Yahoo Tuning Groups Ultimate Backup tuning reply to rick mcgowan (discussion moved from makemicromusic)

warning -- this post contains lots of numbers!

--- In MakeMicroMusic@yahoogroups.com, Rick McGowan <rick@u...> wrote:

> > I've heard Easley Blackwood's music in 13 through 24 note
> > equal temperament.
>
> Just FYI, we generally distinguish temperament from tuning. So
these are
> 13-24 equal tunings. They're not properly temperaments of anything;
they
> just consist of a number of equal sized intervals per octave.

why would you say that? certainly the way blackwood used them, many
of these tunings *are* temperaments. in the abstract, you can look at
the chart and table on this page

http://www.sonic-arts.org/dict/eqtemp.htm

and determine how to derive *most* of these equal tunings as 5-limit
temperaments.

presumably it's familiar to many readers that 12-equal can be derived
by tempering out the usual "comma" that's tempered out in all
meantones -- the syntonic comma or 81:80 -- and then tempering out an
additional small interval representing the ratio between "enharmonic
equivalents" like G#:Ab -- such as 648:625 (the "major diesis"),
128:125 (the usual "diesis"), 2048:2025 (the "diaschisma"),
32805:32768 (the "schisma"), or 531441:524288 (the "pythagorean
comma"). on the chart, you'll see that the green lines corresponding
to these intervals, labeled 'meantone', 'diminished', 'augmented',
'schismic', and 'aristoxenean' respectively (in accord with the table
below the chart), all intersect at the big number *12*, making it
easy to read this information off the chart.

one instance of *13* is at the intersection of the 'magic' and 'beep'
lines, so 13-equal can be derived from 5-limit JI by tempering out
3125:3072 and 27:25.

another instance of *13* is at the intersection of the 'father'
and 'tetracot' lines, so 13-equal can be derived from 5-limit JI by
tempering out 16:15 (missing from monz's table) and 20000:19683.

27:25 and 16:15 are unlikely intervals to temper out when using
harmonic timbres, but the next two are solidly derived as
temperaments and reflect the patterns of blackwood's highly triadic
usage:

*15* is at the intersection of
the 'blackwood', 'augmented', 'kleismic', and 'porcupine' lines, so
15-equal can be derived from 5-limit JI by tempering out any two of
256:243, 128:125, 15625:15552, and 250:243.

*16* is at the intersection of the 'diminished', 'pelogic',
and 'magic' lines, so 16-equal can be derived from 5-limit JI by
tempering out any two of 648:625, 135:128, and 3125:3072.

one instance of *17* is at the intersection of the 'schismic'
and 'dicot' lines, so 17-equal can be derived from 5-limit JI by
tempering out 32805:32768 and 25:24. blackwood's usage actually
corresponds to the *another* instance of 17, on the extreme left on
the 'meantone' line.

*18* is at the intersection of the 'augmented' and 'semisuper' lines,
so 18-equal can be derived from 5-limit JI by tempering out 128:125
and 6115295232:6103515625. admittedly, situations where the latter
interval needs to be tempered out will probably be quite rare, and
when they occur, will usually be handled with more accurate
temperaments.

*19* is at the intersection of
the 'meantone', 'negri', 'magic', 'kleismic', and 'semisixths' lines,
so 19-equal can be derived from 5-limit JI by tempering out any two
of 81:80, 16875:16384, 3125:3072, 15625:15552, and 78732:78125.

one instance of *20* is at the intersection of
the 'blackwood', 'diminished', and 'tetracot' lines, so 20-equal can
be derived from 5-limit JI by tempering out any two of 256:243,
648:625, and 20000:19683.

one instance of *21* is at the intersection of the 'augmented'
and 'escapade' lines, so 21-equal can be derived from 5-limit JI by
tempering out 128:125 and 4294967296:4271484375. admittedly,
situations where the latter interval needs to be tempered out will
probably be quite rare, and when they occur, will usually be handled
with more accurate temperaments.

*22* is at the intersection of the 'diaschismic', 'magic',
and 'porcupine' lines, so 22-equal can be derived from 5-limit JI by
tempering out any two of 2048:2025, 3125:3072, and 250:243.

*23* is at the intersection of the 'pelogic' and 'kleismic' lines, so
23-equal can be derived from 5-limit JI by tempering out 135:128 and
15625:15552.

*14* and *24* appear on the chart too, superimposed upon 7 and 12
respectively, since they temper out the same commas as their halfling
kin. but only half the notes of these larger tunings can be
reasonably said to derive from a single instance of 5-limit JI -- the
other half are "incompatible" and truly result, as you imply, from
constructing equal-sized intervals bisecting those of 7- and 12-equal
respectively.

i'll be more than happy to answer questions if anyone's confused (i
assume someone must be!), and also to describe how the *next* ("small
5-limit intervals") chart on that webpage helps one to understand the
information above when used in conjuction with the "honeycomb"
lattices of the ETs:

/tuning/files/perlich/15.gif
/tuning/files/perlich/22.gif

(there are more like these on the tuning-math list)

🔗Jon Szanto <JSZANTO@ADNC.COM>

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:

> Paul,
>
> To a man with a hammer, everything looks like a nail.

watch your fingers around him!

> To a tuning theorist with Excel, Maple, or MatLab, every equal
>splitting of the octave looks like a temperament.

not all of them do reasonably look like temperaments, as i tried to
point out. and i'm hoping that the "honeycomb" charts (which require
nothing more than clock addition to put together) make the facts i
presented visually immediate. the only thing is that one still has to
multiply various simple ratios to get the commas i posted. even this
amount of arithmetic can be avoided if i had expressed the commas in
vector form, as they are in one column of monz's chart -- then one
simply has to be able to *count* to arrive at all the facts i posted.

so there's really no math beyond addition and subtraction involved.
if that isn't clear to anyone, i'd like to make it clear, so please
let me know!

> I do see what you are getting at. I hope you can also imagine that
>other people would approach a division of the octave space into some
>discrete number of equal portions as strictly that, and *not* a
>tempering of something else.

yes, this would seem to apply to some atonalists' use of 12-equal,
and a bit more recently, of other equal temperaments, typically
subdivisions of 12-equal. before and well into the rise of atonality,
though, the use of 12-equal by the great composers was intimately
entwined with which commas it tempers out -- w. a. mathieu's book
_the harmonic experience_, published by shambhala, is a beautiful
exposition of this thesis:

http://www.gotoit.com/titles/harexp.htm

(of course i don't agree with everything in this book!)

> Probably the main point is that not everyone would approach an ET
>the way Blackwood would.

of course -- that's why i only said "the way blackwood used them,
many of these tunings *are* temperaments." i didn't mean to imply
that anyone else needed to approach them this way. i didn't even say
blackwood approached them *all* this way. and certainly one need not
be cognizant of this sort of derivation (i don't even think blackwood
was) to end up using these tunings in ways which reflect/exploit the
concomitant properties. but rick was talking specifically about
blackwood, and it was within and around that context that my reply
was intended. i posted it here because i thought it contained some
interesting information that someone might like to see.

thanks for your salutary remarks. and if anything still seems
overly "theoristic" or like it involves higher math or math software,
please let me know -- making it clear to you in simple terms will
probably also help make it clear to many others reading along, or at
least will help develop *my* ability to explain things so that my
future writings will come out clearer and simpler!

🔗Jon Szanto <JSZANTO@ADNC.COM>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > To a man with a hammer, everything looks like a nail.
>
> watch your fingers around him!

And he installs windows too.

> not all of them do reasonably look like temperaments, as i tried to
> point out.

It was a somewhat light-hearted analogy. Guess that didn't come through; you needn't have further documented.

> > Probably the main point is that not everyone would approach an ET
> >the way Blackwood would.
>
> of course -- that's why i only said "the way blackwood used them,
> many of these tunings *are* temperaments." i didn't mean to imply
> that anyone else needed to approach them this way.

Got it. Does this mean that there should be a semantic movement towards defining things as "16tet" vs "16eq" or something? Doesn't matter to me, but...

> thanks for your salutary remarks. and if anything still seems
> overly "theoristic" or like it involves higher math or math software,
> please let me know -- making it clear to you in simple terms will
> probably also help make it clear to many others reading along, or at
> least will help develop *my* ability to explain things so that my
> future writings will come out clearer and simpler!

I'm cool on this one.

Cheers,
Jon

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_47160.html#47160

>
> i'll be more than happy to answer questions if anyone's confused (i
> assume someone must be!),

***Hi Paul,

Actually, this post was almost the necessary "gentle introduction" to
this that I needed. It also shows the value of *repetition* since I
believe it's the first time I've gotten a glimmer of what this is all
about!

I didn't understand that temperament is, in a way, a *process* and,
it seems, temperament implies the tempering of *ONLY ONE* comma: at
least that's how you show it in regard to the temperament names.

And it seems that EQUAL temperament needs the tempering of *two*
commas in every instance. So, actually, equal temperament is a kind
of "two dimensional" temperament.

Well, I may have some of this wrong, but at least it's seeming
to "gell" a little bit more than it did before...

and also to describe how the *next* ("small
> 5-limit intervals") chart on that webpage helps one to understand
the
> information above when used in conjuction with the "honeycomb"
> lattices of the ETs:
>
> /tuning/files/perlich/15.gif
> /tuning/files/perlich/22.gif
>

***Yes, please continue, since this is helping a lot (at least it
seems to *me* it is! :)

Joseph

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> warning -- this post contains lots of numbers!

You need to work harder to scare me off. :)

I think below we should keep the various vals in question
distinguised, and look at the TM reduced basis.

12: <81/80, 128/125>

> one instance of *13* is at the intersection of the 'magic' and 'beep'
> lines, so 13-equal can be derived from 5-limit JI by tempering out
> 3125:3072 and 27:25.

This '13' is [13, 20, 30], with TM basis <27/25, 1125/1024>

> another instance of *13* is at the intersection of the 'father'
> and 'tetracot' lines, so 13-equal can be derived from 5-limit JI by
> tempering out 16:15 (missing from monz's table) and 20000:19683.

This '13' is [13, 21, 31] with TM basis <16/15, 6561/6250>

Another 13 possibility is the "standard" [13, 21, 30]; this has
<25/24, 2560/2187> as a TM basis.

If anyone is getting the feeling that 13-et is a crappy 5-limit
system, go with it.

> *15* is at the intersection of
> the 'blackwood', 'augmented', 'kleismic', and 'porcupine' lines, so
> 15-equal can be derived from 5-limit JI by tempering out any two of
> 256:243, 128:125, 15625:15552, and 250:243.

TM basis <128/125, 250/243>.

> *16* is at the intersection of the 'diminished', 'pelogic',
> and 'magic' lines, so 16-equal can be derived from 5-limit JI by
> tempering out any two of 648:625, 135:128, and 3125:3072.

TM basis <135/128, 648/625>

> one instance of *17* is at the intersection of the 'schismic'
> and 'dicot' lines, so 17-equal can be derived from 5-limit JI by
> tempering out 32805:32768 and 25:24.

The "standard" 17, [17, 27, 39], with TM basis <25/24, 20480/19683>.

blackwood's usage actually
> corresponds to the *another* instance of 17, on the extreme left on
> the 'meantone' line.

The meantone val, [17, 27, 40], with TM basis <81/80, 2048/1875>.

> *18* is at the intersection of the 'augmented' and 'semisuper' lines,
> so 18-equal can be derived from 5-limit JI by tempering out 128:125
> and 6115295232:6103515625.

It makes a lot more sense with the TM basis, <128/125, 800/729>.

admittedly, situations where the latter
> interval needs to be tempered out will probably be quite rare, and
> when they occur, will usually be handled with more accurate
> temperaments.
>
> *19* is at the intersection of
> the 'meantone', 'negri', 'magic', 'kleismic', and 'semisixths' lines,
> so 19-equal can be derived from 5-limit JI by tempering out any two
> of 81:80, 16875:16384, 3125:3072, 15625:15552, and 78732:78125.

TM basis <81/80, 3125/3072>

> one instance of *20* is at the intersection of
> the 'blackwood', 'diminished', and 'tetracot' lines, so 20-equal can
> be derived from 5-limit JI by tempering out any two of 256:243,
> 648:625, and 20000:19683.

This is the [20, 32, 47] val, with TM basis <256/243, 648/625>.
There's also the 20-et which is two 10-ets.

> one instance of *21* is at the intersection of the 'augmented'
> and 'escapade' lines, so 21-equal can be derived from 5-limit JI by
> tempering out 128:125 and 4294967296:4271484375. admittedly,
> situations where the latter interval needs to be tempered out will
> probably be quite rare, and when they occur, will usually be handled
> with more accurate temperaments.

The TM basis <128/125, 2187/2000> makes more sense of it.

> *22* is at the intersection of the 'diaschismic', 'magic',
> and 'porcupine' lines, so 22-equal can be derived from 5-limit JI by
> tempering out any two of 2048:2025, 3125:3072, and 250:243.

TM basis <250/243, 2048/2025>.

> *23* is at the intersection of the 'pelogic' and 'kleismic' lines, so
> 23-equal can be derived from 5-limit JI by tempering out 135:128 and
> 15625:15552.

TM basis <135/128, 6561/6250>.

The comma 6561/6250 appears in the TM basis of both 13 and 23, and
should probably be added to your chart.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

hi Jon,

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:

> > > [Jon Szanto:]
> > > Probably the main point is that not everyone would
> > > approach an ET the way Blackwood would.
> >
> > [paul erlich:]
> > of course -- that's why i only said "the way blackwood
> > used them, many of these tunings *are* temperaments."
> > i didn't mean to imply that anyone else needed to approach
> > them this way.
>
> [Jon:]
> Got it. Does this mean that there should be a semantic
> movement towards defining things as "16tet" vs "16eq"
> or something? Doesn't matter to me, but...

that's *exactly* why i began using "EDO" instead of
(or in addition to) "ET" a few years back, and documented
the difference in the Tuning Dictionary:

http://sonic-arts.org/dict/edo.htm

Dan Stearns was the person who first mentioned that i
should put something about this difference in the Dictionary.

another webpage that might help readers who are struggling
to understand what paul wrote about EDOs-as-temperaments is:

http://sonic-arts.org/dict/bingo.htm

that page actually does have two examples of paul's
"honeycomb" graphs: one at the beginning showing the
commas which are tempered out, and one example ET under 31edo.
i'd like to use more of his honeycombs, but haven't had
the time to do the extra graphical work.

anyway, my bingo-card lattices *do* show all of the
5-limit commas which vanish in any given temperament
(within the exponent-limits of the lattice). if you look
at the bingo-card diagrams while you read paul's post
explaining Blackwood's use of 13-to-24-ET, you'll see them.

/tuning/topicId_47160.html#47160

-monz

🔗monz <monz@attglobal.net>

hi Joe,

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

> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_47160.html#47160
>
> >
> > i'll be more than happy to answer questions if anyone's
> > confused (i assume someone must be!),
>
> ***Hi Paul,
>
> Actually, this post was almost the necessary
> "gentle introduction" to this that I needed. It also
> shows the value of *repetition* since I believe it's
> the first time I've gotten a glimmer of what this is all
> about!
>
> I didn't understand that temperament is, in a way, a
> *process* and, it seems, temperament implies the tempering
> of *ONLY ONE* comma: at least that's how you show it
> in regard to the temperament names.
>
> And it seems that EQUAL temperament needs the tempering
> of *two* commas in every instance. So, actually,
> equal temperament is a kind of "two dimensional" temperament.
>
> Well, I may have some of this wrong, but at least it's seeming
> to "gell" a little bit more than it did before...

yes, Joe, you are indeed beginning to understand this better.

i just wanted to mention to you (and everyone else here)
that all of this process will be made obvious, and very
visual, when using my software. the date of release 1.0
is now expected for February 2004.

-monz

🔗monz <monz@attglobal.net>

hi Joe,

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

> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_47160.html#47162
>
> >
> > yes, this would seem to apply to some atonalists' use
> > of 12-equal, and a bit more recently, of other equal
> > temperaments, typically subdivisions of 12-equal.
> > before and well into the rise of atonality,
>
>
> ***It seems as though in the early use of atonality,
> the "free" period of Schoenberg and such like, such
> derivative approaches from temperament were probably
> still coming into play... (??)

i certainly think so.

the very rudimentary form (so far) of my work on this
(which will eventually be transformed into my second book,
_Searching for Schoenberg's Pantonality_) is here:

http://sonic-arts.org/monzo/schoenberg/harm/1911-1922.htm

from the time of his first "atonal" (pantonal) work
in 1908 until his unfinished _Jakobsleiter_ during
World War 1, Schoenberg did indeed believe that his
free use of 12edo was implying 11-limit ratios.

he made a comment in "Twelve Tone Composition"
(a paper from 1923 which is in _Style and Idea_, p 207)
that during this first "atonal" period he and his students
deliberately avoided familiar harmonies -- which i take
to mean 3-, 5-, and possibly 7-limit ratios -- and
emphasized the unfamiliar higher-limit ones.

in his lecture "Problems of Harmony" (originally given
in IIRC 1927, printed in _Style and Idea_), Schoenberg
also invokes 13-limit ratios in his explanation of the
"rational implications" of 12edo. but in the 1911 version
of _Harmonielehre_ his explanation only goes as high as
11-limit. so his comment about "emphasizing the
unfamiliar overtones" refers to ratios of 11.

i wrote about this before in greater detail ...

/tuning/topicId_32256.html#32407

/tuning/topicId_32256.html#32409

... and you responded to those posts, Joe, so you've seen
them before.

(BTW, have you gotten your copy of _Style and Idea_ yet?
... i really think that that's one book *you'd* like to have!)

-monz

🔗Paul Erlich <paul@stretch-music.com>

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

> > and also to describe how the *next* ("small
> > 5-limit intervals") chart on that webpage helps one to understand
> the
> > information above when used in conjuction with the "honeycomb"
> > lattices of the ETs:
> >
> > /tuning/files/perlich/15.gif
> > /tuning/files/perlich/22.gif
> >
>
> ***Yes, please continue, since this is helping a lot (at least it
> seems to *me* it is! :)
>
> Joseph

well, why don't i make this a bit of an exercise. firstly, look at
the honeycombs above for 15-equal and 22-equal. can you see that
these are simply the 5-limit lattices for these tunings? observe how
the numbers along any direction just increment by an additive
constant, except when the number gets below 0 or above the number of
notes per octave, in which case it "wraps around" (this is called
*clock arithmetic*).

using the honeycombs above, and looking at the "small 5-limit
intervals" honeycomb chart (the second chart on monz's equal
temperament dictionary page), but not looking at any of the other
charts or tables there or my previous postings, can you identify any
of the commas that vanish in 22-equal? in 15-equal? give it a try.

🔗Paul Erlich <paul@stretch-music.com>

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

> > > [paul erlich:]
> > > of course -- that's why i only said "the way blackwood
> > > used them, many of these tunings *are* temperaments."
> > > i didn't mean to imply that anyone else needed to approach
> > > them this way.
> >
> > [Jon:]
> > Got it. Does this mean that there should be a semantic
> > movement towards defining things as "16tet" vs "16eq"
> > or something? Doesn't matter to me, but...
>
>
> that's *exactly* why i began using "EDO" instead of
> (or in addition to) "ET" a few years back, and documented
> the difference in the Tuning Dictionary:
>
> http://sonic-arts.org/dict/edo.htm
>
> Dan Stearns was the person who first mentioned that i
> should put something about this difference in the Dictionary.
[...]
> anyway, my bingo-card lattices *do* show all of the
> 5-limit commas which vanish in any given temperament
> (within the exponent-limits of the lattice). if you look
> at the bingo-card diagrams while you read paul's post
> explaining Blackwood's use of 13-to-24-ET, you'll see them.
>
> /tuning/topicId_47160.html#47160

some of them don't correspond, because you're constructing your
diagrams without regard for what the best major sixth and minor third
in a particular equal tuning might be, and as a result you have only
one "instance", as i called it (gene would say "val"), for each equal
tuning.

but, worst of all, they say "how N-edo really works", contradicting
the intended dichotomy between edo and et as you presented it to jon
above! i could find other such deviations from your intended usage
throughout your webpages . . . but i can't very well complain given
the *quantity* of work you've put into all these pages . . .

🔗monz <monz@attglobal.net>

hi paul,

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > anyway, my bingo-card lattices *do* show all of the
> > 5-limit commas which vanish in any given temperament
> > (within the exponent-limits of the lattice). if you look
> > at the bingo-card diagrams while you read paul's post
> > explaining Blackwood's use of 13-to-24-ET, you'll see them.
> >
> > /tuning/topicId_47160.html#47160
>
> some of them don't correspond, because you're constructing
> your diagrams without regard for what the best major sixth
> and minor third in a particular equal tuning might be, and
> as a result you have only one "instance", as i called it
> (gene would say "val"), for each equal tuning.
>
> but, worst of all, they say "how N-edo really works",
> contradicting the intended dichotomy between edo and et
> as you presented it to jon above! i could find other such
> deviations from your intended usage throughout your
> webpages . . . but i can't very well complain given
> the *quantity* of work you've put into all these pages . . .

thanks, paul. your objections are duly noted.
of course i know this stuff too, but have never gotten
around to updating the webpages to reflect that i know it!

actually it would be fairly easy to add a few other
examples of bingo-cards for those ETs that have alternate
mappings. someday ...

meanwhile, work on the software continues.
i have had to make that my main priority right now.

-monz

🔗monz <monz@attglobal.net>

🔗Paul Erlich <paul@stretch-music.com>

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

> I didn't understand that temperament is, in a way, a *process* and,
> it seems, temperament implies the tempering of *ONLY ONE* comma: at
> least that's how you show it in regard to the temperament names.

those are 5-limit *linear* temperaments, meantone of course being the
most famous example, schismic possibly second (but far less famous).
diaschismic seems to be less well-known, perhaps because the period
is 1/2 octave instead of the usual octave . . .

a 7-limit linear temperament, such as miracle for example, requires
the tempering of *two* independent commas (you could use 225:224 and
2401:2400, or 225:224 and 1029:1024).

if you think of the 5-limit as 2-dimensional, and the 7-limit as 3-
dimensional, etc., each independent comma that you temper out reduces
the dimensionality of the temperament by 1, so to end up with a 1-
dimensional (linear) temperament, you need to temper out one comma in
the 5-limit, two in the 7-limit, etc.

> And it seems that EQUAL temperament needs the tempering of *two*
> commas in every instance.

that's only true in the 5-limit case. in the 7-limit, you'd need to
temper out *three* independent commas to get an equal temperament. in
the 3-limit, you only need *one* -- that's why 12-equal results
immediately from tempering out the pythagorean comma, 531441:524288 --
note that the only prime factors there are 2 and 3. this would
correspond to the "chinese" derivation of 12-equal, or perhaps even
an "aristoxenean" one -- ratios of 5 and above are not recognized as
consonances in these frameworks.

> So, actually, equal temperament is a kind
> of "two dimensional" temperament.

rather, it's a kind of "zero dimensional" temperament, since you've
reduced the number of infinite dimensions in the lattice down to
zero, and end up with a finite set of pitches. it's the original 5-
limit lattice that's two dimensional, really.

🔗monz <monz@attglobal.net>

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_47160.html#47191

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > > and also to describe how the *next* ("small
> > > 5-limit intervals") chart on that webpage helps one to
understand
> > the
> > > information above when used in conjuction with the "honeycomb"
> > > lattices of the ETs:
> > >
> > > /tuning/files/perlich/15.gif
> > > /tuning/files/perlich/22.gif
> > >
> >
> > ***Yes, please continue, since this is helping a lot (at least it
> > seems to *me* it is! :)
> >
> > Joseph
>
> well, why don't i make this a bit of an exercise. firstly, look at
> the honeycombs above for 15-equal and 22-equal. can you see that
> these are simply the 5-limit lattices for these tunings? observe
how
> the numbers along any direction just increment by an additive
> constant, except when the number gets below 0 or above the number
of
> notes per octave, in which case it "wraps around" (this is called
> *clock arithmetic*).
>

***Sure, that would make sense since each instance of an interval
going in one direction on the lattice would have the same number of
steps...

> using the honeycombs above, and looking at the "small 5-limit
> intervals" honeycomb chart (the second chart on monz's equal
> temperament dictionary page), but not looking at any of the other
> charts or tables there or my previous postings, can you identify
any
> of the commas that vanish in 22-equal? in 15-equal? give it a try.

***I really love it, Paul, when you present exercises like this.
This is great fun, a bit like a comma "Easter Egg" hunt!

Well, I get the following:

15-tET:

maximal diesis
limma
diesis (great/minor)

22-tET:

diaschisma
maximal diesis
small diesis
semicomma

I didn't "cheat" but I may go look at the rest of the Monz page now.

Speaking of which, I was wondering how you created the "main" graphic
on that page with the comma lines.

I understand that one axis is the just fifths, and the other is the
just thirds, so I guess you could plot the "slope" of the lines by
the deviations of both of those.

But, how did you determine the *specific* place on the line where the
individual ETs would land? I guess you could determine the
percentage deviation of the fifths and thirds in each case and figure
out where the ET would go on the line in each case. You must have
done that, or something like that...??

Thanks again!

Joseph

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_47160.html#47197

> if you think of the 5-limit as 2-dimensional, and the 7-limit as 3-
> dimensional, etc., each independent comma that you temper out
reduces the dimensionality of the temperament by 1, so to end up with
a 1-dimensional (linear) temperament, you need to temper out one
comma in the 5-limit, two in the 7-limit, etc.
>
> > And it seems that EQUAL temperament needs the tempering of *two*
> > commas in every instance.
>
> that's only true in the 5-limit case. in the 7-limit, you'd need to
> temper out *three* independent commas to get an equal temperament.
in the 3-limit, you only need *one* -- that's why 12-equal results
> immediately from tempering out the pythagorean comma,
531441:524288 --note that the only prime factors there are 2 and 3.
this would correspond to the "chinese" derivation of 12-equal, or
perhaps even an "aristoxenean" one -- ratios of 5 and above are not
recognized as consonances in these frameworks.
>

***Paul, why is it that the *equal* temperaments make it necessary to
temper out an *additional* comma? (i.e. 5-limit linear has *one*,
but 5-limit ET needs *two..* etc.??)

> > So, actually, equal temperament is a kind
> > of "two dimensional" temperament.
>
> rather, it's a kind of "zero dimensional" temperament, since you've
> reduced the number of infinite dimensions in the lattice down to
> zero, and end up with a finite set of pitches. it's the original 5-
> limit lattice that's two dimensional, really.

***That's fascinating to try to visualize... Wouldn't it be great if
there were some kind of video that would show a lattice morphing
somehow into a set of pitches?? I'm not sure how that would be
done...

Thanks!

Joseph

🔗monz <monz@attglobal.net>

hi Joe,

i'm sure paul will explain in greater detail ... and
probably before he realizes that i've already answered you! :)

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

> I didn't "cheat" but I may go look at the rest of the Monz page now.
>
> Speaking of which, I was wondering how you created the
> "main" graphic on that page with the comma lines.
>
> I understand that one axis is the just fifths, and the other
> is the just thirds, so I guess you could plot the "slope"
> of the lines by the deviations of both of those.
>
> But, how did you determine the *specific* place on the line
> where the individual ETs would land? I guess you could
> determine the percentage deviation of the fifths and thirds
> in each case and figure out where the ET would go on the
> line in each case. You must have done that, or something
> like that...??

paul used a triangular plotting convention similar to the
one used by John Chalmers on the cover of his _Divisions
of the Tetrachord_ ... it's also here (#4 is the good one):

http://sonic-arts.org/chalmers/diagrams.htm

there are actually *3* lines representing just ratios,
not 2. one is the "4th/5th", one is the "major-3rd/minor-6th",
the third is the "minor-3rd/major-6th". these are the
three pairs of basic concordant 5-limit ratios.

since those ratios are co-dependant (i.e., the "major-3rd"
plus the "minor-3rd" equals the "perfect 5th"), then for
any given mapping of JI to ET, each ET will obviously
have to be plotted at one specific point where the errors
for all three dimensions meet.

for those ETs that have multiple possible mappings, the
numbers appear in more than one place on the graphs.

hope that helps.

-monz

🔗monz <monz@attglobal.net>

🔗Paul Erlich <paul@stretch-music.com>

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

> ***I really love it, Paul, when you present exercises like this.
> This is great fun, a bit like a comma "Easter Egg" hunt!
>
> Well, I get the following:
>
> 15-tET:
>
> maximal diesis
> limma
> diesis (great/minor)
>
> 22-tET:
>
> diaschisma
> maximal diesis
> small diesis
> semicomma
>
> ??
>
> I didn't "cheat" but I may go look at the rest of the Monz page now.

looking at the first two pages of this database,

/tuning/database?
method=reportRows&tbl=10&sortBy=3

i was able to verify that you're right!

> Speaking of which, I was wondering how you created the "main"
graphic
> on that page with the comma lines.
>
> I understand that one axis is the just fifths, and the other is the
> just thirds, so I guess you could plot the "slope" of the lines by
> the deviations of both of those.

no, that's simply how i plotted the ETs themselves.

> But, how did you determine the *specific* place on the line where
the
> individual ETs would land?

see above. i drew the green lines *by hand* (though i could have
plotted them mathematically) -- all i had to do was find two ETs that
were examples of the linear temperament and draw a straight line
passing through both; it then automatically passes through all the
other ETs that are examples of that temperament. also note that each
line has the same slope as the arrow representing the corresponding
comma in the other ("small 5-limit intervals") graph!

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_47160.html#47197
>
> > if you think of the 5-limit as 2-dimensional, and the 7-limit as
3-
> > dimensional, etc., each independent comma that you temper out
> reduces the dimensionality of the temperament by 1, so to end up
with
> a 1-dimensional (linear) temperament, you need to temper out one
> comma in the 5-limit, two in the 7-limit, etc.
> >
> > > And it seems that EQUAL temperament needs the tempering of
*two*
> > > commas in every instance.
> >
> > that's only true in the 5-limit case. in the 7-limit, you'd need
to
> > temper out *three* independent commas to get an equal
temperament.
> in the 3-limit, you only need *one* -- that's why 12-equal results
> > immediately from tempering out the pythagorean comma,
> 531441:524288 --note that the only prime factors there are 2 and 3.
> this would correspond to the "chinese" derivation of 12-equal, or
> perhaps even an "aristoxenean" one -- ratios of 5 and above are not
> recognized as consonances in these frameworks.
> >
>
> ***Paul, why is it that the *equal* temperaments make it necessary
to
> temper out an *additional* comma? (i.e. 5-limit linear has *one*,
> but 5-limit ET needs *two..* etc.??)

each comma you temper out reduces the number of infinite dimensions
in the lattice by one. so if you start with the 5-limit JI lattice,
which has two infinite dimensions, and temper out one comma, you end
up with a linear temperament, with only one infinite dimension. in
the direction of the comma you tempered out, you no longer have
infinitude, just repetition. hopefully _the forms of tonality_ makes
this clear. now if you temper out another comma, you can get very
close to *any* point in the lattice by moving some number of the
first comma and some number of the second comma, which of course
amounts to *zero* pitch movement, and then a few small steps will get
you to the point you want. hopefully the "honeycombs" of various ETs
you were just looking at make this clear. thus the entire lattice now
contains only a small, finite number of intervals, with repetition in
*every* direction that you look. assuming you did the tempering
regularly, this means an equal temperament. in two dimensions, it
requires the tempering-out of two independent commas to ge this
condition.

if you're used to thinking of linear temperaments in terms of
generator, note that the generator of a linear temperament can, in
general, be stacked upon itself an infinite number of times and never
yield the same (octave-reduced) pitch twice. thus, again, linear
temperaments have one dimension of infinitude. now if you temper out
*another* comma (not the one which defined the linear temperament in
the first place), which will at this point have to correspond to some
number of generators (and periods) in the linear temperament, you're
turning the infinite chain of generators into a circle; i.e., an
equal temperament.

> > > So, actually, equal temperament is a kind
> > > of "two dimensional" temperament.
> >
> > rather, it's a kind of "zero dimensional" temperament, since
you've
> > reduced the number of infinite dimensions in the lattice down to
> > zero, and end up with a finite set of pitches. it's the original
5-
> > limit lattice that's two dimensional, really.
>
> ***That's fascinating to try to visualize... Wouldn't it be great
if
> there were some kind of video that would show a lattice morphing
> somehow into a set of pitches?? I'm not sure how that would be
> done...
>
> Thanks!
>
> Joseph

i have an exercise you can do with scissors and tape that can help
you visualize at least the first step, where you temper out one
comma . . . wait, didn't we do this already?

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_47160.html#47235
>
> if you're used to thinking of linear temperaments in terms of
> generator, note that the generator of a linear temperament can, in
> general, be stacked upon itself an infinite number of times and
never
> yield the same (octave-reduced) pitch twice. thus, again, linear
> temperaments have one dimension of infinitude. now if you temper
out
> *another* comma (not the one which defined the linear temperament
in
> the first place), which will at this point have to correspond to
some
> number of generators (and periods) in the linear temperament,
you're
> turning the infinite chain of generators into a circle; i.e., an
> equal temperament.
>

***Well, I guess I'm getting confused again... :(

Is quarter comma meantone a *linear* temperament? It rather looks as
though it is on the chart. Each of the fifths is tempered in
meantone and every octave comes out the same...

But wait... maybe that's just because the octave-reduced pitches
aren't carried out fully... ?

I guess it would have to be 31-tET in order to be the "fully
circulating" version of this...

Or am I just getting mixed up again...

Thanks!

Joseph

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_47160.html#47235
> >
> > if you're used to thinking of linear temperaments in terms of
> > generator, note that the generator of a linear temperament can,
in
> > general, be stacked upon itself an infinite number of times and
> never
> > yield the same (octave-reduced) pitch twice. thus, again, linear
> > temperaments have one dimension of infinitude. now if you temper
> out
> > *another* comma (not the one which defined the linear temperament
> in
> > the first place), which will at this point have to correspond to
> some
> > number of generators (and periods) in the linear temperament,
> you're
> > turning the infinite chain of generators into a circle; i.e., an
> > equal temperament.
> >
>
> ***Well, I guess I'm getting confused again... :(
>
> Is quarter comma meantone a *linear* temperament?

yes . . .

> It rather looks as
> though it is on the chart. Each of the fifths is tempered in
> meantone and every octave comes out the same...
>
> But wait... maybe that's just because the octave-reduced pitches
> aren't carried out fully... ?

which chart are you looking at? i don't know what you mean :(

> I guess it would have to be 31-tET in order to be the "fully
> circulating" version of this...

yes, or 12, or 19, or 43, or 50, or 55 . . .

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_47160.html#47358
> > >
> >
> > ***Well, I guess I'm getting confused again... :(
> >
> > Is quarter comma meantone a *linear* temperament?
>
> yes . . .
>
> > It rather looks as
> > though it is on the chart. Each of the fifths is tempered in
> > meantone and every octave comes out the same...
> >
> > But wait... maybe that's just because the octave-reduced pitches
> > aren't carried out fully... ?
>
> which chart are you looking at? i don't know what you mean :(
>

***Hi Paul,

I was just looking at the chart on the Monz ET page that shows the
commas and the linear temperament names...

> > I guess it would have to be 31-tET in order to be the "fully
> > circulating" version of this...
>
> yes, or 12, or 19, or 43, or 50, or 55 . . .

***Well, I meant a "fully circulating" version of 1/4 comma meantone,
so I believe 31-tET would be the only one to qualify for *that*, yes??

And, if I recall correctly, 31-tET is *still* slightly off from a
truly circulating 1/4 comma meantone, but by only a minuscule amount.

Do I remember that correctly??

Thanks!

Joseph

🔗monz <monz@attglobal.net>

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_47160.html#47358
> > > >
> > >
> > > ***Well, I guess I'm getting confused again... :(
> > >
> > > Is quarter comma meantone a *linear* temperament?
> >
> > yes . . .
> >
> > > It rather looks as
> > > though it is on the chart. Each of the fifths is tempered in
> > > meantone and every octave comes out the same...
> > >
> > > But wait... maybe that's just because the octave-reduced
pitches
> > > aren't carried out fully... ?
> >
> > which chart are you looking at? i don't know what you mean :(
> >
>
> ***Hi Paul,
>
> I was just looking at the chart on the Monz ET page that shows the
> commas and the linear temperament names...

well then, i don't know where you're getting "every octave comes out
the same" and "the octave-reduced pitches aren't carried out
fully" . . .

> > > I guess it would have to be 31-tET in order to be the "fully
> > > circulating" version of this...
> >
> > yes, or 12, or 19, or 43, or 50, or 55 . . .
>
> ***Well, I meant a "fully circulating" version of 1/4 comma
meantone,
> so I believe 31-tET would be the only one to qualify for *that*,
yes??

almost, but not exactly, as you yourself state below:

> And, if I recall correctly, 31-tET is *still* slightly off from a
> truly circulating 1/4 comma meantone, but by only a minuscule
amount.
>
> Do I remember that correctly??

yes. i could have included marks for 1/4-comma meantone and all sorts
of other regular 5-limit temperaments on the chart, but didn't. if i
did, you'd see that the 1/4-comma meantone point would be really
close to the 31-equal point, and even closer to the 205-equal
point . . .

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_47160.html#47390

> hi Joe,
>
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > ***Well, I meant a "fully circulating" version of 1/4 comma
> > meantone, so I believe 31-tET would be the only one to qualify
> > for *that*, yes??
> >
> > And, if I recall correctly, 31-tET is *still* slightly off
> > from a truly circulating 1/4 comma meantone, but by only a
> > minuscule amount.
> >
> > Do I remember that correctly??
>
>
> yes. i document the discrepancies here:
>
> http://sonic-arts.org/dict/1-4cmt.htm
>
> http://sonic-arts.org/dict/31edo.htm
>
>
> note that, since 31edo is *not* exactly the same as
> true 1/4-comma meantone, there are other ETs which also
> approximate 1/4-comma meantone, and in fact approximate
> it better ... but they have higher cardinalities.
> 31edo is the simplest.
>
>
>
> -monz

***Hmm... according to these, it looks as though there is a little
interval "left over" after the 31 chain... hence a "32..."

Oh, by the way, Monz, while I have you "on the line," I've been
listening again to Schoenberg's Second String Quartet. I think it's
one of the best pieces of music ever written. More on Metatuning...

J. Pehrson

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_47160.html#47394

> >
> > ***Hi Paul,
> >
> > I was just looking at the chart on the Monz ET page that shows
the
> > commas and the linear temperament names...
>
> well then, i don't know where you're getting "every octave comes
out
> the same" and "the octave-reduced pitches aren't carried out
> fully" . . .
>

***I'm really not expressing myself very well here... I was just
thinking about the standard 12-note meantone and how every octave of
the piano (or whatever) is tuned with octave equivalence. Obviously,
the meantone chain goes further than that in scale generation with
more pitches, if one would let it, into the split keys of, I guess,
19 and on even further to close to 31-tET... and for the 12-note
one, if you would take all 12 pitches of the chain and bring them
back into an octave, they are not, obviously as "carried out" as if
one were to take 31 fifths of meantone and bring *those* back into an
octave of 31 pitches. Nothing particularly profound here, just
poorly expressed... :)

🔗Kurt Bigler <kkb@breathsense.com>

on 9/30/03 12:24 PM, Paul Erlich <paul@stretch-music.com> wrote:

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

>>>> I guess it would have to be 31-tET in order to be the "fully
>>>> circulating" version of this...
>>>
>>> yes, or 12, or 19, or 43, or 50, or 55 . . .
>>
>> ***Well, I meant a "fully circulating" version of 1/4 comma
> meantone,
>> so I believe 31-tET would be the only one to qualify for *that*,
> yes??
>
> almost, but not exactly, as you yourself state below:
>
>> And, if I recall correctly, 31-tET is *still* slightly off from a
>> truly circulating 1/4 comma meantone, but by only a minuscule
> amount.
>>
>> Do I remember that correctly??
>
> yes. i could have included marks for 1/4-comma meantone and all sorts
> of other regular 5-limit temperaments on the chart, but didn't. if i
> did, you'd see that the 1/4-comma meantone point would be really
> close to the 31-equal point, and even closer to the 205-equal
> point . . .

Hmm.. I coudn't find "circulating" in monz's dictionary. I'm trying to
relate this connection between meantone and ET approximations to circulating
12-tone temperaments.

Maybe a brief entry for "circulating", perhaps including "fully circulating"
would be a worth entry, Monz? (With a link from "temperament".)

Thanks.

-Kurt