music is an interesting balance between art and science, and
> between manipulating emotional feelings (right-brain) and manipulating the
> mathematics of pattern recognition (left-brain). in my book, Mahler
> stands supreme.

i should go pull out my mahler discs and see if i can hear more in them than
the last time i listened. i think (good) hip-hop plays this line well:
hopefully the beats are interesting as well as rhythmically engaging; and
rapping is a strong synthesis of structure and lyrical expression.

> (coming down off soapbox now)

thanks for the lesson ;)

aaron.

- | | | | | | | | | | -
- -
- -
- aaron brick -
- aa@lithic.org -
- -
- -
- | | | | | | | | | | -

hi Aaron,

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:

> hiya monz,
>
> > i view logarithmic pitch-height space and prime-space
> > as two entirely different measurements of the pitch continuum,
> > and probably irreconcilable. so using them *both* together
> > is very reasonable.
>
> accepting a small amount of imprecision we can write one
> of the "prime-space" values as a fraction or decimal, or
> a log thereof. so as far as i can tell, they are
> reconcilable

ah, but that's exactly the crux of it!

the two different approaches can only be reconciled *if*
i'm willing to accept "a small amount of imprecision".
but how exactly how small is "small"?

if i'm *not* willing to accept any imprecision,
then i have to use both types of measurement, because
they the two approaches are indeed, as i said, irreconcilable.

> (though i bet it's more computationally
> expensive to translate from a decimal to prime notation
> than vice versa).

nah, i'm pretty sure it takes the same amount of
computation to go either way.

> > simple 5-limit JI is 2-dimensional, 7-limit JI is 3-D,
> > and 11-limit is 4-D.
>
> i understand now. how can you leverage these prime
> components - what is the use in keeping track of them?

my theory is that each prime-factor gives a unique and
distinctive flavor to the sound of the harmony ... at least,
within the limits described by paul erlich's "harmonic entropy"
concept. see

http://tonalsoft.com/enc/harmentr.htm

(i haven't finished updating this page yet ...
the graphs on it will only be visible if you join and log in
to the harmonic-entropy Yahoo group)

/harmonic_entropy/

the whole point of using lattice diagrams is to have
an easy-to-understand visual representation of the
mathematics of prime-factored ratios. see:

http://tonalsoft.com/enc/lattice.htm

http://tonalsoft.com/monzo/lattices/lattices.htm

> > "great" music is an interesting balance between
> > art and science, and between manipulating emotional
> > feelings (right-brain) and manipulating the mathematics
> > of pattern recognition (left-brain). in my book,
> > Mahler stands supreme.
>
> i should go pull out my mahler discs and see if i can
> hear more in them than the last time i listened.

since i go on so much about Mahler, you might like to
listen to some of the work i've done on his music.
these are two of the best:

http://tonalsoft.com/monzo/mahler/mahler7th.htm

http://tonalsoft.com/monzo/mahler/mahlerdaslied.htm

(i'm currently working on the website, so if these
links don't work right now, just try them again later
or tomorrow.)

> i think (good) hip-hop plays this line well:
> hopefully the beats are interesting as well as
> rhythmically engaging; and rapping is a strong
> synthesis of structure and lyrical expression.

unfortunately i haven't kept up with any hip-hop or
rap since the demise of Public Enemy, and i used to
think they were really fantastic. (i totally missed
the whole Eminem thing.) but based on the *good* rap
music i know from the 1980s and early 1990s, i agree.

but this kind of thing could be argued for The Beatles as well.
i think it applies to all "good" music.

-monz

In a message dated 2004:01:04 07:15:33 PM, monz@attglobal.net quotes _moi_
quoting him, etc. & writes:

>> In a message dated 2004:01:03 09:55:18 PM, monzo bonzo writes:
>> > music gives us a way of
>> > expressing patterns in sound, and the more deeply a
>> > composer/performer can embed different layers of
>> > recognizability in the patterns of his/her music,
>> > the more satisfying a listener finds it to be.
>> > at least that's my theory.
>
>> Good one. But that doesn't even begin to explain
>> the attraction to the juicy, spicy-spikey-ness of the
>> asymmetrical, the non-harmonic/atonal, the accidental,
>> etc. - the "Dark Side" in music (as W.A. Mathieu calls it),
>> the stuffings "off to the side or in[to] 'outer space'..."
>> my personal pet theory ;) is that both order and chaos
>> make for intriguing musics. Us MonkeyBrained Ones love
>> surprises, right, lotsa "highper" brain stimulation, right?
>> We hate "mindless monotony and repetition," right? Even
>> Muzak tech-geeks know this... Am I right or I am right?
>
>good points. ... but i'd bet that even in "chaotic"
>music listeners will be trying to find the patterns!

Like I said/say: >> both order and chaos
>> make for intriguing musics. Us MonkeyBrained Ones love
>> surprises, right, lotsa "highper" brain stimulation, right?
>> We hate "mindless monotony and repetition

_datsuzoku_: Japanese Romaji, astonishing surprise;
"surprise in creativity"
the most elusive Zen (Daoist as well)
aesthetic value.
_yugen_: mystery, profundity, subtle, occult

But being the Curious George MonkeyMinded Types we are, we hafta take a
gamble and risk reachin' for the moon(s) we see reflected on the shiney,
sometimes lethal, surfaces we barely scratch... Gotta scratch that aesthetic Itch,
Komraden...push the envelope, etc.

>> >"great" music is an interesting balance between art
>> >and science, and between manipulating emotional feelings
>> >(right-brain) and manipulating the mathematics of
>> >pattern recognition (left-brain). in my book,
>> >Mahler stands supreme.
>
>> ::begs to seriously differ with one of his "on-line"
>> music teachers, but keeps damn mouth shut as this is not
>> the contextual time or place... proud to know his place ;)
>> time enuff for friendly headbanging, brainstungunning debate
>> later, hehe...::
>
>are you saying that you don't like Mahler? >:-(

not at all, I like some stuff he did ("Marz") since my childhood (drove
my Mum bonkers with me playing & re-playin "Marz" and the space alien jazz
cantina scene in _Star Wars_... over & over & over... but she, a nurse educator &
Montesorrian and sworn Int'l-Congress-of-Nurses enemy of the slimey
penis-headed-doctor-elitist AMA, allowed me to play with all the kitchen - aluminium
sink included ;) and went face-to-face with puritannical neighbors who made a
bloody issue of this and my runnin' all creatively amok and "hyperactive" (ADD)
in the 'hood... (oddly strikingly like Grateful Dead drummer Mikey Hart's mom
did... Thanx Mums! and thanx too to the fiercest human [& animal] force on
Earth: a mother's maternal instinct)

I grew up hearin' South East Asian music (Indonesian mainly) besides
Classical Euro and Trad Chinese (and some Hindi Folk Musics and Hippie Rock)

::mock chagrinnie: is my percussion background & "bias" slippin' n' showin'?

as to reigning supreme, in my book, hehe, I may like lotta people for various
reasons or gut-feelings... somedays I feel Varesean or just plain goofy for
soundscapes & toy piano & toy music & music boxes... other days Xenakisian or
Tan Dun-ean... and still other days nostalgic for Bartok, Debussy & Co... or
aching for Indonesian/Chinese/Japanese/Indian homeland spiciness.... just like
sometimes I am a Daoist or Buddhist or Quaker or Sufi or Jesuit or Hindu... and
even Atheistic Nihilist...
or simulanteous multiplex _combinatoire frisson_ thereof...

I play no favourites, I mindshift mindwarp and shapeshift constantly...
besides in this 21st Schizo-Postmodernist age the ol' hoary ideas of
authoritive EuroAmericancentricist Canons is under justifiable MultiCulti attack just
as Imperialistic EuroRomanticism was in the first half or so of the
Collective-Mind-Expanding 20th C.

Enuff said, never ever enuff done...

>... we'd better continue this on metatuning ...

no need now :)

---|-----|--------|-------------|---------------------|
Hanuman Zhang, musical mad scientist
"Space is a practiced place." -- Michel de Certeau
"Space is the Place for the Human Race." -- William S. Burroughs

"... simple, chaotic, anarchic and menacing.... This is what people of today
have lost and need most - the ability to experience permanent bodily and
mental ecstasy, to be a receiving station for messages howling by on the ether from
other worlds and nonhuman entities, those peculiar short-wave messages which
come in static-free in the secret pleasure center in the brain." - Slava Ranko
(Donald L. Philippi)

The German word for "noise" _Geräusch_ is derived from _rauschen_ "the
sound of the wind," related to _Rausch_ "ecstasy, intoxication" hinting at some
of the possible aesthetic, bodily effects of noise in music. In Japanese
Romaji: _uchu_ = "universe"... _uchoten_ = "ecstasty," "rapture"..._uchujin_ =
[space] alien!

"When you're trying to do something you should feel absolutely alone, like a
spark in the blackness of the universe."-Xenakis

"For twenty-five centuries, Western knowledge has tried to look upon the
world. It has failed to understand that the world is not for the beholding. It
is for the hearing. It is not legible, but audible. ... Music is a herald,
for change is inscribed in noise faster than it transforms society. ...
Listening to music is listening to all noise, realizing that its appropriation and
control is a reflection of power, that is essentially political." - Jacques
Attali, _Noise: The Political Economy of Music_

"The sky and its stars make music in you." - Dendera, Egypt wall
inscription

"Sound as an isolated object of reproduction, call it our collective memory
bank... Any sound can be you." - DJ Spooky that Subliminal Kid (a.k.a. Paul D.
Miller)

"Overhead, without any fuss, the stars were going out."
--Arthur C. Clarke, _The Nine Billion Names of God_

hi hanuman,

--- In tuning@yahoogroups.com, czhang23@a... wrote:
>
> In a message dated 2004:01:04 07:15:33 PM, monz@a... quotes _moi_
> quoting him, etc. & writes:
>
> > are you saying that you don't like Mahler? >:-(
>
> not at all, I like some stuff he did ("Marz") since
> my childhood (drove my Mum bonkers with me playing &
> re-playin "Marz" and the space alien jazz cantina scene
> in _Star Wars_... over & over & over...

you've got the wrong Gustav there ... "Mars, The Bringer
of War" is the first piece in _The Planets_, by Gustav *Holst*.

Mahler's most famous pieces are his 2nd Symphony
("Resurrection"), 8th ("Symphony of a Thousand"), and
"Das Lied von der Erde" ... and of course, the most
famous of all, the "Adagietto" from the 5th Symphony.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> Mahler's most famous pieces are his 2nd Symphony
> ("Resurrection"), 8th ("Symphony of a Thousand"), and
> "Das Lied von der Erde" ... and of course, the most
> famous of all, the "Adagietto" from the 5th Symphony.

My impression is that #4 gets the most performances, and that both
#1 and #3 are quite popular.

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:
>
> > > strange indeed. :) i haven't yet been convinced of any reason
to use
> > > irrational ratios other than ET stepping in Hz or precise
jumping in a
> > > log scale.
> >
> > Aaron, are you familiar with meantone temperament, perhaps the
most
> > important tuning system in the history of Western music?
>
> hello paul,
>
> i have just read up on it. please correct me, but it doesn't sound
like
> mathematical innovation, instead just a set of intervals which are
somewhat
> more functional than JI. could you please elaborate on why it's
still
> relevant?
>
> thanks,
> aaron.

Hi Aaron,

You wrote "i haven't yet been convinced of any reason to use
irrational ratios other than ET stepping in Hz or precise jumping in
a log scale." I guess I read this wrong as "i haven't yet been
convinced of any reason to use irrational ratios in tuning systems
other than ET". So could you please tell me what you really meant?

Thanks,
Paul

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Aaron,
>
> > (though i bet it's more computationally
> > expensive to translate from a decimal to prime notation
> > than vice versa).
>
>
>
> nah, i'm pretty sure it takes the same amount of
> computation to go either way.

Hmm . . . I can go from prime notation to decimal values in one line
of code. If you can do the reverse in one line of code, hats off to
you . . . but I don't believe you can.

> the whole point of using lattice diagrams is to have
> an easy-to-understand visual representation of the
> mathematics of prime-factored ratios.

And, more importantly, I would say, to have an easy-to-understand
visual representation of the appealing harmonies available in any
given set of pitches.

on 1/4/04 11:12 AM, monz <monz@attglobal.net> wrote:

> hi hanuman,
>
>
> --- In tuning@yahoogroups.com, czhang23@a... wrote:
>
>>
>> ::steals Monz's soapbox before someone else does::
>>
>> In a message dated 2004:01:03 09:55:18 PM, monzo bonzo writes:
>>
>>> music gives us a way of
>>> expressing patterns in sound, and the more deeply a
>>> composer/performer can embed different layers of
>>> recognizability in the patterns of his/her music,
>>> the more satisfying a listener finds it to be.
>>> at least that's my theory.
>>
>> Good one. But that doesn't even begin to explain
>> the attraction to the juicy, spicy-spikey-ness of the
>> asymmetrical, the non-harmonic/atonal, the accidental,
>> etc. - the "Dark Side" in music (as W.A. Mathieu calls it),
>> the stuffings "off to the side or in[to] 'outer space'..."
>> my personal pet theory ;) is that both order and chaos
>> make for intriguing musics. Us MonkeyBrained Ones love
>> surprises, right, lotsa "highper" brain stimulation, right?
>> We hate "mindless monotony and repetition," right? Even
>> Muzak tech-geeks know this... Am I right or I am right?
>
> good points. ... but i'd bet that even in "chaotic"
> music listeners will be trying to find the patterns!

The order/chaos balance is one way to describe it, however I think of it
more like old order versus "new order" if you will excuse the expression.
As I am beginning to approach composing for real I have to keep asking
myself whether I really have anything to say, or do I need to rehash what
others have said because it is so intriguing to me.

This in itself is still not something I should judge myself by, however.
Better to do what I am inspired to do regardless. However the question of
whether I have anything to say is an interesting one, not because of the
presence or absense of something to say but because of how I suspect this
will relate to my process of learning to have a clearer sense of myself.

I hope that was not too chaotic (or too off-topic).

-Kurt

In a message dated 2004:01:05 06:43:32 PM, monz@attglobal.net writes:

>hi hanuman,

saluta monz!

>--- In tuning@yahoogroups.com, czhang23@a... wrote:

>> In a message dated 2004:01:04 07:15:33 PM, monz@a... quotes _moi_
>> quoting him, etc. & writes:
>
>> > are you saying that you don't like Mahler? >:-(
>
>> not at all, I like some stuff he did ("Marz") since
>> my childhood (drove my Mum bonkers with me playing &
>> re-playin "Marz" and the space alien jazz cantina scene
>> in _Star Wars_... over & over & over...
>
>you've got the wrong Gustav there ... "Mars, The Bringer
>of War" is the first piece in _The Planets_, by Gustav *Holst*.

me BAAAAD! Entschuldigen Sie mein schlechtes Deutsch...
::bonks self on head::
I knew was committing a brain-fart somewhere
'fore I impulzif push the SEND button dichte... lately I have
mangled or Mixedup & misMatched more names than I wanna
(i.e. David Rosenberg's and Jerome Rothenberg... big diff there,
both famous in 2 differin' fields...redichte, Entschuldigen Sie mein
schlechtes Deutsch...)

(I have been listenin' to MP3s I DLed today** (finally got somethin' to
work on my phuqedup desktop sized PDA of a Mac)... for a truly HyperMonkey
Brain like me, biwa music is a brain-stunner ... and the musician she's one Day
Glo Orange hot cutie, no less a brain-stunner (a la koto-player/microtonalist
Yagi Michiyo):

http://www.os.rim.or.jp/~kenichii/biwa.html

... yeah, I am distracted, _mea culpa_... **& I can't stop playing
Pehrson's _Blacklight_ for cello & electronics, it's in book IMMHO as some the best
electroacoustic work this side of Richard Teitelbaum's _Blends_ [don't even
get me arguin' against Stockhausen... music profs who foolishly clashed with me
have rued that decision majorly...suffice to say I agree with Cardew's polemic
IIRC titled "Stockhausen is an cultural imperialist"and that I much prefer
Xenakis or post-industrial apoca-rock group Einstuzende Neubauten to either
Stockhaus-neu-agey-baby or Ligeti: all just surface & not enuff style-substance or
spine to back it all up...::stumbles off rant box::])

>Mahler's most famous pieces are his 2nd Symphony
>("Resurrection"), 8th ("Symphony of a Thousand"), and
>"Das Lied von der Erde"

damn yes I like that one, even just the title sends my neckhairs up...just as
Xenakis' _La Legended'Eer_ does... and the sounds of loud cicadae and birds...

> ... and of course, the most famous of all, the "Adagietto" from the 5th
Symphony.

don't dead faint on me, but I am not at all familiar with that one AFAIR... I
mighta heard it... but was sorely distracted by something ... or some one,
hehe...

::scampers off to music files to make note of the "Adagietto" by Gustav
MAHLER... not space-case Holst::

---
Hanuman Zhang, _Gomi no sensei_ [Master of junk]

"...There is life feeding on this dead heap. ... life will complete its
cycle, teeming within this lump of death." - Bela Bartok

"I'd never dare attack anyone who doesn't think the way I do. Thought is the
property of the person who has it. No one else has a right to even touch it."
- Erik Satie

"There are two means of refuge from the miseries of life: music and cats." -
Albert Schweitzer
Please: Cats are not disposable. Adopt and love your cat for life...

"Life'n'art, art'n'life. There is _no_ separation. Each is a manifestation of
every aspect of the other. This is a difficult game... A difficult game to
play. It seems, to me, like the only game that needs playing at the moment." -
Warren Burt

--- In tuning@yahoogroups.com, czhang23@a... wrote:

> ... yeah, I am distracted, _mea culpa_... **& I can't stop
playing
> Pehrson's _Blacklight_ for cello & electronics, it's in book IMMHO
as some the best
> electroacoustic work this side of Richard Teitelbaum's _Blends_

Score one for Blacky (that's you, Joe)!

--- In tuning@yahoogroups.com, czhang23@a... wrote:

/tuning/topicId_38724.html#51108

for a truly HyperMonkey
> Brain like me, biwa music is a brain-stunner ...

***That works for *me* czhang...

>
> http://www.os.rim.or.jp/~kenichii/biwa.html
>
> ... yeah, I am distracted, _mea culpa_... **& I can't stop
playing
> Pehrson's _Blacklight_ for cello & electronics, it's in book IMMHO
as some the best
> electroacoustic work this side of Richard Teitelbaum's _Blends_

***Thanks so much czhang! Glad you're enjoying it!

JP

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

/tuning/topicId_38724.html#51128

> --- In tuning@yahoogroups.com, czhang23@a... wrote:
>
> > ... yeah, I am distracted, _mea culpa_... **& I can't stop
> playing
> > Pehrson's _Blacklight_ for cello & electronics, it's in book
IMMHO
> as some the best
> > electroacoustic work this side of Richard Teitelbaum's _Blends_
>
> Score one for Blacky (that's you, Joe)!

***Yessim...boss... Ol' Black Joe...(whoops, the PC police are at the
door...)(to check my PC...)

[entschuldigen...]

JP

> Hi Aaron,
>
> You wrote "i haven't yet been convinced of any reason to use
> irrational ratios other than ET stepping in Hz or precise jumping in
> a log scale." I guess I read this wrong as "i haven't yet been
> convinced of any reason to use irrational ratios in tuning systems
> other than ET". So could you please tell me what you really meant?
>
> Thanks,
> Paul

hi paul!

from a microtonal perspective, i have only identified two kinds of
intervals: ET-style stepping and JI-style jumping. in a hertz scale,
stepping requires an irrational (eg, the twelfth root of 2); in log2 scale,
it's jumping that does (eg, log2(3/2)).

having found these two types and determined that using a logscale would be
computationally equivalent to hertz measurement (there is one irrational and
one real coefficient in each), i have privately arrived at the idea that i
would like to use a dynamic, microtonal, log2 scale. i hope this answers
your question - please comment!

best,

aaron.

| | | | | | | |
/ \
- -
- aaron brick -
- aa@lithic.org -
- -
\ /
| | | | | | | |

hello monz,

> the two different approaches can only be reconciled *if*
> i'm willing to accept "a small amount of imprecision".
> but how exactly how small is "small"?

why, as small as you like, of course! let's consider, say, the 10th decimal
digit of an interval. it is of the order 10^-10, which would represent only
1/10^5 of a cycle even at 10,000 hz. that sort of error sounds to me small
enough to be insubstantial.

> if i'm *not* willing to accept any imprecision,
> then i have to use both types of measurement, because
> they the two approaches are indeed, as i said, irreconcilable.

i think the irreconcilability is only in one direction:

> nah, i'm pretty sure it takes the same amount of
> computation to go either way.

i think paul is right about this. additionally, the decimal notation
can include some rounding, which makes derviation of its factors a lot
harder.

> http://tonalsoft.com/enc/harmentr.htm

fascinating. i'm very excited to have found you guys who can comment and
theorize about this in such depth.

so having the factors (and their powers) at hand is revealing in terms of
the harmonic possibilities. i'm going to read up on the lattice ideas now.
you guys rock, thank you very much for engaging me.

aaron.

| | | | | | | |
/ \
- -
- aaron brick -
- aa@lithic.org -
- -
\ /
| | | | | | | |

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:
>
> > Hi Aaron,
> >
> > You wrote "i haven't yet been convinced of any reason to use
> > irrational ratios other than ET stepping in Hz or precise jumping
in
> > a log scale." I guess I read this wrong as "i haven't yet been
> > convinced of any reason to use irrational ratios in tuning
systems
> > other than ET". So could you please tell me what you really meant?
> >
> > Thanks,
> > Paul
>
> hi paul!
>
> from a microtonal perspective, i have only identified two kinds of
> intervals: ET-style stepping and JI-style jumping.

I see these as two extremes, between which there are "middle path"
approaches.

> in a hertz scale,
> stepping requires an irrational (eg, the twelfth root of 2); in
log2 scale,
> it's jumping that does (eg, log2(3/2)).
>
> having found these two types and determined that using a logscale
would be
> computationally equivalent to hertz measurement (there is one
irrational and
> one real coefficient in each), i have privately arrived at the idea
that i
> would like to use a dynamic, microtonal, log2 scale.

Sure, that's fine. If you then multiply that by 1200, you get cents,
which is what is most commonly used to specify and compare microtonal
intervals.

> > from a microtonal perspective, i have only identified two kinds of
> > intervals: ET-style stepping and JI-style jumping.
>
> I see these as two extremes, between which there are "middle path"
> approaches.

oo! what is the axis that connects them? do you have an equation which
represents this set of possible intervals?

> > i have privately arrived at the idea that i
> > would like to use a dynamic, microtonal, log2 scale.
>
> Sure, that's fine. If you then multiply that by 1200, you get cents,
> which is what is most commonly used to specify and compare microtonal
> intervals.

cool, i didn't realize that cents were logarithmic. however, why would they
use base 10? i think i'm willing to assume octave equivalence on the basis
of our perceptive systems using it so heavily.

i think i would rather use decimals than integers (a la cents) because
precision is arbitrary. in addition, not coming from a 12-tone background, i
am not awfully fond of the number 1200.

best,

aaron.

| | | | | | | |
\ /
- -
- aaron brick -
- aa@lithic.org -
- -
/ \
| | | | | | | |

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:
> > > from a microtonal perspective, i have only identified two kinds
of
> > > intervals: ET-style stepping and JI-style jumping.
> >
> > I see these as two extremes, between which there are "middle
path"
> > approaches.
>
> oo! what is the axis that connects them?

Temperament (or rather regular temperament) in general. Meantones
would be one set of examples. Like JI, meantones are infinite (except
an infinitesimal minority which are ETs), but unlike JI, they only
have two dimensions of infinitude, or just one if you think in octave-
equivalent terms -- thus we call meantone an example of "linear
temperament". There are others, like MIRACLE, and then there are
planar temperaments, etc . . .

> do you have an equation which
> represents this set of possible intervals?

Well, a typical example would be in 1/4-comma meantone temperament,
which is simply called "meantone temperament" if you ask John
Chalmers, where the perfect fifth can be written as 5^(1/4), the
major second as (5/4)^(1/2) or 5^(1/2)/2, and hence two major seconds
equals a major third = 5/4. An interval in common with JI. In
general, though, meantone temperaments don't have to have any
intervals in common with JI -- Golden meantone and LucyTuning are
examples.

> > > i have privately arrived at the idea that i
> > > would like to use a dynamic, microtonal, log2 scale.
> >
> > Sure, that's fine. If you then multiply that by 1200, you get
cents,
> > which is what is most commonly used to specify and compare
microtonal
> > intervals.
>
> cool, i didn't realize that cents were logarithmic. however, why
would they
> use base 10?

don't know what you mean -- you mean because 1200 is twice divisible
by 10?

> i think i would rather use decimals than integers (a la cents)
because
> precision is arbitrary. in addition, not coming from a 12-tone
background, i
> am not awfully fond of the number 1200.

That's understandable, but unfortunately, you pick up the New Grove
and start comparing cultures, and all you see is cents, cents, cents.
It's an accepted standard and helps people to communicate with one
another a lot more quickly and effectively.

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:

> > Sure, that's fine. If you then multiply that by 1200, you get
cents,
> > which is what is most commonly used to specify and compare
microtonal
> > intervals.
>
> cool, i didn't realize that cents were logarithmic. however, why
would they
> use base 10?

It's not base 10, it's base 2^(1/1200). The idea is to divide the 12-
equal semitone into 100 parts.

> > > > from a microtonal perspective, i have only identified two kinds of
> > > > intervals: ET-style stepping and JI-style jumping.
> > >
> > > I see these as two extremes, between which there are "middle
> path" approaches.
> >
> > oo! what is the axis that connects them?
>
> Temperament (or rather regular temperament) in general.

it sounds like the set of temperaments aren't so much points on an axis but
individual schemes with attributes that differentiate them: all i can think
of are finity, rationality and dimensionality. i'd like to see that
matrix....

> > cool, i didn't realize that cents were logarithmic. however, why would
> > they use base 10?
>
> don't know what you mean -- you mean because 1200 is twice divisible by
> 10?

the equation given in monz's dictionary shows them derived with a base 10
logarithm of the (hertz) ratio; there are 10^2 cents between the 12 notes. i
assume that's is why it ended up this way, being abstracted from the 12-tone
scale and intending to retain readability back to it.

> That's understandable, but unfortunately, you pick up the New Grove
> and start comparing cultures, and all you see is cents, cents, cents.
> It's an accepted standard and helps people to communicate with one
> another a lot more quickly and effectively.

of course. i have no problem with learning my way around them. :)

aaron.

_______________
/\ /\
| |
| aaron brick |
| aa@lithic.org |
| |
\/_______________\/

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:
> > > > > from a microtonal perspective, i have only identified two
kinds of
> > > > > intervals: ET-style stepping and JI-style jumping.
> > > >
> > > > I see these as two extremes, between which there are "middle
> > path" approaches.
> > >
> > > oo! what is the axis that connects them?
> >
> > Temperament (or rather regular temperament) in general.
>
> it sounds like the set of temperaments aren't so much points on an
>axis

Well, in the 5-limit, you can make a diagram that shows JI in the
center, various ETs as points, and meantone and other linear
temperaments as lines in a plane -- see the first chart on
http://tonalsoft.com/enc/eqtemp.htm

> but
> individual schemes with attributes that differentiate them: all i
can think
> of are finity, rationality and dimensionality. i'd like to see that
> matrix....

There's much information on each of those (green-line) linear
temperament in the table below the chart you were just looking
at . . . and more in the current posts on this list.

> > > cool, i didn't realize that cents were logarithmic. however,
why would
> > > they use base 10?
> >
> > don't know what you mean -- you mean because 1200 is twice
divisible by
> > 10?
>
> the equation given in monz's dictionary shows them derived with a
base 10
> logarithm of the (hertz) ratio;

That's unnecessary and more complex. cents = log2(ratio) * 1200.

> i
> assume that's is why it ended up this way, being abstracted from
the 12-tone
> scale and intending to retain readability back to it.

Yes.

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:
> > > > > from a microtonal perspective, i have only identified two
kinds of
> > > > > intervals: ET-style stepping and JI-style jumping.
> > > >
> > > > I see these as two extremes, between which there are "middle
> > path" approaches.
> > >
> > > oo! what is the axis that connects them?
> >
> > Temperament (or rather regular temperament) in general.
>
> it sounds like the set of temperaments aren't so much points on an
> axis

Well, by the axis I meant, essentially, the dimensionality of the
system. Say you start with JI with some number of dimensions. Then as
you temper out more and more independent "commas", you'll be stepping
down on the dimensionality axis, until you get to a planar
temperament, a linear temperament, and finally an equal temperament.

hi Aaron,

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:

> > > from a microtonal perspective, i have only
> > > identified two kinds of intervals: ET-style stepping
> > > and JI-style jumping.
> >
> > I see these as two extremes, between which there are
> > "middle path" approaches.
>
> oo! what is the axis that connects them? do you have
> an equation which represents this set of possible intervals?

that's pretty much what the Tonalsoft software is all about.

and it doesn't give you only the math, but also the visuals
that go along with the math. (and of course, the music
composition tools as well.)

at this point, it looks like the beta release will be out
around April and version 1.0 around July (of this year).

> > > i have privately arrived at the idea that i
> > > would like to use a dynamic, microtonal, log2 scale.
> >
> > Sure, that's fine. If you then multiply that by 1200,
> > you get ents, which is what is most commonly used to
> > specify and compare microtonal intervals.
>
> cool, i didn't realize that cents were logarithmic.

Aaron, you would profit from examining the
'table showing advocates of various "octave"-based ETs'
beginning 1/3 of the way down on my "equal temperament" page:

http://tonalsoft.com/enc/eqtemp.htm#edo-table

i list there every EDO/ET that i know of which has
been advocated or used by a theorist or composer.

(with the exception of a small handful of lower-cardinality
EDOs which have probably been used by Brian McLaren and
Marc Jones.)

> however, why would they use base 10? i think i'm
> willing to assume octave equivalence on the basis
> of our perceptive systems using it so heavily.
>
> i think i would rather use decimals than integers
> (a la cents) because precision is arbitrary.

that's a very good opinion!

> in addition, not coming from a 12-tone background,
> i am not awfully fond of the number 1200.

on the table referenced above, you'll find
"millioctaves" at 1000edo.

this is exactly the same as taking your measurement
to a precision of 3 decimal places.

but i do like your idea.

-monz

> at this point, it looks like the beta release will be out
> around April and version 1.0 around July (of this year).

Hey, monz, do you know about the technique of plotting an estimate of
remaining work against the projected release date, to look for a trend by
which a better prediction can be made?

(I'm serious, I think it is a good technique.)

(I'm also kidding you at the same time. ;)

-Kurt

hi paul and Aaron,

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

> --- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:
>
> > > > cool, i didn't realize that cents were logarithmic.
> > > > however, why would they use base 10?
> > >
> > > don't know what you mean -- you mean because 1200 is
> > > twice divisible by 10?
> >
> > the equation given in monz's dictionary shows them derived
> > with a base 10 logarithm of the (hertz) ratio;
>
> That's unnecessary and more complex. cents = log2(ratio) * 1200.

yes, i realize that it's more complex than the most
elegant mathematical description, but i wrote it that
way deliberately, because i've using Excel for years
to do my tuning calculations, and in Excel, just plain
old "log" means log10.

to get the log2 of a ratio in Excel, you must use
the formula: log(ratio)/log(2) .

i should probably amend every "how to calculate"
explanation in all of my webpages to show the
more elegant formula as well.

-monz

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> > at this point, it looks like the beta release will be out
> > around April and version 1.0 around July (of this year).
>
> Hey, monz, do you know about the technique of plotting
> an estimate of remaining work against the projected release
> date, to look for a trend by which a better prediction
> can be made?
>
> (I'm serious, I think it is a good technique.)
>
> (I'm also kidding you at the same time. ;)

thanks for the suggestion, but ... at this point, i'm
more concerned about using my time to get the software
done than to open up Excel and plot blah blah blah.

-monz
(whose tongue is also firmly in cheek)

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul and Aaron,
>
...
>
> yes, i realize that it's more complex than the most
> elegant mathematical description, but i wrote it that
> way deliberately, because i've using Excel for years
> to do my tuning calculations, and in Excel, just plain
> old "log" means log10.
>
> to get the log2 of a ratio in Excel, you must use
> the formula: log(ratio)/log(2) .
>
> i should probably amend every "how to calculate"
> explanation in all of my webpages to show the
> more elegant formula as well.
>
>
>
> -monz

Strange it seems to me...

Which version of Excel have you got?

Mine (Office 2000) has ln, log10 and log(arg;base).

hi Maximiliano,

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi paul and Aaron,
> >
> ...
> >
> > yes, i realize that it's more complex than the most
> > elegant mathematical description, but i wrote it that
> > way deliberately, because i've using Excel for years
> > to do my tuning calculations, and in Excel, just plain
> > old "log" means log10.
> >
> > to get the log2 of a ratio in Excel, you must use
> > the formula: log(ratio)/log(2) .
> >
> > i should probably amend every "how to calculate"
> > explanation in all of my webpages to show the
> > more elegant formula as well.
> >
> >
> >
> > -monz
>
> Strange it seems to me...
>
> Which version of Excel have you got?
>
> Mine (Office 2000) has ln, log10 and log(arg;base).

thanks! i have Office 2000 too, but i never knew
about that log function.

i've always done it like this:

cents = log(ratio)*(1200/log(2))

but i see now that

cents = log(ratio,2)*1200

gives the same result, and is much more elegant.

nice.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

/tuning/topicId_38724.html#51336

> hi Maximiliano,
>
>
>
> --- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
> <giordanobruno76@y...> wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > > hi paul and Aaron,
> > >
> > ...
> > >
> > > yes, i realize that it's more complex than the most
> > > elegant mathematical description, but i wrote it that
> > > way deliberately, because i've using Excel for years
> > > to do my tuning calculations, and in Excel, just plain
> > > old "log" means log10.
> > >
> > > to get the log2 of a ratio in Excel, you must use
> > > the formula: log(ratio)/log(2) .
> > >
> > > i should probably amend every "how to calculate"
> > > explanation in all of my webpages to show the
> > > more elegant formula as well.
> > >
> > >
> > >
> > > -monz
> >
> > Strange it seems to me...
> >
> > Which version of Excel have you got?
> >
> > Mine (Office 2000) has ln, log10 and log(arg;base).
>
>
>
> thanks! i have Office 2000 too, but i never knew
> about that log function.
>
>
> i've always done it like this:
>
> cents = log(ratio)*(1200/log(2))
>
>
>
> but i see now that
>
> cents = log(ratio,2)*1200
>
> gives the same result, and is much more elegant.
>

***And more confusing for some of us whose "specialty" is high school
algebra...

JP

hi monz,

> > > with a base 10 logarithm of the (hertz) ratio;
> >
> > That's unnecessary and more complex. cents = log2(ratio) * 1200.

> yes, i realize that it's more complex than the most
> elegant mathematical description, but i wrote it that
> way deliberately, because i've using Excel for years
> to do my tuning calculations, and in Excel, just plain
> old "log" means log10.

i don't understand the schism. log2 and log10 of a ratio are different
numbers, and if cents are a standardized measure, there can't be confusion
about the logarithmic base. so which is it? 2 makes more sense to me because
of 8ve equivalence but this may be a fallacious connection.

aaron.

_______________
/\ /\
| |
| aaron brick |
| aa@lithic.org |
| |
\/_______________\/

monz,

> > > > from a microtonal perspective, i have only
> > > > identified two kinds of intervals: ET-style stepping
> > > > and JI-style jumping.
> > >
> > > I see these as two extremes, between which there are
> > > "middle path" approaches.
> >
> > oo! what is the axis that connects them? do you have
> > an equation which represents this set of possible intervals?
>
> that's pretty much what the Tonalsoft software is all about.

i look forward to seeing it! i also look forward to learning it only works
on windows.... :/

> Aaron, you would profit from examining the
> 'table showing advocates of various "octave"-based ETs'
> beginning 1/3 of the way down on my "equal temperament" page:

i will read this in more detail. there's a lot on that page!

> > in addition, not coming from a 12-tone background,
> > i am not awfully fond of the number 1200.
>
>
> on the table referenced above, you'll find
> "millioctaves" at 1000edo.
>
> this is exactly the same as taking your measurement
> to a precision of 3 decimal places.
>
> but i do like your idea.

nice. thanks for the reference.

aaron.

_______________
/\ /\
| |
| aaron brick |
| aa@lithic.org |
| |
\/_______________\/

--- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:

> i don't understand the schism. log2 and log10 of a ratio are
different
> numbers, and if cents are a standardized measure, there can't be
confusion
> about the logarithmic base.

There isn't any confusion about the base--cents are logarithms base
2^(1/1200). However, you can calculate logarithms to any base using
logarithms to any other base. If loga is log base a, and logb is log
base b, then

logb(x) = loga(x)/loga(b)

If you apply that to the situation where b = 2^(1/1200), you get
formulas for whatever your favorite base is--base 10, base 2, base e,
or whatever.

whoops, i'm an idiot. i missed the second log10 term. thanks for
straightening me out, gene. i knew it should be a log2. how about making
that explicit in the encyclopedia, monz?

aaron.

so said gwsmith@svpal.org in 1.5K bytes on Mon, Jan 12, 2004:

> --- In tuning@yahoogroups.com, Aaron Brick <aaron@l...> wrote:
>
> > i don't understand the schism. log2 and log10 of a ratio are
> different
> > numbers, and if cents are a standardized measure, there can't be
> confusion
> > about the logarithmic base.
>
> There isn't any confusion about the base--cents are logarithms base
> 2^(1/1200). However, you can calculate logarithms to any base using
> logarithms to any other base. If loga is log base a, and logb is log
> base b, then
>
> logb(x) = loga(x)/loga(b)
>
> If you apply that to the situation where b = 2^(1/1200), you get
> formulas for whatever your favorite base is--base 10, base 2, base e,
> or whatever.
>
>
>
>
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_______________
/\ /\
| |
| aaron brick |
| aa@lithic.org |
| |
\/_______________\/