Jorgensen....'The equal-beating temperaments' book

...I got this from the library yesterday. There is a Pythagorean tuning of
Grammateus mentioned in the same breath as John Bull, which I though was
interesting, considering we had that debate about 'Ut Re Mi Fa Sol La' back
when I realized it in 19-equal.

Jorgensen writes:

"...The virute os Grammateus's temperament is that all the chromatic keys are
meantones between adjacent natural keys, In this way, ten of the semitones
are all the same size, and the chromatic scale sounds like that of equal
temperament. All twenty-four major and minor triads are good, so this
satisfies the requirements of early composers such as John Bull who modulated
through all the keys..."

The temperament is simply pure fifths from F to B, and then in the reference
first inversion minor chord D-F#-B, tune F# pure to D, then sharpen F# until
D-F# beats at the same rate as F#-B; from there tune F#-C#-G#-D#-A# all as
pure fifths.

-Aaron.

Hi Gene, Yahya, etc.,

I'd like to create a Tonescape tuning and tonespace
for this. What's the math? And the best generators
for the lattice?

-monz

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:

>
>
> ...I got this from the library yesterday. There is a
> Pythagorean tuning of Grammateus mentioned in the
> same breath as John Bull, which I though was interesting,
considering we had that debate about 'Ut Re Mi Fa Sol La'
> back when I realized it in 19-equal.
>
> Jorgensen writes:
>
> "...The virute os Grammateus's temperament is that
> all the chromatic keys are meantones between adjacent
> natural keys, In this way, ten of the semitones are
> all the same size, and the chromatic scale sounds like
> that of equal temperament. All twenty-four major and
> minor triads are good, so this satisfies the requirements
> of early composers such as John Bull who modulated
> through all the keys..."
>
> The temperament is simply pure fifths from F to B, and
> then in the reference first inversion minor chord D-F#-B,
> tune F# pure to D, then sharpen F# until D-F# beats at the
> same rate as F#-B; from there tune F#-C#-G#-D#-A# all as
> pure fifths.
>
> -Aaron.

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

It's in the Scala scale repository.

-C.

> I'd like to create a Tonescape tuning and tonespace
> for this. What's the math? And the best generators
> for the lattice?
>
> -monz
>
>
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
> wrote:
> > ...I got this from the library yesterday. There is a
> > Pythagorean tuning of Grammateus mentioned in the
> > same breath as John Bull, which I though was interesting,
> considering we had that debate about 'Ut Re Mi Fa Sol La'
> > back when I realized it in 19-equal.
> >
> > Jorgensen writes:
> >
> > "...The virute os Grammateus's temperament is that
> > all the chromatic keys are meantones between adjacent
> > natural keys, In this way, ten of the semitones are
> > all the same size, and the chromatic scale sounds like
> > that of equal temperament. All twenty-four major and
> > minor triads are good, so this satisfies the requirements
> > of early composers such as John Bull who modulated
> > through all the keys..."
> >
> > The temperament is simply pure fifths from F to B, and
> > then in the reference first inversion minor chord D-F#-B,
> > tune F# pure to D, then sharpen F# until D-F# beats at the
> > same rate as F#-B; from there tune F#-C#-G#-D#-A# all as
> > pure fifths.
> >
> > -Aaron.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

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

> It's in the Scala scale repository.
>
> -C.
>
> > I'd like to create a Tonescape tuning and tonespace
> > for this. What's the math? And the best generators
> > for the lattice?

I used Aaron's formula (equal beating between
D:F# and F#:B) and assumed a 1/1 A of 420 Hz. So:

* The B is a 9/8 above A-420, making it 472.5 Hz.

* The D is a 3/2 below A-420, making it 280 Hz.

* The F# which is a 5/4 above D-280 is 350 Hz.

* The tempered frequency for F# which i calculated,
which gives the two intervals (one above and one below F#)
an equal beat ratio, is 352.1875 Hz. This is
~397.1002537 cents above D.

Thus, the tuning is constructed of two generators,
one axis representing 3 (= 1901.955 cents in the
Tonescape file), and the other a tempered "5" of
~2797.1 cents.

The 5 tempered notes in the Tonescape file are
all ~1.01 cents higher than those in the Scala file.
???

GRAMMATEUS (1518) TUNING
TABLE OF CENTS VALUES

..... --- Tonescape ---- ... Scala
..... 1/1=A .......1/1=C ... 1/1=C .. difference

A ...... 0 ....... 905.865 ... 27/16
A# ... 102.965 .. 1008.83 .. 1007.82 ... -1.01
B .... 203.91 ... 1109.775 .. 243/128
C .... 294.135 ..... 0 ........ 1/1
C# ... 397.1 ..... 102.965 .. 101.955 .. -1.01
D .... 498.045 ... 203.91 ..... 9/8
D# ... 601.01 .... 306.875 .. 305.865 .. -1.01
E .... 701.955 ... 407.82 .... 81/64
F .... 792.18 .... 498.045 .... 4/3
F# ... 895.145 ... 601.01 ... 600.0 .... -1.01
G .... 996.09 .... 701.955 .... 3/2
G# .. 1099.055 ... 804.92 ... 803.91 ... -1.01

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

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

> The 5 tempered notes in the Tonescape file are
> all ~1.01 cents higher than those in the Scala file.
> ???

I suppose the Scala file assumes an exactly equal
division of the 9/8s, as illustrated in Margo's
article:

http://www.medieval.org/emfaq/harmony/pyth5.html

But Jorgensen's procedure (described in Aaron's post)
is slightly different:

* the two "white-key" diatonic-semitones are pythagorean,
~90 cents,

* the rest of the diatonic-semitones are ~101 cents,

* the chromatic-semitones are ~103 cents.

So now my question is: exactly how did Grammateus
specify his tuning? Is Jorgensen's procedure simply
a practical method for attaining an approximation to
the results desired by Grammateus?

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

On Tuesday 23 August 2005 5:15 pm, monz wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > The 5 tempered notes in the Tonescape file are
> > all ~1.01 cents higher than those in the Scala file.
> > ???
>
> I suppose the Scala file assumes an exactly equal
> division of the 9/8s, as illustrated in Margo's
> article:
>
> http://www.medieval.org/emfaq/harmony/pyth5.html
>
> But Jorgensen's procedure (described in Aaron's post)
> is slightly different:
>
> * the two "white-key" diatonic-semitones are pythagorean,
> ~90 cents,
>
> * the rest of the diatonic-semitones are ~101 cents,
>
> * the chromatic-semitones are ~103 cents.
>
>
> So now my question is: exactly how did Grammateus
> specify his tuning? Is Jorgensen's procedure simply
> a practical method for attaining an approximation to
> the results desired by Grammateus?

I would assume the equal-beting method of Jorgensen *is* an approximation,
however close. Joorgensen seems to make the distinction often between
'theoretically correct' methods, and 'equal beating' methods. Since my
description came from 'The Equal-Beating Temperaments', I assume it's the
latter, i.e. an approximation.

Best,
Aaron.

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

> So now my question is: exactly how did Grammateus
> specify his tuning? Is Jorgensen's procedure simply
> a practical method for attaining an approximation to
> the results desired by Grammateus?

Yes. Jorgensen likes approximating various tuning systems using equal-
beating strategies, because they can be accomplished by ear. But if I
remember my Barbour correctly, Grammateus was one of the first to use
Euclid's method for finding a mean proportional between two lengths
(which would divide a musical interval into two intervals equal in
cents) in specifying a tuning system -- so the Scala file is in fact
more correct.

Thanks, guys. I'll create two separate Tonescape
files, one labeled as Grammateus 1518 and the other
as Jorgensen's approximation to it.

-monz

-- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > So now my question is: exactly how did Grammateus
> > specify his tuning? Is Jorgensen's procedure simply
> > a practical method for attaining an approximation to
> > the results desired by Grammateus?
>
> Yes. Jorgensen likes approximating various tuning systems using equal-
> beating strategies, because they can be accomplished by ear. But if I
> remember my Barbour correctly, Grammateus was one of the first to use
> Euclid's method for finding a mean proportional between two lengths
> (which would divide a musical interval into two intervals equal in
> cents) in specifying a tuning system -- so the Scala file is in fact
> more correct.