reply to Paul

Dear Paul,

SNIP

[PA]
Then I presume the notation given my Scala above can be safely
ignored for the purposes of this discussion, and has nothing to do
with your requirements/conclusions?

[OZ]
Not entirely. The default diatonical major scale is correctly notated to
begin with, and so are the sharps and flats, which are interchangable.

> What I require is a system which is
> cyclic,

A great many generators are capable of yielding a complete circle of
152 tones.

> where the chain has no wolves,

Isn't that the same as "cyclic"? Can you be a bit more explicit?

[OZ]
I was rather thinking about perfect fifths, or else, fourths, major thirds,
etc...

> the circle closes on itself after a
> managable number of tones,

If it's "cyclic", that implies that all the tones are visited before
the circle closes. But if you want smaller circles, 152-equal
certainly has them in spades.

[OZ]
I'm sure it does. But it doesn't serve my purposes exactly.

> a 19-tET subset could be achieved on any degree,

Obviously this is the case in 152-equal, since 152 = 19*8.

> etc...

More explicit conditions would help, and it wouldn't hurt if you show
exactly how your proposal (with 159-equal) satisfies them.

[OZ]
I'm not sure if my tuning vocabulary is sufficient to explain the `murky`
details in my thinking to an expert like you.

> > > Can you extract a circular 50 or so tones without disrupting the
> > cycle of
> > > fifths?
> >
> > Again I have no idea what you mean by "disrupting". Sure, I can
find
> > plenty of intervals that give you a circle of 38 or 76 notes in
152-
> > equal. So?

[OZ]
Well, unfortunately, I haven't been able to discover any cycle with a fifth
generator matching my requirements.

> >
>
>
> Please provide me a system out of 152-tET that has half the number
of tones
> and where the cycle closes without any wolf fifths.

Obviously, 76-equal does it. I'm assuming you're talking specifically
about a cycle of *fifths* here since you mention wolf fifths. The
fifths here would be 710.5263 cents (which I like because, in
conjuction with pure octaves, they work great for Superpythagorean
and Pajara purposes).

[OZ]
Yes, you noticed the catch-22 proposed above. However, 76-equal is a
terrible tuning, as terrible as all the 19-tET subsets in it. I am not sure
anymore if 171-tET is a good universal choice either. The best fifths I have
heard so far arise from a 1/4 P-comma temperament.

> > > Can you also preserve the 19-tET subset at the same time?
> >
> > 152-equal will always have eight 19-equal subsets, no matter what
> > else you do with it. I don't know what could fail to "preserve"
it.
>
> Can you preserve the 19 equal subset in my request above?

76-equal contains four different transpositions of 19-equal in it. So
as best I can understand your question, 19-equal is trivially
preserved.

[OZ]
Well, I won't be delving into that anymore then.

Cordially,
Oz.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Dear Paul,
>
> SNIP
>
> [PA]
> Then I presume the notation given my Scala above can be safely
> ignored for the purposes of this discussion, and has nothing to do
> with your requirements/conclusions?
>
> [OZ]
> Not entirely. The default diatonical major scale is correctly notated to
> begin with,

Is this a Pythagoraen or meantone conception of the diatonic scale?

> and so are the sharps and flats, which are interchangable.

You mean G-sharp
equals A-flat? To me, that seems to utterly contradict what you wrote right above that, given the temperaments that we're discussing. Correctly preserving the default (say according to Zarlino's concept, which favored 2/7-comma meantone) diatonic scale in the notation would mean, in 152-equal, mapping it, and all the sharps and flats, to a 19-equal subset, but then, you get B-sharp equals C-flat and E-sharp equals F-flat, instead of the enharmonic relations we get in 12-equal.
>
> > What I require is a system which is
> > cyclic,
>
> A great many generators are capable of yielding a complete circle of
> 152 tones.
>
> > where the chain has no wolves,
>
> Isn't that the same as "cyclic"? Can you be a bit more explicit?
>
>
> [OZ]
> I was rather thinking about perfect fifths, or else, fourths, major thirds,
> etc...

No problems there. There won't be any wolves if you go all the way around the circle, which will contain 152 notes or a factor of. For example, the circle of near-6:5 minor thirds closes after 19 of them.

> > the circle closes on itself after a
> > managable number of tones,
>
> If it's "cyclic", that implies that all the tones are visited before
> the circle closes. But if you want smaller circles, 152-equal
> certainly has them in spades.
>
>
> [OZ]
> I'm sure it does. But it doesn't serve my purposes exactly.

I still don't have anything like a clear picture of your purposes. You are welcome to share them in more detail, or keep them to yourself, as you feel appropriate.
>
> > a 19-tET subset could be achieved on any degree,
>
> Obviously this is the case in 152-equal, since 152 = 19*8.
>
> > etc...
>
> More explicit conditions would help, and it wouldn't hurt if you show
> exactly how your proposal (with 159-equal) satisfies them.
>
>
> [OZ]
> I'm not sure if my tuning vocabulary is sufficient to explain the `murky`
> details in my thinking to an expert like you.
>
>
> > > > Can you extract a circular 50 or so tones without disrupting the
> > > cycle of
> > > > fifths?
> > >
> > > Again I have no idea what you mean by "disrupting". Sure, I can
> find
> > > plenty of intervals that give you a circle of 38 or 76 notes in
> 152-
> > > equal. So?
>
> [OZ]
> Well, unfortunately, I haven't been able to discover any cycle with a fifth
> generator matching my requirements.

Please share those requirements, as explicitly as you can, showing how they're fulfilled in your proposed tuning -- but only if you see fit.

> >
> >
> > Please provide me a system out of 152-tET that has half the number
> of tones
> > and where the cycle closes without any wolf fifths.
>
> Obviously, 76-equal does it. I'm assuming you're talking specifically
> about a cycle of *fifths* here since you mention wolf fifths. The
> fifths here would be 710.5263 cents (which I like because, in
> conjuction with pure octaves, they work great for Superpythagorean
> and Pajara purposes).
>
>
> [OZ]
> Yes, you noticed the catch-22 proposed above. However, 76-equal is a
> terrible tuning, as terrible as all the 19-tET subsets in it.

What do you mean by terrible?

I am not sure
> anymore if 171-tET is a good universal choice either. The best fifths I have
> heard so far arise from a 1/4 P-comma temperament.

What do you mean, exactly? Fifths narrowed from 3:2 by 1/4 of a Pyth. Comma? If so, these are awfully close to the fifths of 19-equal, so I'm even more puzzled by your statement above.

>
> > > > Can you also preserve the 19-tET subset at the same time?
> > >
> > > 152-equal will always have eight 19-equal subsets, no matter what
> > > else you do with it. I don't know what could fail to "preserve"
> it.
> >
> > Can you preserve the 19 equal subset in my request above?
>
> 76-equal contains four different transpositions of 19-equal in it. So
> as best I can understand your question, 19-equal is trivially
> preserved.
>
>
> [OZ]
> Well, I won't be delving into that anymore then.

I wish you'd give those of us who have shown interest an opportunity to understand your ideas. I'm not being deliberately obstinate, just trying to "get it" . . .
> Cordially,
> Oz.

Likewise.

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 21 �ubat 2006 Sal� 8:20
Subject: [tuning] Re: reply to Paul

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Dear Paul,
> >
> > SNIP
> >
> > [PA]
> > Then I presume the notation given my Scala above can be safely
> > ignored for the purposes of this discussion, and has nothing to do
> > with your requirements/conclusions?
> >
> > [OZ]
> > Not entirely. The default diatonical major scale is correctly notated to
> > begin with,
>
> Is this a Pythagoraen or meantone conception of the diatonic scale?
>

Neither. It is a well-temperament, is it not?

> > and so are the sharps and flats, which are interchangable.
>
> You mean G-sharp
> equals A-flat?

Not in the least! But they can be made enharmonical too if one wishes it.

>To me, that seems to utterly contradict what you wrote right above that,
>given the temperaments that we're discussing.

Certainly not.

>Correctly preserving the default (say according to Zarlino's concept, which
>favored 2/7-comma meantone) diatonic scale in the notation would mean, in
>152-equal, mapping it, and all the sharps and flats, to a 19-equal subset,
but >then, you get B-sharp equals C-flat and E-sharp equals F-flat, instead
of the >enharmonic relations we get in 12-equal.

So?

> >
> > [OZ]
> > I was rather thinking about perfect fifths, or else, fourths, major
thirds,
> > etc...
>
> No problems there. There won't be any wolves if you go all the way around
the circle, which will contain 152 notes or a factor of. For example, the
circle of near-6:5 minor thirds closes after 19 of them.

You cannot preserve enharmonical sharps and flats in a 12-tone framework
even if you wanted in 152-edo unless you alter the notation drastically.

> > [OZ]
> > I'm sure it does. But it doesn't serve my purposes exactly.
>
> I still don't have anything like a clear picture of your purposes. You are
welcome to share them in more detail, or keep them to yourself, as you feel
appropriate.

Everyone who is following my responses on this list knows that I am not
keeping anything to myself on this matter. If I'm handicapped, blame it on
my choice of words, or else, your lack of attention.

> > [OZ]
> > Well, unfortunately, I haven't been able to discover any cycle with a
fifth
> > generator matching my requirements.
>
> Please share those requirements, as explicitly as you can, showing how
they're fulfilled in your proposed tuning -- but only if you see fit.

Allah knows I'm trying to do so.

> >
> > [OZ]
> > Yes, you noticed the catch-22 proposed above. However, 76-equal is a
> > terrible tuning, as terrible as all the 19-tET subsets in it.
>
> What do you mean by terrible?

Distasteful, horrible, putrid, perplexing, agitating, disconcerting,
hurtful, foreboding... do you want me to go on?

>
> I am not sure
> > anymore if 171-tET is a good universal choice either. The best fifths I
have
> > heard so far arise from a 1/4 P-comma temperament.
>
> What do you mean, exactly? Fifths narrowed from 3:2 by 1/4 of a Pyth.
Comma? If so, these are awfully close to the fifths of 19-equal, so I'm even
more puzzled by your statement above.

Awfully close? 694.737 cents compared to 696.09 cents makes a -1.35 cents
difference, which causes dysfunctional beats all too clearly in several
chords.

> > [OZ]
> > Well, I won't be delving into that anymore then.
>
> I wish you'd give those of us who have shown interest an opportunity to
understand your ideas. I'm not being deliberately obstinate, just trying to
"get it" . . .

Likewise, I'm not being deliberately obstinate, just trying to `explain it`.

Cordially,
Oz.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> > > Yes, you noticed the catch-22 proposed above. However, 76-equal is a
> > > terrible tuning, as terrible as all the 19-tET subsets in it.
> >
> > What do you mean by terrible?

> Distasteful, horrible, putrid, perplexing, agitating, disconcerting,
> hurtful, foreboding... do you want me to go on?

The way I'd treat 76-et is via the extension of 19edo meantone to the
7-limit obtained by tempering out 16875/16384 in addition to 81/80.
This temperament involves chains of 19edo, separated by another
generator which gives the 7th partial (and 11 and 13, if you extend
it.) 76 doesn't do so badly here; the 7-limit is in far better tune
than in 19 by itself. In the 11-limit and beyond, other ways of
organizing things suggest themselves.

Of course the obvious equal temperament in this price range is 72-et.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@...>
> To: <tuning@yahoogroups.com>
> Sent: 21 Þubat 2006 Salý 8:20
> Subject: [tuning] Re: reply to Paul
>
>
> > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> > >
> > > Dear Paul,
> > >
> > > SNIP
> > >
> > > [PA]
> > > Then I presume the notation given my Scala above can be safely
> > > ignored for the purposes of this discussion, and has nothing to do
> > > with your requirements/conclusions?
> > >
> > > [OZ]
> > > Not entirely. The default diatonical major scale is correctly notated to
> > > begin with,
> >
> > Is this a Pythagoraen or meantone conception of the diatonic scale?
> >
>
>
> Neither. It is a well-temperament, is it not?

The diatonic scale has only 7 notes, while a well-temperament normally has 12. The conception behind the diatonic scale in the historical well-temperaments is a meantone one, where three fifths give you a consonant ~5:3 major sixth and four fifths give you a consonant ~5:4 major third. The most commonly used key signatures held close to typical meantone proportions, while the notes in more distant keys were distorted in order to obtain a closed circle, in the historical well-temperaments.

>
> > > and so are the sharps and flats, which are interchangable.
> >
> > You mean G-sharp
> > equals A-flat?
>
>
> Not in the least!

So what do you mean by "interchangable?

>But they can be >made enharmonical >too if one wishes >it.

What does that mean?

>
>
>
>
> >Correctly preserving the default (say according to Zarlino's concept, which
> >favored 2/7-comma meantone) diatonic scale in the notation would mean, in
> >152-equal, mapping it, and all the sharps and flats, to a 19-equal subset,
> but >then, you get B-sharp equals C-flat and E-sharp equals F-flat, instead
> of the >enharmonic relations we get in 12-equal.
>
>
> So?

So that's another reason I propose notating 152-equal as I did.

> > >
> > > [OZ]
> > > I was rather thinking about perfect fifths, or else, fourths, major
> thirds,
> > > etc...
> >
> > No problems there. There won't be any wolves if you go all the way around
> the circle, which will contain 152 notes or a factor of. For example, the
> circle of near-6:5 minor thirds closes after 19 of them.
>
>
>
> You cannot preserve enharmonical sharps and flats in a 12-tone framework
> even if you wanted in 152-edo unless you alter the notation drastically.
>

As above, you'll have to explain the difference between "enharmonical sharps and flats" and "G-sharp equals A-flat", because you're saying you want the former but not the latter.
>
> > > [OZ]
> > > I'm sure it does. But it doesn't serve my purposes exactly.
> >
> > I still don't have anything like a clear picture of your purposes. You are
> welcome to share them in more detail, or keep them to yourself, as you feel
> appropriate.
>
>
>
> Everyone who is following my responses on this list knows that I am not
> keeping anything to myself on this matter.

You've said, "you'll have to wait for my dissertation" to a number of people.

> If I'm handicapped, blame it on
> my choice of words, or else, your lack of attention.
>
>
> > > [OZ]
> > > Well, unfortunately, I haven't been able to discover any cycle with a
> fifth
> > > generator matching my requirements.
> >
> > Please share those requirements, as explicitly as you can, showing how
> they're fulfilled in your proposed tuning -- but only if you see fit.
>
>
> Allah knows I'm trying to do so.
>
>
> > >
> > > [OZ]
> > > Yes, you noticed the catch-22 proposed above. However, 76-equal is a
> > > terrible tuning, as terrible as all the 19-tET subsets in it.
> >
> > What do you mean by terrible?
>
>
> Distasteful, horrible, putrid, perplexing, agitating, disconcerting,
> hurtful, foreboding... do you want me to go on?

Sure.
>
> >
> > I am not sure
> > > anymore if 171-tET is a good universal choice either. The best fifths I
> have
> > > heard so far arise from a 1/4 P-comma temperament.
> >
> > What do you mean, exactly? Fifths narrowed from 3:2 by 1/4 of a Pyth.
> Comma? If so, these are awfully close to the fifths of 19-equal, so I'm even
> more puzzled by your statement above.
>
>
> Awfully close? 694.737 cents compared to 696.09 cents makes a -1.35 cents
> difference, which causes dysfunctional beats all too clearly in several
> chords.

I trust that that's what your ears are telling you, you're entitled to your opinion, and I won't dispute this point. But I wonder, which chords?
>
> > > [OZ]
> > > Well, I won't be delving into that anymore then.
> >
> > I wish you'd give those of us who have shown interest an opportunity to
> understand your ideas. I'm not being deliberately obstinate, just trying to
> "get it" . . .
>
>
> Likewise, I'm not being deliberately obstinate, just trying to `explain it`.
>
>
> Cordially,
> Oz.
>

[PA]
The diatonic scale has only 7 notes, while a well-temperament normally has
12. The conception behind the diatonic scale in the historical
well-temperaments is a meantone one, where three fifths give you a consonant
~5:3 major sixth and four fifths give you a consonant ~5:4 major third. The
most commonly used key signatures held close to typical meantone
proportions, while the notes in more distant keys were distorted in order to
obtain a closed circle, in the historical well-temperaments.

[OZ]
A well-temperament is not confined to 12 tones, is it? You can very well
have such a temperament with as much as 79 tones. With my 79 MOS 159-tET
proposal, I get very good representation of 5 limit intervals, especially on
such keys as E, F and F#/Gb. The rest are also consonant enough for me. So,
the diatonical scale is well-tempered the way I see it.

>
> > > and so are the sharps and flats, which are interchangable.
> >
> > You mean G-sharp
> > equals A-flat?
>
>
> Not in the least!

[PA]
So what do you mean by "interchangable?

[OZ]
You can have them apart by a comma, or you can either make the sharps
enharmonical with the flats, or the flats enharmonical with the sharps.
Either of the latter options close the cycle in 12 tones.

SNIP

> Everyone who is following my responses on this list knows that I am not
> keeping anything to myself on this matter.

[PA]
You've said, "you'll have to wait for my dissertation" to a number of
people.

[OZ]
That's because I didn't have a ready answer to give at that time.

>
> Distasteful, horrible, putrid, perplexing, agitating, disconcerting,
> hurtful, foreboding... do you want me to go on?

[PA]
Sure.

[OZ]
Ok. Terrible, invidious, stupefying, ghastly, foul, morbid, deplorable,
woeful, grievous, distressing, burdensome, pitiable, ugly, coarse,
repellent, odious, loathsome, hideous, stagnant, insalubrious, stale,
noxious, nauseous, revolting. I think that's enough already.

>
> Awfully close? 694.737 cents compared to 696.09 cents makes a -1.35 cents
> difference, which causes dysfunctional beats all too clearly in several
> chords.

[PA]
I trust that that's what your ears are telling you, you're entitled to your
opinion, and I won't dispute this point. But I wonder, which chords?

[OZ]
Even the simplest major chord beats horribly in 19-tET.

I like 76-edo, not a specific val, because allowing different vals it supports not only meantone, but also injera, pajara, superpyth, and mavila, allowing for a wealth of omnitetrachordal scales. Can any other edo really compete in this regard?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
>
> > > > Yes, you noticed the catch-22 proposed above. However, 76-equal is a
> > > > terrible tuning, as terrible as all the 19-tET subsets in it.
> > >
> > > What do you mean by terrible?
>
> > Distasteful, horrible, putrid, perplexing, agitating, disconcerting,
> > hurtful, foreboding... do you want me to go on?
>
> The way I'd treat 76-et is via the extension of 19edo meantone to the
> 7-limit obtained by tempering out 16875/16384 in addition to 81/80.
> This temperament involves chains of 19edo, separated by another
> generator which gives the 7th partial (and 11 and 13, if you extend
> it.) 76 doesn't do so badly here; the 7-limit is in far better tune
> than in 19 by itself. In the 11-limit and beyond, other ways of
> organizing things suggest themselves.
>
> Of course the obvious equal temperament in this price range is 72-et.
>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@...> wrote:
>
> I like 76-edo, not a specific val, because allowing different vals
> it supports not only meantone, but also injera, pajara, superpyth,
and mavila...

Sometimes contortedly. Moreover, the only one of these it does a very
good job with is superpyth, where it is an excellent tuning choice.

> allowing for a wealth of omnitetrachordal scales. Can any other edo
> really compete in this regard?

Yes, so long as size does not concern you. As n->infinity, the number
of lower complexity temperaments n supports increases, until we reach
the point where n does a fine job with everything on your list.
Anyway, tastes differ; some people are not going to be interested in
these tunings at all.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
>
> [PA]
> The diatonic scale has only 7 notes, while a well-temperament
normally has
> 12. The conception behind the diatonic scale in the historical
> well-temperaments is a meantone one, where three fifths give you a
consonant
> ~5:3 major sixth and four fifths give you a consonant ~5:4 major
third. The
> most commonly used key signatures held close to typical meantone
> proportions, while the notes in more distant keys were distorted in
order to
> obtain a closed circle, in the historical well-temperaments.
>
> [OZ]
> A well-temperament is not confined to 12 tones, is it?

Not on this list! Certainly, well-temperaments with 17, 26, and other
larger numbers of notes have been constructed.

> You can very well
> have such a temperament with as much as 79 tones. With my 79 MOS
159-tET
> proposal, I get very good representation of 5 limit intervals,
especially on
> such keys as E, F and F#/Gb. The rest are also consonant enough for
me. So,
> the diatonical scale is well-tempered the way I see it.

The diatonic scale has 7 notes, so I'm not sure I follow you.

Setting aside the thirds and sixths for now, I can't see how you can
say this given your reaction to the 19-equal fifth. The way I'm
reading it, each of the diatonic scales in your proposal contain a
number of fifths that are even worse. What am I missing?

> > > > and so are the sharps and flats, which are interchangable.
> > >
> > > You mean G-sharp
> > > equals A-flat?
> >
> >
> > Not in the least!
>
> [PA]
> So what do you mean by "interchangable?
>
> [OZ]
> You can have them apart by a comma,

How does that make them "interchangeable"?

> or you can either make the sharps
> enharmonical with the flats, or the flats enharmonical with the
sharps.
> Either of the latter options close the cycle in 12 tones.

What do the two latter options mean? And now I'm confused -- is the
well-temperament you're referring to one of 12 tones, or much more
than 12 tones?

>
> SNIP
>
> > Everyone who is following my responses on this list knows that I
am not
> > keeping anything to myself on this matter.
>
> [PA]
> You've said, "you'll have to wait for my dissertation" to a number
of
> people.
>
> [OZ]
> That's because I didn't have a ready answer to give at that time.

Well, we're all waiting, as I can say from a number of off-list e-
mails I've received on this topic.

> > Distasteful, horrible, putrid, perplexing, agitating,
disconcerting,
> > hurtful, foreboding... do you want me to go on?
>
> [PA]
> Sure.
>
> [OZ]
> Ok. Terrible, invidious, stupefying, ghastly, foul, morbid,
deplorable,
> woeful, grievous, distressing, burdensome, pitiable, ugly, coarse,
> repellent, odious, loathsome, hideous, stagnant, insalubrious,
stale,
> noxious, nauseous, revolting. I think that's enough already.

Would you make these statements without qualification to composers
and performers of 19-equal music such as Neil Haverstick, Easley
Blackwood, and the dozens of others that Monz lists? Have you
listened to any of the 19-equal examples that have been shared on the
MMM list or elsewhere?

> > Awfully close? 694.737 cents compared to 696.09 cents makes a -
1.35 cents
> > difference, which causes dysfunctional beats all too clearly in
several
> > chords.
>
> [PA]
> I trust that that's what your ears are telling you, you're entitled
to your
> opinion, and I won't dispute this point. But I wonder, which chords?
>
> [OZ]
> Even the simplest major chord beats horribly in 19-tET.

That's your opinion, and I respect it, and won't try to convince you
otherwise. Don't state it as a universal, though. We've had
testimonials as to how readily audiences and even musicians accepted
Neil Haverstick's 19-equal guitar when playing triadic music. Not to
mention dozens of examples by other composers in 19-equal which have
been listened to and discussed here. 19-equal has an illustrious
history in the West, starting as far back as Costeley or further. To
my ears and many other, the discordance of the major chord in 12-
equal is quite a bit worse than in 19-equal.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@> wrote:
> >
> > I like 76-edo, not a specific val, because allowing different
vals
> > it supports not only meantone, but also injera, pajara,
superpyth,
> and mavila...
>
> Sometimes contortedly.

Not a problem -- "supports" is different from "arises in its entirety
from". And I forgot to mention one more very important
temperament/scale system it supports: diminished.

> Moreover, the only one of these it does a very
> good job with is superpyth, where it is an excellent tuning choice.

Why do you say that? It seems to do a very good job with all of them.

> > allowing for a wealth of omnitetrachordal scales. Can any other
edo
> > really compete in this regard?
>
> Yes, so long as size does not concern you.

Size?

> As n->infinity,

What's n?

> the number
> of lower complexity temperaments n supports increases, until we
reach
> the point where n does a fine job with everything on your list.

Obviously. But in my question, I meant for you to penalize the total
number of notes as you typically do.

> Anyway, tastes differ; some people are not going to be interested in
> these tunings at all.

Indeed.

> > You can very well
> > have such a temperament with as much as 79 tones. With my 79 MOS
> 159-tET
> > proposal, I get very good representation of 5 limit intervals,
> especially on
> > such keys as E, F and F#/Gb. The rest are also consonant enough for
> me. So,
> > the diatonical scale is well-tempered the way I see it.
>
> The diatonic scale has 7 notes, so I'm not sure I follow you.
>

Why can you not? The 12-tone cycle of either:

0: 1/1 C
1: 90.554 cents C# Db\
2: 196.200 cents D
3: 286.753 cents D# Eb\
4: 392.399 cents E
5: 498.045 cents F
6: 588.599 cents F# Gb\
7: 701.955 cents G
8: 792.509 cents G# Ab\
9: 898.155 cents A
10: 988.708 cents A# Bb\
11: 1094.354 cents B
12: 1200.000 cents C

or:

0: 1/1 C unison, perfect prime
1: 105.646 cents C#/ Db
2: 196.200 cents D
3: 301.845 cents D#/ Eb
4: 392.399 cents E
5: 498.045 cents F
6: 603.691 cents F#/ Gb
7: 701.955 cents G
8: 807.601 cents G#/ Ab
9: 898.155 cents A
10: 1003.800 cents A#/ Bb
11: 1094.354 cents B
12: 1200.000 cents C

or other alternating sharps and flats within 79 MOS 159-tET yield reasonable
well-temperaments, am I wrong? If I'm applying the wrong concept here, what
would you call it?

> Setting aside the thirds and sixths for now, I can't see how you can
> say this given your reaction to the 19-equal fifth. The way I'm
> reading it, each of the diatonic scales in your proposal contain a
> number of fifths that are even worse. What am I missing?
>

The pure fifths for one thing. The 12-tone cycle with enharmonical sharps
and flats for another. But enough of that, I have repeated myself enough
already.

SNIP

> > [OZ]
> > That's because I didn't have a ready answer to give at that time.
>
> Well, we're all waiting, as I can say from a number of off-list e-
> mails I've received on this topic.
>

What are you waiting for? And what off-list messages are you referring to
that I should know about?

> > [OZ]
> > Ok. Terrible, invidious, stupefying, ghastly, foul, morbid,
> deplorable,
> > woeful, grievous, distressing, burdensome, pitiable, ugly, coarse,
> > repellent, odious, loathsome, hideous, stagnant, insalubrious,
> stale,
> > noxious, nauseous, revolting. I think that's enough already.
>
> Would you make these statements without qualification to composers
> and performers of 19-equal music such as Neil Haverstick, Easley
> Blackwood, and the dozens of others that Monz lists? Have you
> listened to any of the 19-equal examples that have been shared on the
> MMM list or elsewhere?
>

Hey! I'm not criticizing any composers who might have written excellent
works for 19-tET. For one thing, I have listened to some excellent
microtonal music by Neil. But I'm sure he (if not others) benefits from the
acoustic flexibilities of his instrument much more than a 19-tET theoretical
framework allows for.

Besides, my argument is against the consonance of the 19-tET fifth, not the
beauty of any music that utilizes it.

> >
> > [OZ]
> > Even the simplest major chord beats horribly in 19-tET.
>
> That's your opinion, and I respect it, and won't try to convince you
> otherwise. Don't state it as a universal, though.

Have I stated anything other than my personal views here?

We've had
> testimonials as to how readily audiences and even musicians accepted
> Neil Haverstick's 19-equal guitar when playing triadic music.

I applaud his achievements. But to say that he remains faithful entirely to
the 19-tET pitches to an exact precision is going too far. Even the
slightest vibrato, glissando, pluck and stretch would make a whole
difference in the performance of the tuning. My criticism is not against the
music-making process, but the theoretical properties of 19-tET alone.

Not to
> mention dozens of examples by other composers in 19-equal which have
> been listened to and discussed here. 19-equal has an illustrious
> history in the West, starting as far back as Costeley or further. To
> my ears and many other, the discordance of the major chord in 12-
> equal is quite a bit worse than in 19-equal.
>

Did I ever say that 12-tET is superior to 19-tET? Nae! On the contrary, I
agree with you that 19-equal is much more appealing in several ways. But
this does not mean that I consent to its flawlessness.

Cordially,
Oz.