Yahoo Tuning Groups Ultimate Backup tuning probable use of 7/5, 7/4, 7/6 in common practice

This harks back to 2007 (if not 1667), but almost whenever you hear an
augmented or diminished interval in standard meantone (~1/4 comma, or
very slightly less flat) -tuned music, you are basically hearing a
7-limit interval. This was noticed already in the 17th century by
Huyghens, who proposed that the 7th harmonic should be allowed into
the list of theoretically 'harmonic' intervals. As a mathematical
fact, 7/5 works practically frictionlessly (i.e. pitch shifts small or
nonexistent) as a tritone, 7/4 as an aug6, and 7/6 almost as well as
an aug2, if the harmonic roots are tuned to meantone.

Apropos, here are some golden oldies - Handelian Harmonies, no less:

http://www.thphys.uni-heidelberg.de/~dent/Since_JI.MP3

- what sort of tuning did I use, and how 'valid' can it be?

- also at
/tuning/files/tomd/
for the more curious-minded.

Martin Vogel wrote whole books about the use of the 7th harmonic, and
while much of it is speculative, there is a lot of historical
information there, which one should study before trying to pronounce
on the 'validity' of any use.
~~~T~~~

🔗Andreas Sparschuh <a_sparschuh@...>

🔗Marcel de Velde <m.develde@...>

Hi Tom,

This harks back to 2007 (if not 1667), but almost whenever you hear an
> augmented or diminished interval in standard meantone (~1/4 comma, or
> very slightly less flat) -tuned music, you are basically hearing a
> 7-limit interval. This was noticed already in the 17th century by
> Huyghens, who proposed that the 7th harmonic should be allowed into
> the list of theoretically 'harmonic' intervals. As a mathematical
> fact, 7/5 works practically frictionlessly (i.e. pitch shifts small or
> nonexistent) as a tritone, 7/4 as an aug6, and 7/6 almost as well as
> an aug2, if the harmonic roots are tuned to meantone.
>

Yes I know Huyghens, and Fokker (am Dutch myself.)
I've found many instances where you play an augmented or dimished interval
you're augmenting or diminishing by 16/15.
This makes for instance 75/64, not 7/6. 45/32, not 7/5. 225/128, not 7/4.
Yes I'm aware 7/5 has a much smaller vibration ratio than 45/32, but my
point is that this may not be the thing relevant to the structure of music
the way we hear and use it. It may just be relevant for consonance.
Also a chord we don't hear well what it's made up, just some consonance
which isn't very exact.
It is when we do something with that chord that it's makup is reveiled.

Apropos, here are some golden oldies - Handelian Harmonies, no less:
>
> http://www.thphys.uni-heidelberg.de/~dent/Since_JI.MP3
>
> - what sort of tuning did I use, and how 'valid' can it be?
>

It sounds slightly off to me. Not sure if that's because of the 7th harmonic
use or because something else is going a little bit wrong here.
But such examples are not very helpfull in this discussion.
Could you please post (written) a small part of this song in JI ratios so we
can investigate and discuss it better?

Marcel

🔗caleb morgan <calebmrgn@...>

🔗Tom Dent <stringph@...>

My explication of 'common-practice' 7-limit is here
/tuning/topicId_68501.html#68501

& please read the ensuing discussions from the archives.
(Searching the archives for 'Handel' and/or 'Messiah' will bring up
the whole thing.)

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> many instances where you play an augmented or dimished interval
> you're augmenting or diminishing by 16/15.

I don't know what you mean here. I play a harpsichord tuned
(sometimes) in meantone where there isn't any 16/15. More like 15/14,
actually.

But, in fact, the difference between the major or perfect interval and
the augmented one is a *chromatic* semitone, which won't be 16/15 anyway.

The meantone tritone is 5/4 * sqrt(5/4) = 1.398, pretty damn close to
7/5 = (3/2)/(15/14) (cf. 45/32=1.406);

the meantone aug6 (5/4)^(5/2) = 1.747 (225/128=1.758)

meantone aug2 is (5/4)^(5/2)/5^(1/4) = 1.168 (75/64=1.172)

and if one makes major thirds very slightly wider than 5/4, as some
historical instructions indicate, the aug4 and aug6 become even closer
to 7's. There is an English meantone instruction that asks for
Eb-G-Bb-Db to be checked as a 'good chord' when Db is actually tuned
as C#: i.e. Eb-G-Bb-C# should be audibly 'good', which in meantone can
only indicate 7/4. There is also a clavichord making instruction which
asks for diatonic semitones to be 15/14...

> 7/5 has a much smaller vibration ratio than 45/32, but my
> point is that this may not be the thing relevant to the structure of
music
> the way we hear and use it.

I don't know what you mean. What could be the definition of 'relevant'
here? And what is 'music the way we hear and use it' - who exactly is
doing the playing or hearing or 'using'? Obviously, 7-limit has no
relevance or useful relation to Glenn Gould playing Sweelinck on a
Steinway. However, I am stating that 7-limit *did* have relevance to
some 17th- and 18th century musicians, at the very least because they
heard intervals which were very close to it, whether or not they were
aware of the fact. If your only concern is 'music the way WE hear and
use it' then please don't try to enter any discussion of historical
practices.

> It may just be relevant for consonance.

If 'consonance' does not qualify as 'relevant to the structure of
music' you may find it difficult to have any discussion about tuning
at all.

> Also a chord we don't hear well what it's made up, just some consonance
> which isn't very exact.

It depends what performance you are listening to. My consonances were
pretty damn exact, I think. If JI, including 7-limit, has any point
for me, it is hearing consonant chords as such, and not as something
'which isn't very exact' i.e. more or less tempered.

> Could you please post (written) a small part of this song in JI
ratios so we
> can investigate and discuss it better?
>

In the message first cited. It was even synthesized by Joe Monzo at
the time.
~~~T~~~

🔗Marcel de Velde <m.develde@...>

Hi Tom,

> I don't know what you mean here. I play a harpsichord tuned
> (sometimes) in meantone where there isn't any 16/15. More like 15/14,
> actually.
>
> But, in fact, the difference between the major or perfect interval and
> the augmented one is a *chromatic* semitone, which won't be 16/15 anyway.
>

Ah yes sorry. I was wrong here in saying augmented or dimished is by 16/15.
I'm not that great with note names.
Indeed should be chromatically which means 135/128 or 25/24 depending on the
case.

> The meantone tritone is 5/4 * sqrt(5/4) = 1.398, pretty damn close to
> 7/5 = (3/2)/(15/14) (cf. 45/32=1.406);
>
> the meantone aug6 (5/4)^(5/2) = 1.747 (225/128=1.758)
>
> meantone aug2 is (5/4)^(5/2)/5^(1/4) = 1.168 (75/64=1.172)
>
> and if one makes major thirds very slightly wider than 5/4, as some
> historical instructions indicate, the aug4 and aug6 become even closer
> to 7's. There is an English meantone instruction that asks for
> Eb-G-Bb-Db to be checked as a 'good chord' when Db is actually tuned
> as C#: i.e. Eb-G-Bb-C# should be audibly 'good', which in meantone can
> only indicate 7/4. There is also a clavichord making instruction which
> asks for diatonic semitones to be 15/14...
>

Yes this may all be so but I'm not talking about meantone, I'm talking about
pure JI.

> > 7/5 has a much smaller vibration ratio than 45/32, but my
> > point is that this may not be the thing relevant to the structure of
> music
> > the way we hear and use it.
>
> I don't know what you mean. What could be the definition of 'relevant'
> here? And what is 'music the way we hear and use it' - who exactly is
> doing the playing or hearing or 'using'? Obviously, 7-limit has no
> relevance or useful relation to Glenn Gould playing Sweelinck on a
> Steinway. However, I am stating that 7-limit *did* have relevance to
> some 17th- and 18th century musicians, at the very least because they
> heard intervals which were very close to it, whether or not they were
> aware of the fact. If your only concern is 'music the way WE hear and
> use it' then please don't try to enter any discussion of historical
> practices.
>

Ok we're talking about 2 different things here then.
I'm talking about the structure of music.
The way I see it music theory is a mess, music tuning is a mess.
Of neither there is a good coherent theory on how things work.
I beleive if you solve one you solve the other.
I don't beleive tuning is adaptive, so 45/32 is a different note with a
different meaning than 7/5 or any other note no matter how close.

> It may just be relevant for consonance.
>
> If 'consonance' does not qualify as 'relevant to the structure of
> music' you may find it difficult to have any discussion about tuning
> at all.
>

I do offcourse qualify it as relevant to music, but not nessecarily the only
or single most important thing in music to which all other things bow.

> Also a chord we don't hear well what it's made up, just some consonance

> > which isn't very exact.
>
> It depends what performance you are listening to. My consonances were
> pretty damn exact, I think. If JI, including 7-limit, has any point
> for me, it is hearing consonant chords as such, and not as something
> 'which isn't very exact' i.e. more or less tempered.
>

Yes but can you tell in music which one of many enharmonically equivalent
chords a certain chord is?
You will be able to tell by how you got to that chord and by how it resolves
etc.
Not by how it's tuned.
This is why when 2 different chords when they are tuned equally in 12tet
they are still 2 different chords as you get the meaning from the chord not
from how the chord sounds but from how you got there and how it resolves.