Meantone diesis

Occasionally we find names not just for JI intervals, but for
intervals in a temperament, such as "secor". Meantone is particularly
well-supplied, with names for everything from 1 to 11 generators; the
same names can be used for schismatic, of course.

I'm wondering if "meantone diesis" should be the name for 12
generators, or if someone has a better one to suggest. It is the
meantone version of 128/125 (the standard or minor diesis) and 648/625
(the major diesis.)
It's also the meantone version of the diaschisma, 2048/2025, and of
6561/6250 which has been called a diesis also. It's also the meantone
version for the reciprocals of the Pythagorean comma and the schisma,
but we can ignore that, I hope.

If we go to the 7-limit we get it also as 36/35, sometimes called the
septimal diesis, and 50/49, the tritonic diesis. There is also a whole
raft of things called a diesis which are not 12 genetators of
meantone, though sometimes they are one step of 31 equal. For instance
245/243, the minor BP diesis, 525/512, the Avicenna enharmonic diesis,
3125/3072, the small diesis are all -19 generators of meantone, making
them the same in 31-et at least.

Hmmm...augmented unison--how's that's for a name? Meanharmonica? :)

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> Occasionally we find names not just for JI intervals, but for
> intervals in a temperament, such as "secor". Meantone is particularly
> well-supplied, with names for everything from 1 to 11 generators; the
> same names can be used for schismatic, of course.
>
> I'm wondering if "meantone diesis" should be the name for 12
> generators, or if someone has a better one to suggest.

Yes. This is an accepted name for it.

> Hmmm...augmented unison--how's that's for a name?

Diminished second or super unison would be valid, but not augmented
unison. In meantone, "augmented" means raised by a chromatic semitone
(from the perfect or major interval). This is consistent with 10
meantone fifths (octave-reduced) (an approx 4:7) being called either
an augmented sixth or a subminor seventh.

meantone
G#:Ab diesis (diminished second, super unison)
G#:Gx chromatic semitone (subminor second, augmented unison)

See
http://dkeenan.com/Music/IntervalNaming.htm

-- Dave Keenan

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

> > I'm wondering if "meantone diesis" should be the name for 12
> > generators, or if someone has a better one to suggest.
>
> Yes. This is an accepted name for it.

Your naming scheme does not make that clear, since it is a purely
31-et system. Hence it conflates two different meantone intervals, one
for -12 fifths, the other for +19 fifths:

-12 fifths: 36/35, 50/49, 64/63, 128/125, 648/625, 2048/2025 etc.

+19 fifths: 49/48, 245/243, 525/512, 686/675, 16875/16384 etc.

What would be different names for these two? Of course, when it comes
to 11-limit intervals it gets even dicier, with two different systems
which 31 also conflates. 45/44 is a -12 under one system, and a +19
under the other, and so forth. Hence sticking to the 7-limit for names
seems like a good plan.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > > I'm wondering if "meantone diesis" should be the name for 12
> > > generators, or if someone has a better one to suggest.
> >
> > Yes. This is an accepted name for it.
>
> Your naming scheme does not make that clear, since it is a purely
> 31-et system. Hence it conflates two different meantone intervals, one
> for -12 fifths, the other for +19 fifths:
>
> -12 fifths: 36/35, 50/49, 64/63, 128/125, 648/625, 2048/2025 etc.
>
> +19 fifths: 49/48, 245/243, 525/512, 686/675, 16875/16384 etc.

Good point. Please treat the names I gave as being names for -15 to
+15 generators. When generalising to a larger meantone, you could add
the word "wide" or "narrow" as the case may be, to the ones outside of
-15 to +15 generators.

How often does anyone want more than +-15 generators of meantone
anyway? I guess you want 50-ET.

I did a similar thing here
http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt
but that time it was taking 31-ET as a Miracle temperament and
generalising it to 72-ET.

-- Dave

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

> --- In tuning@yahoogroups.com,
> "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > I'm wondering if "meantone diesis" should be the name for 12
> > generators, or if someone has a better one to suggest.
>
> Yes. This is an accepted name for it.

i like it, so you have one more vote for it.

> > Hmmm...augmented unison--how's that's for a name?
>
> Diminished second or super unison would be valid,
> but not augmented unison. In meantone, "augmented"
> means raised by a chromatic semitone (from the
> perfect or major interval). This is consistent
> with 10 meantone fifths (octave-reduced) (an
> approx 4:7) being called either an augmented sixth
> or a subminor seventh.
>
> meantone
> G#:Ab diesis (diminished second, super unison)
> G#:Gx chromatic semitone (subminor second, augmented unison)
>
> See
> http://dkeenan.com/Music/IntervalNaming.htm

Dave is correct about the "augmented unison", which
is also often called the "augmented prime" ... in fact
i think the latter might be more common.

-monz
http://tonalsoft.com
microtonal music software

For all members who are interested in hear theories
and as a contribution to the discussions between
"place theory" and "frequency theory".

I possess an expander with church organ sounds.
Its heighest effective performed frequency is by a
1 /1/3 aliquote the Note C = effective tone G with
about 12,543 Hz. Some years ago I heared this note
still distinctly. Actually - as a result of my age -
the highest audible tone for me is E with 10,560 Hz.

In order not to fall in depressions, I decided to profit
by this shortcoming by a hear experiment.
I selected the two notes Eb (audible with 9,956 Hz)
and the G (inaudible with 12,543 Hz) and tuned
the G to 12,445 Hz so that both tones have been tuned
to a perfect major third.
Then I played alternately either only the Eb or both the
Eb and G in common. I listened whether I could hear a
change in the sound as a result of the resulting difference
tones.
At the start of the experiment I had only the impression that
the sound changed, but I was not sure. Therefore I held both
tones and tuned the upper and inaudible G gliding to one
octave deeper and then back to its original height. In this
way I could follow with my ears some occuring and
disappearing gliding difference tones.

After about 15 minutes my ears have been more sensible
and I was sure to hear the difference tone(s) even when
the G sounded in its original height of for me in principle
inaudible 12,445 Hz.
At the last step of my experiment a second person
behind my back played either the Eb alone or alternately both
tones Eb and G. I had to lift my hand as soon as both tones
had been "on" and I percieved this without any error.

This result cannot become explained by the traditional
"place theory".

I hope that some members have a suitable equipment for
imitating this experiment. But please weight: The effect is
very subtile and the ears require about 15 minutes until
they will produce optimal results.

From: "monz" <monz@...>

> "Dave Keenan" <d.keenan@b...> wrote:
>
>> --- In tuning@yahoogroups.com,
>> "Gene Ward Smith" <gwsmith@s...>
>> wrote:
>>
>> > I'm wondering if "meantone diesis" should be the name for 12
>> > generators, or if someone has a better one to suggest.
>>
>> Yes. This is an accepted name for it.
>
> i like it, so you have one more vote for it.

My vote too.

In fact, I like to classify equal temperaments based on whether 12 consecutive fifths results in a positive or negative interval (the latter would octave-reduce to under an octave): 7, 19, 31, 43 and 55 are negative-diesis; 5, 17, 22, 29, 34, 41, 46 and 53 are positive-diesis, and 12 is zero-diesis.

~Danny~

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
>
> My vote too.
>
> In fact, I like to classify equal temperaments based on whether 12
> consecutive fifths results in a positive or negative interval (the
latter
> would octave-reduce to under an octave): 7, 19, 31, 43 and 55 are
> negative-diesis; 5, 17, 22, 29, 34, 41, 46 and 53 are
positive-diesis, and
> 12 is zero-diesis.
>
> ~Danny~

A little drawing of fourth-sizes:
/tuning/files/jacob/meantree.pdf

Nothing revolutionary, but I like how this piece of the scale tree
shows all of these. And I'm particularly impressed that you can use
this quasi-fibonacci process to get, exhaustively, every single equal
temperment whose fifth is within whatever range you're looking at.
(Excepting multiples, but who needs them?)

From: "Jacob" <jbarton@...>

> A little drawing of fourth-sizes:
> /tuning/files/jacob/meantree.pdf
>
> Nothing revolutionary, but I like how this piece of the scale tree
> shows all of these. And I'm particularly impressed that you can use
> this quasi-fibonacci process to get, exhaustively, every single equal
> temperment whose fifth is within whatever range you're looking at.
> (Excepting multiples, but who needs them?)

Yeah, I noticed early on that all the good ETs conform to this formula: n = 5a + 7b, where a and b are whole numbers. (Obviously, this is because the first two small Pythagorean intervals are 2^8/3^5 and 3^7/2^11.)

Now when you get into ETs with more notes per octave, you'd want to abandon 5 and 7 and mpve on to the smaller Pythagorean commas, which involve higher powers of three, such as 12, 41, 53, 306 and 665.

And I just now discovered a ridiculously high-order equal temperament: 12,276-tone. Pretty accurate up to 11-limit (and the Pythagorean comma is 240 degrees in this tuning, or 15 seconds in my own measurement system for small intervals). 12,276 = 665*18 + 306, by the way.

~Danny~

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
>
> A little drawing of fourth-sizes:
> /tuning/files/jacob/
meantree.pdf
>
> Nothing revolutionary, but I like how this piece of the scale tree
> shows all of these. And I'm particularly impressed that you can use
> this quasi-fibonacci process to get, exhaustively, every single
equal
> temperment whose fifth is within whatever range you're looking at.
> (Excepting multiples, but who needs them?)

my version (delete the line-break):

/tuning/files/monz/edo-mos-
scale-tree.gif

-monz
http://tonalsoft.com
microtonal music software

It is most pleasing to see 29 and 41 tones per octave in those drawings.

Cordially,
Ozan
----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 05 Mayıs 2005 Perşembe 0:32
Subject: [tuning] Re: Meantone diesis

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
>
> A little drawing of fourth-sizes:
> /tuning/files/jacob/
meantree.pdf
>
> Nothing revolutionary, but I like how this piece of the scale tree
> shows all of these. And I'm particularly impressed that you can use
> this quasi-fibonacci process to get, exhaustively, every single
equal
> temperment whose fifth is within whatever range you're looking at.
> (Excepting multiples, but who needs them?)

my version (delete the line-break):

/tuning/files/monz/edo-mos-
scale-tree.gif

-monz
http://tonalsoft.com
microtonal music software

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> From: "monz" <monz@>
>
> > "Dave Keenan" <d.keenan@b...> wrote:
> >
> >> --- In tuning@yahoogroups.com,
> >> "Gene Ward Smith" <gwsmith@s...>
> >> wrote:
> >>
> >> > I'm wondering if "meantone diesis" should be the name for 12
> >> > generators, or if someone has a better one to suggest.
> >>
> >> Yes. This is an accepted name for it.
> >
> > i like it, so you have one more vote for it.
>
> My vote too.
>
> In fact, I like to classify equal temperaments based on whether 12
> consecutive fifths results in a positive or negative interval (the
latter
> would octave-reduce to under an octave): 7, 19, 31, 43 and 55 are
> negative-diesis; 5, 17, 22, 29, 34, 41, 46 and 53 are
positive-diesis, and
> 12 is zero-diesis.

Oops. No. What you are talking about here is the pythagorean comma.
You're looking at a more general case.

It's only in meantone (possibly including 12-equal as one extreme)
that the tempered Pythagorean comma is also the tempered minor diesis
and the tempered major diesis.

These dieses are 5-limit intervals. One is the octave residue of 3
major thirds and the other is the octave residue of 4 minor thirds. No
one seems able to agree which diesis is minor and which is major but
it makes sense to me that the smaller one (when in just intonation)
should be called minor.

The other reason it makes sense to call this a "diesis" and not the
Pythagorean comma, when talking typical meantones, is because with the
tempering, it is diesis-sized (an approximate quartertone) rather than
comma-sized (an approximate eight-tone).

-- Dave

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> Yeah, I noticed early on that all the good ETs conform to this
formula: n =
> 5a + 7b, where a and b are whole numbers. (Obviously, this is
because the
> first two small Pythagorean intervals are 2^8/3^5 and 3^7/2^11.)

I think a better way to see it than concerning yourself with
Pythagorean intervals is to note that both v5 = <5 8 11| and v7 =
<7 11 16| are meantone vals, annihilating 81/80. Hence any linear
combination with positive integer coefficients a and b, with a and b
relatively prime, a*v5+b*v7, will also annihilate 81/80 and define a
meantone val. The tuning will be intermediate between 5 and 7.

> Now when you get into ETs with more notes per octave, you'd want to
abandon
> 5 and 7 and mpve on to the smaller Pythagorean commas, which involve
higher
> powers of three, such as 12, 41, 53, 306 and 665.

Using combinations of 12 and 19 puts things in a more reasonable range.

> And I just now discovered a ridiculously high-order equal temperament:
> 12,276-tone. Pretty accurate up to 11-limit (and the Pythagorean
comma is
> 240 degrees in this tuning, or 15 seconds in my own measurement
system for
> small intervals). 12,276 = 665*18 + 306, by the way.

It's OK, but not really one to write home about.

In a message dated 5/4/2005 8:58:56 PM Eastern Standard Time,
gwsmith@svpal.org writes:
I think a better way to see it than concerning yourself with
Pythagorean intervals is to note that both v5 = <5 8 11| and v7 =
<7 11 16| are meantone vals, annihilating 81/80. Hence any linear
combination with positive integer coefficients a and b, with a and b
relatively prime, a*v5+b*v7, will also annihilate 81/80 and define a
meantone val. The tuning will be intermediate between 5 and 7.
I'm sorry, Gene, but shouldn't this be on the microtonal math list? As you
have pointed out there are differences.

best, Johnny

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 5/4/2005 8:58:56 PM Eastern Standard Time,
> gwsmith@s... writes:

> I'm sorry, Gene, but shouldn't this be on the microtonal math list?
As you
> have pointed out there are differences.

Since the topic originated here, I posted it here. On the tuning-math
list someone could complain it was too obvious to bother stating, and
here someone could complain it was too mathematical to state here. But
if someone posts a topic here, it hardly makes sense to respond there
unless it really was seriously mathematical in a way in which what I
said wasn't. The sad fact of life to those allergic to math is that
the topic of tuning is inherently and ineradicably mathematical.

From: "Dave Keenan" <d.keenan@...>

>> In fact, I like to classify equal temperaments based on whether 12
>> consecutive fifths results in a positive or negative interval (the
> latter
>> would octave-reduce to under an octave): 7, 19, 31, 43 and 55 are
>> negative-diesis; 5, 17, 22, 29, 34, 41, 46 and 53 are
> positive-diesis, and
>> 12 is zero-diesis.
>
> Oops. No. What you are talking about here is the pythagorean comma.
> You're looking at a more general case.

I meant "enharmonic" then, whatever you call an augmented seventh/diminished second.

From: "Gene Ward Smith" <gwsmith@...>

> I think a better way to see it than concerning yourself with
> Pythagorean intervals is to note that both v5 = <5 8 11| and v7 =
> <7 11 16| are meantone vals, annihilating 81/80. Hence any linear
> combination with positive integer coefficients a and b, with a and b
> relatively prime, a*v5+b*v7, will also annihilate 81/80 and define a
> meantone val. The tuning will be intermediate between 5 and 7.

You're just talking about traditional meantones; I'm talking about equal temperaments in general.

>> Now when you get into ETs with more notes per octave, you'd want to
> abandon
>> 5 and 7 and mpve on to the smaller Pythagorean commas, which involve
> higher
>> powers of three, such as 12, 41, 53, 306 and 665.
>
> Using combinations of 12 and 19 puts things in a more reasonable range.

The list of increasingly-smaller Pythagorean commas are chains of these numbers of fifths:

1 2 5 12 41 53 306 665 15601 31867 79335 111202

The powers of three alternate between positive and negative, and the ETs with the better fifths mix positives and negatives. 19 = 12 + 5 + 2, so it's a good composite.

Danny does not read tuning-math, so I am responding here. If you are
allergic to math, don't read this. In my opinion, it is in any case
on-topic for this group.

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> The list of increasingly-smaller Pythagorean commas are chains of these
> numbers of fifths:
>
> 1 2 5 12 41 53 306 665 15601 31867 79335 111202
>
> The powers of three alternate between positive and negative, and the
ETs
> with the better fifths mix positives and negatives. 19 = 12 + 5 + 2,
so it's
> a good composite.

Those are the denominators of the continued fraction for log2(3), and
it follows from the theory of continued fractions that they will be
alternatly greater than and less than log2(3); consequently, the
powers of three alternate between positive and negative. It's an
interesting idea to take sums of three successive convergents, which
gets us out of the semiconvergent game:

8 19 58 106 400 1024 16572 48133...

What this nets us I can't say.

From: "Gene Ward Smith" <gwsmith@...>

> Danny does not read tuning-math, so I am responding here. If you are
> allergic to math, don't read this. In my opinion, it is in any case
> on-topic for this group.

Actually I do read tuning-math; I just don't post much there. It's not my area of expertise, though I'm one of those apparent few that doesn't get turned off by math. It does fascinate me.

> Those are the denominators of the continued fraction for log2(3), and
> it follows from the theory of continued fractions that they will be
> alternatly greater than and less than log2(3); consequently, the
> powers of three alternate between positive and negative. It's an
> interesting idea to take sums of three successive convergents, which
> gets us out of the semiconvergent game:

I never go to continued fractions in high school or college, and I regret that.

> 8 19 58 106 400 1024 16572 48133...
>
> What this nets us I can't say.

ETs of 19 and 106, and also 24 (8 * 3), 65 (8 + 19 * 3) and 72 (8 * 9), are all I can see useful out of that series, superficially. But how did 1024 (2^10, of course) get in there?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> > In a message dated 5/4/2005 8:58:56 PM Eastern Standard Time,
> > gwsmith@s... writes:
>
> > I'm sorry, Gene, but shouldn't this be on the microtonal math list?
> As you
> > have pointed out there are differences.
>
> Since the topic originated here, I posted it here. On the tuning-math
> list someone could complain it was too obvious to bother stating,

Aw, c'mon Gene. No one has ever made such a complaint about anything
posted to tuning-math.

> and
> here someone could complain it was too mathematical to state here.

They not only could. They have. Many people. Many times.

> But
> if someone posts a topic here, it hardly makes sense to respond there
> unless it really was seriously mathematical in a way in which what I
> said wasn't.

Just ask yourself before you post to this list, "Is there anything in
this post other than math?" "Is it basic, high-school math with little
jargon?" If not, please post it to tuning-math. That's why tuning-math
exists. People often redirect topics to tuning math when they become
heavily mathematical. That's why tuning-math exists.

> The sad fact of life to those allergic to math is that
> the topic of tuning is inherently and ineradicably mathematical.

Assuming that some people _are_ "allergic to math", how about a little
courtesy and consideration for them. Please.

-- Dave Keenan

In a message dated 5/5/2005 9:00:01 AM Eastern Standard Time,
d.keenan@bigpond.net.au writes:
The sad fact of life to those allergic to math is that
> the topic of tuning is inherently and ineradicably mathematical.

Assuming that some people _are_ "allergic to math", how about a little
courtesy and consideration for them. Please.

-- Dave Keenan
The sad fact of life, Gene, is that you are pushing the List into your own
image. Increasingly, I and others are finding this offensive. If you are on
metatuning, we can continue this there. FYI, I have been teaching high school
math for months. As an asthmatic I am quite aware to what I am allergic. It
may be in my Genes. ;) Johnny

"monz" <monz@tonalsoft.com> writes:

> --- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> >
> > A little drawing of fourth-sizes:
> > /tuning/files/jacob/
> meantree.pdf

[...]

> my version (delete the line-break):
>
> /tuning/files/monz/edo-mos-
> scale-tree.gif

Hmm, I guess everyone makes that diagram sooner or later, eh? Here's
my short version (fixed width font required):

^ ^
| |
685.71 7 | 7---------------------------
689.36 47 | | | | | | | 47---
690.00 40 | | | | | | 40---------
690.91 33 | | | | | 33------------
692.31 26 | | | | 26------------------
693.33 45 | | | | | 45------
694.74 19 | | | 19---------------------
696.00 50 | | | | | 50---
696.77 31 | | | | 31---------------
697.67 43 | | | | | 43------
700.00 12 | | 12------------------------
701.89 53 | | | | | | 53
701.96 *****|**|**|**|*****|********|*****|
702.44 41 | | | | | 41------
703.45 29 | | | | 29---------------
704.35 46 | | | | | 46------
705.88 17 | | | 17---------------------
707.69 39 | | | | | 39---------
709.09 22 | | | | 22------------------
710.20 49 | | | | | | 49---
711.11 27 | | | | | 27---------------
712.50 32 | | | | | | 32------------
713.51 37 | | | | | | | 37---------
714.29 42 | | | | | | | | 42------
714.89 47 | | | | | | | | | 47---
715.38 52 | | | | | | | | | | 52
720.00 5 5------------------------------
|
V

The column on the left is the generator size in cents (period =
1200.00 cents assumed) for the EDO given in the second column. Then
in the tree itself the pipes ("|") indicate the range of generators
for which one gets a MOS of the given size: for instance, looking at
the third vertical line from the left, one gets MOS of size 12 with
generators from 685.71 to 720.00 cents. Each such range begins and
ends on a horizontal line denoting an equal temperament (for 12, these
are 7-EDO at 685.71 cents and 5-EDO at 720.00 cents.) Likewise you
get MOS of size 41 from 700.00 cents (12-EDO) to 703.45 cents (29-EDO)
(in this case the vertical line from 41 to the 29-EDO line is omitted
because they're on consecutive ASCII lines). The row of asterisks
marks the JI perfect fifth.

For the long version:
< http://www.richholmes.net/music/xen/linscaleranges.txt >

So which side of the tree does the MOS grow on?

- Rich Holmes

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

> Assuming that some people _are_ "allergic to math", how about a little
> courtesy and consideration for them. Please.

I was trying to be courteous to Danny, whom I thought had said did not
read tuning-math any more. I think it is a sad situation when
something as easy as what I posted, which came in response to a
posting here, is attacked as off-topic for this group. I do not see
anything in the description of this group which says using math is
off-topic, but if it is there are a hell of a lot of off-topic posts here.

The objection I think was not to any actual mathematical difficulties
but to the jargon: "linear combination", "greatest common
denominator", "val". This jargon is not impenetrable any more than any
other examples of jargon; I am sure many people would faint if they
heard "apotome", much less "unidecimal augmented fifth", but they can
find out what those mean if they don't already know. If we are going
to avoid jargon, can we still talk about composing in Blackjack, for
instance? Can we use Partch's neologisms, but nothing more recent, or
what would be the rule, exactly?

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

> The sad fact of life, Gene, is that you are pushing the List into
your own
> image. Increasingly, I and others are finding this offensive. If
you are on
> metatuning, we can continue this there. FYI, I have been teaching
high school
> math for months.

Then you ought to be able to explain what, precisely, you found
offensive. This you have not done. If you could do that, and also
explain *why* you think it is unacceptable to respond in such a way to
a posting made on this group, we might get somewhere.

As for pushing the list by being who I am, people are who they are,
and that includes you. Being the person you are should not be some
kind of thought-crime.

--- In tuning@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

/tuning/files/monz/edo-mos-
> > scale-tree.gif

Could the people complaining about math on this group explain if this
is acceptable, unacceptable, borderline, or what, exactly? What can we
talk about on a group devoted to "exchanging ideas relevant to
alternate musical tunings"?

>> The sad fact of life, Gene, is that you are pushing the List into
>>your own image. Increasingly, I and others are finding this offensive.
>>If you are on metatuning, we can continue this there. FYI, I have
>>been teaching high school math for months.
>
>Then you ought to be able to explain what, precisely, you found
>offensive. This you have not done. If you could do that, and also
>explain *why* you think it is unacceptable to respond in such a way to
>a posting made on this group, we might get somewhere.
>
>As for pushing the list by being who I am, people are who they are,
>and that includes you. Being the person you are should not be some
>kind of thought-crime.

True that.

-Carl

>> Assuming that some people _are_ "allergic to math", how about a little
>> courtesy and consideration for them. Please.
>
>I was trying to be courteous to Danny, whom I thought had said did not
>read tuning-math any more. I think it is a sad situation when
>something as easy as what I posted, which came in response to a
>posting here, is attacked as off-topic for this group. I do not see
>anything in the description of this group which says using math is
>off-topic, but if it is there are a hell of a lot of off-topic posts here.
>
>The objection I think was not to any actual mathematical difficulties
>but to the jargon: "linear combination", "greatest common
>denominator", "val". This jargon is not impenetrable any more than any
>other examples of jargon; I am sure many people would faint if they
>heard "apotome", much less "unidecimal augmented fifth", but they can
>find out what those mean if they don't already know. If we are going
>to avoid jargon, can we still talk about composing in Blackjack, for
>instance? Can we use Partch's neologisms, but nothing more recent, or
>what would be the rule, exactly?

Yes, I'd like to know too. Since complainers tend to win when it comes
to setting the the tone and rules around here, maybe we can just cut to
the chase and have them tell us what goes.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> The sad fact of life, Gene, is that you are pushing the List into
> >>your own image. Increasingly, I and others are finding this
offensive.
> >>If you are on metatuning, we can continue this there. FYI, I have
> >>been teaching high school math for months.
> >
> >Then you ought to be able to explain what, precisely, you found
> >offensive. This you have not done. If you could do that, and also
> >explain *why* you think it is unacceptable to respond in such a
way to
> >a posting made on this group, we might get somewhere.
> >
> >As for pushing the list by being who I am, people are who they are,
> >and that includes you. Being the person you are should not be some
> >kind of thought-crime.
>
> True that.
>
> -Carl

Uh, could you guys do the rest of us a kinda favor and kinda continue
this on metatuning.

Thanx!

--GS

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

> The objection I think was not
> to any actual mathematical
> difficulties but to the jargon:
> "linear combination", "greatest
> common denominator", "val".
> This jargon is not impenetrable
> any more than any other examples
> of jargon; I am sure many people
> would faint if they heard "apotome",
> much less "unidecimal augmented
> fifth", but they can find out
> what those mean if they don't
> already know. If we are going
> to avoid jargon, can we still
> talk about composing in Blackjack,
> for instance? Can we use Partch's
> neologisms, but nothing more recent,
> or what would be the rule, exactly?

i created a Yahoo group specifically for this:

/tuning-jargon/

but aside from me, Gene, and Robert, not many here seem
interested.

-monz

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
...
> Diminished second or super unison would be valid, but not augmented
> unison. In meantone, "augmented" means raised by a chromatic
semitone
> (from the perfect or major interval). This is consistent with 10
> meantone fifths (octave-reduced) (an approx 4:7) being called either
> an augmented sixth or a subminor seventh.
>
> meantone
> G#:Ab diesis (diminished second, super unison)
> G#:Gx chromatic semitone (subminor second, augmented unison)
>
> See
> http://dkeenan.com/Music/IntervalNaming.htm
>
> -- Dave Keenan

I find your work on interval names great, Dave, except for the fact
that I disapprove of using super/sub with octaves, unisons, etc.

Since super and sub are prefixes, and not adjectives, I find no logic
in a "super unison". I'd rather "superperfect unison".

In Spanish, one would have "unísono superjusto" where
justo=perfect.

:) Max

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

> Since super and sub are prefixes, and not adjectives, I find no logic
> in a "super unison". I'd rather "superperfect unison".

I'm happy with "meantone diesis". Since both 128/125 and 648/625
are called a diesis and both translate to the same meantone interval,
it seems a good choice. We also have 36/35, 50/49, and 3125/3087 being
called a diesis, and they are the same interval in septimal meantone.
This is the -12 fifths interval of meantone.

For meantones between 12 and 19 we may also want to consider the +19
fifths interval. 3125/3072 is called a small diesis; in meantone it is
the same as the kleisma, so one possible name would be the meantone
kleisma. However, a kleisma suggests a small interval of around 8
cents, and this is usually not so small, since we have to get down to
a fifth of 695.16 cents to make it equal a kleisma. That's flatter
than Lucy tuning. Equating it to 49/48 is dubious since 49/48 is
practically the same size as 50/49 in just intonation. If we had a
better name for 245/243 than "minor BP diesis" we could use that, I
suppose. We can't call it "narrow" or "minor", because it probably
isn't narrow. The meantone diesis is the interval C:Dbb, and this
interval is C:B##, so I could make an awful pun and suggest "biesis"
for it.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> I'm happy with "meantone diesis". Since both 128/125 and 648/625
> are called a diesis and both translate to the same meantone interval,
> it seems a good choice....
>
> For meantones between 12 and 19 we may also want to consider the +19
> fifths interval. 3125/3072 is called a small diesis....

I'd prefer to better distinguish among the dieses. For the three 5-
limit intervals here quoted I suggest the following:

intervals cents name
========= ===== ====
3125/3072 29.61 small diesis (quoted by Gene)
.128/125. 41.06 lesser diesis (Grove's Encyclopedia)
.648/625. 62.57 greater diesis (ibid)

In the quarter-tone meantone system, Ab:G# is this same lesser diesis.

--- In tuning@yahoogroups.com, "Glenn Leider" <GlennLeider@n...> wrote:

> I'd prefer to better distinguish among the dieses.

I'm looking for a word for meantone specifically, where the lesser and
greater diesis are not distinguished; hence, meantone diesis.

For the three 5-
> limit intervals here quoted I suggest the following:
>
> intervals cents name
> ========= ===== ====
> 3125/3072 29.61 small diesis (quoted by Gene)
> .128/125. 41.06 lesser diesis (Grove's Encyclopedia)
> .648/625. 62.57 greater diesis (ibid)
>
> In the quarter-tone meantone system, Ab:G# is this same lesser diesis.

In 1/4 comma Ab:G# is exactly 128/125, but in 1/3 comma exactly
648/625. In any meantone system, it will be 128/f^12, where f is the
fifth; this is actually the meantone version of the inversion of the
Pythagorean comma. We can make this interval be various things; for
instance if we use a fifth of (126)^(1/12), or 697.7 cents, we can
make the meantone diesis exactly 64/63.

--- In tuning@yahoogroups.com, "Glenn Leider" <GlennLeider@n...> wrote:

> 3125/3072 29.61 small diesis (quoted by Gene)

If there was another name than "small diesis" for this interval, it
would be nice. The +19 fifths interval of meantone is f^19/2048; and if
f^19/2048 = 3125/3072, we get 5/19 comma meantone, which is an
interesting meantone. It is in the middle of 5-limit poptimal range,
with 1/4 comma at the sharp end and 5/26 more to the flat end. It is
also very close to 81-et. Putting in 81 equal in Monzo's equal
temperament table as an approxmimation to 5/19 meantone would make sense.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Glenn Leider" <GlennLeider@n...> wrote:
>
> > 3125/3072 29.61 small diesis (quoted by Gene)
>
> If there was another name than "small diesis" for this interval, it
> would be nice.

Magicomma?

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

> --- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
>
> > The list of increasingly-smaller Pythagorean commas are chains of
these
> > numbers of fifths:
> >
> > 1 2 5 12 41 53 306 665 15601 31867 79335 111202
> >
> > The powers of three alternate between positive and negative.
>
> Those are the denominators of the continued fraction for log2(3), and
> it follows from the theory of continued fractions that they will be
> alternatly greater than and less than log2(3); consequently, the
> powers of three alternate between positive and negative. It's an
> interesting idea to take sums of three successive convergents, which
> gets us out of the semiconvergent game:
>
> 8 19 58 106 400 1024 16572 48133...
>
That corresponds to old chinese:

Y0:=3/2 5th Yang(male,light) principle: positive powers of 3
y0:=4/3 4th yin(female,dark) principle: negative powers of 3

Y1:=Y0/y0=3^2/2^3=9:8 2nd, male tone
y1:=y0/Y1^2=256/243 limma, female (semi)tone

Y2:=Y1/y1=3^12/2^19=531441:524288 PC, male comma
y2:=y1/Y2^3=2^65/3^41=.../...(Philolaos?)female comma

Y3:=Y2/y2=3^53/2^84 Jing Fangs male schisma (Mercator?,Holder?)
y3:=y2/Y3^5=2^485:306 Chien lo Chics female schisma (I.Newton 612EDO?)

Y4:=Y3/y3^2=3^665/2^1054 M.W.Drobisch m. "satanic" comma ~1/13 Cents
y4:=y3/Y4^22=2^24727/3^15601 f. ~0.013...Cents, ~1/80 Cents

Y5:=Y4/y4^2=3^31867/2^50502 m.
y5:=Y4:y5^2=2^125743/3^78335 f. ~0.0065...Cents, ~1/154 Cents

Y6:=Y5/y5=3^111202/2^176202 m. ~0.0062...Cents, ~1/165 Cents
........

more coefficents in:
http://www.ericr.nl/wondrous/cycles.html
or
http://www.math.niu.edu/~rusin/known-math/99/farey

intersexual EDOs:=(Evil-Dis-Orders) like
7 EDO: 1,2^(1/7),2^(2/7),.....,2 instead Pyth. heptatonics:
C-9/8-D-9/8-E-256/243-F-9/8-G-9/8-A-9/8-

53 EDO: in equal asexual 2^(1/53) Holderian steps
instead distincting and observing
Y2 and y2 as different commas differring just Y3.

665 EDO:
instead distincting the steps y3*Y4=3^359/2^569
and y3.

even such high valued EDOs as like
31867, 111202,.....
would destroy and mix the intended gender type
in above infinte iteration approximation process
and prohibit the next step to access following generations,
preserving the gender.