Yahoo Tuning Groups Ultimate Backup tuning Re: Interval names: rational 'tritones'

There's been much back-and-forth here lately about the
rational definitions for 'augmented 4th' and 'diminished 5th',
which are synonymous in 12-EDO/tET tuning. The latest
installments were Jerry Eskelin in TD 524.9, Dave Keenan
in TD 525.12, and the following:

> [Bob Valentine, TD 525.22]
> I see the point that B up to F should be termed a diminished
> fifth and that F up to B should be termed an augmented fourth
> if ones terms are going to match the traditional diatonic scale.
> However, for F over B to be a 7/5 and B over F a 10/7, implies
> the belief that the fourth should always be flattened as part
> of a cadence. Perhaps that is what singers do, although I'd
> like to see some numerical data to back that. Without that
> assumption I would suspect that the diminished fifth is 64/45
> and the augmented fourth 45/32, which are very close to the
> exact opposite view!

I offer this explanation.

The terminology (in words) derives from 'common-practice' theory,
which was/is based on a diatonic scale in meantone tuning which
represents 5-limit JI.

Here is a lattice for the 'major scale' that is the
'standard' paradigm for 5-limit 'common-practice' theory:

A --- E --- B
5/3 5/4 15/8
/ \ / \ / \
/ \ / \ / \
F --- C --- G --- D
4/3 1/1 3/2 9/8

The 'dominant 7th' chord in this context, which most of us
would agree stems from the pitches available here, would
be formed as follows:

'7th' 4/3
'5th' 9/8
'3rd' 15/8
'root' 3/2

giving the proportions 36:45:54:64. (It can also be described
utonally as 1/(240:192:160:135), but the otonal description
gives smaller numbers and is the one commonly used.)

This chord, in 'root' position with the exact proportions
given above (i.e., no 'octave' displacements), contains a
'diminished 5th' with the ratio 45:64. This is defined by
Ellis (see Helmholtz 1954, p 455) as 3^-2 * 5^-1, and has a
size of ~610 cents. The septimal version, 7:10, is quite
close at ~618 cents; it does not appear in Ellis's list.

If the notes appear in a different position, with the 'B'
above the 'F', then the chord contains an 'augmented 4th'
(which is synonymous with 'tritone'), with the ratio 32:45.
This is defined on the same page by Ellis as 3^2 * 5^1,
and has a size of ~590 cents. The septimal version, 5:7,
is quite close at ~583 cents and is called by Ellis the
'septimal or subminor Fifth', a term (or terms) which I
find wholly inadequate.

The difference between the 5-limit intervals and their septimal
cousins is the 64:63 (= ~27 cents), which, following Ellis,
I have defined as the 'septimal comma'; see
http://www.ixpres.com/interval/dict/septcom.htm

I hope that has cleared up a few things. (More likely,
around here, it's opened up more questions than it answered...)

PS -
Clarence Barlow (Barlow 1987, p 46-48) gives a detailed
exploration of different possibilities for rational
interpretations of the 'tritone', according to his harmonicity
theories.

REFERENCES
----------

Helmholtz, Hermann L.F. von. 1954.
_Die Lehre von den Tonempfindungen als physiologische
Grundlage f�r die Theorie der Musik_.
_On the Sensations of Tone as a Psychological basis for
the Theory of Music_.
2nd English edition translated by Alexander James Ellis,
based on the 4th German edition of 1877
with extensive notes, foreword and afterword: 1885.
Reprint by Dover Publications.

Barlow, Clarence. 1987.
"Two Essays On Theory".
_Computer Music Journal_ vol. 11 no. 1, p 44-60.

-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 <stg3music@earthlink.net>

Joe Monzo noted:
>
> There's been much back-and-forth here lately about the
> rational definitions for 'augmented 4th' and 'diminished 5th',
> which are synonymous in 12-EDO/tET tuning.

And offered an interesting explanation. He began:

> The terminology (in words) derives from 'common-practice' theory,
> which was/is based on a diatonic scale in meantone tuning which
> represents 5-limit JI.

Is "common practice theory" the same as "common practice"? If so, why limit
it to "meantone tuning which represents 5-limit JI. Since the ear can hear
higher relationships, why discount them in a "theory" of tuning? Remember,
not all historical singers and string players read the theory books. They
simply adjusted pitches to fit their aural concepts of "best."

I"m beginning to suspect that this "problem" is grounded in theorists (and
perhaps microtonalists, as well) intent on providing a rational system of
fixed pitches that relate directly to a tonal center. Interest in allowing
for "scale step" deviations to accommodate root changes seems minimal, and
appears to result in compromised intervals having high-number ratios. Help
me on this, Monz. Is there any truth to that? Or am I out on a shaky and
naive limb here?

Well into his explanation, Joe states:

> If the notes appear in a different position, with the 'B'
> above the 'F', then the chord contains an 'augmented 4th'
> (which is synonymous with 'tritone'), with the ratio 32:45.

To me, the functional "augmented fourth" in a dominant 7th chord is _not
synonymous with "tritone." The former is larger and specifically has the
leading tone on top and the latter is the keyboard's compromised ambiguous
version of both "inversions."

> This is defined on the same page by Ellis as 3^2 * 5^1,
> and has a size of ~590 cents. The septimal version, 5:7,
> is quite close at ~583 cents and is called by Ellis the
> 'septimal or subminor Fifth',...

Why not 600 cents? Aren't we talking keyboard here? In any case, is 7 cents
considered "quite close"? The difference between a keyboard "tritone" and an
acoustic 5:7 diminished fifth or 7:10 augmented fourth is aurally quite
considerable.

> ...a term (or terms) which I
> find wholly inadequate
.
I don't know that "subminor fifth" is _wholly inadequate, but I do like
"diminished fifth" better. At least it is more likely to be understood by a
wider audience.

> The difference between the 5-limit intervals and their septimal
> cousins is the 64:63 (= ~27 cents),...

Sounds like a whopping difference. I'll go with the one that best describes
what ears hear. It appears to me that the 5-limit concept is simply a
compromised apology to 12-tET tradition.

> which, following Ellis,
> I have defined as the 'septimal comma'; see
> http://www.ixpres.com/interval/dict/septcom.htm

Does the mincrotonalist's term "comma" relate to my question above regarding
high-number compromises?

> I hope that has cleared up a few things. (More likely,
> around here, it's opened up more questions than it answered...)

For me it did both. My most pressing questions are included above. I look
forward to seeing your responses, Joe.

Jerry

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

Jerry wrote,

>I"m beginning to suspect that this "problem" is grounded in theorists (and
>perhaps microtonalists, as well) intent on providing a rational system of
>fixed pitches that relate directly to a tonal center.

Even in the fixed-pitch interpretation of the diatonic scale that Monz gave,
the D-A fifth is 27:40, and the D-F minor third is 27:32. Clearly this is an
example of what you are talking about. There had to be a way to sidestep
this problem on Renaissance and Baroque (and later!) keyboard instruments
(necessarily of fixed pitch); the solution was meantone temperament.
Unfortunately for Jerry, meantone temperament has very dissonant dominant
seventh chord. During the meantone period, a keyboard continuo was quite
common and it is doubtful that many vocalists would have made the ~35-cent
change necessary to make 4:5:6:7 dominant seventh chords. But I guess we'll
never know for sure.

>Interest in allowing
>for "scale step" deviations to accommodate root changes seems minimal, and
>appears to result in compromised intervals having high-number ratios.

The type of theory you're describing here is one that I've always lashed out
against. You'll find a lot of theorists doing that sort of thing, including
Yasser of all people, but those high-number ratios are almost always
irrelevant. The adaptive tuning algorithms John deLaubenfels has been
implementing and I and others have been discussing aim to tune almost all
simultaneities in simple-integer ratios; each "scale step" will fluctuate
over the course of the piece, though John's program tries to keep sudden,
drastic changes (which would be disturbing in performance) to a minimum.
Meanwhile, John Link and others seems to like sudden, drastic changes,
particularly in scale step 2 in the major scale, but in my opinion they're
clinging too strongly to an essentially fixed-pitch interpretation like the
one Monz gave (augmented with an additional "D" at 10/9 to make a consonant
D minor triad).

>> If the notes appear in a different position, with the 'B'
>> above the 'F', then the chord contains an 'augmented 4th'
>> (which is synonymous with 'tritone'), with the ratio 32:45.

>To me, the functional "augmented fourth" in a dominant 7th chord is _not
>synonymous with "tritone." The former is larger and specifically has the
>leading tone on top and the latter is the keyboard's compromised ambiguous
>version of both "inversions."

"Tritone" literally means three tones, or three major seconds, or an
augmented fourth.

>> This is defined on the same page by Ellis as 3^2 * 5^1,
>> and has a size of ~590 cents. The septimal version, 5:7,
>> is quite close at ~583 cents and is called by Ellis the
>> 'septimal or subminor Fifth',...

>Why not 600 cents? Aren't we talking keyboard here?

Ellis was one of the "high-number ratio" people we both take issue with. His
590 cents is the 45:32. For practical purposes, the keyboard tuning would
indeed be relevant -- as it happens, one variety of meantone, 1/6-comma
meantone, has exactly this 590-cent augmented fourth.

>In any case, is 7 cents
>considered "quite close"? The difference between a keyboard "tritone" and
an
>acoustic 5:7 diminished fifth or 7:10 augmented fourth is aurally quite
>considerable.

There are meantone tunings, though, where the augmented fourth would
coincide with Jerry's "diminished fifth"; namely, 583 cents. Most of the
historical meantone tunings come closer to this than 1/6-comma meantone.

>Sounds like a whopping difference. I'll go with the one that best describes
>what ears hear. It appears to me that the 5-limit concept is simply a
>compromised apology to 12-tET tradition.

The 5-limit concept is much older than 12-tET tradition. The 5-limit
concept, combined with the consensus ears of musicians, brought about the
adoption of meantone tuning in the late 15th century. The dominant seventh
chord was born in this 5-limit, meantone environment. In that environment,
it was the augmented sixth chord, not the dominant seventh chord, that
approximated the 4:5:6:7 chord. In fact, if you look at the historical
literature, you'll find people discussing ratios of 7 in the context of
augmented sixth chords and on their own merits long before ratios of 7 were
ever discussed in the context of dominant seventh chords. And this wasn't
the result of any over-abstract theory -- Tartini could play the ratios of 7
on his violin (we know because he reported hearing the then-unknown
difference tones in the correct places) and yet considered these ratios to
be foreign (in a good way!) to diatonic music rather than part of an
already-existing practice involving dominant seventh chords.

🔗John Link <johnlink@con2.com>

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

🔗Joe Monzo <monz@juno.com>

> [Jerry Eskelin, TD 527.25]
> Is "common practice theory" the same as "common practice"?
> If so, why limit it to "meantone tuning which represents
> 5-limit JI. Since the ear can hear higher relationships, why
> discount them in a "theory" of tuning? Remember, not all
> historical singers and string players read the theory books.
> They simply adjusted pitches to fit their aural concepts of
> "best."

OK - you make a really good point here.

I was simply presenting the 'standard' view of the 'dominant 7th'
as it may have been generally recognized in the 1800s. (During
most of the 1900s, I would say that '12-EDO thinking' monopolized
most trained musicians's view of things. I know for sure that
in my own very extensive musical schooling I never learned one
thing about ratios. It's all been a result of personal study
inspired by Partch's book.)

You are certainly correct to note that at all times in history
and in all places, music was and is conceived and performed
by people who have no conscious training or mathematical
awareness of their intonation - and sometimes it is very good!.

> [Jerry]
> I"m beginning to suspect that this "problem" is grounded in
> theorists (and perhaps microtonalists, as well) intent on
> providing a rational system of fixed pitches that relate
> directly to a tonal center.

I'll say that the grain of truth in what you say is this:
that theorists often assume that a musical performance makes
use of a set of pitches which is far smaller and less
sophisticated than the one actually being used.

Partly (perhaps mainly) this is a result of cultural conditioning.
Very often, if you're not aware of some kind of subtlety in
music you're listening to, it will be very subtle indeed: you
won't notice it at all.

About 12 years ago, I stunned myself one night because I had
been partying late, and the radio played _All My Loving_ by
the Beatles. A huge Beatle fan in my teens and 20s, I'd heard
the tune probably thousands of times and knew it by heart...
I thought. Well, this time I was listening so closely that
I noticed something I'd never heard before: on the last verse
(after the guitar solo), when Paul & John sing 'miss you', Paul
plays two wrong notes on his bass. I was so stoned that even
with my extremely minimal knowledge of bass-guitar technique I
figured out what he did wrong: he played those two notes on the
next higher string.

Point is: no matter how familiar you think something is, a
true masterpiece of a performance has so much packed into it
that you hear something new every time if you really pay
attention. And a *lot* of this subtlety in good musicians
has to do with variations in pitch ...and apparently, some
of it is unintentional! :)

> [Jerry]
> Interest in allowing for "scale step" deviations to
> accommodate root changes seems minimal, and appears to
> result in compromised intervals having high-number ratios.
> Help me on this, Monz. Is there any truth to that? Or am
> I out on a shaky and naive limb here?

The kind of deviations you're talking about here do indeed
involve high-number ratios, but as I like to point out on
a lattice diagram, the relationships may actually be much
simpler than they would appear (by looking only at the numbers).

I've become very interested in what I call 'cross-exponent'
connections, where the tuning of a note in a blues performance
suggests a ratio which is several rungs away from the chord-root
on the lattice, often in several different prime-dimensions.
But functionally, the notes are acting as valid chord-members,
as if they had a direct connection with (i.e., vector to) the
'root' (in the way, for instance, that 5/4, 3/2, and 7/4 all
have a direct connection to 1/1). One interesting example is
the 80:99 'low 3rd', which is ~369 cents and functions as a
'3rd' in the chord. The most relevant work I've done on this
is the old post I just referenced yesterday:
http://www.onelist.com/messages/tuning?archive=132
TD 132.2, an analysis of an Etta James MIDI-file.

On the contrary to what you say, it seems from my reading
of these Digests every day that interest in *lots* of
different aspects of tuning theory is very high, and
increasing all the time. Quite a few subscribers here
have posted much interesting information on 'deviations
to accommodate root changes' - unless I'm really missing
something about all the discussion surrounding John
deLaubenfels's work.

> [me, monz]
> If the notes appear in a different position, with the 'B'
> above the 'F', then the chord contains an 'augmented 4th'
> (which is synonymous with 'tritone'), with the ratio 32:45.

> [Jerry]
> To me, the functional "augmented fourth" in a dominant 7th
> chord is _not synonymous with "tritone." The former is larger
> and specifically has the leading tone on top and the latter
> is the keyboard's compromised ambiguous version of both
> "inversions."

You are entirely correct to point out that 'tritone' has
this meaning. My dictionary still lacks an entry for this
important term, and I see right away that there are at least
2 distinct definitions.

One is yours: the tritone does indeed specify the 'compromised
ambiguous version', 2^(6/12) = 600 cents = exactly 1/2-'octave'.

But the original meaning, which is obvious from the word's
etymology, is to define an interval composed of *3 tones*.
The term developed at a time which stipulated a Pythagorean
context (help me on the history, Margo), and so thus the
interval was (9/8)^3 = 512:729 = ~611.73 cents. This is
exactly synonymous with 'augmented 4th' in Pythagorean tuning.

Of course, 'tritone' has also been put into service to
mean the 5-limit varieties I discussed in my post, as well
as a myriad of other rational and irrational pitches.
In fact, it is probably the most malleable of all the
interval categories. Barlow too noticed this, and that's
why he chose precisely this interval to illustrate in his
paper.

> [me, monz]
> This is defined on the same page by Ellis as 3^2 * 5^1,
> and has a size of ~590 cents. The septimal version, 5:7,
> is quite close at ~583 cents and is called by Ellis the
> 'septimal or subminor Fifth',...

> [Jerry]
> Why not 600 cents? Aren't we talking keyboard here?

Partch complained a lot (in 1949) that keyboards had forced
12-tET onto everyone and everything, but today that's no
longer true. Lots of synthesizers are microtunable, and lots
of people experiment with 'alternate tunings'. So I wouldn't
equate the two anymore.

And to answer your question: No. *I* wasn't 'talking keyboard
here'! - or more precisely, I wasn't talking 12-EDO/tET here.
My lattice explictly portrayed 5-limit JI!

I assumed that we all knew the lovable old 600-cent tritone,
and were looking for alternate rational interpretations.

I was just trying to provide a backdrop that was based
on *theoretical* tunings for say the last 500 years or so.
Of course in practice this was translated into meantone
until about 1850 and into 12-EDO since then (until recently ;-).
But until the total dominance of 12-EDO in Euro-centric
theory around 1900, the 5-limit JI paradigm usually played
at last a small part in concepts concerning harmony/tonality.

> [Jerry]
> In any case, is 7 cents considered "quite close"?

Well, Fokker considered the 224:225 [= ~7.7 cents] to be
'imperceptible', and used it in his 1949 book as what I would
call a '5==7 xenharmonic bridge', that is, a small interval
that related a pitch that listeners familiar with 5-limit
practice would know, with the 7th harmonic, so that singers
could 'hear' the 'new' musical element.

I disagree with Fokker, and indeed have chosen (in _3 Plus 4_)
one pitch over another that were separated by exactly this
interval, because to my ears they made a big difference in the
harmony, and I preferred the 5-limit version, while the 7-limit
one (which my non-absolute-pitch mind thought it wanted) sounded
totally out-of-tune.

But in any case, by this difference of opinion, it seemed
to me that it was OK to characterize two intervals separated
by this distance as 'quite close'. (BTW, since the naming game
is going on, this category is a 'kleisma'.)

> [Jerry]
> The difference between a keyboard "tritone" and an acoustic
> 5:7 diminished fifth or 7:10 augmented fourth is aurally quite
> considerable.

2^(6/12) / (7/5) or (10/7) / 2^(6/12) = ~17.48 cents.

That's quite a bit greater - and *usually* *much* more
perceptible - than 7 cents. But again, *context can play
tricks on you!!!*. In the very piece I was talking about
above, I *heard* a much larger difference between 75/64
and 7/6 [= the ~7.7 cent 'bridge' discussed above] than I
did between 19/16 and 75/64 [= ~22.93 cents].

> [Jerry]
> I'll go with the one that best describes what ears hear.

Hah! - What do you think it is that we Tuning List addicts
try to thrash out day after day?! No-one is quite sure
exactly what 'ears' hear, and indeed, it's certain that
everyone hears a little differently. Let me know when
you figure it out. ;-)

> [Jerry]
> It appears to me that the 5-limit concept is simply a
> compromised apology to 12-tET tradition.

HARDLY !!!! The 5-limit concept far preceded the '12-tET
tradition'. That 'tradition' is really only a product of the
last century! At most, it goes back to the Renaissance lutenists.

The 5-limit concept is at least as old, perhaps stretching
all the way back to ancient Greece. I have written/will
eventually write a few papers speculating on the use of
5-limit tuning with the Franks, Romans, and Greeks, but most
theorists accept it as beginning in earnest around 1430.

At any rate, various meantones 'won out' first as a popular
fixed tuning which implied 5-limit JI. Well-temperaments
came later, *then* 12-tET carried the day (for a while).

> [Jerry]
> Does the mincrotonalist's term "comma" relate to my question
> above regarding high-number compromises?

Well, yes... Take a look at the various entries in my Dictionary.

The desideratum is usually to have various commas 'vanish',
so that pitches that would be a comma apart are both represented
by a single pitch, generally to prevent 'commatic drift'.

The pitch chosen need not be a 'high-number compromise'. In
many musical styles, particularly folk-music, the performer
simply chooses (or is forced by his instrument to choose) one
of the tones and uses it all the time, in which case it will
be a comma 'off' when the other pitch is the one that 'belongs'
in the harmony or melody.

Or, as in meantone, the pitch chosen is not a ratio at all,
but rather an irrational division of the syntonic comma.
For example, in '1/4-comma meantone', that 1/4-comma is
calculated as (9/8)^(1/4), or 'the 4th root of 9/8'.
This results in a number which has a non-repeating decimal
part, and which thus cannot be considered an integer ratio,
no matter how high the terms are multiplied. Of course,
at some point the ear/brain system fails to distinguish
a difference, but these limits of perception are another
thing no-one is sure of.

Anyway, I think the most important point you made was that
of the variability of tuning, especially among untrained
performers.

-monz

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🔗David C Keenan <d.keenan@uq.net.au>

Dear Jerry,

As Paul E. mentioned, there are (at least) two important "pulls" on the diminished fifth that is contained within a chord functioning as a dominant seventh chord in a diatonic scale. These apply whether we are talking about fixed tuning or adaptive tuning (e.g. by singers).

I earlier only gave one of them which is that the chord contains two stacked (approximate) minor thirds 5:6|5:6. This would make the dim 5th 25:36 (631 c).

The other, and probably more important "pull", mentioned by Paul E. (and Mark Nowitzky in his web page) is that there is a very important note in the scale, but not in the chord, that needs to be an approximate 3:4 below the high note and an approximate 3:4 above the low note (of the dim 7th chord). Of course that important note is the tonic. This relationship is required to prevent tonic drift in the common-as-dirt I IV V7 I chord progression, with notes sustained between chords. e.g.

F---F
E | E
| D
C---C C
B |
A |
G G---G

That implies that G:F is 9:16, and assuming we also want G:B to be 4:5, we get a size of 45:64 (610 c) for the B:F dim 5th.

In 5-limit JI the D:F minor third is broken anyway and so we go with the latter result. In meantone we compromise between the two. And I expect singers make a similar compromise, even when singing a cappella (except for the very distinctive "barbershop" style).

Diminished fifth candidates:

Ratio Cents Description
----------------------------------------
25:36 631.3 1/3-comma meantone (Just minor thirds 5:6)
620.5 1/4-comma meantone (Just major thirds 4:5)
7:10 617.5
614.1 1/5-comma meantone
45:64 609.8 5-limit JI (Just fourths 3:4 and major thirds 4:5)
600.0 12-tET

588.3 Pythagorean (Just fourths 3:4)
5:7 582.5 Barbershop (extreme dynamic tuning)

So if we want a low numbered (or 7 limit) ratio to represent the diminished fifth, 7:10 is way more representative of common diatonic practice for the last 400 years.

If you want to teach your singers to sing barbershop style, that's fine. In that case your diminished fifths _will_ be 5:7. But don't kid yourself that this is the "natural" size for a dim 5th. They must somehow hide the not inconsiderable difference between the 9:16 and the 4:7 (27.3 cents). With the above progression, I think that the best they can do is to gliss the C and F down by 9.1 cents and the G up by 9.1 cents.

Of course a V7 ii progression would present a _much_ harder problem. That of hiding the difference between a 6:7 and a 5:6 (48.8 c) in just one move. Best case would be to gliss the F up by 24.4 cents and the D down by 24.4 cents. I expect this would be quite noticeable.

And what about IV V7 ii IV?

Frankly Jerry, I don't trust that you know what ratios you're hearing. When you get the "high third" analysis over with, maybe you want to look at some dominant 7th chords as sung in some of the above progressions with sustained notes.

Thanks for helping me hone my arguments.

Regards,

-- Dave Keenan
http://dkeenan.com

🔗Gerald Eskelin <stg3music@earthlink.net>

David Keenan wrote:
>
> Dear Jerry,
>
> As Paul E. mentioned, there are (at least) two important "pulls" on the
> diminished fifth that is contained within a chord functioning as a dominant
> seventh chord in a diatonic scale. These apply whether we are talking about
> fixed tuning or adaptive tuning (e.g. by singers).
>
> I earlier only gave one of them which is that the chord contains two
> stacked (approximate) minor thirds 5:6|5:6. This would make the dim 5th
> 25:36 (631 c).
>
> The other, and probably more important "pull", mentioned by Paul E. (and
> Mark Nowitzky in his web page) is that there is a very important note in
> the scale, but not in the chord, that needs to be an approximate 3:4 below
> the high note and an approximate 3:4 above the low note (of the dim 7th
> chord). Of course that important note is the tonic. This relationship is
> required to prevent tonic drift in the common-as-dirt I IV V7 I chord
> progression, with notes sustained between chords. e.g.
>
> F---F
> E | E
> | D
> C---C C
> B |
> A |
> G G---G

Dave, I think your assumption that singer repeat the "same" pitches common
to neighboring chords leads to your "objection." Correct me if I'm wrong.

While the C common to I and IV is likely constant, the F common to IV and V7
clearly is not. Singers are more likely to sense the perfect-fifth root of
the V7 in relation to the remembered tonic pitch than to adjust the G, B,
and D to a firmly repeated F. As soon as the chord changes to V, the F
becomes less stable and finds "security" in relation to the root G. All of
this returns with no big problem to the same C on which the first tonic
chord was based.

Apparently, your conclusion that singers would sing a dominant seventh chord
with a 5:6 third from fifth to seventh is a result of your need to minimize
"drift." Believe me, drift does not occur when singers tune the dominant
seventh chord as 4:5:6:7. The "adjustment" occurs melodically in the
"less-than-half-step" resolution of the chord seventh.
>
> That implies that G:F is 9:16, and assuming we also want G:B to be 4:5, we
> get a size of 45:64 (610 c) for the B:F dim 5th.
>
> In 5-limit JI the D:F minor third is broken anyway and so we go with the
> latter result. In meantone we compromise between the two. And I expect
> singers make a similar compromise, even when singing a cappella (except for
> the very distinctive "barbershop" style).

Why the distinction? Classical singers can achieve wonderfully expressive
results when sensitive to acoustic tuning. To my ears, Brahms could never
have sounded better--even with a piano accompaniment.
>
> Diminished fifth candidates:
>
> Ratio Cents Description
> ----------------------------------------
> 25:36 631.3 1/3-comma meantone (Just minor thirds 5:6)
> 620.5 1/4-comma meantone (Just major thirds 4:5)
> 7:10 617.5
> 614.1 1/5-comma meantone
> 45:64 609.8 5-limit JI (Just fourths 3:4 and major thirds 4:5)
> 600.0 12-tET
>
> 588.3 Pythagorean (Just fourths 3:4)
> 5:7 582.5 Barbershop (extreme dynamic tuning)
>
> So if we want a low numbered (or 7 limit) ratio to represent the diminished
> fifth, 7:10 is way more representative of common diatonic practice for the
> last 400 years.
>
> If you want to teach your singers to sing barbershop style, that's fine. In
> that case your diminished fifths _will_ be 5:7. But don't kid yourself that
> this is the "natural" size for a dim 5th. They must somehow hide the not
> inconsiderable difference between the 9:16 and the 4:7 (27.3 cents). With
> the above progression, I think that the best they can do is to gliss the C
> and F down by 9.1 cents and the G up by 9.1 cents.

The C is not involved in the V7 tuning and the F will take care of itself
when tuned to the G.
>
> Of course a V7 ii progression would present a _much_ harder problem. That
> of hiding the difference between a 6:7 and a 5:6 (48.8 c) in just one move.
> Best case would be to gliss the F up by 24.4 cents and the D down by 24.4
> cents. I expect this would be quite noticeable.

Huh?
>
> And what about IV V7 ii IV?

V7 might legitimately go to ii but it had better get back to V before going
anywhere else. ii seldom goes to IV.
>
> Frankly Jerry, I don't trust that you know what ratios you're hearing.

Thanks for your frankness, Dave. Frankly, I haven't bothered to "prove" what
I'm hearing. It just make so much sense and seems to verify acoustic "truth"
that I don't feel the need to do so. If anything I say verifies anything you
have experienced then I'm glad. If you haven't experienced anything like
what I'm describing, then you have to decide whether my words are worth
anything. Or better yet, try it out with real humans.

> When
> you get the "high third" analysis over with, maybe you want to look at some
> dominant 7th chords as sung in some of the above progressions with sustained
notes.

An excellent suggestion. I shall.
>
> Thanks for helping me hone my arguments.

And me mine.
>
> Regards,

Likewise,
>
> -- Dave Keenan

Jerry Eskelin
> http://dkeenan.com