Re: Interval names

Because of the confusion between augmented fourths and diminished fifths,
I suggest 5:7 and 7:10 be called lesser and greater tritones repectively.
Otherwise, I agree with Dave Keenan (we did discuss this off-list).

If "augmented" and "diminished" are going to be used, the standard should
be something like 5-limit JI, ie something like quarter-comma meantone,
but we'll never agree on this. In reality, 12-note enharmonies apply.

Because of this, I suggest here:

http://x31eq.com/schismic.htm

writing schismic temperament using the usual naturals, sharps and flats to
mean a 12 note scale, and / and \ to mean comma shifts from that. So you
don't have to worry about whether C# is higher or lower than Db. For many
purposes, this notation is good enough for 7-limit JI.

Call me naive, but to me it seems that referring to JI intervals by names
that confuse many seems counterproductive and at least potentially
ambiguous. Call a 6:7 a 6:7, I say... And really the same could go for
various equal temperaments, referring to the number of steps in an interval,
for example: 5of22. For other types of intervals, cents values tell the
whole story. In each of these cases, numbers convey an unambiguous interval.
To me, names are much less clear and more arbitrary, and typically rooted in
a particular harmonic practice. As such, they can tend to limit one's
thinking when composing... Even the term "octave" bugs me, since it assumes
a heptatonic scale; I have long advocated "double" instead.

Of course, when trying to communicate with those versed in traditional music
theory, I suppose it can be helpful to use terminology based on other terms
they already know... But still, explaining the numbers is not much tougher
than explaining exactly which "augmented second" one means.

- just my 865:866 worth of ranting...

> -----Original Message-----
> <manuel.op.de.coul@ezh.nl> wrote:
>
> He did give names to JI ratios too, in _Rekenkundige bespiegeling der
> muziek_ (Mathematical reflection on music), albeit without any
> accompanying
> explanation. This book hardly goes into 31-tone equal temperament.
> But Dave is right. He even named 6:7 an augmented second (harmonische
> vergrote
> sekunde) and 7:8 a diminished third (harmonische verkleinde terts).
>
David Canright http://www.mbay.net/~anne/david/

[Paul Erlich, TD 521.22]
>The only sensible solutions for 22-tET seem to be a consistent application
>of the former [#'s and b's based on chain of fifths],
>in which it is understood that all ratios involving 5 will
>involve the sub or super modifiers;

No way. I't wouldn't do to have an approximate 4:5 being called a major third in most tunings but called a neutral (or submajor, or narrow) third in 22-tET.

>or a version of decatonic notation described in my paper.

For those unfamiliar: This is where, for example, Paul refers to an approximate 4:5 as a major 4th and an approximate 2:3 as a perfect 7th (with "10" subscripts to indicate decatonic).

Well it works, but I just can't see it ever catching on. I can just imagine conversations like:
"Blah blah blah the decatonic major fourth blah blah."
"Oh yeah, what does that sound like?"
"It's an approximate 4:5."
"Oh you mean a major third."

>Although 22-tET is consistent with the set of
>11-limit ratios, it is not consistent with conventional diatonic
>nomenclature, because the syntonic comma does not vanish. In your scheme,
>the interval between the "major second" and the "major sixth" is not a
>"perfect fifth" but a "sub fifth". I find that screwy. If you don't, at
>least you're in good company -- Ben Johnston's JI notation of C major is
>similar.

I _do_ find that screwy and I don't like Ben Johnston's 5-limit JI notation. Thanks for spotting it.

Since 22-tET is 11-limit consistent then it must be possible to name it consistently when the names simply correspond to 11-limit ratios. It was Graham breed who convinced me the names should be tied to the 11-limit ratios rather than the intervals of 31-tET. Thanks Graham.

So my list of names was wrong. There are two ways of dealing with this.

1. Deny that 22-tET has any usable approximations to major 2nds (and hence 9ths and minor 7ths). They _are_ pretty bad. This is what I did in a (spreadsheet generated) list I sent to Paul Erlich in December. Without thinking too carefully I added major second and minor seventh as alternate names when I posted it to the list. My bad (as Joe says).

2. The correct way to include these is to give different names to 8:9 and 9:10 since they correspond to different number of 22-tET steps (as you say, the syntonic comma doesn't disappear).

0 C perfect unison
1 C/ super unison
2 Db/ minor second
3 D\ neutral second, narrow major second (9:10)
4 D supermajor second, wide major second (8:9)
5 Eb subminor third
6 Eb/ minor third, neutral third
7 E\ major third
8 E supermajor third
9 F perfect fourth
10 F/ super fourth
11 F#\, Gb/ augmented fourth, diminished fifth
12 G\ sub fifth
13 G perfect fifth
14 Ab subminor sixth
15 Ab/ minor sixth
16 A\ neutral sixth, major sixth
17 A supermajor sixth
18 Bb subminor seventh, narrow minor seventh (9:16)
19 Bb/ neutral seventh, wide minor seventh (5:9)
20 B\ major seventh
21 C\ sub octave
22 C perfect octave

I hope you will now find it to be consistent.

Regards,

-- Dave Keenan
http://dkeenan.com

[Paul Erlich, TD 521.22]
>The only sensible solutions for 22-tET seem to be a consistent application
>of the former [#'s and b's based on chain of fifths],
>in which it is understood that all ratios involving 5 will
>involve the sub or super modifiers;

No way. I't wouldn't do to have an approximate 4:5 being called a major third in most tunings but called a neutral (or submajor, or narrow) third in 22-tET.

>or a version of decatonic notation described in my paper.

For those unfamiliar: This is where, for example, Paul refers to an approximate 4:5 as a major 4th and an approximate 2:3 as a perfect 7th (with "10" subscripts to indicate decatonic).

Well it works, but I just can't see it ever catching on. I can just imagine conversations like:
"Blah blah blah the decatonic major fourth blah blah."
"Oh yeah, what does that sound like?"
"It's an approximate 4:5."
"Oh you mean a major third."

>Although 22-tET is consistent with the set of
>11-limit ratios, it is not consistent with conventional diatonic
>nomenclature, because the syntonic comma does not vanish. In your scheme,
>the interval between the "major second" and the "major sixth" is not a
>"perfect fifth" but a "sub fifth". I find that screwy. If you don't, at
>least you're in good company -- Ben Johnston's JI notation of C major is
>similar.

I _do_ find that screwy and I don't like Ben Johnston's 5-limit JI notation. Thanks for spotting it.

Since 22-tET is 11-limit consistent then it must be possible to name it consistently when the names simply correspond to 11-limit ratios. It was Graham breed who convinced me the names should be tied to the 11-limit ratios rather than the intervals of 31-tET. Thanks Graham.

So my list of names was wrong. There are two ways of dealing with this.

1. Deny that 22-tET has any usable approximations to major 2nds (and hence 9ths and minor 7ths). They _are_ pretty bad. This is what I did in a (spreadsheet generated) list I sent to Paul Erlich in December. Without thinking too carefully I added major second and minor seventh as alternate names when I posted it to the list. My bad (as Joe says).

2. The correct way to include these is to give different names to 8:9 and 9:10 since they correspond to different number of 22-tET steps (as you say, the syntonic comma doesn't disappear).

0 C perfect unison
1 C/ super unison
2 Db/ minor second
3 D\ neutral second, narrow major second (9:10)
4 D supermajor second, wide major second (8:9)
5 Eb subminor third
6 Eb/ minor third, neutral third
7 E\ major third
8 E supermajor third
9 F perfect fourth
10 F/ super fourth
11 F#\, Gb/ augmented fourth, diminished fifth
12 G\ sub fifth
13 G perfect fifth
14 Ab subminor sixth
15 Ab/ minor sixth
16 A\ neutral sixth, major sixth
17 A supermajor sixth
18 Bb subminor seventh, narrow minor seventh (9:16)
19 Bb/ neutral seventh, wide minor seventh (5:9)
20 B\ major seventh
21 C\ sub octave
22 C perfect octave

I hope you will now find it to be consistent.

Regards,

-- Dave Keenan
http://dkeenan.com

>Well it works, but I just can't see it ever catching on.

Can you see the decatonic scale ever catching on?

>Since 22-tET is 11-limit consistent then it must be possible to name it
>consistently when the names simply correspond to 11-limit ratios.

That doesn't mean you can commensurate the ratios with diatonic scale degrees.

-Carl

In-Reply-To: <DB17FBA383E4D211A9AF00A0C99E01694831B7@monterey.nps.navy.mil>
I said yesterday that schismic notation is good for 7-limit music.
Actually, it's only in the 5- and 9-limits that it's significantly
better than meantone, although those 9-limit intervals will occur in
7-limit music. The utility in the 5-limit is that a computer can read
the notation with less than 0.3 cent errors from JI. In the 9-limit,
it removes the ambiguity between 9/8 and 10/9, and also allows for a
closer approximation to JI (you can still get better than meantone in
the 7-limit (<7 cent errors) but the difference is clearer in the
9-limit).

So here are the "important" 9-limit related intervals, showing the
number of steps from 41 pitch classes, some words, and the way of
writing the intervals relative to C and E.

20 7:5 lesser tritone F# Bb\
17 4:3 perfect fourth F A
15 9:7 supermajor third E// G#/
14 (-6 4)H Pythagorean major third E/ G#
13 5:4 major third E G#\
11 6:5 minor third Eb/ G
10 32:27 Pythagorean minor third Eb G\
9 7:6 subminor third Eb\ G\\
8 8:7 supermajor second D/ F#/
7 9:8 major tone D F#
6 10:9 minor tone D\ F#\
4 16:15 semitone C#/ F
3 (8 -5)H Pythagorean limma C# F\
3 21:20 septimal limma C# F\
2 25:24 chromatic semitone C#\ F\\
2 (7 0 -3)H some kind of diesis C// E//
2 36:35 septimal diesis C// C//
1 64:63 septimal comma C/ E/
1 81:80 syntonic comma C/ E/
0 1:1 unison C E

The names aren't consistent, but I'm working on that.

And you can still get schismic notation to work in the 11-limit,
although I don't see any real advantages over meantone (31-based)
notation. Here are the important intervals:

20 7:5 lesser tritone F# Bb\
19 11:8 super fourth F// A//
17 4:3 perfect fourth F A
15 9:7 supermajor third E// GA/
14 14:11 ? E/ GA
14 (-6 4)H Pythagorean major third E/ GA
13 5:4 major third E GA\
12 11:9 neutral third Eb// G/
11 6:5 minor third Eb/ G
10 32:27 Pythagorean minor third Eb G\
9 7:6 subminor third Eb\ G\\
8 8:7 supermajor second D/ F#/
7 9:8 major tone D F#
6 10:9 minor tone D\ F#\
6 11:10 greater neutral second D\ F#\
5 12:11 lesser neutral second D\\ F#\\
4 16:15 semitone C#/ F
3 (8 -5)H Pythagorean limma C# F\
3 21:20 septimal limma C# F\
3 22:21 ? C# F\
2 25:24 chromatic semitone C#\ F\\
2 33:32 diesis C// E//
2 (7 0 -3)H C// E//
2 36:35 septimal diesis C// E//
1 81:80 syntonic comma C/ E/
0 1:1 unison C E

That uses the additional commas 56/55 and 55/54. As it's only
consistent with 41=, you may as well throw in 45/44, and use the
enharmonies as well.

I did work out some ratios for 16 and 18 steps, but can't remember
them now.

Anyway, David Canright wrote:

> Call me naive, but to me it seems that referring to JI intervals by
> names
> that confuse many seems counterproductive and at least potentially
> ambiguous.

I'd prefer to find names that don't confuse most ...

> Call a 6:7 a 6:7, I say...

Like it or not, a lot of people will be turned off the subject if they
see a lot of numbers.

> And really the same could go for
> various equal temperaments, referring to the number of steps in an
> interval,
> for example: 5of22.

Yes, but it may not be intuitively obvious to all that, say, 15of46 is
the nearest approximation in that temperament to 5:4. Saying 15of46
is the 5-limit major third is a simple way to state this. Say that
you're playing a 5-limit major third and nobody need care what
temperament your using.

> For other types of intervals, cents values tell the
> whole story.

Oh no they don't! The major third has a different size in different
meantones, but the set of all meantone major thirds can be referred to
as "meantone major thirds".

You may also want to describe a note with so much vibrato that it
doesn't have a single cents value.

> In each of these cases, numbers convey an unambiguous
> interval.
> To me, names are much less clear and more arbitrary, and typically
> rooted in
> a particular harmonic practice. As such, they can tend to limit
one's
> thinking when composing...

Ambiguity is bad for mathematicians, but can be good for artists.
There's also a general human tendency to name things. Ratios are
strongly linked to one harmonic practice: just intonation.

> Of course, when trying to communicate with those versed in
traditional
> music
> theory, I suppose it can be helpful to use terminology based on
other
> terms
> they already know...

Indeed.

> But still, explaining the numbers is not much
> tougher
> than explaining exactly which "augmented second" one means.

Assuming you have one particular augmented second in mind. A lot of
the time, calling it a minor or subminor third would be good enough
for me. I like a bit of ambiguity, although not so much as to use 12
pitch classes for everything.

If I want my augmented second tuned to 7:6, I'll call it a 7:6
subminor third. If you've never met a 7:6 before, you will at least
know that I'm talking about a 7-limit consonance. You could work out
what size it is, but you already know it's slightly flat of a normal
minor third. Redundancy is good in general communication. It makes
things easier to understand.

In-Reply-To: <memo.484078@cix.compulink.co.uk>
I wrote:

> closer approximation to JI (you can still get better than meantone
in
> the 7-limit (<7 cent errors) but the difference is clearer in the

That should be <5 cent errors.

Dave Keenan wrote,

>So my list of names was wrong. There are two ways of dealing with this.

>1. Deny that 22-tET has any usable approximations to major 2nds (and hence
9ths >and minor 7ths). They _are_ pretty bad. This is what I did in a
(spreadsheet >generated) list I sent to Paul Erlich in December. Without
thinking too >carefully I added major second and minor seventh as alternate
names when I >posted it to the list. My bad (as Joe says).

>2. The correct way to include these is to give different names to 8:9 and
9:10 >since they correspond to different number of 22-tET steps (as you say,
the >syntonic comma doesn't disappear).

It seems you did both (1) and (2).

> 0 C perfect unison
> 1 C/ super unison
> 2 Db/ minor second
> 3 D\ neutral second, narrow major second (9:10)
> 4 D supermajor second, wide major second (8:9)
> 5 Eb subminor third
> 6 Eb/ minor third, neutral third
> 7 E\ major third
> 8 E supermajor third
> 9 F perfect fourth
>10 F/ super fourth
>11 F#\, Gb/ augmented fourth, diminished fifth
>12 G\ sub fifth
>13 G perfect fifth
>14 Ab subminor sixth
>15 Ab/ minor sixth
>16 A\ neutral sixth, major sixth
>17 A supermajor sixth
>18 Bb subminor seventh, narrow minor seventh (9:16)
>19 Bb/ neutral seventh, wide minor seventh (5:9)
>20 B\ major seventh
>21 C\ sub octave
>22 C perfect octave

>I hope you will now find it to be consistent.

The notated symbols (2nd column) still don't agree with the wordy names (3rd
column).

Carl Lumma wrote,

>Can you see the decatonic scale ever catching on?

Well, I guess that's partially dependent on my own effort. Aside from the
"Decatonic Waltz" I performed at Microthon, and some tinkering by Steve
Reszutek, there hasn't been any music in the air to allow it to catch on.
But on the other side, the "conventional" 22-tET examples by fine composers
such as Blackwood and Fortuin have been bombs (to my ears) and speak loudly
against heptatonic (diatonic) thinking in that tuning.

Dave Keenan wrote,

>>Since 22-tET is 11-limit consistent then it must be possible to name it
>>consistently when the names simply correspond to 11-limit ratios.

Carl Lumma wrote,

>That doesn't mean you can commensurate the ratios with diatonic scale
degrees.

That's exactly right.

>Well, I guess that's partially dependent on my own effort. Aside from the
>"Decatonic Waltz" I performed at Microthon, and some tinkering by Steve
>Reszutek, there hasn't been any music in the air to allow it to catch on.
>But on the other side, the "conventional" 22-tET examples by fine composers
>such as Blackwood and Fortuin have been bombs (to my ears) and speak loudly
>against heptatonic (diatonic) thinking in that tuning.

Gee, I really liked both Blackwood's and Fortuin's efforts. Especially the
Fortuin/Peck work. None of it is decatonic, though.

I wouldn't worry about decatonic music catching on. Maybe not in our
lifetimes, but if the scales represent a type that is as rare as it seems
to be, then it will surely catch on at some point.

-Carl

[Carl Lumma, TD 523.1]
>Can you see the decatonic scale ever catching on?

Yes, as one of several.

[Dave Keenan]
>>Since 22-tET is 11-limit consistent then it must be possible to name it
>>consistently when the names simply correspond to 11-limit ratios.

[Carl Lumma]
That doesn't mean you can commensurate the ratios with diatonic scale degrees.

{Paul Erlich]
That's exactly right.

I'm not sure I'm following. What counterexample do you have in mind? In my way of thinking, any particular tuning of a diatonic scale has to take its turn just like 22-tET. The meantones will end up with B:F as a diminished fifth (approx 7:10) and the Pythagorean tunings will have it as a subdiminished fifth (approx 5:7). Of course it can be argued that 7-limit distinctions are irrelevant to diatonic scales and in this case we should simply call them all tritones (as per Graham Breed).

[Paul H. Erlich, TD523.16]
>It seems you did both (1) and (2).

Yes. As I said, I did (1) in December, I'm doing (2) now. Or do you mean I'm doing them simultaneously now, or that I did them simultaneously in the past? I don't get it.

>> 0 C perfect unison
>> 1 C/ super unison
>> 2 Db/ minor second
>> 3 D\ neutral second, narrow major second (9:10)
>> 4 D supermajor second, wide major second (8:9)
>> 5 Eb subminor third
>> 6 Eb/ minor third, neutral third
>> 7 E\ major third
>> 8 E supermajor third
>> 9 F perfect fourth
>>10 F/ super fourth
>>11 F#\, Gb/ augmented fourth, diminished fifth
>>12 G\ sub fifth
>>13 G perfect fifth
>>14 Ab subminor sixth
>>15 Ab/ minor sixth
>>16 A\ neutral sixth, major sixth
>>17 A supermajor sixth
>>18 Bb subminor seventh, narrow minor seventh (9:16)
>>19 Bb/ neutral seventh, wide minor seventh (5:9)
>>20 B\ major seventh
>>21 C\ sub octave
>>22 C perfect octave
>
>>I hope you will now find it to be consistent.
>
>The notated symbols (2nd column) still don't agree with the wordy names >(3rd column).

But each column is now consistent in its own way. One is a list of note names, the other a list of interval names. They don't _have_ to be consistent across all tunings. Why must the major third from C always be called E? Somethings gotta give (between chains-of-fifths and low-limit ratios), so I say let it give here.

The chains-of-fifths note-names ensure that familiar things happen when you modulate by fifths or fourths (the most common modulations by far), and the interval names tell you roughly what they sound like. Both of these properties are independent of the tuning (assuming it _has_ fifths and other JI approximations).

What better way to point out to someone (versed in standard music theory only) that the syntonic comma doesn't dissapear in 22-tET, than by saying "In this tuning, the major third from C isn't E, it's a step lower at E\."?

When we know we're dealing with decatonic scales, the \'s can be absorbed into some kind of key signature so that the major third from C _does_ look like E on the staff and if we need the supermajor third we use an accidental natural E|.

We can almost have our cake and eat it too.

Much of the following I wrote to Paul off-list in December.

A minimum-modifiers symmetrical decatonic is

B\ C C#\ Eb E\ F F#\ G A\ Bb (B\)

We also want to use certain enharmonics from time to time, to give the right spelling. The full set of these is

D#\\ A#\\ E#\\ Cb/ Gb/ Db/ instead of
Eb Bb F B\ F#\ C#\

Note that these enharmonics can also have their \\ and / removed (and indicated only by the key signature) without ambiguity. So, on the staff we need use no \ or / for in-key notes.

Here's that symmetrical decatonic on a standard 7 position per octave treble staff complete with key signature. The parenthesis around a sharp in the key signature indicates that the \ applies to the sharpened note but the sharp itself isn't part of the key sig. Similarly for the flats. Other keys would have unparenthesised sharps or flats in their key sig.

The key signature below is way too ugly, but all the information is there. It would be better to have a blank key signature for this decatonic key and show only what differs from this.

-----------------------------------------------------------------
\ (#)\\
---(#)\\------(b)/-----------------------------------------------
(#)\ (b)/
--\------------------------------------------------------Ob--O---
\ (#)\\ O
-----------(b)/----------------------------------O---------------
(#)\ O O#
--------------------------------Ob---O---------------------------

--- -O- -O#-
O

Once we do this, an approximate 4:5:6:7 (subminor 7th) chord looks, on the staff, like a dominant 7th would in meantone. A 1/6:1/5:1/4:2/7 (minor supermajor 6th) looks like a minor major 6th (half diminished 7th). A 10:12:15:18 looks like a minor 7th (which it is). A 8:10:12:15 looks like a major 7th (which it is). Likewise all the 1,5,7,9 (Paul's terminology) chords look like what they are and so do the 1,4,6,9's.

As Paul pointed out, a lot of dissonant chords look consonant under this scheme. This is similar to the diminished chord in diatonic but there are more to keep track of.

The above notation implies a (12-note per octave) keyboard mapping. A symmetrical decatonic would omit the D and G#Ab keys.

The keyboard mapping suggests a way of adaptively tuning decatonics in 22-tET. The algorithm could be constantly figuring out whether you were more likely to modulate up or down (by a fifth) and would be retuning those keys to either D and G#\ or D\ and Ab. For any more complicated modulation it would have to retune the note _after_ you hit the key.

The pentachordal scale would not use the D or A keys but these would be tuned to D and A\. An upward modulation would retune Eb to D#\. A downward modulation would retune G#\ to Ab. A pair of footswitches could _force_ modulations. Getting a bit off-topic here.

If we were introducing decatonics into a vacuum, your [Paul E's] notation would be fine. But I aim to make maximum use of what people are already familiar with.

No matter their origins, you'd have to agree that the terms "octave" and "fifth" at least, are stuck way harder to the ideas of 1:2 and 2:3 than they are to the ideas of spanning 7 and 4 steps of a scale.

Regards,

-- Dave Keenan
http://dkeenan.com

I had expressed concern regarding:

>>> 5:7 augmented fourth
>>> 7:10 diminished fifth
>>
>>Aren't these last two backward?

And Dave Keenan responded:
>
> I just knew you were gonna say that. :-)

Why? Because my ear seems to be limited to JI? Because my brain seems to be
limited to common practice. Because I appear to be limited in general? :-)

Appararently, (if you anticipated my response) you must have thought about
it before.
>
>>The 5:7 tritone is smaller than the 7:10 tritone,
>
> Yes.
>
>>thus the 5:7 "contracts" (E-Bb) in resolution (diminished fifth)
>>and the 7:10 (E-A#) expands in resolution (augmented fourth).
>
> I'd say the 7:10 (E-Bb) contracts in resolution (diminished fifth)
> and the 5:7 (E-A#) expands in resolution (augmented fourth).

But....but.....but.....but 7:10 fits the partial series (of C) as Bb _up_ to
E--an augmented fourth in a dominant seventh chord belonging to the key of
F. In the C dominant seventh chord "acoustically" tuned, the Bb is closer to
the E below--a diminished fifth.

What's that you say? Common practice is not the only music system in
existence? Well, I'll be hornswaggled! You mean some composers could care
less about acoustic reality and four hundred years of practice? I'll be
damned.
>
> Or putting them both in C major, B:F (7:10) contracts in resolution to C:E
> (4:5) and F:B (5:7) expands in resolution to E:C (5:8), not because of
> their initial sizes but just because of where the nearest _diatonic_ notes
are.

Hold on there, Mr. K. B-F in the key of C is not 7:10 over a C fundamental;
it's 7:10 over a _G_ fundamental--the dominant of C. Therefore, when B is
the upper member it is the 10 of 7:10, an augmented fourth (F up to B) and
goes to C; and when F is the upper member it is 7 of 5:7, a diminished fifth
(B up to F) and goes down to E. Therefore, since 7:10 is larger than 5:7,
their initial sizes are related to where they will resolve.

> Of course whether B:F is closer to 5:7 or 7:10 depends on your tuning.

Not on _my_ tuning--on "nature's" tuning (if one accepts the principles of
"functional harmony"). If you want to resolve them the other way, you
certainly can. To me, the resolution of an augmented fourth is predetermined
by its "larger" tuning.

> I have assumed something tending toward meantone above (since these are
> optimum for diatonic). For tunings tending toward (or beyond) the
> Pythagorean, B:F would be closer to 5:7 which one might then call a
> "Pythagorean diminished fifth", or (as I'd prefer) a "subdiminished fifth".

I'm sorry. I don't speak Pythagorean. (And I suspect my ears don't either.)
>
>>Remember, Paul and Joe hold that "context is everything."
>
> It's certainly very important, and the system, as explained on my web page,
> http://dkeenan.com/Music/IntervalNaming.htm allows for alternative
> names for the same ratio in different contexts, as with 5:7 = aug 4th =
> subdim 5th above.

Perhaps we could simply agree to disagree on this.
>
>>(You don't suppose Mr. Fokker would reconsider these, do you?
>
> No.
>
>> Too late, you say?
>
> As in "the late Mr Fokker", yes.

As you correctly interpreted, I was being facetious here. It might be
considered that during Fokker's productive years, the principle of tonality
was being seriously challenged by Schoenberg and friends. It seems likely
that his use of these interval terms were intentionally "nonfunctional."
>
>>Okay. Do you suppose he'd mind if we messed with it a bit?)

(Still being facetious, in the sense that it is out of touch with JI
principles.)
>
> Yes. The whole system

And this is the rub. We're talking about different musical systems here, as
you clearly point out as this paragraph continues...

> would be screwed if we swapped these two. To be fair
> to Mr Fokker, strictly speaking, he gave the names to the intervals of
> 31-tET (a meantone), not to the JI ratios which they approximate. The
> latter seems like an obvious move to many people, but I understand that
> Paul Erlich is still dubious about it because of the apparent disagreement,
> in the case of 22-tET (and other super-Pythagorean tunings), between the
> A-G, #, b, /, \ names for the notes, based on chains of approximate 2:3's,
> and the Fokker-style names for the intervals, based on the 11-limit ratios
> most nearly approximated.
>
> 22-tET intervals from C to:
>
> C perfect unison
> C/ super unison
> Db/ minor second
> D\ neutral second
> D major second, supermajor second
> Eb subminor third
> Eb/ minor third, neutral third
> E\ major third
> E supermajor third
> F perfect fourth
> F/ super fourth
> F#\, Gb/ augmented fourth, diminished fifth
> G\ sub fifth
> G perfect fifth
> Ab subminor sixth
> Ab/ minor sixth
> A\ neutral sixth, major sixth
> A supermajor sixth
> Bb subminor seventh, minor seventh
> Bb/ neutral seventh
> B\ major seventh
> C\ sub octave
> C perfect octave
>
> Personally, I don't have a problem with that. But one does need to know (at
> least) whether the tuning being used has G# < Ab or G# > Ab.

Exactly. To me, there is no question about it. But, of course, it is not
always _up_ to me.

In any case, thanks again for providing alternative language for the rather
undignified term "dinky third."

Jerry

>I'm not sure I'm following. What counterexample do you have in mind?

I just meant that the diatonic scale is not 11-limit consistent, so no
naming scheme based on diatonic degrees will be either.

>But each column is now consistent in its own way. One is a list of note
>names, the other a list of interval names.

How are you using consistency here? The wordy names in your table aren't
even close to consistent with 22. For example, stacking two of your thirds
would wind one up with, variously, types of fourths, fifths, and sixths.

>I't wouldn't do to have an approximate 4:5 being called a major third in
>most tunings but called a neutral (or submajor, or narrow) third in 22-tET.

There are very good reasons for Paul's position when it comes to diatonic
naming in 22. The 9:7 and 7:6 are the natural diatonic major and minor
thirds in this tuning, based on the way they fall in the scale. The o- and
-utonal 6:7:9 chords function as 1-3-5 triads exactly like the 4:5:6's
function in a meantone diatonic.

Dave, I hope you don't take me wrong on this issue. When working with the
diatonic scale, diatonic names are appropriate. When working with the
decatonic scale they are not.

For composers working with the diatonic scale and 11-limit chromatic
harmony, your naming scheme (or something like it) will be essential.
Inconsistency is just something that composers interested in this setup
will have to deal with. By the way, diatonic music with high-limit
chromatic harmony is one of the most fertile grounds in tuning theory in my
opinion, and I have plans to explore it in depth one day, when your naming
scheme will very likely be handy to me.

-Carl

> -----Original Message-----
> (Graham Breed wrote:)
[snip]
> Anyway, David Canright wrote:
>
> > Call a 6:7 a 6:7, I say...
>
> Like it or not, a lot of people will be turned off the subject if they
> see a lot of numbers.
>
> > And really the same could go for
> > various equal temperaments, referring to the number of steps in an
> > interval, for example: 5of22.
>
> Yes, but it may not be intuitively obvious to all that, say, 15of46 is
> the nearest approximation in that temperament to 5:4. Saying 15of46
> is the 5-limit major third is a simple way to state this. Say that
> you're playing a 5-limit major third and nobody need care what
> temperament your using.
>
> > For other types of intervals, cents values tell the whole story.
>
> Oh no they don't! The major third has a different size in different
> meantones, but the set of all meantone major thirds can be referred to
> as "meantone major thirds".
>
> You may also want to describe a note with so much vibrato that it
> doesn't have a single cents value.
>
> > In each of these cases, numbers convey an unambiguous interval.
> > To me, names are much less clear and more arbitrary, and typically
> > rooted in
> > a particular harmonic practice. As such, they can tend to limit
> > one's thinking when composing...
>
> Ambiguity is bad for mathematicians, but can be good for artists.
> There's also a general human tendency to name things. Ratios are
> strongly linked to one harmonic practice: just intonation.
>
> > Of course, when trying to communicate with those versed in
> traditional music theory, I suppose it can be helpful to use terminology
> based on
> other terms they already know...
>
> Indeed.
>
> > But still, explaining the numbers is not much tougher
> > than explaining exactly which "augmented second" one means.
>
> Assuming you have one particular augmented second in mind. A lot of
> the time, calling it a minor or subminor third would be good enough
> for me. I like a bit of ambiguity, although not so much as to use 12
> pitch classes for everything.
>
> If I want my augmented second tuned to 7:6, I'll call it a 7:6
> subminor third. If you've never met a 7:6 before, you will at least
> know that I'm talking about a 7-limit consonance. You could work out
> what size it is, but you already know it's slightly flat of a normal
> minor third. Redundancy is good in general communication. It makes
> things easier to understand.
>
In reply:

Graham, you make a number of good points. But I think the main point is that
we are talking about different things.

I am talking about how to communicate any _particular interval_; as I
understood some of these lists of names, they list a specific interval next
to a specific name. But apparently my interpretation was not necessarily the
intent of such lists, because...

You are talking about _categories_ of intervals, a distinct but related
idea. So in your list where you say, e.g.,
showing the number of steps from 41 pitch classes,
9 7:6 subminor third Eb\ G\\
this would seem to mean that the term "subminor third" would apply to any
interval in pitch class 9of41, (from 248.8c to 278.0c), which category would
include the interval 9of41 (263.4c) and the interval 6:7 (266.9c) and
presumably 5of24 (250c). This would be consistent with your naming of the
neighboring pitch classes 8of41 and 10of41. But where you have more than one
name with the same 41-ET representation (commas to limmas) you must mean
smaller ranges, and perhaps where you skip 41-ET classes you mean the names
cover larger ranges...? Actually, I suspect you sometimes mean categories
(like "major third") and sometimes mean specific intervals (like "septimal
comma").

IMHO, it is important to distinguish categories of intervals (e.g. "thirds")
from particular intervals (e.g. 272.3c). (This distinction disappears as the
categories shrink to the size of human pitch discrimination.) For intervals,
I like numbers (yeah, you got me: a mathematician who composes in JI).
Categorical names are useful, and can be made unambiguous by specifying the
ranges they refer to. Of course, as you point out, ambiguity can be good
too...

David Canright http://www.mbay.net/~anne/david/

[Gerald Eskelin, TD 524.9]
>And Dave Keenan responded:
>>
>> I just knew you were gonna say that. :-)
>
>Why? Because my ear seems to be limited to JI? Because my brain seems to
>be limited to common practice. Because I appear to be limited in
>general? :-)

Much simpler than that. It's because you wrote either "augmented fourth (7:10)" or "diminished fifth (5:7)" (I forget which) in an earlier message. And because it's a common misconception, partly blameable on the ubiquitousness of 12-tET (where of course they are the same size).

>But....but.....but.....but 7:10 fits the partial series (of C) as Bb _up_ to
>E--an augmented fourth in a dominant seventh chord belonging to the key of
>F. In the C dominant seventh chord "acoustically" tuned, the Bb is closer to
>the E below--a diminished fifth.
>
>What's that you say? Common practice is not the only music system in
>existence? Well, I'll be hornswaggled! You mean some composers could care
>less about acoustic reality and four hundred years of practice? I'll be
>damned.

That's not what I'm saying at all.

Ratios with odd-factors greater than 5 are irrelevant to diatonic music, with the possible exception of ratios of 9. 7's don't matter. They are only involved in functional _dissonances_.

This is a somewhat "religious" issue but I contend that a dominant 7th chord is not (should not be) an approximate 4:5:6:7 chord. A 4:5:6:7 is much more consonant and should be called a subminor 7th or (German) augmented 6th chord.

A dominant seventh chord has a _minor_ third stacked on a major triad (not a _subminor_ third).

So what you say above, is true only if you replace "dominant" with "subminor", but we would no longer be in the realm of diatonic scales.

>Hold on there, Mr. K. B-F in the key of C is not 7:10 over a C fundamental;
>it's 7:10 over a _G_ fundamental--the dominant of C.

If B:F (B up to F) is closer to a 7:10 than a 5:7 then that's true no matter what other notes it is over. I assume we're not talking about dynamic retuning and we are both assuming the convention of writing the notes of an interval or chord with pitch ascending from left to right. Yes, the B:F forms part of the G7 chord and is closer to 7:10 than 5:7 in most reasonable diatonic tunings of that chord. So it seems we agree.

>Therefore, when B is
>the upper member it is the 10 of 7:10, an augmented fourth (F up to B) and
>goes to C; and when F is the upper member it is 7 of 5:7, a diminished fifth
>(B up to F) and goes down to E. Therefore, since 7:10 is larger than 5:7,
>their initial sizes are related to where they will resolve.

Ok. We disagree. Same problem. dominant 7th chord is closer to 4:5:6|5:6 than 4:5:6:7. 5:6 + 5:6 is closer to 7:10 than 5:7.

>> Of course whether B:F is closer to 5:7 or 7:10 depends on your tuning.
>
>Not on _my_ tuning--on "nature's" tuning (if one accepts the principles of
>"functional harmony"). If you want to resolve them the other way, you
>certainly can. To me, the resolution of an augmented fourth is predetermined
>by its "larger" tuning.

We don't disagree on how the notes resolve, or what the intervals are called. Our only disagreement is what JI ratio they best approximate. The only way you could be right is if the scale was tuned with fifths _wider_ than 700 cents (on the Pythagorean side of 12-tET, a "negative" meantone). And as you probably know, this would give very bad major and minor thirds (assuming we're still trying to be diatonic).

>Perhaps we could simply agree to disagree on this.

Perhaps.

>It might be
>considered that during Fokker's productive years, the principle of tonality
>was being seriously challenged by Schoenberg and friends. It seems likely
>that his use of these interval terms were intentionally "nonfunctional."

No. Absolutely not. They are perfectly sensible terms for extended meantone tunings. Even Paul and Carl agree with that.

>> Yes. The whole system
>
>And this is the rub. We're talking about different musical systems here, as
>you clearly point out as this paragraph continues...
>
>> would be screwed if we swapped these two. To be fair
>> to Mr Fokker, strictly speaking, he gave the names to the intervals of
>> 31-tET (a meantone), not to the JI ratios which they approximate. The
>> latter seems like an obvious move to many people

Not different systems. One is merely a superset of the other. 31-tET is an extended meantone and contains perfectly ordinary diatonic subsets. Their Fokker interval names are completely standard and uncontrovertial.

>> But one does need to know (at
>> least) whether the tuning being used has G# < Ab or G# > Ab.
>
>Exactly. To me, there is no question about it.

So which is it, G# < Ab (meantone) or G# > Ab (Pythagorean)

Show me a tuning of a diatonic scale that has a reasonable approximation to 4:5:6:7 as its "dominant 7th" (while all the major and minor triads still work). It can't be done. One must simultaneously distribute the syntonic comma 80:81 (so 8:9 = 9:10) 21.5c and the huge septimal diesis 35:36 (so 6:7 = 5:6) 48.8c.

Regards,

-- Dave Keenan
http://dkeenan.com

[Carl Lumma:]
> For composers working with the diatonic scale and 11-limit chromatic
harmony, your naming scheme (or something like it) will be essential.
Inconsistency is just something that composers interested in this
setup will have to deal with.

Yeah, this really strikes me as a good and important point to make: it
can be dealt with.

Dan

On Wed, 09 Feb 2000 16:09:20 -0800, David C Keenan <d.keenan@uq.net.au>
wrote:

>This is a somewhat "religious" issue but I contend that a dominant 7th
>chord is not (should not be) an approximate 4:5:6:7 chord. A 4:5:6:7 is
>much more consonant and should be called a subminor 7th or (German)
>augmented 6th chord.
>
>A dominant seventh chord has a _minor_ third stacked on a major triad (not
>a _subminor_ third).

I prefer the term "major-minor seventh" for this chord, since "dominant"
implies V of I (i.e., the fifth degree of the traditional diatonic scale).

--
see my music page ---> +--<http://www.io.com/~hmiller/music/music.html>--
Thryomanes /"If all Printers were determin'd not to print any
(Herman Miller) / thing till they were sure it would offend no body,
moc.oi @ rellimh <-/ there would be very little printed." -Ben Franklin

David Canright wrote:

> But where you have more
> than one
> name with the same 41-ET representation (commas to limmas) you must mean
> smaller ranges, and perhaps where you skip 41-ET classes you mean the
> names
> cover larger ranges...? Actually, I suspect you sometimes mean
> categories
> (like "major third") and sometimes mean specific intervals (like
> "septimal
> comma").

Yes, what exactly do I mean? The idea is to give a unique name to each
"interesting" 11-limit related interval, and also to name each of the 41
pitch classes. Where two intervals share the same pitch class, I gave
them more specific names. However, each pitch class also corresponds to a
schismic interval, and the generic name is more precisely for that
interval.

So, here are some names of schismic intervals, with their position in 41=
and my schismic notation:

20 narrow tritone F# Bb\
19 super fourth F// A//
18 wide fourth F/ A/
17 perfect fourth F A
16 narrow fourth F\ A\
15 sub fourth F\\ A\\
15 supermajor third E// G#/
14 wide major third E/ G#
13 narrow major third E G#\
12 neutral third E\,Eb// G/,G#\\
11 wide minor third Eb/ G
10 narrow minor third Eb G\
9 subminor third Eb\ G\\
8 supermajor second D/ F#/
7 wide major second D F#
6 narrow major second D\ F#\
5 neutral second D\\,C#// F/,F#\\
4 semitone (minor second) C#/ F
3 limma (narrow minor second) C# F\
2 subminor second C#\ F\\
2 diesis (super unison) C// E//
1 comma (wide unison) C/ E/
0 unison C E

This list is now as consistent as I want it. Here's the logic. First,
the following intervals are fixed:

of41 of29
20 14 narrow tritone F# Bb\ 7:5
17 12 perfect fourth F A 4:3
13 9 (narrow) major third E G#\ 5:4
11 8 (wide) minor third Eb/ G 6:5
6 4 narrow major second D\ F#\ 10:9
4 3 (wide) minor second C#/ F 16:15
0 0 unison C E 1:1

I'm giving the sizes in both 41 and 29=. You can work them out in any
schismic temperament from that. Each interval is defined as the nearest
approximation to the ratio on the right. This ensures consistency between
different temperament classes. The equivalent intervals in the upper
tritone can be chosen through simple transformations:

major <--> minor
super <--> sub
wide <--> narrow

41 29 octave C E 2:1
37 26 (narrow) major seventh B Eb\ 15:8
35 25 wide minor seventh Bb/ D 9:5
30 21 (narrow) major sixth A C#\ 5:3
28 20 (wide) minor sixth GA/ C 8:5
24 17 perfect fifth G B 3:2
21 15 wide tritone F#/ Bb 10:7

Then, we have to decide what "wide" and "narrow" mean. The classic
"wideness" interval is the syntonic comma, 81/80. This is one step in
both 29 and 41=, and is associated in schismic temperaments with the
modifier /.

For "sub" and "super" the classic interval is the septimal diesis of
36:35. This is 2 steps in 41=, and associated with two / modifiers in
schismic notation. The general pattern of modifiers is as follows:

+2 super //
+1 wide /
0 perfect
-1 narrow \
-2 sub \\

+2 supermajor //
+1 wide major /
0 narrow major
- neutral
0 wide minor
-1 narrow minor \
-2 subminor \\

So major and minor intervals come in wide/narrow pairs whereas perfect
intervals have narrow and wide alternatives. The "supermajor" has to be
super of "narrow major" so that 8/7 is a "supermajor second".

Tritones are a law unto themselves.

For the notation to be consistent, the difference between "narrow major"
and "wide minor" should always be the same, and so the approximation to
25:24. This follows from 5-limit consistency. These names don't fit
5-limit inconsistent tunings very well.

If the 25:24 happens to approximate to an even number of steps, there are
uniquely defined neutral intervals. Otherwise, there will either be no or
ambiguous neutral intervals. (25/24 is *not* the template I would use for
augmentation or diminution, if I were to use such things.)

For 7-limit inconsistent tunings, "sub" and "super" are not clearly
defined. As David Canright implied, they would equate to 1 step in 24= as
anything else would be silly.

In the 11-limit, "super" has to mean 33:32 for 11:8 to be a super fourth.
Where this is not the same as 36:35, some compromise will have to be made.
Either distinguish between "septimal super" and "undecimal super", or
sacrifice consistency and go for the nearest interval by default. 55:54,
56:55 and 45:44 can variously be used for "wide" and "narrow" in a manner
only consistent with 41=.

Anyway, applying these rules to 22= gives the core intervals:

11 tritone 7:5,10:7
9 perfect fourth 4:3
7 narrow major third 5:4
6 wide minor third 6:5
3 narrow major second 10:9
2 wide minor second 16:15
0 unison 1:1

The syntonic comma is 1 step, so we can fill the whole scale with wide and
narrow intervals:

11 tritone 7:5,10:7
10 wide fourth 11:8
9 perfect fourth 4:3
8 wide major third 9:7
7 narrow major third 5:4
6 wide minor third 6:5
5 narrow minor third 7:6
4 wide major second 9:8
3 narrow major second 10:9
2 wide minor second 16:15
1 narrow minor second 25:24
0 unison 1:1

Although these are not the usual names for the 7- and 11-limit intervals,
it looks like the best way to consistently name 22= in a diatonic manner.
Comparing it with David Keenan's latest:

11 F#\, Gb/ augmented fourth, diminished fifth
10 F/ super fourth
9 F perfect fourth
8 E supermajor third
7 E\ major third
6 Eb/ minor third, neutral third
5 Eb subminor third
4 D supermajor second, wide major second (8:9)
3 D\ neutral second, narrow major second (9:10)
2 Db/ minor second
1 C/ super unison
0 C perfect unison

"Super" and "wide" are synonyms, as are "sub" and "narrow". Then it
works. I'd prefer "neutral" not be used in this context. The symbols are
perfectly consistent: all kinds of minor thirds are kinds of Eb, and so
on.

With 46=, the 11-limit comes out very similar to how it does in 41=:

23 tritone
22 narrow tritone 7:5
21 super fourth 11:8
20 wide fourth
19 perfect fourth 4:3
18 narrow fourth
17 supermajor third 9:7
16 wide major third
15 major third 5:4
14 wide neutral third 14:11
13 narrow neutral third 11:9
12 minor third 6:5
11 narrow minor third
10 subminor third 7:6
9 supermajor second 8:7
8 wide major second 9:8
7 narrow major second 10:9
6 wide neutral second 11:10,12:11
5 narrow neutral second
4 wide minor second 16:15
3 narrow minor second 25:24
1 wide unison
0 unison 1:1

All we have to do is alter the laws of harmony so that 12:11 becomes a
narrow neutral second, and everything will be perfect :)

So there we are, a general way of naming pitch classes in meantone,
schismic and diaschismic temperaments. Sometime, I'll try and get a
computer program together that will assign the names automatically. The
next challenge is getting it to work with 17=.

Dave Keenan explained:

> It's because you wrote either "augmented fourth
> (7:10)" or "diminished fifth (5:7)" (I forget which) in an earlier message.
> And because it's a common misconception, partly blameable on the
> ubiquitousness of 12-tET (where of course they are the same size).

I'm afraid you missed my point. The fact that you can't remember which is
which verifies it. While the 12-tET tritone is symmetrically invertible, the
acoustics tritones are not. To the contrary, 12-tET tends to promote the
idea that #4 and b5 are essentially the same pitch. They are _not_--
particularly in performance practice of common practice music.
>
>>But....but.....but.....but 7:10 fits the partial series (of C) as Bb _up_ to
>>E--an augmented fourth in a dominant seventh chord belonging to the key of
>>F. In the C dominant seventh chord "acoustically" tuned, the Bb is closer to
>>the E below--a diminished fifth.
>>
>>What's that you say? Common practice is not the only music system in
>>existence? Well, I'll be hornswaggled! You mean some composers could care
>>less about acoustic reality and four hundred years of practice? I'll be
>>damned.
>
> That's not what I'm saying at all.
>
> Ratios with odd-factors greater than 5 are irrelevant to diatonic music,
> with the possible exception of ratios of 9. 7's don't matter. They are only
> involved in functional _dissonances_.

ONLY involved with functional dissonance?????????? What do you think pushes
the dynamic forces of functional harmony? It's the tritone, baby. And
well-tuned tritones do the job better than any piano could ever dream of.
>
> This is a somewhat "religious" issue

(I think I just illustrated that point with my passionate outburst. :-)

> but I contend that a dominant 7th
> chord is not (should not be) an approximate 4:5:6:7 chord.

Why not? What do you have against consonance? Did you _believe_ your theory
teacher when he/she said "The seventh partial is quite unusable in
traditional music"? Wow! What a waste!

> A 4:5:6:7 is
> much more consonant and should be called a subminor 7th or (German)
> augmented 6th chord.

I have no trouble with "subminor 7th," but a 4:5:6:7 has nothing to do with
an augmented sixth chord (spell Ab-C-EborD#-F#). Listen to a good string
quartet play an augmented sixth chord and you will never bring up that
argument again.
>
> A dominant seventh chord has a _minor_ third stacked on a major triad (not
> a _subminor_ third).

You've go to be kidding! Why would a pitch sensitive singer/player be
satisfied with a 12-tET seventh when an acoustic one is much more
expressive?
>
> So what you say above, is true only if you replace "dominant" with
> "subminor", but we would no longer be in the realm of diatonic scales.

Ta-dah!!!! That statement shines a light on the problem. While some (perhaps
most) musicians tend to think in terms of a scale of pitches directly
related to the tonic pitch, string players and singers (among others) tend
to think (perhaps intuitively) of tuning in terms of the prevailing harmony.
In other words, the "diatonic scale steps"actually _move_ in relation to the
changing fundamental. Tuning is a dynamic function. The ear provides instant
correction of related pitches as the music proceeds. No need for math. No
need for 43 pitch scales. (Easy folks. I _do_ dig your microtonal
creations.)

Here's an example. When a singer is on the tonic pitch in tonic harmony
(C-E-G) and the harmony changes to a V7-of-ii (D-F#-A-C), if the singer
"repeats" the C, it will be slightly lower (4:7 over D) in preparation for
the resolution to B, the third of the G major target chord. Don't tell me
this is not true. I've witnessed it hundreds of times. (Never on a piano,
however.)
>
>>Hold on there, Mr. K. B-F in the key of C is not 7:10 over a C fundamental;
>>it's 7:10 over a _G_ fundamental--the dominant of C.
>
> If B:F (B up to F) is closer to a 7:10 than a 5:7

Hooooooooold it! That's _not_ "closer to 7:10 than 5:7." You've still got it
backwards. Also, it _is_ 5:7; _not_ 7:10. And "closer to" has nothing to do
with it.

> then that's true no
> matter what other notes it is over. I assume we're not talking about
> dynamic retuning

I can't speak for you, but I am definitely talking about dynamic tuning
(forget the "re-"). That's the only tuning I know anything about.

> and we are both assuming the convention of writing the
> notes of an interval or chord with pitch ascending from left to right.

I'm not sure what you mean by "left to right." Keyboard left to right?

> Yes,
> the B:F forms part of the G7 chord and is closer to 7:10 than 5:7 in most
> reasonable diatonic tunings of that chord.

You still have it backwards.

> So it seems we agree.

I'm afraid not.
>
>>Therefore, when B is
>>the upper member it is the 10 of 7:10, an augmented fourth (F up to B) and
>>goes to C; and when F is the upper member it is 7 of 5:7, a diminished fifth
>>(B up to F) and goes down to E. Therefore, since 7:10 is larger than 5:7,
>>their initial sizes are related to where they will resolve.
>
> Ok. We disagree. Same problem. dominant 7th chord is closer to 4:5:6|5:6
> than 4:5:6:7. 5:6 + 5:6 is closer to 7:10 than 5:7.

No doubt about that. But why do you buy into the notion that 4:7 is not
usable in diatonic music?
>
>>> Of course whether B:F is closer to 5:7 or 7:10 depends on your tuning.
>>
>>Not on _my_ tuning--on "nature's" tuning (if one accepts the principles of
>>"functional harmony"). If you want to resolve them the other way, you
>>certainly can. To me, the resolution of an augmented fourth is predetermined
>>by its "larger" tuning.
>
> We don't disagree on how the notes resolve, or what the intervals are
> called. Our only disagreement is what JI ratio they best approximate. The
> only way you could be right is if the scale was tuned with fifths _wider_
> than 700 cents (on the Pythagorean side of 12-tET, a "negative" meantone).
> And as you probably know, this would give very bad major and minor thirds
> (assuming we're still trying to be diatonic).

Don't forget about what you refer to as "dynamic tuning" and I as "moving
pitches."
>
>>Perhaps we could simply agree to disagree on this.
>
> Perhaps.

I think we have. As long as you prefer a keyboard-influenced dominant chord,
we have little in common.
>
>>It might be
>>considered that during Fokker's productive years, the principle of tonality
>>was being seriously challenged by Schoenberg and friends. It seems likely
>>that his use of these interval terms were intentionally "nonfunctional."
>
> No. Absolutely not. They are perfectly sensible terms for extended meantone
> tunings. Even Paul and Carl agree with that.

I know nothing of "extended meantone tunings." I only know what I hear.
(And, as you may have noticed, Paul and Carl have caused me to be very
careful about _that_. LOL)
>
>>> Yes. The whole system
>>
>>And this is the rub. We're talking about different musical systems here, as
>>you clearly point out as this paragraph continues...
>>
>>> would be screwed if we swapped these two. To be fair
>>> to Mr Fokker, strictly speaking, he gave the names to the intervals of
>>> 31-tET (a meantone), not to the JI ratios which they approximate. The
>>> latter seems like an obvious move to many people
>
> Not different systems. One is merely a superset of the other. 31-tET is an
> extended meantone and contains perfectly ordinary diatonic subsets. Their
> Fokker interval names are completely standard and uncontrovertial.

Does "standard and uncontroversial" for meantone include naked and raw
acoustic tuning?
>
>>> But one does need to know (at
>>> least) whether the tuning being used has G# < Ab or G# > Ab.
>>
>>Exactly. To me, there is no question about it.
>
> So which is it, G# < Ab (meantone) or G# > Ab (Pythagorean)

Probably, neither.
>
> Show me a tuning of a diatonic scale that has a reasonable approximation to
> 4:5:6:7 as its "dominant 7th" (while all the major and minor triads still
> work). It can't be done. One must simultaneously distribute the syntonic
> comma 80:81 (so 8:9 = 9:10) 21.5c and the huge septimal diesis 35:36 (so
> 6:7 = 5:6) 48.8c.

It's called "live performance." It's really quite exciting. Join a good
choir or play in a fine string group. Ears and nature get along very well
when left to themselves.

Our "differences" are clearly due to our different experience with music.
Neither is "better" than the other. They are simply different. I am learning
an enormous amount of valuable information by participating on the List. I
hope my "foreign" contribution is of value.

Thanks for the exchange of ideas, David.

Jerry

[Carl Lumma, TD 525.2]
>I just meant that the diatonic scale is not 11-limit consistent, so no
>naming scheme based on diatonic degrees will be either.

I agree with the premise (of course), but not the conclusion. How would one tell if an interval naming scheme was 11-limit consistent? What is a counterexample to this test in the scheme under discussion?

>>But each column is now consistent in its own way. One is a list of note
>>names, the other a list of interval names.
>
>How are you using consistency here?

I mean that each column is internally consistent. The first (the note names) consistently represents fifths and half-octaves. The second (the interval names) consistently represents 11-limit ratios being approximated (at least I hope so). However, you will notice I have given some intervals multiple names and one must be allowed to choose the name which is most appropriate in any given context (since more than one ratio is approximated by one 22-tET interval).

>The wordy names in your table aren't
>even close to consistent with 22. For example, stacking two of your thirds
>would wind one up with, variously, types of fourths, fifths, and sixths.

Hmm. I guess you haven't read the full paper.
http://dkeenan.com/Music/IntervalNaming.htm

As well as multiple names that are specific to a particular tuning, multiple names are always available. For those that don't read web pages, here's the most relevant table (slightly abridged).

The least-preferred names are in parenthesis. Context will often dictate the use of a less-preferred term.

31-tET Ratios Names
degree
----- ---------- --------------------------------------------
0 1:1 ..................... unison
1 48:49 44:45 35:36 32:33 (dimin. second) super unison
2 27:28 24:25 20:21 subminor second (augmented unison)
3 15:16 14:15 minor second .....................
4 11:12 10:11 neutral second
5 9:10 8:9 major second .....................
6 7:8 supermajor second (diminished third)
7 6:7 (augmented second) subminor third
8 5:6 ..................... minor third
9 9:11 ..................... neutral third
10 4:5 (subdiminished fourth) major third
11 11:14 7:9 (diminished fourth) supermajor third
12 16:21 sub fourth (augmented third)
13 3:4 perfect fourth
14 8:11 super fourth
15 5:7 augmented fourth (subdiminished fifth)
16 7:10 (superaugmented fourth) diminished fifth
17 11:16 ..................... sub fifth
18 2:3 ..................... perfect fifth
19 21:32 (diminished sixth) super fifth
20 9:14 7:11 subminor sixth (augmented fifth)
21 5:8 minor sixth (superaugmented fifth)
22 11:18 neutral sixth
23 3:5 major sixth
24 7:12 supermajor sixth (diminished seventh)
25 4:7 (augmented sixth) subminor seventh
26 9:16 5:9 ..................... minor seventh
27 11:20 6:11 ..................... neutral seventh
28 8:15 ..................... major seventh
29 14:27 (diminished octave) supermajor seventh
30 18:35 sub octave (augmented seventh)
31 1:2 octave
32 22:45 16:33 super octave (diminished ninth)
33 12:25 10:21 (augmented octave) subminor ninth
34 15:32 7:15 ..................... minor ninth
35 11:24 5:11 ..................... neutral ninth
36 9:20 4:9 ..................... major ninth
37 7:16 (diminished tenth) supermajor ninth
38 3:7 subminor tenth (augmented ninth)
39 5:12 minor tenth
40 9:22 neutral tenth
41 2:5 major tenth (subdiminished eleventh)
42 11:28 7:18 supermajor tenth (diminished eleventh)
43 8:21 (augmented tenth) sub eleventh
44 3:8 ..................... perfect eleventh
45 4:11 ..................... super eleventh
46 5:14 (subdiminished twelfth) augmented eleventh
47 7:20 diminished twelfth (superaugmented eleventh)
48 11:32 sub twelfth
49 1:3 perfect twelfth

Of course the series for any nth can be continued to "double-diminished", "sub-double-diminished" etc., if it's really necessary in some context.

>>I't wouldn't do to have an approximate 4:5 being called a major third in
>>most tunings but called a neutral (or submajor, or narrow) third in 22-tET.
>
>There are very good reasons for Paul's position when it comes to diatonic
>naming in 22. The 9:7 and 7:6 are the natural diatonic major and minor
>thirds in this tuning, based on the way they fall in the scale. The o- and
>-utonal 6:7:9 chords function as 1-3-5 triads exactly like the 4:5:6's
>function in a meantone diatonic.

I know what you mean, but for me this is stretching the meaning of "diatonic" beyond the breaking point. As others have said, a 7:9 is a car horn, not a major third. It can _function like_ a diatonic major third but it sure doesn't _sound like_ one. I want names that stick to how they sound. Like octave and fifth undoubtedly already do (unless one prefaces them with the word "formal" or suchlike). So by all means call an approx 7:9 a "formal major third" in the specific context of playing in a pseudo-diatonic scale in 22-tET.

Of course, another take on that is:

Sure a 7:9 can be perceived as a major third (just barely), but 4:5 will be perceived as one too. So what kind of major third is a 7:9? Why it's a _super_ major-third, since clearly the 4:5 is the standard one we are all used to.

>Dave, I hope you don't take me wrong on this issue. When working with the
>diatonic scale, diatonic names are appropriate. When working with the
>decatonic scale they are not.
>
>For composers working with the diatonic scale and 11-limit chromatic
>harmony, your naming scheme (or something like it) will be essential.
>Inconsistency is just something that composers interested in this setup
>will have to deal with. By the way, diatonic music with high-limit
>chromatic harmony is one of the most fertile grounds in tuning theory in my
>opinion, and I have plans to explore it in depth one day, when your naming
>scheme will very likely be handy to me.

OK. So long as we agree on _what it is_ we are disagreeing on.

[David Beardsley, TD525.10]
>>"Canright, David" wrote:
>> but to me it seems that referring to JI intervals by names
>> that confuse many seems counterproductive and at least potentially
>> ambiguous. Call a 6:7 a 6:7, I say...
>
>I agree. If you look closely at the Partch instruments,
>you'll see ratios painted on the instruments. When I got
>my guitar with Catler JI Tuning II, he gave me a
>chart with the ratios on the neck not a list of descriptive names.

I'm proposing these as names of (categories of) intervals, not pitches. To be used in addition to the numbers, not instead of them. It's great to have the JI pitches written on the instruments.

>Names like supraminor
>third are useless to communicate a ratio to a musican.
>It's a nice description but why make it more complicated?

No one's proposing "supraminor" anything. I'll assume you meant subminor third (i.e. 6:7 and its vicinity).

Surely it would be useful for indicate what the interval _sounds_ like (say to a classically trained (or untrained) singer or violinist).
"Sing/play me a 6:7 above this note. Wah..."
"What the heck is a 6:7?"
"Its a subminor third."
"Ok. I guess that means it's narrower than a minor third. Is it the same as a major second?"
No, it's in between, but closer to a minor third. Just sing a minor third and then lower it gradually until you hear the beats cancel again".

>Those points aside...if you really do have to always refer to
>an orange as "a round fruit with a pulpy inside and seeds with a
>a peelable skin" instead of just calling it an orange:
>
>Using terms like "supermajor third", "subminor third" and
>"supermajor second" help to thoroughly confuse the issue.
>Why not use a common term like septimal to describe these
>7 limit ratios?

Please read my paper:
http://dkeenan.com/Music/IntervalNaming.htm

Because "septimal" doesn't tell you whether they are wider or narrower than the nearby (more familiar to most) 5-limit intervals. In fact, if you look at the preferred names I've given above (and in previous posts), supermajor and subminor always imply septimal (except for some ratios which use both 7 and 11), and neutral always implies undecimal (11-limit).

>Message: 21
> Date: Wed, 09 Feb 2000 22:39:44 -0600
> From: Herman Miller <hmiller@io.com>
>Subject: Re: Re: Interval names
>
>On Wed, 09 Feb 2000 16:09:20 -0800, David C Keenan <d.keenan@uq.net.au>
>wrote:
>
>>This is a somewhat "religious" issue but I contend that a dominant 7th
>>chord is not (should not be) an approximate 4:5:6:7 chord. A 4:5:6:7 is
>>much more consonant and should be called a subminor 7th or (German)
>>augmented 6th chord.
>>
>>A dominant seventh chord has a _minor_ third stacked on a major triad (not
>>a _subminor_ third).
>
>I prefer the term "major-minor seventh" for this chord, since "dominant"
>implies V of I (i.e., the fifth degree of the traditional diatonic scale).

[Dan Stearns, TD 525.13]
>I'm probably just chronically prone to
>claustrophobic episodes when exposed to certain strains of
>incontestable assertions.

Sorry Dan. I know a categorical statement such as "It can't be done" is seldom true. But sometimes it would just distract from the point at issue if I were to include all the caveats. Often, I'd be just as happy if someone proved me wrong. We all just want to know the truth dont we. This might be called the red-rag-to-a-bull approach.

[Herman Miller, TD 525.21]
>I prefer the term "major-minor seventh" for this chord, since "dominant"
>implies V of I (i.e., the fifth degree of the traditional diatonic
>scale).

Me too. But I've never been brave enough to use it on this list before. Thanks for the encouragement.

If you're going to hyphenate it wouldn't "major minor-seventh" make more sense?

I assume you agree that in strictly diatonic music the dominant seventh function is always performed by a major minor 7th chord (approx 4:5:6|5:6), not a major subminor 7th (approx 4:5:6:7)

Regards,

-- Dave Keenan
http://dkeenan.com

Sorry about the mess with the previous post (Herman's reply quoted twice). Also...

I did it again Dan. I wrote in response to Herman:
>>I assume you agree that in strictly diatonic music the dominant seventh
>>function is always performed by a major minor 7th chord >(approx 4:5:6|5:6), not a major subminor 7th (approx 4:5:6:7).

Sorry about the "always".

Of course there's Barbershop with its tricky adaptive tuning. So let's just say "nearly always".

-- Dave Keenan
http://dkeenan.com

> [Dave Keenan, TD 526.15]
> How would one tell if an interval naming scheme was 11-limit
> consistent?

Paul Hahn has developed the concept of *levels* of consistency,
and has charts of ETs showing these levels.
http://library.wustl.edu/~manynote/music.html

> [Dave]
> I'm proposing these as names of (categories of) intervals,
> not pitches.
> <snip>
> Surely it would be useful for indicate what the interval
> _sounds_ like (say to a classically trained (or untrained)
> singer or violinist).
> "Sing/play me a 6:7 above this note. Wah..."
> "What the heck is a 6:7?"
> "Its a subminor third."
> "Ok. I guess that means it's narrower than a minor third.
> Is it the same as a major second?"
> No, it's in between, but closer to a minor third. Just sing
> a minor third and then lower it gradually until you hear the
> beats cancel again".

What a great practical illustration of harmonic entropy!

I got an idea from this: I'd think that those involved in
this discussion of interval names would want to correlate the
names in some way with Paul's harmonic entropy mathmetics.
For info on harmonic entropy, see
http://www.ixpres.com/interval/td/Erlich/entropy.htm

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Gerald Eskelin wrote,

>But....but.....but.....but 7:10 fits the partial series (of C) as Bb _up_
to
>E--an augmented fourth in a dominant seventh chord belonging to the key of
>F. In the C dominant seventh chord "acoustically" tuned, the Bb is closer
to
>the E below--a diminished fifth.

>What's that you say? Common practice is not the only music system in
>existence? Well, I'll be hornswaggled! You mean some composers could care
>less about acoustic reality and four hundred years of practice? I'll be
>damned.

Your "four hundred years of music practice" is a myth. In the meantone era,
roughly 1480-1780, diminished fifths were tuned closer to 7:10 and augmented
fourths were tuned closer to 5:7, making dominant seventh chords very
dissonant by modern standards. But these dissonant dominant sevenths are
highly appropriate for the music of that period.

>As you correctly interpreted, I was being facetious here. It might be
>considered that during Fokker's productive years, the principle of tonality
>was being seriously challenged by Schoenberg and friends. It seems likely
>that his use of these interval terms were intentionally "nonfunctional."

On the contrary, Fokker kept all the features of meantone tuning intact, and
remember, meantone is the tuning in which functional harmony was born.

Graham, you applied a schismic naming scheme to a non-schismic (but
diaschismic) temperament, 22-equal. Do you see any problems with that?

(Since this was returned to me, I'll suppose it never got posted on the
List. If it has, I apologize for taking up the space twice. GRE)

Dave Keenan explained:

> It's because you wrote either "augmented fourth
> (7:10)" or "diminished fifth (5:7)" (I forget which) in an earlier message.
> And because it's a common misconception, partly blameable on the
> ubiquitousness of 12-tET (where of course they are the same size).

I'm afraid you missed my point. The fact that you can't remember which is
which verifies it. While the 12-tET tritone is symmetrically invertible, the
acoustics tritones are not. To the contrary, 12-tET tends to promote the
idea that #4 and b5 are essentially the same pitch. They are _not_--
particularly in performance practice of common practice music.
>
>>But....but.....but.....but 7:10 fits the partial series (of C) as Bb _up_ to
>>E--an augmented fourth in a dominant seventh chord belonging to the key of
>>F. In the C dominant seventh chord "acoustically" tuned, the Bb is closer to
>>the E below--a diminished fifth.
>>
>>What's that you say? Common practice is not the only music system in
>>existence? Well, I'll be hornswaggled! You mean some composers could care
>>less about acoustic reality and four hundred years of practice? I'll be
>>damned.
>
> That's not what I'm saying at all.
>
> Ratios with odd-factors greater than 5 are irrelevant to diatonic music,
> with the possible exception of ratios of 9. 7's don't matter. They are only
> involved in functional _dissonances_.

ONLY involved with functional dissonance?????????? What do you think pushes
the dynamic forces of functional harmony? It's the tritone, baby. And
well-tuned tritones do the job better than any piano could ever dream of.
>
> This is a somewhat "religious" issue

(I think I just illustrated that point with my passionate outburst. :-)

> but I contend that a dominant 7th
> chord is not (should not be) an approximate 4:5:6:7 chord.

Why not? What do you have against consonance? Did you _believe_ your theory
teacher when he/she said "The seventh partial is quite unusable in
traditional music"? Wow! What a waste!

> A 4:5:6:7 is
> much more consonant and should be called a subminor 7th or (German)
> augmented 6th chord.

I have no trouble with "subminor 7th," but a 4:5:6:7 has nothing to do with
an augmented sixth chord (spell Ab-C-EborD#-F#). Listen to a good string
quartet play an augmented sixth chord and you will never bring up that
argument again.
>
> A dominant seventh chord has a _minor_ third stacked on a major triad (not
> a _subminor_ third).

You've go to be kidding! Why would a pitch sensitive singer/player be
satisfied with a 12-tET seventh when an acoustic one is much more
expressive?
>
> So what you say above, is true only if you replace "dominant" with
> "subminor", but we would no longer be in the realm of diatonic scales.

Ta-dah!!!! That statement shines a light on the problem. While some (perhaps
most) musicians tend to think in terms of a scale of pitches directly
related to the tonic pitch, string players and singers (among others) tend
to think (perhaps intuitively) of tuning in terms of the prevailing harmony.
In other words, the "diatonic scale steps"actually _move_ in relation to the
changing fundamental. Tuning is a dynamic function. The ear provides instant
correction of related pitches as the music proceeds. No need for math. No
need for 43 pitch scales. (Easy folks. I _do_ dig your microtonal
creations.)

Here's an example. When a singer is on the tonic pitch in tonic harmony
(C-E-G) and the harmony changes to a V7-of-ii (D-F#-A-C), if the singer
"repeats" the C, it will be slightly lower (4:7 over D) in preparation for
the resolution to B, the third of the G major target chord. Don't tell me
this is not true. I've witnessed it hundreds of times. (Never on a piano,
however.)
>
>>Hold on there, Mr. K. B-F in the key of C is not 7:10 over a C fundamental;
>>it's 7:10 over a _G_ fundamental--the dominant of C.
>
> If B:F (B up to F) is closer to a 7:10 than a 5:7

Hooooooooold it! That's _not_ "closer to 7:10 than 5:7." You've still got it
backwards. Also, it _is_ 5:7; _not_ 7:10. And "closer to" has nothing to do
with it.

> then that's true no
> matter what other notes it is over. I assume we're not talking about
> dynamic retuning

I can't speak for you, but I am definitely talking about dynamic tuning
(forget the "re-"). That's the only tuning I know anything about.

> and we are both assuming the convention of writing the
> notes of an interval or chord with pitch ascending from left to right.

I'm not sure what you mean by "left to right." Keyboard left to right?

> Yes,
> the B:F forms part of the G7 chord and is closer to 7:10 than 5:7 in most
> reasonable diatonic tunings of that chord.

You still have it backwards.

> So it seems we agree.

I'm afraid not.
>
>>Therefore, when B is
>>the upper member it is the 10 of 7:10, an augmented fourth (F up to B) and
>>goes to C; and when F is the upper member it is 7 of 5:7, a diminished fifth
>>(B up to F) and goes down to E. Therefore, since 7:10 is larger than 5:7,
>>their initial sizes are related to where they will resolve.
>
> Ok. We disagree. Same problem. dominant 7th chord is closer to 4:5:6|5:6
> than 4:5:6:7. 5:6 + 5:6 is closer to 7:10 than 5:7.

No doubt about that. But why do you buy into the notion that 4:7 is not
usable in diatonic music?
>
>>> Of course whether B:F is closer to 5:7 or 7:10 depends on your tuning.
>>
>>Not on _my_ tuning--on "nature's" tuning (if one accepts the principles of
>>"functional harmony"). If you want to resolve them the other way, you
>>certainly can. To me, the resolution of an augmented fourth is predetermined
>>by its "larger" tuning.
>
> We don't disagree on how the notes resolve, or what the intervals are
> called. Our only disagreement is what JI ratio they best approximate. The
> only way you could be right is if the scale was tuned with fifths _wider_
> than 700 cents (on the Pythagorean side of 12-tET, a "negative" meantone).
> And as you probably know, this would give very bad major and minor thirds
> (assuming we're still trying to be diatonic).

Don't forget about what you refer to as "dynamic tuning" and I as "moving
pitches."
>
>>Perhaps we could simply agree to disagree on this.
>
> Perhaps.

I think we have. As long as you prefer a keyboard-influenced dominant chord,
we have little in common.
>
>>It might be
>>considered that during Fokker's productive years, the principle of tonality
>>was being seriously challenged by Schoenberg and friends. It seems likely
>>that his use of these interval terms were intentionally "nonfunctional."
>
> No. Absolutely not. They are perfectly sensible terms for extended meantone
> tunings. Even Paul and Carl agree with that.

I know nothing of "extended meantone tunings." I only know what I hear.
(And, as you may have noticed, Paul and Carl have caused me to be very
careful about _that_. LOL)
>
>>> Yes. The whole system
>>
>>And this is the rub. We're talking about different musical systems here, as
>>you clearly point out as this paragraph continues...
>>
>>> would be screwed if we swapped these two. To be fair
>>> to Mr Fokker, strictly speaking, he gave the names to the intervals of
>>> 31-tET (a meantone), not to the JI ratios which they approximate. The
>>> latter seems like an obvious move to many people
>
> Not different systems. One is merely a superset of the other. 31-tET is an
> extended meantone and contains perfectly ordinary diatonic subsets. Their
> Fokker interval names are completely standard and uncontrovertial.

Does "standard and uncontroversial" for meantone include naked and raw
acoustic tuning?
>
>>> But one does need to know (at
>>> least) whether the tuning being used has G# < Ab or G# > Ab.
>>
>>Exactly. To me, there is no question about it.
>
> So which is it, G# < Ab (meantone) or G# > Ab (Pythagorean)

Probably, neither.
>
> Show me a tuning of a diatonic scale that has a reasonable approximation to
> 4:5:6:7 as its "dominant 7th" (while all the major and minor triads still
> work). It can't be done. One must simultaneously distribute the syntonic
> comma 80:81 (so 8:9 = 9:10) 21.5c and the huge septimal diesis 35:36 (so
> 6:7 = 5:6) 48.8c.

It's called "live performance." It's really quite exciting. Join a good
choir or play in a fine string group. Ears and nature get along very well
when left to themselves.

Our "differences" are clearly due to our different experience with music.
Neither is "better" than the other. They are simply different. I am learning
an enormous amount of valuable information by participating on the List. I
hope my "foreign" contribution is of value.

Thanks for the exchange of ideas, David.

Jerry

>> [Dave Keenan, TD 526.15]
>> How would one tell if an interval naming scheme was 11-limit
>> consistent?

Joe Monzo wrote,

>Paul Hahn has developed the concept of *levels* of consistency,
>and has charts of ETs showing these levels.
>http://library.wustl.edu/~manynote/music.html

That has nothing to do with naming schemes, as far as I can tell.

>I got an idea from this: I'd think that those involved in
>this discussion of interval names would want to correlate the
>names in some way with Paul's harmonic entropy mathmetics.
>For info on harmonic entropy, see
>http://www.ixpres.com/interval/td/Erlich/entropy.htm

Again, I think categorical perception of intervals, especially melodic
intervals, is very different from the operation of the brain's central
pitch processor, and its contribution to the sensation of dissonance, as
modeled by harmonic entropy.

>
> Your "four hundred years of music practice" is a myth. In the meantone
era,
> roughly 1480-1780, diminished fifths were tuned closer to 7:10 and
augmented
> fourths were tuned closer to 5:7, making dominant seventh chords very
> dissonant by modern standards. But these dissonant dominant sevenths are
> highly appropriate for the music of that period.

Specifically, in 1/4 comma meantone, the diminished fifth each had 621 cents
while the augmented fourths had 579 cents. The diminished fifths are very
close to 10:7 (617.5), reasonably close to 64:45 (609.8). Compare this with
the 1/5 comma temperament with a diminished fifth of 614 cents and 1/6
comma, where the diminished fifth of 610 cents.

>
> >As you correctly interpreted, I was being facetious here. It might be
> >considered that during Fokker's productive years, the principle of
tonality
> >was being seriously challenged by Schoenberg and friends. It seems likely
> >that his use of these interval terms were intentionally "nonfunctional."
>
> On the contrary, Fokker kept all the features of meantone tuning intact,
and
> remember, meantone is the tuning in which functional harmony was born.
>

Indeed, Fokker and the musicians around him used 31tet as a substitute for
an extended meantone for playing early music. Fokker's own theoretical
speculations and compositional examples may in fact be heard as an effort to
present a tonal alternative to contemporary non-tonal approaches.

Paul Erlich wrote:

> Graham, you applied a schismic naming scheme to a non-schismic (but
> diaschismic) temperament, 22-equal. Do you see any problems with that?

I'm not sure if that's what I did do. But anyway, no problems with that.
I have some problems with applying the traditional names to
non-meantones, but I don't have any better ideas. 46= works very will
with the system I had previously worked out for 31 and 41=. It works as
well with 22= as any other diatonic-derived system.

>>I just meant that the diatonic scale is not 11-limit consistent, so no
>>naming scheme based on diatonic degrees will be either.
>
>I agree with the premise (of course), but not the conclusion.

Can you demonstrate otherwise?

>How would one tell if an interval naming scheme was 11-limit consistent?

I'm open to suggestions. The sure-fire method would be to check all triads
for consistency, just like with ET's. Remembering, of course, to check all
modes of the (un-even) scale. Checking only one mode is an ET-only shortcut.

One thing that may be worth looking at: the consistency results for the ET
with the same number of notes as the highest rank in the scale's rank-order
matrix (of the un-even scale on which the naming scheme is based). They
may be the same as the un-even scale itself. That is: the smallest ET
capable of tuning the scale may share it's consistency value. That would
mean consistency is an invariant of equivalence...

>What is a counterexample to this test in the scheme under discussion?

Take the 7:9:11 chord in 22. That tunes 0:8:14. In your scheme, that
makes two thirds equal a sixth.

>However, you will notice I have given some intervals multiple names and one
>must be allowed to choose the name which is most appropriate in any given
>context (since more than one ratio is approximated by one 22-tET interval).

If you need multiple names, you are admitting that your scheme is not
consistent (not that your scheme isn't unique, as you claim). Tempered
intervals may each approximate multiple just intervals without requiring
multiple names. I believe, for example, that diatonic naming is 7-limit
consistent even though the minor third must serve as both a 6:5 and a 7:6.

-Carl

Carl Lumma wrote,

>One thing that may be worth looking at: the consistency results for the ET
>with the same number of notes as the highest rank in the scale's rank-order
>matrix (of the un-even scale on which the naming scheme is based). They
>may be the same as the un-even scale itself.

Not following. The diatonic scale in a linear temperament other than 12-tET
has 13 different intervals less than an octave; are you suggesting we look
at 13-tET?

In-Reply-To: <4.0.1.20000213150239.01aee410@lumma.org>
Carl Lumma wrote:

> Take the 7:9:11 chord in 22. That tunes 0:8:14. In your scheme, that
> makes two thirds equal a sixth.

The offending interval is 7:11. The octave complement is 14:11, which is
on my charts. This is interesting, as I had previously seen arguments for
calling it a diminished fourth rather than a kind of third. Lemmeesee

14/11*4/5=56/55. This is much more like a comma than a diesis, so the
term "supermajor third" doesn't really apply. In 41=, for consistency
with it being such in 31=, we would have a "narrow supermajor third".
Quite a mouth full.

4/3*11/14 = 2/3*11/7 = 22/21. That's comparable with 21:20, which is a
limma-like interval. So "diminished fourth" fits well, if we're going to
use words like "diminished". In 31=, a diminished fourth is the same as a
supermajor third, so that works. In 41= it works: a limma flat of 4:3.
In 46= it works, consistent with 7:5 being an augmented fourth (the One
True Mapping). It doesn't work in 22= so you'd have to call it a narrow
fourth.

> If you need multiple names, you are admitting that your scheme is not
> consistent (not that your scheme isn't unique, as you claim). Tempered
> intervals may each approximate multiple just intervals without requiring
> multiple names. I believe, for example, that diatonic naming is 7-limit
> consistent even though the minor third must serve as both a 6:5 and a
> 7:6.

Not if that diatonic naming's applied to 12=. It would mean the interval
of 6 steps requires multiple names: augmented fourth and diminished fifth.
However, you could say that 7:5 is always the diminished fifth and 10:7
the augmented fourth. That is consistent. Wrong, but consistent.

In 22=, the interval of 8 steps can be a wide (or super) major third when
it's 9:7 and a narrow (but not diminished) fourth when it's a 14:11. It's
consistent in the same way. In 31=, the same intervals would be a
supermajor third and diminished fourth respectively, and be 11 steps.

In 41 and 46=, the diminished fourth becomes equivalent to a wide major
third, which could be the Pythagorean third 81:64.

[Paul Erlich]
>>One thing that may be worth looking at: the consistency results for the
>>ET with the same number of notes as the highest rank in the scale's rank-
>>order matrix (of the un-even scale on which the naming scheme is based).
>>They may be the same as the un-even scale itself.
>
>Not following. The diatonic scale in a linear temperament other than 12-tET
>has 13 different intervals less than an octave; are you suggesting we look
>at 13-tET?

You've shown that my speculation was wrong. BTW, that would be 13
intervals including the octave.

[Graham Breed]
>Not if that diatonic naming's applied to 12=. It would mean the interval
>of 6 steps requires multiple names: augmented fourth and diminished fifth.
>However, you could say that 7:5 is always the diminished fifth and 10:7
>the augmented fourth. That is consistent. Wrong, but consistent.

Hmm. Looks like defining consistency over un-even scales is trickier than
I thought... I'll have to get back to you.

-Carl

This may have been beaten to death on this list before, but what is
the definitive reference on all the different intervals named
"diesis", "limma", "apotome", and so on? I ask because I'm cleaning up
the respective Wikipedia articles.

Keenan

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...> wrote:
>
> This may have been beaten to death on this list before, but what is
> the definitive reference on all the different intervals named
> "diesis", "limma", "apotome", and so on? I ask because I'm cleaning up
> the respective Wikipedia articles.

You may be writing it.

I would check John Chalmers' definitions at
Tonalsoft.com

-- Mark Rankin

--- Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Keenan Pepper
> <keenanpepper@g...> wrote:
> >
> > This may have been beaten to death on this list
> before, but what is
> > the definitive reference on all the different
> intervals named
> > "diesis", "limma", "apotome", and so on? I ask
> because I'm cleaning up
> > the respective Wikipedia articles.
>
> You may be writing it.
>
>
>
>

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
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Check *my* definitions at Tonalsoft ...
they're the most comprehensive that i'm aware of:

http://tonalsoft.com/enc/d/diesis.aspx

http://tonalsoft.com/enc/a/apotome.aspx

http://tonalsoft.com/enc/l/limma.aspx

... and here's some of the "and so on":

http://tonalsoft.com/enc/c/comma.aspx

http://tonalsoft.com/enc/a/anomaly.aspx

http://tonalsoft.com/enc/s/schisma.aspx

http://tonalsoft.com/enc/d/diaschisma.aspx

http://tonalsoft.com/enc/k/kleisma.aspx

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

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@y...> wrote:
>
> I would check John Chalmers' definitions at
> Tonalsoft.com
>
> -- Mark Rankin
>
>
>
>
> --- Gene Ward Smith <gwsmith@s...> wrote:
>
> > --- In tuning@yahoogroups.com, Keenan Pepper
> > <keenanpepper@g...> wrote:
> > >
> > > This may have been beaten to death on
> > > this list before, but what is the definitive
> > > reference on all the different intervals named
> > > "diesis", "limma", "apotome", and so on? I ask
> > > because I'm cleaning up the respective Wikipedia
> > > articles.
> >
> > You may be writing it.

Check at the Tonalsoft Encyclopedia under S for
Skhisma, K for Kleisma, etc.

--- Mark Rankin <markrankin95511@yahoo.com> wrote:

> I would check John Chalmers' definitions at
> Tonalsoft.com
>
> -- Mark Rankin
>
>
>
>
> --- Gene Ward Smith <gwsmith@svpal.org> wrote:
>
> > --- In tuning@yahoogroups.com, Keenan Pepper
> > <keenanpepper@g...> wrote:
> > >
> > > This may have been beaten to death on this list
> > before, but what is
> > > the definitive reference on all the different
> > intervals named
> > > "diesis", "limma", "apotome", and so on? I ask
> > because I'm cleaning up
> > > the respective Wikipedia articles.
> >
> > You may be writing it.
> >
> >
> >
> >
>
>
> __________________________________________________
> Do You Yahoo!?
> Tired of spam? Yahoo! Mail has the best spam
> protection around
> http://mail.yahoo.com
>

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...> wrote:
>
> This may have been beaten to death on this list before, but what is
> the definitive reference on all the different intervals named
> "diesis", "limma", "apotome", and so on? I ask because I'm cleaning up
> the respective Wikipedia articles.
>
> Keenan

There are several sources, not all consistent with one another. One is:

http://www.xs4all.nl/~huygensf/doc/intervals.html
(Manuel Op de Coul)

Another is:
http://www.kylegann.com/Octave.html
(Kyle Gann)

Manuel and Kyle are both greats in the field and they both did a very
careful job here. When they disagree, it's not their fault -- we have
names coming down from Rameau and from Helmholtz that conflict with one
another!