7:5 tritone in Common Practice JI : Handel Messiah

I am continuing to investigate how the 7:5 tritone or 10:7 dim5 can
fit into common practice harmony. It seems to provide a neat solution
to the JI problem of the supertonic in some contexts. That is, rather
than choose between 9:8 and 10:9, one may use 28:25, which is derived
via 5:4 and 7:5, if the chord includes a dim5.

To my current opinion 5-limit JI (adaptive or not) is optimal only for
triadic harmony. When the dim5 and dim7 enter as essential elements or
unprepared 'dissonances', in the later Baroque period, one needs a
further principle for these extra tones.

The septimal tritone tuning is sufficiently close to the tritone in
various historical types of meantone that I believe it would have been
recognisable by ear and tunable in the context of performance by
voices or instruments of flexible tuning.

My suggestion to tune the dim7 chord is 5:6:50/7:60/7 which contains
two pure m3's and the pure 12/7 dim7. The middle interval is 25:21
which is very close to a 12ET minor 3rd.

To see how it works out in a well-known piece I take the chorus 'Since
by man came death' from Messiah. This has two short, slow passages for
unaccompanied choir in which choral tuning is totally exposed and
essential to the effect.

I have reduced the chords (including all passing 'dissonances') to
root position and will simply state the ratios of each note above the
root. Then what remains is to specify the relations of each root to
the tonic. In adaptive JI the roots would not be in JI relations to
each other, they would be tempered by some meantone. However in this
passage I did not find it necessary to temper the roots.

Excerpt 1. Key A minor

Chord A-C-E, 6/5 3/2

Chord D-F-A, 6/5 3/2 (A may be held over)

Chord B-D-F, 6/5 10/7 (F may be held over)
... or perhaps better 25/21 10/7 (both D and F may be held over)

Chord A-C-E, 6/5 3/2

Chord C#-E-G-Bb, 6/5 10/7 12/7

Chord D-F#-A, 5/4 3/2

Chord F-A-(B)-D#, 5/4 7/5 7/4 (B is a passing note, A is held over)

Chord A-C-E, 6/5 3/2 (A may be held over)

Chord E-G#-B, 5/4 3/2 (E held over)

Chord F#-A-E, 6/5 9/5 (passing dissonance, E held over)
then E-G#-B again

Excerpt 2. Key D minor

Chord G-Bb-D, 6/5 3/2

Chord C#-E-G, 6/5 10/7 (G may be held over)

Chord D-F-A, 6/5 3/2

Chord Bb-D-A, 5/4 15/8 (A held over, resolving to
Chord Bb-D-G, 5/4 5/3

Chord E-G-Bb-D, 25/21 10/7 25/14 (G, Bb, D held over)

Chord D-F-A, 6/5 3/2

Chord A-C#-E, 5/4 3/2

Chord B-D-A, 6/5 9/5 (A held over)
then A-C#-E again

The roots may be tuned as follows:

Ex.1 A 1, D 4/3, B 28/25, A 1, C# 5/4, D 4/3, F 8/5, A 1, E 3/2, F# 5/3

Ex.2 G 4/3, C# 28/15, D 1, Bb 8/5, E 28/25, D 1, A 3/2, Bnat 5/3.

In each excerpt the leading-note and supertonic are treated as
'septimal' notes in that they are tuned as 14/15 and 28/25 whenever
the chord includes a dim5.

Now I need to know how it sounds!!

~~~T~~~

Hi Tom,

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> I am continuing to investigate how the 7:5 tritone or
> 10:7 dim5 can fit into common practice harmony.

Because "common-practice" harmonic theory evolved within
the context of, first, pythagorean (3-limit), and subsequently,
"just-intonation" and meantone (5-limit) tuning math,
and never really progressed beyond that, how one associates
interval-name labels with ratios of 7-limit and higher
prime-limits is somewhat arbitrary.

However, in general, tuning theorists tend to label these
two particular 7-limit ratios in exactly the opposite way to
what you wrote: 10:7 as the septimal tritone (aug-4th)
and 7:5 as the septimal dim-5th:

* 7:5 is the 5:4 major-3rd below the 7:4 septimal minor-7th,
which makes it a diminished-5th;

* 10:7 is conversely the 5:4 above the 8:7 septimal major-2nd,
making it the augmented-4th.

The "diminished triad" is generally considered to be
rendered in 7-limit form as the proportion 5:6:7,
vividly illustrating the diminished-5th as a 7:5 ratio.

If i had time, i could probably find a more legible
7-limit lattice than this one:

http://sonic-arts.org/monzo/lattices/pitch-bend-lattice.htm

but it does illustrate my points. Paul Erlich's _The Forms
of Tonality_ is in color and his lattices are structured the
same way as mine, but not labeled as profusely:

http://lumma.org/tuning/erlich/erlich-tFoT.pdf

(it's embarassing that i don't have a beautiful Tonescape
lattice of this to link to! ... if i were at home right now
i'd just create one on the spot and upload it ...)

Of course, the matter is complicated somewhat by the fact
that in meantone tunings, the augmented-6th is acoustically
very close in pitch and similar in sound to the 7:4 ratio,
with similar correspondences for all other meantone and
7-limit intervals.

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

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> I am continuing to investigate how the 7:5 tritone or 10:7 dim5 can
> fit into common practice harmony.

I'm not clear what you mean by "common practice harmony", since this
usually entails that you are *not* using JI and certainly not 7-limit
JI. Are you referring to adaptive tunings of common practice music?

In a meantone system, 7/5 is representable by an augmented fourth, and
10/7 by a diminished fifth, and the two are distinguished. However, no
distinction is made between 10/9, 9/8, and 28/25; they are all just a
tone. (9/8)/(28/25) = 225/224 and (28/25)/(10/9) = 126/125, and if you
temper out both small intervals, the result is (septimal) meantone.
It's also quite practical to just temper out one of them, and for
example merge 9/8 with 28/25 but leave 10/9 as a distinct interval. If
you detune 5-limit JI slightly (for example, by using 72-et, or even
228-et, the 12x19 system) you get an excellent practical method for
working a lot of septimal harmony into an initially 5-limit JI scale.

But, of course, this is a fixed tuning I am talking about.

> It seems to provide a neat solution
> to the JI problem of the supertonic in some contexts. That is, rather
> than choose between 9:8 and 10:9, one may use 28:25, which is derived
> via 5:4 and 7:5, if the chord includes a dim5.

That becomes a meantone system if you carry it out consistently. That
is, if you are going to make 56/25 an exact interval, and also the
result of two fifths in succession, then your fifth should be exactly
sqrt(56/25) = (2/5)*sqrt(14) = 698.099 cents. That's about 2/11 comma,
an excellent meantone tuning if you like the 1/5 to 1/6 comma range.

> To my current opinion 5-limit JI (adaptive or not) is optimal only for
> triadic harmony. When the dim5 and dim7 enter as essential elements or
> unprepared 'dissonances', in the later Baroque period, one needs a
> further principle for these extra tones.

What do you mean by "optimal"?

> The septimal tritone tuning is sufficiently close to the tritone in
> various historical types of meantone that I believe it would have been
> recognisable by ear and tunable in the context of performance by
> voices or instruments of flexible tuning.

In fact, if you use a fifth of 2^(1/2) 5^(-1/6) 7^(1/6) = 697.085
cents the augmented fourth is exactly 7/5, and this is another good
meantone value.

> My suggestion to tune the dim7 chord is 5:6:50/7:60/7 which contains
> two pure m3's and the pure 12/7 dim7. The middle interval is 25:21
> which is very close to a 12ET minor 3rd.

That's certainly one way to do it. Another way is to temper out
126/125, so that 25/21 is the same as 6/5, and of course there is the
ever-popular 17-limit option.

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

> However, in general, tuning theorists tend to label these
> two particular 7-limit ratios in exactly the opposite way to
> what you wrote: 10:7 as the septimal tritone (aug-4th)
> and 7:5 as the septimal dim-5th:

That's not right.

The Pythagorean tritone is 729/512, which is 4/3 augmented by an
apotome of 2187/2048. It is therefore also the Pythagorean augmented
fourth. Since it is (9/8)^3, it is three tones, hence "tritone". The
rule is that augmenting moves you +7 on the chain of fifths, and
diminishing moves you -7.

Tempering this via meantone produces the meantone augmented fourth or
tritone, which depending on the tuning will be close to or even
exactly equal to 7/5; in any case, 7/5 is mapped by septimal meantone
to the same interval, as the ratio (729/512)/(7/5) = 3645/3584 =
81/80 * 225/224.

> * 7:5 is the 5:4 major-3rd below the 7:4 septimal minor-7th,
> which makes it a diminished-5th;

That's an augmented sixth, not a minor seventh, hence dropping by a
major third leads to an augmented fourth.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > However, in general, tuning theorists tend to label these
> > two particular 7-limit ratios in exactly the opposite way to
> > what you wrote: 10:7 as the septimal tritone (aug-4th)
> > and 7:5 as the septimal dim-5th:
>
> That's not right.

Yes it is ... please read on ...

> The Pythagorean tritone is 729/512, which is 4/3 augmented
> by an apotome of 2187/2048. It is therefore also the Pythagorean
> augmented fourth. Since it is (9/8)^3, it is three tones, hence
> "tritone". The rule is that augmenting moves you +7 on the
> chain of fifths, and diminishing moves you -7.
>
> Tempering this via meantone produces the meantone
> augmented fourth or tritone, which depending on the tuning
> will be close to or even exactly equal to 7/5; in any case,
> 7/5 is mapped by septimal meantone to the same interval,
> as the ratio (729/512)/(7/5) = 3645/3584 = 81/80 * 225/224.
>
> > * 7:5 is the 5:4 major-3rd below the 7:4 septimal minor-7th,
> > which makes it a diminished-5th;
>
> That's an augmented sixth, not a minor seventh, hence dropping
> by a major third leads to an augmented fourth.

I didn't have time to go into the kind of detail you did,
but my last paragraph, which you didn't quote:

>> Of course, the matter is complicated somewhat by the fact
>> that in meantone tunings, the augmented-6th is acoustically
>> very close in pitch and similar in sound to the 7:4 ratio,
>> with similar correspondences for all other meantone and
>> 7-limit intervals.

was put there precisely to convey the meantone-to-septimal-JI
correspondences which you laid out in detail.

The part of my post which you did quote, where i did go
into some detail, refers to 7-limit JI, and it is correct.
In 7-limit JI, the diminished-5th is indeed the 7:5 ratio.

I suspect that the meantone correspondences are more relevant
to Tom's question than the septimal JI, since "common-practice"
would almost certainly be within a meantone context, so you
were right to go into it at some length ... but i'm a real
stickler for exactitude in terminology. ;-)

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

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > I am continuing to investigate how the 7:5 tritone or 10:7 dim5 can
> > fit into common practice harmony.
>
> I'm not clear what you mean by "common practice harmony", since this
> usually entails that you are *not* using JI and certainly not 7-limit
> JI. Are you referring to adaptive tunings of common practice music?

I guess I am. Or in fact, in the example I gave, 7-limit JI tuning of
common practice music. There I was somewhat lucky, in that strict JI
did not entail any shifts as large as a syntonic comma in the passages
I quoted.

This sounds theoretically crazy, but I hope to show that it might be
quite consonant with the actual musical practice of the time. I am
well aware that almost all common practice *theorists* strongly
disapproved of 7-limit. From what I understand of formal
common-practice harmony, so far as any well-defined tuning scheme was
imagined (which was rarely!) it would be adaptive 5-limit JI a la
Vicentino. This theory also forbade unprepared dissonance such as
dominant 7ths, let alone unprepared augmented intervals.

However, it is also clear that very many keyboard instruments in the
English Baroque (for example) would be in a meantone very close to the
ones you describe, in which aug 6ths and 4ths, for example, would turn
out as near-exact 7-limit intervals. So I deduce that good
representations of the 7-limit *were* part of the late Baroque
sound-world, whenever these intervals were introduced. Whether or not
theorists mentioned them, people heard them!

(There is for example one English meantone tuning instruction that
says to check if Bb-D-F-Ab (tuned as G#) and Eb-G-Bb-Db (C#) are 'good
chords'.)

Also, as I noted in my experience of singing the Schumann Mass, if a
chord is voiced to have a tritone between the lowest parts, and the
voices are free to adjust to one another, it is difficult to *avoid*
singing a 7:5. (E.g. with voicing F-B-D-G.) It is simply easier to
tune! At least, that is how I interpreted what happened when this
chord sounded unexpectedly 'together'. Now I simply hypothesize that
Baroque singers may also have found the 7:5 tritone to be an attractor
- without taking them far out of the way of familiar meantones, and
without even knowing they were singing a septimal interval.

The proof of the pudding would be in the eating - if the 7-limit JI
tunings I propose for unaccompanied choir sound (in some currently
ill-defined way) like Handel, then they should be accepted as a
possible common practice tuning. This would imply that previous
performances of those bits of Handel had been approximating 7-limit
without knowing it! But, if they sound like barbershop or Turkish
music instead, then by all means throw them out!

> > To my current opinion 5-limit JI (adaptive or not) is optimal only for
> > triadic harmony. (...)
>
> What do you mean by "optimal"?

Very simply, that every consonant chord has a tuning which is
obviously as consonant as possible. But if your repertoire of chords
extends to unprepared dominant 7ths, the tuning of the 7th becomes
doubtful between 9/5 and 16/9. I am proposing that 50/28 can also be a
solution, in certain contexts, due to the 10/7 ratio with the major
3rd. The rationale becomes stronger in aug6 chords, such as F-A-B-D#
in my example. Also dim5 or dim7: the dim 5th and dim 7th intervals
become difficult to tune by ear, or involve large comma shifts (or
deviations from a prevailing meantone) if done by superimposed pure
m3's. Whereas if done by interlocking pure m3 and 7:5's they are both
tunable by ear and stay quite close to historically common meantones.

My underlying thought is that singers would instinctively 'want' to
sing pure intervals, and a meantone continuo might suggest to them
pure septimal dim 5ths, tritones and aug 6ths - such that they might
have simply found these intervals easy to sing in tune, if and when a
continuo chord died away and they were sufficiently close. Just as a
meantone continuo would suggest a near-pure 5th and the singers could
adjust it minutely towards a pure one.

As a first aural step towards seeing whether this makes sense, I would
accept (in place of 7-limit JI) the compromise meantone in which the
aug6 is pure 7:4 and thus 5:4 and 7:5 have equal (and tiny)
deviations. This is also extremely close to 31-EDO. Given, say, the
Handel excerpt performed in 31-EDO, it would be a relatively tiny step
to get to my JI solution.

~~~T~~~

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

> The part of my post which you did quote, where i did go
> into some detail, refers to 7-limit JI, and it is correct.
> In 7-limit JI, the diminished-5th is indeed the 7:5 ratio.

In 7-limit JI, the diminished fifth isn't anything--it is undefined.
Huyghens-Fokker call all of 13/9, 22/15, 36/25 and 1024/729 a
"diminished fifth" of one kind or another, for whatever that is worth.
If we take 36/25 to be a diminished fifth, that is sharper than 10/7
by 126/125 and so the two are equated when we are tempering out
126/125, as for instance in septimal meantone.

> I suspect that the meantone correspondences are more relevant
> to Tom's question than the septimal JI, since "common-practice"
> would almost certainly be within a meantone context, so you
> were right to go into it at some length ... but i'm a real
> stickler for exactitude in terminology. ;-)

I think claiming 10/7 is an augmented fourth is not only not precise,
it's completely wrong.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > The part of my post which you did quote, where i did go
> > into some detail, refers to 7-limit JI, and it is correct.
> > In 7-limit JI, the diminished-5th is indeed the 7:5 ratio.
>
> In 7-limit JI, the diminished fifth isn't anything--it is
> undefined. Huyghens-Fokker call all of 13/9, 22/15, 36/25
> and 1024/729 a "diminished fifth" of one kind or another,
> for whatever that is worth. If we take 36/25 to be a
> diminished fifth, that is sharper than 10/7 by 126/125 and so
> the two are equated when we are tempering out 126/125,
> as for instance in septimal meantone.

Umm ... i *did* open my original post to Tom with this:

>> Because "common-practice" harmonic theory evolved within
>> the context of, first, pythagorean (3-limit), and subsequently,
>> "just-intonation" and meantone (5-limit) tuning math,
>> and never really progressed beyond that, how one associates
>> interval-name labels with ratios of 7-limit and higher
>> prime-limits is somewhat arbitrary.

> > I suspect that the meantone correspondences are more relevant
> > to Tom's question than the septimal JI, since "common-practice"
> > would almost certainly be within a meantone context, so you
> > were right to go into it at some length ... but i'm a real
> > stickler for exactitude in terminology. ;-)
>
> I think claiming 10/7 is an augmented fourth is not only
> not precise, it's completely wrong.

Here's my argument for:

http://sonic-arts.org/monzo/lattices/pitch-bend-lattice.htm

What's your argument against?

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

> To my current opinion 5-limit JI (adaptive or not) is optimal
> only for triadic harmony.

There are some nice tetrads ("7th chords") in 5-limit JI.

> When the dim5 and dim7 enter as
> essential elements or unprepared 'dissonances', in the later
> Baroque period, one needs a further principle for these extra
> tones.

Both chords can be done with 3-limit ratios. If you insist on
using ratios of 5, a passible the dim5 triad can be reached via
a stack of two 6:5s, depending on your application.

Of course 5-limit JI was not in use in the late Baroque, at
least not on the major fixed-pitch instruments like the organ
and harpsichord, which were deemed suitable continuo
instruments. So any performance of this music in 5-limit JI
is at least speculative.

> The septimal tritone tuning is sufficiently close to the tritone
> in various historical types of meantone that I believe it would
> have been recognisable by ear and tunable in the context of
> performance by voices or instruments of flexible tuning.

I think Baroque music would sound cool in septimal JI, and for
all we know it could have been performed this way. However, I
don't think it was. Here's why:

. When and why was the practice lost?
. Why didn't they tune septimal intervals on their continuo
instruments?
. Though septimal intervals are sometimes heard in isolated
cases in performances today, their sound is geneally very
foreign in Western music.
. As you point out, 7-limit intervals aren't needed unless
you want to tune diminished chords pure. But there seems
little motivation to do this in Baroque music, where such
chords are treated as dissonances.

> To see how it works out in a well-known piece I take the
> chorus 'Since by man came death' from Messiah.

How seasonal!

> Excerpt 1. Key A minor
>
> Chord A-C-E, 6/5 3/2
> Chord D-F-A, 6/5 3/2 (A may be held over)
> Chord B-D-F, 6/5 10/7 (F may be held over)
> Chord A-C-E, 6/5 3/2
> Chord C#-E-G-Bb, 6/5 10/7 12/7
> Chord D-F#-A, 5/4 3/2
> Chord F-A-(B)-D#, 5/4 7/5 7/4
> Chord A-C-E, 6/5 3/2 (A may be held over)
> Chord E-G#-B, 5/4 3/2 (E held over)
> Chord F#-A-E, 6/5 9/5 (passing dissonance, E held over)
> then E-G#-B again
>
> Excerpt 2. Key D minor
>
> Chord G-Bb-D, 6/5 3/2
> Chord C#-E-G, 6/5 10/7 (G may be held over)
> Chord D-F-A, 6/5 3/2
> Chord Bb-D-A, 5/4 15/8 (A held over, resolving to
> Chord Bb-D-G, 5/4 5/3
> Chord E-G-Bb-D, 25/21 10/7 25/14 (G, Bb, D held over)
> Chord D-F-A, 6/5 3/2
> Chord A-C#-E, 5/4 3/2
> Chord B-D-A, 6/5 9/5 (A held over)
> then A-C#-E again
>
> The roots may be tuned as follows:
>
> Ex.1 A 1, D 4/3, B 28/25, A 1, C# 5/4, D 4/3, F 8/5,
> A 1, E 3/2, F# 5/3
>
> Ex.2 G 4/3, C# 28/15, D 1, Bb 8/5, E 28/25, D 1,
> A 3/2, Bnat 5/3.
>
> In each excerpt the leading-note and supertonic are treated as
> 'septimal' notes in that they are tuned as 14/15 and 28/25
> whenever the chord includes a dim5.
>
> Now I need to know how it sounds!!

Me too! Does anyone have time to synthesize this before
the 'big day' (also my son's birthday)?

-Carl

I don't have a horse in this race (which seems to be another fight
about definitions), but some clarification would be welcome.

Who gets to decide what a dim5 is or isn't in 7-limit JI? Some guy who
typed out the list of intervals at the Huygens-Fokker website?
(For what it's worth, they call 10:7 "Euler's tritone"... so was Euler
completely wrong there?)

Barbershop is 7-limit JI and uses 5:7 as dim5, doesn't it? And if the
third of the barbershop chord is voiced above the 7th, you get a 10:7
aug4.

To my ears, depending on harmonic & voice-leading context, 10:7 can
also be a perfectly good dim5.

I'm also not sure I understand what any 7-limit JI theorists mean by
giving an unique ratio for each interval. Even in 5-limit this may be
undesirable, if one wants to allow for 32:27 minor thirds.

And I'm not sure what Gene is getting at by referring to meantones
that temper out differences between JI versions of the same interval
(type). 12tet tempers out 50/49, but that doesn't persuade me that 7:5
and 10:7 are interchangeable.

~~~T~~~

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > In 7-limit JI, the diminished-5th is indeed the 7:5 ratio.
>
> In 7-limit JI, the diminished fifth isn't anything--it is undefined.
> (...)

> If we take 36/25 to be a diminished fifth, that is sharper than 10/7
> by 126/125 and so the two are equated when we are tempering out
> 126/125, as for instance in septimal meantone.
>
> (...) i'm a real
> > stickler for exactitude in terminology. ;-)
>
> I think claiming 10/7 is an augmented fourth is not only not precise,
> it's completely wrong.
>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > To my current opinion 5-limit JI (adaptive or not) is optimal
> > only for triadic harmony.
>
> There are some nice tetrads ("7th chords") in 5-limit JI.
>

OK ... yes, there is not much reason to go to 7-limit for simpler
chords such as dominant 7th in root position or 2nd inversion, where
the 7th can be tuned from the 5th as a pure 6:5 third. My thinking was
more for the situation where dim5, dim7, aug4, aug6 have to be tuned
above the bass, or occur prominently between neighbouring voice parts.
This is relatively rare until the later Baroque period.

Concerning simply the sound I have nothing against 36/25 or 45/32 etc.
- just that they are either difficult to tune by ear, or are too far
from the probable generic English Baroque meantone tuning which I
estimate as circa 2/9 comma.

> 5-limit JI was not in use in the late Baroque, at
> least not on the major fixed-pitch instruments like the organ
> and harpsichord, which were deemed suitable continuo
> instruments.

Ha... The second part of the sentence is true, but not altogether to
the point, unless you believe that singers and instrumentalists were
incapable to intone any pitch outside the keyboard. I chose an
unaccompanied choral passage both because it had interesting dim/aug
harmonies, and because it was free of continuo questions.

There existed whole compositions for unaccompanied voices, 'catches'
etc., somewhat trivial music with mostly simple (triadic) harmony. I
don't see any plausible tuning for these other than 5-limit JI (or
adaptive when needed) - assuming that the singers knew what a pure 5th
and a pure 3rd sounded like. Temperament was and is not necessary.

> So any performance of this music in 5-limit JI
> is at least speculative.

For unaccompanied choral passages, I think *any* specific tuning
system is speculative, and it is up to you, or anyone else, to argue
what is more or less likely. I have argued that, for example, fifths
would have been tuned pure, because that is the only way they can be
sung in tune by choirs.

> I think Baroque music would sound cool in septimal JI, and for
> all we know it could have been performed this way. However, I
> don't think it was. Here's why:
>
> . When and why was the practice lost?

When keyboards stopped being tuned mainly in 1/4- or 1/5- comma meantone.

Alternatively, the practice has never been entirely lost, since (as
per my previous messages) I found the 7:5 turning up of its own
accord, in singing the 3rd inversion of the dominant 7th. Let me
specify that 'practice' doesn't mean that people knew it was happening
and consciously practised it; just that it happened.

> . Why didn't they tune septimal intervals on their continuo
> instruments?

As per previous messages, they did, if these instruments were in 1/4-
or 1/5- comma meantone. Which is not to say they tuned *by* septimal
intervals.

> their sound is generally very foreign in Western music.

But it would not be to those who played music on those meantone
keyboards, if the intervals were introduced in an appropriate manner.
(Not to get into the barbershop question...)

> you want to tune diminished chords pure. But there seems
> little motivation to do this in Baroque music, where such
> chords are treated as dissonances.

Even dissonances have to be tuned somehow. The rules of counterpoint
are well crafted to allow this: for example a (diatonic) semitone
dissonance must be prepared by a suspension, which allows the
performer to simply keep the note in place - e.g. tenor part of
'Surely he hath borne our griefs'.

But I am considering places where 1) such dim/aug 'dissonances' occur
unprepared and 2) have to be tuned above the bass or above a
neighboring voice. Even though these intervals were treated as
dissonances, it is plausible that they could most easily and stably be
tuned in practice by pure ratios.

> > Now I need to know how it sounds!!
>
> Me too! Does anyone have time to synthesize this before
> the 'big day' (also my son's birthday)?

I could try doing something with my spinet when / if I get back to UK
for Christmas... pray for absence of fog!

~~~T~~~

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

> > I think claiming 10/7 is an augmented fourth is not only
> > not precise, it's completely wrong.

> Here's my argument for:

> http://sonic-arts.org/monzo/lattices/pitch-bend-lattice.htm

> What's your argument against?

I wouldn't call that an argument. You diagram the 7-limit lattice, and
then put names on it based on 72-et. Hence, if C is 0 cents, F is 500
cents, F# is 600 cents, and F#+, the + being a comma, is 616 2/3
cents, which is a good approximation to 10/7, which is 617.5 cents.

None of this has anything to do with traditional musical terminology,
which not only is not based on 72-et, it's not even based on 12-et. An
"augmented fourth" is, by definition, a fourth increased by a
semitone, not a fourth increased by a semitone and then by a comma;
commas aren't even a part of the system of nomenclature here. It will
simply utterly confuse things to twist the old terminology around like
this, and it certainly does not make things more precise.

Why are you using 72 at all? It's a terrible choice if the names are
supposed to relate to traditional uaage based on a chain of fifths,
because it doesn't *have* a single circle of fifths. It has six of
them in parallel.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> I think Baroque music would sound cool in septimal JI, and for
> all we know it could have been performed this way. However, I
> don't think it was. Here's why:
>
> . When and why was the practice lost?

Augmented sixths were occasionally used in Baroque music, and given a
meantone tuning would certainly sound septimal. If the tuning was
adapted, what would it adapt to? I agree it doesn't seem very
characteristic of Baroque style.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> Who gets to decide what a dim5 is or isn't in 7-limit JI?

Anybody, since it isn't a 7-limit JI interval. However, when you make
your decision, you must fit it into a framework where a dim5 is a
fifth lowered by a chromatic semitone, since it is a chromatic
adjustment of a fifth. Hence if you call 10/7 a septimal diminished
fifth, you are in effect saying 21/20 is a septimal chromatic
semitone, and are putting it in the same bin as 25/24, the classical
5-limit chromatic semitone. If you say 7/5 is a diminished fifth, then
you are saying that 15/14 should not be classed with 16/15 as a kind
of diatonic semitone, as one might have imagined, but should be put
with 25/24 instead.

Classing both 15/14 and 16/15 (225/224 apart) and 21/20 and 25/24
(126/125 apart) as the same interval leads to meantone, as 126/125
times 225/224 is 81/80. This is the system of septimal meantone, which
seems the overwhelmingly obvious way of extending traditional musical
terminology to the 7-limit.

> Barbershop is 7-limit JI and uses 5:7 as dim5, doesn't it?

Barbershop is adaptive, and I presume would adapt chords in different
ways, depending.

> To my ears, depending on harmonic & voice-leading context, 10:7 can
> also be a perfectly good dim5.

So I'm right. Which means this method of assigning names won't work.

> I'm also not sure I understand what any 7-limit JI theorists mean by
> giving an unique ratio for each interval.

In JI, all intervals have unique ratios by definition.

> And I'm not sure what Gene is getting at by referring to meantones
> that temper out differences between JI versions of the same interval
> (type). 12tet tempers out 50/49, but that doesn't persuade me that 7:5
> and 10:7 are interchangeable.

Neither exists on a 12-et keyboard, and to the extent there is such an
interval, it is the same interval. Therefore whether or not they are
interchangable in whatever sense you mean, they are being interchanged.

Hi Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> > Excerpt 1. Key A minor
> >
> > Chord A-C-E, 6/5 3/2
> > Chord D-F-A, 6/5 3/2 (A may be held over)
> > Chord B-D-F, 6/5 10/7 (F may be held over)
> > Chord A-C-E, 6/5 3/2
> > Chord C#-E-G-Bb, 6/5 10/7 12/7
> > Chord D-F#-A, 5/4 3/2
> > Chord F-A-(B)-D#, 5/4 7/5 7/4
> > Chord A-C-E, 6/5 3/2 (A may be held over)
> > Chord E-G#-B, 5/4 3/2 (E held over)
> > Chord F#-A-E, 6/5 9/5 (passing dissonance, E held over)
> > then E-G#-B again
> >
> > Excerpt 2. Key D minor
> >
> > Chord G-Bb-D, 6/5 3/2
> > Chord C#-E-G, 6/5 10/7 (G may be held over)
> > Chord D-F-A, 6/5 3/2
> > Chord Bb-D-A, 5/4 15/8 (A held over, resolving to
> > Chord Bb-D-G, 5/4 5/3
> > Chord E-G-Bb-D, 25/21 10/7 25/14 (G, Bb, D held over)
> > Chord D-F-A, 6/5 3/2
> > Chord A-C#-E, 5/4 3/2
> > Chord B-D-A, 6/5 9/5 (A held over)
> > then A-C#-E again
> >
> > The roots may be tuned as follows:
> >
> > Ex.1 A 1, D 4/3, B 28/25, A 1, C# 5/4, D 4/3, F 8/5,
> > A 1, E 3/2, F# 5/3
> >
> > Ex.2 G 4/3, C# 28/15, D 1, Bb 8/5, E 28/25, D 1,
> > A 3/2, Bnat 5/3.
> >
> > In each excerpt the leading-note and supertonic are treated as
> > 'septimal' notes in that they are tuned as 14/15 and 28/25
> > whenever the chord includes a dim5.
> >
> > Now I need to know how it sounds!!
>
> Me too! Does anyone have time to synthesize this before
> the 'big day' (also my son's birthday)?
>
> -Carl

I'll be finished with school on Thursday, and over the weekend
should have time to work this up quickly in Tonescape. I'll
export MIDI files and post them.

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

> Concerning simply the sound I have nothing against 36/25 or
> 45/32 etc. - just that they are either difficult to tune by
> ear, or are too far from the probable generic English Baroque
> meantone tuning which I estimate as circa 2/9 comma.

I thought you didn't want a comparison to meantone.

Both 45/32 and 36/25 can be easily tuned as compounds.

> There existed whole compositions for unaccompanied voices, 'catches'
> etc., somewhat trivial music with mostly simple (triadic) harmony. I
> don't see any plausible tuning for these other than 5-limit JI (or
> adaptive when needed) - assuming that the singers knew what a pure
> 5th and a pure 3rd sounded like. Temperament was and is not
> necessary.

Agree.

> > I think Baroque music would sound cool in septimal JI, and for
> > all we know it could have been performed this way. However, I
> > don't think it was. Here's why:
> >
> > . When and why was the practice lost?
>
> When keyboards stopped being tuned mainly in 1/4- or 1/5- comma
> meantone.

There are very few examples of the aug 6th being used in a
septimal 7th sort of way.

> > their sound is generally very foreign in Western music.
>
> But it would not be to those who played music on those meantone
> keyboards,

I still think it would be.

> But I am considering places where 1) such dim/aug 'dissonances'
> occur unprepared and 2) have to be tuned above the bass or above
> a neighboring voice. Even though these intervals were treated as
> dissonances, it is plausible that they could most easily and
> stably be tuned in practice by pure ratios.

If you can find a recording of a late baroque piece with a
septimal dim5 used more than once I'd love to hear it.

> pray for absence of fog!

Very well; I will. :)

-Carl

> > Me too! Does anyone have time to synthesize this before
> > the 'big day' (also my son's birthday)?
>
> I'll be finished with school on Thursday, and over the weekend
> should have time to work this up quickly in Tonescape. I'll
> export MIDI files and post them.

Awesome! -Carl

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

> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > > Excerpt 1. Key A minor
> > >
> > > Chord A-C-E, 6/5 3/2
> > > Chord D-F-A, 6/5 3/2 (A may be held over)
> > > Chord B-D-F, 6/5 10/7 (F may be held over)
> > > Chord A-C-E, 6/5 3/2
> > > Chord C#-E-G-Bb, 6/5 10/7 12/7
> > > Chord D-F#-A, 5/4 3/2
> > > Chord F-A-(B)-D#, 5/4 7/5 7/4
> > > Chord A-C-E, 6/5 3/2 (A may be held over)
> > > Chord E-G#-B, 5/4 3/2 (E held over)
> > > Chord F#-A-E, 6/5 9/5 (passing dissonance, E held over)
> > > then E-G#-B again
>
> <snip>
>
> > > The roots may be tuned as follows:
> > >
> > > Ex.1 A 1, D 4/3, B 28/25, A 1, C# 5/4, D 4/3, F 8/5,
> > > A 1, E 3/2, F# 5/3
>
> <snip>
>
> I'll be finished with school on Thursday, and over the weekend
> should have time to work this up quickly in Tonescape. I'll
> export MIDI files and post them.

Here's "Excerpt 1" in MIDI:

/tuning/files/monz/tuning68501_messiah-1_7-limit-ji.mid

It took me all of 15 minutes to do in Tonescape.

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

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

> Here's "Excerpt 1" in MIDI:
>
>
/tuning/files/monz/tuning68501_messiah-1_7-limit-ji.mid
>
> It took me all of 15 minutes to do in Tonescape.

And here's a screenshot of the Score and Lattice in Tonescape:

/tuning/files/monz/tuning68501_messiah-1_7-limit-ji_tonescape-screenshot.gif

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

> Here's "Excerpt 1" in MIDI:
//
> It took me all of 15 minutes to do in Tonescape.

Sweet!

I have to admit this sounds remarkably normal.

Here's the excerpt from Taverner / Parrot (one of my
favorite Messiah recordings).

http://lumma.org/stuff/SinceByMan.wav

(I equalized the max amplitudes of the left and right
channels, ran the thing through a moderate compressor,
then mixed to mono.)

Here's Tom's tuning again:

Since by..Chord A-C-E, 6/5 3/2
man.......Chord D-F-A, 6/5 3/2 (A may be held over)
came......Chord B-D-F, 6/5 10/7 (F may be held over) *1
death.....Chord A-C-E, 6/5 3/2
Since by..Chord C#-E-G-Bb, 6/5 10/7 12/7 *2
man.......Chord D-F#-A, 5/4 3/2
came......Chord F-A-(B)-D#, 5/4 7/5 7/4 *3
death.....Chord A-C-E, 6/5 3/2 (A may be held over)

According to my analysis in Transcribe! (which isn't the
best tool for this sort of thing, but it is fast), *1 is
tuned pretty close to 12-ET, *2 is perhaps tuned like a
12:14:17:20 chord, and *3 is pretty close to 12 also.

One shouldn't put too much stock in Transcribe! results,
though. It's great for transcribing music in 12; not so
great for microtonal analysis.

-Carl