Diatonic Semitones

Hello, all. I have seen the interval 15/14 referred to as a "major
diatonic semitone" and 16/15 as a "minor diatonic semitone". My
contention is that this is incorrect. In the traditional (Plato's
diatonic syntonon) 5-limit JI tuning there exists only the 16/15
semitone. The 15/14 interval would be obtained by introducing the
7-limit and in that case would be a chromatic, not diatonic, semitone.
For example, 7/4 (Bb) x 15/14 = 15/8 (B). One could substitute 15/14
for 16/15 in the Plato tuning but that would throw off the 3rds and
5ths from their just values. Thanks for any clarification you can
provide. Sincerely,

ham_45242 wrote:

> The 15/14 interval would be obtained by introducing the
> 7-limit and in that case would be a chromatic, not diatonic, semitone.
> For example, 7/4 (Bb) x 15/14 = 15/8 (B).

You can't explicitely say that one is right and the other is wrong because it depends completely on which temperament are you working with. The way you view it is just one of more possible ways and it's the same as the one of Huygens or Tartini or Fokker. It uses what we now call "dominant temperament" which maps the 4:5:6:7 to C-E-G-Bb. But if you use, for example, the 7-limit schismatic temperament, then the mapping is completely different and the 4:5:6:7 is then written as C-Fb-G-Cbb. Another way, which is the one I prefer, is to use standard 7-limit meantone temperament which maps the 4:5:6:7 to C-E-G-A#, not Bb. Another possibility, with which perhaps someone like Ozan might have more experience than I do, is superpyth temperament which spells the 4:5:6:7 as C-D#-G-Bb. One more suggestion could be a 7-limit version of mavila (which I think people like Herman or Kraig could say more about) that does the 4:5:6:7 as C-Eb-G-B. As you can see, there are more ways of mapping the basic harmonic factors and its just upon your personal choice which temperament you decide to use.

> One could substitute 15/14
> for 16/15 in the Plato tuning but that would throw off the 3rds and
> 5ths from their just values.

Is there anything wrong about that? I don't think so. It just depends on whether you want to temper any intervals or not. If you wish all the 5-limit intervals sounded just, then, of course, it's undesirable to use a 7-limit one as a replacement for one of these. If you don't mind a slight detuning, then you can use 7-limit intervals in place of 5-limit ones. For example, if you have downloaded Manuel's scale archive, one of the files you find there is called "parizek_7lqmtd2.scl". It's a scale I made in 2004 right in order it were easily tunable just by ear. The 1/1 is meant to be D there, which means that Bb-D-F-G# is tuned to 4:5:6:7.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> ham_45242 wrote:
>
>
>
> > The 15/14 interval would be obtained by introducing the
> > 7-limit and in that case would be a chromatic, not diatonic,
semitone.
> > For example, 7/4 (Bb) x 15/14 = 15/8 (B).
>
>
>
> You can't explicitely say that one is right and the other is wrong
because it depends completely on which temperament are you working
with. The way you view it is just one of more possible ways and it's
the same as the one of Huygens or Tartini or Fokker. It uses what we
now call "dominant temperament" which maps the 4:5:6:7 to C-E-G-Bb.
But if you use, for example, the 7-limit schismatic temperament, then
the mapping is completely different and the 4:5:6:7 is then written as
C-Fb-G-Cbb. Another way, which is the one I prefer, is to use standard
7-limit meantone temperament which maps the 4:5:6:7 to C-E-G-A#, not
Bb. Another possibility, with which perhaps someone like Ozan might
have more experience than I do, is superpyth temperament which spells
the 4:5:6:7 as C-D#-G-Bb. One more suggestion could be a 7-limit
version of mavila (which I think people like Herman or Kraig could say
more about) that does the 4:5:6:7 as C-Eb-G-B. As you can see, there
are more ways of mapping the basic harmonic factors and its just upon
your personal choice which temperament you decide to use.
>
>
>
> > One could substitute 15/14
> > for 16/15 in the Plato tuning but that would throw off the 3rds and
> > 5ths from their just values.
>
>
>
> Is there anything wrong about that? I don't think so. It just
depends on whether you want to temper any intervals or not. If you
wish all the 5-limit intervals sounded just, then, of course, it's
undesirable to use a 7-limit one as a replacement for one of these. If
you don't mind a slight detuning, then you can use 7-limit intervals
in place of 5-limit ones. For example, if you have downloaded Manuel's
scale archive, one of the files you find there is called
"parizek_7lqmtd2.scl". It's a scale I made in 2004 right in order it
were easily tunable just by ear. The 1/1 is meant to be D there, which
means that Bb-D-F-G# is tuned to 4:5:6:7.
>
>
>
> Petr
>
Thanks for replying but that just adds to my confusion. Intervals of
the type (n+1)/n where n is an integer are generally associated with
some form of JI. A diatonic interval has to be associated with a
diatonic, not chromatic scale. Also, JI in the usual sense refers to
scales constructed at least in part from JI perfect fifths and JI
major thirds. The issue here is that in traditional JI tunings of the
diatonic scale you obtain only one type of semitone (e.g. 16/15 in the
5-limit or 28/27 in the 7-limit) The concept of major and minor
diatonic semitones doesn't seem to make sense. Major and minor
diatonic (whole) tones (e.g. 9/8 & 10/9 in 5-limit) do occur, however.
Sincerely,

ham_45242 wrote:

> Thanks for replying but that just adds to my confusion.

Okay, let's see if we can sort it out.

> Intervals of
> the type (n+1)/n where n is an integer are generally associated with
> some form of JI.

Agreed.

> A diatonic interval has to be associated with a
> diatonic, not chromatic scale.

Agreed.

> Also, JI in the usual sense refers to
> scales constructed at least in part from JI perfect fifths and JI
> major thirds. The issue here is that in traditional JI tunings of the
> diatonic scale you obtain only one type of semitone (e.g. 16/15 in the
> 5-limit or 28/27 in the 7-limit) The concept of major and minor
> diatonic semitones doesn't seem to make sense.

Can you tell me, please, why you think that the only correct ratio for the 7-limit minor second is 28/27 and nothing else? As I've already said, this is true only if you treat the 7/4 as a minor seventh, which is the case of the "dominant temperament". But again, there's no particular reason why the 7/4 should always work as a minor seventh. And as I've also said earlier, there are tunings in which the 7/4 approximation IS used in the function of something else. The reason why dominant temperament works the way it does is that the syntonic comma (which is the distance between 10/9 and 9/8) and the septimal comma (which is the distance between 9/8 and 8/7) turn in to unison (or "vanish") because all the three intervals (10/9, 9/8, 8/7) are approximated by only one size of major second. However, let's také the example of 7-limit meantone where 7/4 works as an augmented sixth, not as a minor seventh. In this tuning, both the syntonic comma (the distance between 10/9 and 9/8) and the septimal kleisma (the distance between 16/15 and 15/14) turn into unison because there's only one major second and only one minor second in meantone tunings, not two. This means that the 4:5:6:7 is approximated here by C-E-G-A#. And then, as I've said in my previous post, there are other tunings which work differently than dominant or meantone.

Petr

the 21/20 crops up also in 7 limit diatonics. but there is something special about 28/27
one greek included it in every scale!

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Petr Par�zek wrote:
>
> ham_45242 <mailto:arl_123@...> wrote:
>
> > Thanks for replying but that just adds to my confusion.
>
> Okay, let�s see if we can sort it out.
>
> > Intervals of
> > the type (n+1)/n where n is an integer are generally associated with
> > some form of JI.
>
> Agreed.
>
> > A diatonic interval has to be associated with a
> > diatonic, not chromatic scale.
>
> Agreed.
>
> > Also, JI in the usual sense refers to
> > scales constructed at least in part from JI perfect fifths and JI
> > major thirds. The issue here is that in traditional JI tunings of the
> > diatonic scale you obtain only one type of semitone (e.g. 16/15 in the
> > 5-limit or 28/27 in the 7-limit) The concept of major and minor
> > diatonic semitones doesn't seem to make sense.
>
> Can you tell me, please, why you think that the only correct ratio for > the 7-limit minor second is 28/27 and nothing else? As I�ve already > said, this is true only if you treat the 7/4 as a minor seventh, which > is the case of the �dominant temperament�. But again, there�s no > particular reason why the 7/4 should always work as a minor seventh. > And as I�ve also said earlier, there are tunings in which the 7/4 > approximation IS used in the function of something else. The reason > why dominant temperament works the way it does is that the syntonic > comma (which is the distance between 10/9 and 9/8) and the septimal > comma (which is the distance between 9/8 and 8/7) turn in to unison > (or �vanish�) because all the three intervals (10/9, 9/8, 8/7) are > approximated by only one size of major second. However, let�s tak� the > example of 7-limit meantone where 7/4 works as an augmented sixth, not > as a minor seventh. In this tuning, both the syntonic comma (the > distance between 10/9 and 9/8) and the septimal kleisma (the distance > between 16/15 and 15/14) turn into unison because there�s only one > major second and only one minor second in meantone tunings, not two. > This means that the 4:5:6:7 is approximated here by C-E-G-A#. And > then, as I�ve said in my previous post, there are other tunings which > work differently than dominant or meantone.
>
> Petr
>
>

"ham_45242" isn't talking about temperaments, but about tunings.

But aren't mixing up the terminologies of different periods here,
anyway?

--- In tuning@yahoogroups.com, "ham_45242" <arl_123@...> wrote:
>
> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@> wrote:
> >
> > ham_45242 wrote:
> >
> >
> >
> > > The 15/14 interval would be obtained by introducing the
> > > 7-limit and in that case would be a chromatic, not diatonic,
> semitone.
> > > For example, 7/4 (Bb) x 15/14 = 15/8 (B).
> >
> >
> >
> > You can't explicitely say that one is right and the other is wrong
> because it depends completely on which temperament are you working
> with. The way you view it is just one of more possible ways and it's
> the same as the one of Huygens or Tartini or Fokker. It uses what we
> now call "dominant temperament" which maps the 4:5:6:7 to C-E-G-Bb.
> But if you use, for example, the 7-limit schismatic temperament,
then
> the mapping is completely different and the 4:5:6:7 is then written
as
> C-Fb-G-Cbb. Another way, which is the one I prefer, is to use
standard
> 7-limit meantone temperament which maps the 4:5:6:7 to C-E-G-A#, not
> Bb. Another possibility, with which perhaps someone like Ozan might
> have more experience than I do, is superpyth temperament which
spells
> the 4:5:6:7 as C-D#-G-Bb. One more suggestion could be a 7-limit
> version of mavila (which I think people like Herman or Kraig could
say
> more about) that does the 4:5:6:7 as C-Eb-G-B. As you can see, there
> are more ways of mapping the basic harmonic factors and its just
upon
> your personal choice which temperament you decide to use.
> >
> >
> >
> > > One could substitute 15/14
> > > for 16/15 in the Plato tuning but that would throw off the 3rds
and
> > > 5ths from their just values.
> >
> >
> >
> > Is there anything wrong about that? I don't think so. It just
> depends on whether you want to temper any intervals or not. If you
> wish all the 5-limit intervals sounded just, then, of course, it's
> undesirable to use a 7-limit one as a replacement for one of these.
If
> you don't mind a slight detuning, then you can use 7-limit intervals
> in place of 5-limit ones. For example, if you have downloaded
Manuel's
> scale archive, one of the files you find there is called
> "parizek_7lqmtd2.scl". It's a scale I made in 2004 right in order it
> were easily tunable just by ear. The 1/1 is meant to be D there,
which
> means that Bb-D-F-G# is tuned to 4:5:6:7.
> >
> >
> >
> > Petr
> >
> Thanks for replying but that just adds to my confusion. Intervals
of
> the type (n+1)/n where n is an integer are generally associated with
> some form of JI. A diatonic interval has to be associated with a
> diatonic, not chromatic scale. Also, JI in the usual sense refers
to
> scales constructed at least in part from JI perfect fifths and JI
> major thirds. The issue here is that in traditional JI tunings of
the
> diatonic scale you obtain only one type of semitone (e.g. 16/15 in
the
> 5-limit or 28/27 in the 7-limit) The concept of major and minor
> diatonic semitones doesn't seem to make sense. Major and minor
> diatonic (whole) tones (e.g. 9/8 & 10/9 in 5-limit) do occur,
however.
> Sincerely,
>

Cameron Bobro wrote:
> "ham_45242" isn't talking about temperaments, but about tunings. > > But aren't mixing up the terminologies of different periods here, > anyway? Absolutely. You can sort it out if you like.

Graham

Petr Par�zek wrote:
> You can�t explicitely say that one is right and the other is wrong > because it depends completely on which temperament are you working with. > The way you view it is just one of more possible ways and it�s the same > as the one of Huygens or Tartini or Fokker. It uses what we now call > �dominant temperament� which maps the 4:5:6:7 to C-E-G-Bb. But if you > use, for example, the 7-limit schismatic temperament, then the mapping > is completely different and the 4:5:6:7 is then written as C-Fb-G-Cbb. > Another way, which is the one I prefer, is to use standard 7-limit > meantone temperament which maps the 4:5:6:7 to C-E-G-A#, not Bb. Another > possibility, with which perhaps someone like Ozan might have more > experience than I do, is superpyth temperament which spells the 4:5:6:7 > as C-D#-G-Bb. One more suggestion could be a 7-limit version of mavila > (which I think people like Herman or Kraig could say more about) that > does the 4:5:6:7 as C-Eb-G-B. As you can see, there are more ways of > mapping the basic harmonic factors and its just upon your personal > choice which temperament you decide to use.

There's some confusion in the 7-limit mavila terminology, but that particular 7-limit version of mavila has another name "hexadecimal" (as it has a 16-note MOS). It works best with stretched octaves (around 1205 to 1209 cents).

There's also flattone, with C-E-G-Bbb, and probably others. Flattone is what you get with meantone when you go past 19-ET, up to around 26-ET or so; A# and Bbb are enharmonic equivalents in 19-ET.

Still, "major diatonic semitone" for an interval (15/14) that doesn't exist in the diatonic scale seems like an unfortunate name. Maybe "large septimal semitone" or something.

I no nothing about the possibility of a 7 limit Mavila.
Maliva is a recurrent sequence

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Herman Miller wrote:
> Petr Par�zek wrote:
> >> You can�t explicitely say that one is right and the other is wrong >> because it depends completely on which temperament are you working with. >> The way you view it is just one of more possible ways and it�s the same >> as the one of Huygens or Tartini or Fokker. It uses what we now call >> �dominant temperament� which maps the 4:5:6:7 to C-E-G-Bb. But if you >> use, for example, the 7-limit schismatic temperament, then the mapping >> is completely different and the 4:5:6:7 is then written as C-Fb-G-Cbb. >> Another way, which is the one I prefer, is to use standard 7-limit >> meantone temperament which maps the 4:5:6:7 to C-E-G-A#, not Bb. Another >> possibility, with which perhaps someone like Ozan might have more >> experience than I do, is superpyth temperament which spells the 4:5:6:7 >> as C-D#-G-Bb. One more suggestion could be a 7-limit version of mavila >> (which I think people like Herman or Kraig could say more about) that >> does the 4:5:6:7 as C-Eb-G-B. As you can see, there are more ways of >> mapping the basic harmonic factors and its just upon your personal >> choice which temperament you decide to use.
>> >
> There's some confusion in the 7-limit mavila terminology, but that > particular 7-limit version of mavila has another name "hexadecimal" (as > it has a 16-note MOS). It works best with stretched octaves (around 1205 > to 1209 cents).
>
> There's also flattone, with C-E-G-Bbb, and probably others. Flattone is > what you get with meantone when you go past 19-ET, up to around 26-ET or > so; A# and Bbb are enharmonic equivalents in 19-ET.
>
> Still, "major diatonic semitone" for an interval (15/14) that doesn't > exist in the diatonic scale seems like an unfortunate name. Maybe "large > septimal semitone" or something.
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>
>
>
>

Maybe the best thing would be, first of all:

http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_t
he_tetrachord/index.html

(copy and paste the whole address if it doesn't "click")

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Cameron Bobro wrote:
> > "ham_45242" isn't talking about temperaments, but about tunings.
> >
> > But aren't mixing up the terminologies of different periods
here,
> > anyway?
>
> Absolutely. You can sort it out if you like.
>
>
> Graham
>

--- In tuning@yahoogroups.com, "ham_45242" <arl_123@...> wrote:
>
> Hello, all. I have seen the interval 15/14 referred to as a
> "major diatonic semitone" and 16/15 as a "minor diatonic
> semitone". My contention is that this is incorrect.
[snip]
> Thanks for any clarification you can provide. Sincerely,

There's no golden standard as to what names like
"diatonic semitone" should mean, or indeed, even what
the "diatonic scale" is. To make them precise we need
definitions. If we define the diatonic scale to be a
5-limit periodicity block with 81/80 and 25/24 as unison
vectors, then 16/15 would be a strong candidate for
something called the "diatonic semitone".

The diatonic scale is not typically seen as a 7-limit
affair. If we use a linear tuning, getting the 7ths to
approximate 7:4 very well means using a 5th sharp of
3:2, and this means making the 3rds bad. If we're using
JI there is more flexibility... John de Laubenfels had
some success retuning diatonic music to 7-limit JI with
his adaptive tuning method. In that case you can wind
up with all sorts of semitones.

100 cents also has a good claim to the name at the moment,
and of course in real performances we do hear things like
adaptive JI and therefore all sorts of semitones.

-Carl

Confusing indeed;-)

Just FYI

I am using the following namings:

A to Bbb = bbII = 54.08¢
28/27 = 62.96¢
A to A# = #I = 68.45
21/20 = 84.47

There's a simple ratio to cents Javascript calculator on this page:

http://www.lucytune.com/new_to_lt/pitch_01.html

and you can play/hear whatever pitches using this application for Mac
or PC.

http://www.versiontracker.com/dyn/moreinfo/macosx/31561

http://www.versiontracker.com/dyn/moreinfo/win/95240

On 30 May 2008, at 03:36, Carl Lumma wrote:

> --- In tuning@yahoogroups.com, "ham_45242" <arl_123@...> wrote:
> >
> > Hello, all. I have seen the interval 15/14 referred to as a
> > "major diatonic semitone" and 16/15 as a "minor diatonic
> > semitone". My contention is that this is incorrect.
> [snip]
> > Thanks for any clarification you can provide. Sincerely,
>
> There's no golden standard as to what names like
> "diatonic semitone" should mean, or indeed, even what
> the "diatonic scale" is. To make them precise we need
> definitions. If we define the diatonic scale to be a
> 5-limit periodicity block with 81/80 and 25/24 as unison
> vectors, then 16/15 would be a strong candidate for
> something called the "diatonic semitone".
>
> The diatonic scale is not typically seen as a 7-limit
> affair. If we use a linear tuning, getting the 7ths to
> approximate 7:4 very well means using a 5th sharp of
> 3:2, and this means making the 3rds bad. If we're using
> JI there is more flexibility... John de Laubenfels had
> some success retuning diatonic music to 7-limit JI with
> his adaptive tuning method. In that case you can wind
> up with all sorts of semitones.
>
> 100 cents also has a good claim to the name at the moment,
> and of course in real performances we do hear things like
> adaptive JI and therefore all sorts of semitones.
>
> -Carl
>
>
>

Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

my Centaur tuning is a 7 limit tuning that has all sorts of 7 limit diatonics
charted out
http://anaphoria.com/centaur.html

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, > "ham_45242" <arl_123@...> wrote:
> >
> > Hello, all. I have seen the interval 15/14 referred to as a
> > "major diatonic semitone" and 16/15 as a "minor diatonic
> > semitone". My contention is that this is incorrect.
> [snip]
> > Thanks for any clarification you can provide. Sincerely,
>
> There's no golden standard as to what names like
> "diatonic semitone" should mean, or indeed, even what
> the "diatonic scale" is. To make them precise we need
> definitions. If we define the diatonic scale to be a
> 5-limit periodicity block with 81/80 and 25/24 as unison
> vectors, then 16/15 would be a strong candidate for
> something called the "diatonic semitone".
>
> The diatonic scale is not typically seen as a 7-limit
> affair. If we use a linear tuning, getting the 7ths to
> approximate 7:4 very well means using a 5th sharp of
> 3:2, and this means making the 3rds bad. If we're using
> JI there is more flexibility... John de Laubenfels had
> some success retuning diatonic music to 7-limit JI with
> his adaptive tuning method. In that case you can wind
> up with all sorts of semitones.
>
> 100 cents also has a good claim to the name at the moment,
> and of course in real performances we do hear things like
> adaptive JI and therefore all sorts of semitones.
>
> -Carl
>
>

Cameron wrote:

> > "ham_45242" isn't talking about temperaments, but about tunings.

I don't know what it changes about the question. The problem is that whether he uses a regular temperament or not, he's still trying to classify 7-limit intervals according to criteria coming from 5-limit harmony, i.e. he's imitating 5-limit pitches with 7-limit ones anyway. As I've said earlier, my "parizek_7lqmtd2.scl" was done with just the same aim even though it's not a temperament. It's a bit like if you replace 3-limit intervals with 5-limit ones -- there's no rule whether you should shrink every 8th fifth by a schisma (which makes 5/4 to work as a diminished fourth) or whether you should shrink every 4th fifth by a syntonic comma like Euler or Ellis (in which case 5/4 works as a major third).

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Cameron wrote:
>
> > > "ham_45242" isn't talking about temperaments, but about
tunings.
>
> I don't know what it changes about the question. The problem is
that whether he uses a regular temperament or not, he's still trying
to classify 7-limit intervals according to criteria coming from 5-
limit harmony, i.e. he's imitating 5-limit pitches with 7-limit ones
anyway. As I've said earlier, my "parizek_7lqmtd2.scl" was done with
just the same aim even though it's not a temperament. It's a bit
like if you replace 3-limit intervals with 5-limit ones -- there's
no rule whether you should shrink every 8th fifth by a schisma
(which makes 5/4 to work as a diminished fourth) or whether you
should shrink every 4th fifth by a syntonic comma like Euler or
Ellis (in which case 5/4 works as a major third).
>
> Petr
>

Heh, I have trouble swallowing the idea that the ancient Greek
tetrachords were based on "5-limit harmony", though of course they've
come to us over the ages after having been thoroughly fingered and
fondled by salesmen whose intentions were willy nilly to foist such
anachronisms on us.

It most certainly DOES matter, when someone introduces the diatonic
with Plato's name attached, that we figure out what (s)he means by
the term. Unless we're in to earnest discussions about which diesel
engine King Tut used on his chariot and so on, which is fine on an
artistic level (Peterbuilt, it's QED!) but not otherwise.

The anachronisms in the root of the whole subject MUST have been
bickered about long ago.... hang on....

http://www.geocities.com/Vienna/Choir/4792/mei.html

(Yeah baby, I'm going for the 11/10 major semitone too!)

as Carl pointed out we'd better define what "world" we're in first.

-Cameron Bobro

The thing that you're missing, which I think has been said, is that a
7/4 ratio, while sounding to our ears like a "dominant 7th", isn't
necessarily a C-Bb in terms of what note names are used. For example,
in 31tet or 19tet (to the extent that 19tet can even do this stuff), a
7/4 interval is actually referred to as an "augmented 6th" rather than
as a dominant 7th. So a 4:5:6:7 chord would be spelled out as
C-E-G-A#. This is the difference back in the day between the
"augmented 6th" chords that you usually see in classical theory
exercises and normal dominant 7th chords. Note that this pattern of
spelling out chords actually applies to all meantone systems.

And then for schismatic systems, the 7/4 is spelled out completely
differently, as is 5/4. For example, a C major chord in a schismatic
temperament is spelled out C-Fb-G. They usually fix this by making Fb
enharmonically equivalent to something like E\ or E-, so it doesn't
look so dumb and counterintuitive.

On the other hand, it does make sense to label 15/14 as a chromatic
semitone because if you're putting together some kind of 4-note chord,
and you have C-E-G, you might view your options as C-E-G-Bb (where Bb
represents a 7/4) or as C-E-G-B (where B represents 15/8). You might
be thinking in terms of different 7ths to take, and so view the
distinction between them would be a chromatic semitone. But always
remember you could just as easily view that 7th harmonic as an
augmented sixth, so that it becomes the difference between C-E-G-A
(where A is 5/3) and C-E-G-A# (where A# is 7/4). Just because it's the
"7th" harmonic doesn't necessarily mean it has to fit in the "7th"
scale degree. You could well have a diatonic scale like C D E F G A# B
C, and so the difference between A# and B would be some kind of a
"diatonic" semitone.

What I think this all is REALLY getting at is how exactly do 7-limit
intervals fit into the current diatonic system? Furthermore, what the
hell IS the current diatonic system, really? It would be nice if we
could all finally agree on some "standard" to go by so that confusions
like this just cease to be. Then we can "expand" whatever system we
come up with to include 7 and 11 and 93 limit intervals.

For example, D-E-F#-G-A-Bb-C-D has some kind of diatonic "flavor" to
it, although in this case it's a mode of the melodic minor scale. But
D-E-F#-G-A#-B-C-D does NOT. Is it because there are two adjacent
diatonic semitones, perhaps, which is a rule theory teachers often
spit out for no apparent reason? Why the hell would that matter? These
are the questions we need to ask.

On Thu, May 29, 2008 at 4:46 PM, ham_45242 <arl_123@...> wrote:
> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>>
>> ham_45242 wrote:
>>
>>
>>
>> > The 15/14 interval would be obtained by introducing the
>> > 7-limit and in that case would be a chromatic, not diatonic,
> semitone.
>> > For example, 7/4 (Bb) x 15/14 = 15/8 (B).
>>
>>
>>
>> You can't explicitely say that one is right and the other is wrong
> because it depends completely on which temperament are you working
> with. The way you view it is just one of more possible ways and it's
> the same as the one of Huygens or Tartini or Fokker. It uses what we
> now call "dominant temperament" which maps the 4:5:6:7 to C-E-G-Bb.
> But if you use, for example, the 7-limit schismatic temperament, then
> the mapping is completely different and the 4:5:6:7 is then written as
> C-Fb-G-Cbb. Another way, which is the one I prefer, is to use standard
> 7-limit meantone temperament which maps the 4:5:6:7 to C-E-G-A#, not
> Bb. Another possibility, with which perhaps someone like Ozan might
> have more experience than I do, is superpyth temperament which spells
> the 4:5:6:7 as C-D#-G-Bb. One more suggestion could be a 7-limit
> version of mavila (which I think people like Herman or Kraig could say
> more about) that does the 4:5:6:7 as C-Eb-G-B. As you can see, there
> are more ways of mapping the basic harmonic factors and its just upon
> your personal choice which temperament you decide to use.
>>
>>
>>
>> > One could substitute 15/14
>> > for 16/15 in the Plato tuning but that would throw off the 3rds and
>> > 5ths from their just values.
>>
>>
>>
>> Is there anything wrong about that? I don't think so. It just
> depends on whether you want to temper any intervals or not. If you
> wish all the 5-limit intervals sounded just, then, of course, it's
> undesirable to use a 7-limit one as a replacement for one of these. If
> you don't mind a slight detuning, then you can use 7-limit intervals
> in place of 5-limit ones. For example, if you have downloaded Manuel's
> scale archive, one of the files you find there is called
> "parizek_7lqmtd2.scl". It's a scale I made in 2004 right in order it
> were easily tunable just by ear. The 1/1 is meant to be D there, which
> means that Bb-D-F-G# is tuned to 4:5:6:7.
>>
>>
>>
>> Petr
>>
> Thanks for replying but that just adds to my confusion. Intervals of
> the type (n+1)/n where n is an integer are generally associated with
> some form of JI. A diatonic interval has to be associated with a
> diatonic, not chromatic scale. Also, JI in the usual sense refers to
> scales constructed at least in part from JI perfect fifths and JI
> major thirds. The issue here is that in traditional JI tunings of the
> diatonic scale you obtain only one type of semitone (e.g. 16/15 in the
> 5-limit or 28/27 in the 7-limit) The concept of major and minor
> diatonic semitones doesn't seem to make sense. Major and minor
> diatonic (whole) tones (e.g. 9/8 & 10/9 in 5-limit) do occur, however.
> Sincerely,
>
>

Just as another thought, I think that some of the diatonic scales
merely coalesced because they were simply just major triads built a
fifth up and a fifth down from the root. What the "modes" truly are I
have no idea, although I know that in 12tet at least there are comma
jumps all over the place for the same parent key and scale but a
different mode. For example, D Dorian's 6th scale degree, a B, is
often played as 27/16, for instance, or as a stack of 3 fifths, while
the same B in C Ionian is usually comma adjusted down so that in
relation to the D it's a 5/3 (15/8 from the C).

But what if we just applied the same principle to septimal chords? For
example, let's say we'll make the I, IV, and V scale degrees all
septimal supermajor third chords. I will use the notation E> to
indicate one septimal diesis over from E (so 9/7).

C D E> F G A> B> C

That translates to C E> G, F A> C, and G B> D. The relative minor of
this scale, A> minor, would be this:

A> B> C D E> F G A>

So the chords are A> C E>, D F A>, and E> G B>. So the modes don't
hold up the right way - the relative minor is NOT 3 subminor triads,
as that D minor chord isn't a subminor. This corresponds to the comma
adjustments found in the normal modes, I think - the D would have to
be septimal diesis adjusted over on the fly for it to work.

Anyone have any ideas on how to make diatonic scale/mode theory extend
out to the 7-limit? Or maybe just how to make it work the way it is
now for non-meantone systems?

On Fri, May 30, 2008 at 4:11 AM, Mike Battaglia <battaglia01@...> wrote:
> The thing that you're missing, which I think has been said, is that a
> 7/4 ratio, while sounding to our ears like a "dominant 7th", isn't
> necessarily a C-Bb in terms of what note names are used. For example,
> in 31tet or 19tet (to the extent that 19tet can even do this stuff), a
> 7/4 interval is actually referred to as an "augmented 6th" rather than
> as a dominant 7th. So a 4:5:6:7 chord would be spelled out as
> C-E-G-A#. This is the difference back in the day between the
> "augmented 6th" chords that you usually see in classical theory
> exercises and normal dominant 7th chords. Note that this pattern of
> spelling out chords actually applies to all meantone systems.
>
> And then for schismatic systems, the 7/4 is spelled out completely
> differently, as is 5/4. For example, a C major chord in a schismatic
> temperament is spelled out C-Fb-G. They usually fix this by making Fb
> enharmonically equivalent to something like E\ or E-, so it doesn't
> look so dumb and counterintuitive.
>
> On the other hand, it does make sense to label 15/14 as a chromatic
> semitone because if you're putting together some kind of 4-note chord,
> and you have C-E-G, you might view your options as C-E-G-Bb (where Bb
> represents a 7/4) or as C-E-G-B (where B represents 15/8). You might
> be thinking in terms of different 7ths to take, and so view the
> distinction between them would be a chromatic semitone. But always
> remember you could just as easily view that 7th harmonic as an
> augmented sixth, so that it becomes the difference between C-E-G-A
> (where A is 5/3) and C-E-G-A# (where A# is 7/4). Just because it's the
> "7th" harmonic doesn't necessarily mean it has to fit in the "7th"
> scale degree. You could well have a diatonic scale like C D E F G A# B
> C, and so the difference between A# and B would be some kind of a
> "diatonic" semitone.
>
> What I think this all is REALLY getting at is how exactly do 7-limit
> intervals fit into the current diatonic system? Furthermore, what the
> hell IS the current diatonic system, really? It would be nice if we
> could all finally agree on some "standard" to go by so that confusions
> like this just cease to be. Then we can "expand" whatever system we
> come up with to include 7 and 11 and 93 limit intervals.
>
> For example, D-E-F#-G-A-Bb-C-D has some kind of diatonic "flavor" to
> it, although in this case it's a mode of the melodic minor scale. But
> D-E-F#-G-A#-B-C-D does NOT. Is it because there are two adjacent
> diatonic semitones, perhaps, which is a rule theory teachers often
> spit out for no apparent reason? Why the hell would that matter? These
> are the questions we need to ask.
>
> On Thu, May 29, 2008 at 4:46 PM, ham_45242 <arl_123@...> wrote:
>> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>>>
>>> ham_45242 wrote:
>>>
>>>
>>>
>>> > The 15/14 interval would be obtained by introducing the
>>> > 7-limit and in that case would be a chromatic, not diatonic,
>> semitone.
>>> > For example, 7/4 (Bb) x 15/14 = 15/8 (B).
>>>
>>>
>>>
>>> You can't explicitely say that one is right and the other is wrong
>> because it depends completely on which temperament are you working
>> with. The way you view it is just one of more possible ways and it's
>> the same as the one of Huygens or Tartini or Fokker. It uses what we
>> now call "dominant temperament" which maps the 4:5:6:7 to C-E-G-Bb.
>> But if you use, for example, the 7-limit schismatic temperament, then
>> the mapping is completely different and the 4:5:6:7 is then written as
>> C-Fb-G-Cbb. Another way, which is the one I prefer, is to use standard
>> 7-limit meantone temperament which maps the 4:5:6:7 to C-E-G-A#, not
>> Bb. Another possibility, with which perhaps someone like Ozan might
>> have more experience than I do, is superpyth temperament which spells
>> the 4:5:6:7 as C-D#-G-Bb. One more suggestion could be a 7-limit
>> version of mavila (which I think people like Herman or Kraig could say
>> more about) that does the 4:5:6:7 as C-Eb-G-B. As you can see, there
>> are more ways of mapping the basic harmonic factors and its just upon
>> your personal choice which temperament you decide to use.
>>>
>>>
>>>
>>> > One could substitute 15/14
>>> > for 16/15 in the Plato tuning but that would throw off the 3rds and
>>> > 5ths from their just values.
>>>
>>>
>>>
>>> Is there anything wrong about that? I don't think so. It just
>> depends on whether you want to temper any intervals or not. If you
>> wish all the 5-limit intervals sounded just, then, of course, it's
>> undesirable to use a 7-limit one as a replacement for one of these. If
>> you don't mind a slight detuning, then you can use 7-limit intervals
>> in place of 5-limit ones. For example, if you have downloaded Manuel's
>> scale archive, one of the files you find there is called
>> "parizek_7lqmtd2.scl". It's a scale I made in 2004 right in order it
>> were easily tunable just by ear. The 1/1 is meant to be D there, which
>> means that Bb-D-F-G# is tuned to 4:5:6:7.
>>>
>>>
>>>
>>> Petr
>>>
>> Thanks for replying but that just adds to my confusion. Intervals of
>> the type (n+1)/n where n is an integer are generally associated with
>> some form of JI. A diatonic interval has to be associated with a
>> diatonic, not chromatic scale. Also, JI in the usual sense refers to
>> scales constructed at least in part from JI perfect fifths and JI
>> major thirds. The issue here is that in traditional JI tunings of the
>> diatonic scale you obtain only one type of semitone (e.g. 16/15 in the
>> 5-limit or 28/27 in the 7-limit) The concept of major and minor
>> diatonic semitones doesn't seem to make sense. Major and minor
>> diatonic (whole) tones (e.g. 9/8 & 10/9 in 5-limit) do occur, however.
>> Sincerely,
>>
>>
>

Mike wrote...

> Anyone have any ideas on how to make diatonic scale/mode theory
> extend out to the 7-limit? Or maybe just how to make it work the
> way it is now for non-meantone systems?

Have you read Paul Erlich's paper on the subject?

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

-Carl

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> The thing that you're missing, which I think has been said, is that
> On the other hand, it does make sense to label 15/14 as a chromatic
> semitone because if you're putting together some kind of 4-note chord,
> and you have C-E-G, you might view your options as C-E-G-Bb (where Bb
> represents a 7/4) or as C-E-G-B (where B represents 15/8). You might
> be thinking in terms of different 7ths to take, and so view the
> distinction between them would be a chromatic semitone. But always
> remember you could just as easily view that 7th harmonic as an
> augmented sixth, so that it becomes the difference between C-E-G-A
> (where A is 5/3) and C-E-G-A# (where A# is 7/4). Just because it's the
> "7th" harmonic doesn't necessarily mean it has to fit in the "7th"
> scale degree. You could well have a diatonic scale like C D E F G A# B
> C, and so the difference between A# and B would be some kind of a
> "diatonic" semitone.
>
Hello, all, and first let me thank everyone for all the replies! And
yes, I'm obviously talking about tunings not temperaments. So if we
can define a 7-note scale of C-D-E-F-G-A#-B-C' as diatonic and assign
pitch ratios of 7/4 and 15/8 to the intervals C-A# and C-B,
respectiveley, we obtain our "diatonic" semitone of 15/14, provided
that A#-B is considered a semitone. One issue here is that if we
assign a 3/2 ratio to C-G, that makes the G-B interval a JI major
third and we then have this interval consisting of a 7/6 septimal
minor third and our 15/14 diatonic semitone. Of course we don't have
to use 3/2 fifths. Sincerely,

> Hello, all, and first let me thank everyone for all the replies! And
> yes, I'm obviously talking about tunings not temperaments. So if we
> can define a 7-note scale of C-D-E-F-G-A#-B-C' as diatonic and assign
> pitch ratios of 7/4 and 15/8 to the intervals C-A# and C-B,
> respectiveley, we obtain our "diatonic" semitone of 15/14, provided
> that A#-B is considered a semitone. One issue here is that if we
> assign a 3/2 ratio to C-G, that makes the G-B interval a JI major
> third and we then have this interval consisting of a 7/6 septimal
> minor third and our 15/14 diatonic semitone. Of course we don't have
> to use 3/2 fifths. Sincerely,

I don't see the issue - the interval consisting of a 7/6 subminor
third and a 15/14 diatonic semitone would be 7/6*15/14 = 5/4. What's
the problem?

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> > Hello, all, and first let me thank everyone for all the replies! And
> > yes, I'm obviously talking about tunings not temperaments. So if we
> > can define a 7-note scale of C-D-E-F-G-A#-B-C' as diatonic and assign
> > pitch ratios of 7/4 and 15/8 to the intervals C-A# and C-B,
> > respectiveley, we obtain our "diatonic" semitone of 15/14, provided
> > that A#-B is considered a semitone. One issue here is that if we
> > assign a 3/2 ratio to C-G, that makes the G-B interval a JI major
> > third and we then have this interval consisting of a 7/6 septimal
> > minor third and our 15/14 diatonic semitone. Of course we don't have
> > to use 3/2 fifths. Sincerely,
>
> I don't see the issue - the interval consisting of a 7/6 subminor
> third and a 15/14 diatonic semitone would be 7/6*15/14 = 5/4. What's
> the problem?
>

Hello, and there's no issue as long as the division of the 5/4 JI
major third is acceptable this way. It does adhere to an(n+1)/n
division (n being a "small" integer) that tends to keep things
harmonically related. Of course this is one interval in the scale.
The real issue is the acceptability of the above 7-tone scale as a
diatonic since it differs from the t-t-s-t-t-s (t=tone, s=semitone)
order of the traditional diatonic scale. Sincerely,

> Hello, and there's no issue as long as the division of the 5/4 JI
> major third is acceptable this way. It does adhere to an(n+1)/n
> division (n being a "small" integer) that tends to keep things
> harmonically related. Of course this is one interval in the scale.
> The real issue is the acceptability of the above 7-tone scale as a
> diatonic since it differs from the t-t-s-t-t-s (t=tone, s=semitone)
> order of the traditional diatonic scale. Sincerely,

You forget that we call it a "subminor third" just because its closest
5-limit match is the 6/5 "minor third." You could just as well call
7/6 a "septimal augmented second", which would make it all work out.
You could just say that a septimal minor third is enharmonically
equivalent to a septimal augmented second.

It's basically the same principle that they use for, say, the harmonic
minor scale:

C D Eb F G Ab B C

The Ab-B is an "augmented second," though that interval might be
closer in distance to a minor third than a major second. We just call
it an augmented second to make the numbering work. We're basically
doing the same thing, except we're adding the "septimal" qualifier to
be specific as to what kind of augmented second we're talking about.

-Mike

> > Also, JI in the usual sense refers to
> > scales constructed at least in part from JI perfect fifths and JI
> > major thirds. The issue here is that in traditional JI tunings of the
> > diatonic scale you obtain only one type of semitone (e.g. 16/15 in the
> > 5-limit or 28/27 in the 7-limit) The concept of major and minor
> > diatonic semitones doesn't seem to make sense.
>
>
>
> Can you tell me, please, why you think that the only correct ratio
for the 7-limit minor second is 28/27 and nothing else?

Hello, and I never said it was. I used "e.g." (for example) not
"i.e."(that is). But we have to make a distinction between a derived
and specified interval. Thus if specify an interval of 7/4 (call it a
septimal minor 7th, harmonic 7th or whatever) along with a Pythagorean
major 6th of 27/16 we derive the 28/27 m2 value. OTOH using a JI
major 6th of 5/3 results in a m2 value of 21/20. Sincerely,

ham_45242 wrote:
> --- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:

>> I don't see the issue - the interval consisting of a 7/6 subminor
>> third and a 15/14 diatonic semitone would be 7/6*15/14 = 5/4. What's
>> the problem?
>>
> > Hello, and there's no issue as long as the division of the 5/4 JI
> major third is acceptable this way. It does adhere to an(n+1)/n
> division (n being a "small" integer) that tends to keep things
> harmonically related. Of course this is one interval in the scale. > The real issue is the acceptability of the above 7-tone scale as a
> diatonic since it differs from the t-t-s-t-t-s (t=tone, s=semitone)
> order of the traditional diatonic scale. Sincerely,

Ain't no t-t-s-t-t-s in JI. You have to have to different sizes of tones. Replace 5/4 with 9/7 and the pattern's the same giving 28:27 as the diatonic semitone. So, yes, there's an argument for 15:14 being the chromatic semitone for 9-limit music. But you can also follow quarter-comma meantone temperament *outside* diatonic scales. Then a 7/4 is well represented by an augmented sixth and 16:15 and 15:14 come out as the same type of interval. So call them both diatonic semitones.

The equivalence 15/14 ~ 16/15 is very common with tempered scales and implies 225:224 as a unison vector. If you don't like the term "diatonic semitone" you can call them "secors" after the generator of miracle temperament.

An alternative terminology is that 16:15 and 15:14 are both "toes". Then, naturally, 25:24, 28:27, and 36:35 are semitoes of some kind. To reach all 7-limit intervals you also need inches of 49:48 or 81:80. I hope that removes any confusion.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>
> Ain't no t-t-s-t-t-s in JI.

Hello, and my bad, Graham. It should have been t-t-s-t-t-t-s. Sincerely,

It might be easier to think in terms of L-L-s-L-L-L-s for a diatonic ionian scale.

This enables you to escape;-) from 12edo mentality.

L=Large
s=small.

see this page:

http://www.lucytune.com/new_to_lt/pitch_03.html

On 2 Jun 2008, at 21:27, ham_45242 wrote:

> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >
> >
> > Ain't no t-t-s-t-t-s in JI.
>
> >>Hello, and my bad, Graham. It should have been t-t-s-t-t-t-s. > Sincerely,
>
>

Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Charles Lucy wrote:
> It might be easier to think in terms of L-L-s-L-L-L-s for a diatonic > ionian scale.
> > This enables you to escape;-) from 12edo mentality.

It doesn't matter what you call them or whether you have a hexachord or the full octave. You can't get JI to work with only two step sizes.

Graham

why settle for one whole tone when you can have two.
Pythagorean works though

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Graham Breed wrote:
>
> Charles Lucy wrote:
> > It might be easier to think in terms of L-L-s-L-L-L-s for a diatonic
> > ionian scale.
> >
> > This enables you to escape;-) from 12edo mentality.
>
> It doesn't matter what you call them or whether you have a
> hexachord or the full octave. You can't get JI to work with
> only two step sizes.
>
> Graham
>
>

Yes Graham, that's another reason why I avoid it like the plague;-)

On 3 Jun 2008, at 03:37, Graham Breed wrote:

> Charles Lucy wrote:
> > It might be easier to think in terms of L-L-s-L-L-L-s for a diatonic
> > ionian scale.
> >
> > This enables you to escape;-) from 12edo mentality.
>
> It doesn't matter what you call them or whether you have a
> hexachord or the full octave. You can't get JI to work with
> only two step sizes.
>
> Graham
>
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Graham: I think he means that having G-A# be a septimal subminor third
somehow disqualifies it as a tone. He's saying that for the major
scale, you have ttsttts so that the "t"s represent something in the
category of a "tone," be it a major or minor tone, and the "s"'s
represent something in the category of a semitone.

And ham: Again, I don't see why the problem can't just be solved by
calling 7/6 an "augmented" second, which it is. That's the point of
"augmented" and "diminished" intervals.

-Mike

On Mon, Jun 2, 2008 at 4:27 PM, ham_45242 <arl_123@...> wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>>
>>
>> Ain't no t-t-s-t-t-s in JI.
>
> Hello, and my bad, Graham. It should have been t-t-s-t-t-t-s. Sincerely,
>
>

Mike Battaglia wrote:
> Graham: I think he means that having G-A# be a septimal subminor third
> somehow disqualifies it as a tone. He's saying that for the major
> scale, you have ttsttts so that the "t"s represent something in the
> category of a "tone," be it a major or minor tone, and the "s"'s
> represent something in the category of a semitone.

I don't know what he means and as he isn't generally clear about what kind of "diatonic" he's talking about I don't like to second guess him. But I least I agree that a 7:6 isn't a tone if that's the point.

So fine, if you want something like a tone when the diatonic semitone's 15:14 then you can use 28:25. This will still give minor thirds of 6:5. I don't know why you'd want to do that but what you'd get looks like a diatonic.

For a more obviously useful diatonic, with 9-limit chords, you can replace the 6:5 minor thirds with 7:6. This naturally leads to a "tone" of 8:7 because 7/6 * 8/7 = 4/3. But that implies a semitone of 28:27, not 15:14. So 28:27 is the septimal semitone that's most likely to be used in a diatonic, and hence a "diatonic semitone". Maybe that's his main point.

> And ham: Again, I don't see why the problem can't just be solved by
> calling 7/6 an "augmented" second, which it is. That's the point of
> "augmented" and "diminished" intervals.

That would make 15:14 a minor second. But has Ham accepted the equivalence between "minor second" and "diatonic semitone"?

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Charles Lucy wrote:
> > It might be easier to think in terms of L-L-s-L-L-L-s for a
diatonic
> > ionian scale.
> >
> > This enables you to escape;-) from 12edo mentality.
>
> It doesn't matter what you call them or whether you have a
> hexachord or the full octave. You can't get JI to work with
> only two step sizes.
>
>
> Graham
>
------------------------------------------------------------------
------------------------------------------------------------------
Chapter IV
THE PIAGUI MUSICAL SCALE
IV.1 THE K AND P SEMITONE FACTORS
An inspection of the cells of the first segment of the progression
shows that part of it is the Pythagorean semitone 256 / 243 as well
as 16/15 and 10/9 of the Just Intonation Scale. So, some cells may
not only be relative frequencies, but also semitone factors of two
consecutive notes of a new musical scale.
The progression may contain two semitone factors that could replace
the tempered T to establish ideal tone frequencies in any octave.
This possibility would generate slight but important changes in
harmony and tone intervals. Let us give the names K and P to the
unknown semitone factors that would rule the new octave.
Both K and P should be progression cells, otherwise, their relative
values with respect to C = 1 would give non-musical results.
It is presumed that the values of both factors to be deduced, when
they lie in the progression, would be placed near the tempered T,
due to the slight imperfection of this scale. Instead of the
relation T12 = 2 of the tempered intonation, the equation Km Pn = 2
is the one that complies with the octave quadrature, when and only
when (m + n) = 12, provided m and n are integer and positive
numbers. Together, K and P should not be higher nor lower than T.
If there is a solution, one of the factors, P, for instance, must be
lower than T and K higher than the tempered factor to establish a
new dodecatonic intonation. No cell equals T, it is a non-musical
number.

Therefore, the following equations can be stated:
(A) K m P n = 2
(B) m + n = 12
The K and P sequence in the octave is a matter to be discussed
should the equations be solved mathematically.
Equations (A) and (B) were supposedly considered in former times.
Hundred of years ago, Ramos de Pareja, Zarlino, W. Holder, Delezenne
and other researchers probably studied this set of equations with
four unknown quantities. Obviously they were never worked out,
since more mathematical information on musical elements was needed.
Now, the required data work with the Natural Progression of Musical
Cells. These equations will be solved in the following pages.
From cell No. 48 to No. 53, having values close to T, we see that
Cells Nos. 48, 49 and 50 are lower than T, just as Cells Nos. 51, 52
and 53 are greater than this parameter. This group is indicated
below.

CELL COMMA F(M,J,U) DECIMAL VALUES
48 M M30 J16 U2 1. 0558784008
49 J M30 J17 U2 1.05707299111
50 J M30 J18 U2 1.05826893295
51 M M31 J18 U2 1.05946387773
52 M M32 J18 U2 1.06066017178
53 M M33 J18 U2 1.06185781663
T = 21/12 = 1.05946309436

T is slightly lower than cell No. 51, therefore P may perhaps be
Cell No. 50 or 49 or 48. Similarly, K may be No. 51 or 52 or 53,
which are slightly greater than T.
If the mentioned equations were solved, then the importance and
validity of cell progression as a link between science and music
would be evident, provided K and P were progression cells. The next
step would be defining the sequence of K and P within the octave in
order to detect the probable theoretical frequencies of the twelve
notes and proceed to further discussion regarding harmony.
Let Pa = (48) Ka = (51)
Pb = (49) Kb = (52)
Pc = (50) Kc = (53)

The next step is to work out equations (A) and (B) by using the
combinations [(Ka)m(Pa)n], [(Ka)m(Pb)n], [(Ka)m(Pc)n], [(Kb)m(Pa)n],
[(Kb)m(Pb)n], [(Kb)m(Pc)n], [(Kc)m(Pa)n], [(Kc)m(Pb)n], [(Kc)m(Pc)
n], since probably one of them may solve the mentioned equations.
The four unknown quantities K, P, m and n may be determined by
applying logarithms. The Neperian logarithm has been chosen, to
work with a sufficient number of digits to reduce errors. By taking
logarithms on both sides of equation (A) we find:
m (ln K) + n (ln P) = ln 2
Therefore, m = [ln 2 – n (ln P)] / ln K
Replacing the above value of m in equation (B), we get:
ln 2 – n (ln P) = 12 (ln K) – n (ln K)
Solving for n:
(C) n = (12 ln K – ln 2) / (ln K – ln P) ln 2
= 0.69314718056

K and P values have to be replaced in equation (C) by cells Nos. 51,
52, 53 and 48, 49, 50 termed Ka, Kb, Kc and Pa, Pb, Pc
respectively. This procedure will set up nine auxiliary equations
similar to equation (C) where Ki and Pi are parameters to yield nine
values for n. When any of these results is an integer number, an
important step in the analysis is accomplished.
The following tabulation of ln Ki , ln Pi , and 12 ln Ki gives the
values that should be replaced in equation (C). Term Ki represents
any of the values Ka, Kb, Kc. Likewise Pi represents Pa or Pb or
Pc.
K i ln K i 12 ln K i
Ka = 1.05946387773 0.05776300444 0.69315605339
Kb = 1.06066017178 0.05889151782 0.70669821393
Kc = 1.06185781663 0.06002003121 0.72024037452

ln Pa = ln 1.0558784008 = 0.05437302789
ln Pb = ln 1.0570729911 = 0.05550375948
ln Pc = ln 1.05826893295 = 0.05663449107

By replacing the known values in (C), nine solutions for n are
obtained:
(1) n1 = (12 ln Ka – ln 2) / (ln Ka – ln Pa) = 0.00261733963
(2) n2 = (12 ln Ka – ln 2) / (ln Ka – ln Pb) = 0.00392729436
(3) n3 = (12 ln Ka – ln 2) / (ln Ka – ln Pc) = 0.00786230826
(4) n4 = (12 ln Kb – ln 2) / (ln Kb – ln Pa) = 2.99901814321
(5) n5 = (12 ln Kb – ln 2) / (ln Kb – ln Pb) = 4
(6) n6 = (12 ln Kb – ln 2) / (ln Kb – ln Pc) = 6.0039311807
(7) n7 = (12 ln Kc – ln 2) / (ln Kc – ln Pa) = 4.79780024071
(8) n8 = (12 ln Kc – ln 2) / (ln Kc – ln Pb) = 5.99901766761
(9) n9 = (12 ln Kc – ln 2) / (ln Kc – ln Pc) = 8.00262079304

The integer number n5 = 4 leads to important conclusions.
If n5 = 4, then m = (12 – 4) = 8, so that the new octave is obtained
by four Pb semitone factors and eight Kb. Auxiliary equation (5)
shows that Kb and Pb together with m = 8 and n = 4 solve equation
(A).
The semitones K and P of a new musical octave have been sought and
worked out:
K = Kb = (Cell No. 52) = (9/8)1/2 = 1.06066017178
P = Pb = (Cell No. 49) = (8/9) ( 21/4 ) = 1.0570729911
K8 P4 = [ (9/8)1/2 ]8 [ (8/9)( 21/4 ) ]4 = 2
Contrasted with T = 21/12 = 1.05946309436, the K and P values show
slight discrepancies.
These remarkable results confirm that the Natural Progression of
Musical Cells establishes the scientific base of the art of music.
However, until it is proved that K and P resolve the harmony
problem, they cannot be accepted as providing the desired solution.
Their validity will depend on chord evaluations.
Whereas Cells Nos. 52 and 49 could be the semitone factors of the
best musical scale, the tone relative frequencies with respect to
note C will depend on their sequence within the octave. The
sequences need to be analyzed, since tone frequencies and chords
will depend on the K and P arrangements within the octave.
IV. 2 THE K AND P SEQUENCE IN THE NEW OCTAVE
The eight K and four P factors comply with proper distribution
within the octave if the same factor sequence is maintained from
note C to B. When an arrangement of twelve semitone factors is
attempted, the idea of four identical groups arises, since there are
four P factors working within the octave.
Elements of each group: K, K, P.
Number of groups = 4
Semitone arrangements: KKP, KPK and PKK.
Following are the three types of Piagui octaves:
I) KKP KKP KKP KKP = 2
II) KPK KPK KPK KPK = 2
III) PKK PKK PKK PKK = 2
Four groups of semitone factors comprised by KKP, KPK and
PKK rule the relations of tone frequencies in the new octaves of
Piagui I, Piagui II and Piagui III scales respectively. However,
the new octaves work simultaneously. In fact, the set
KKPKKPKKPKKPKKP shows that Piagui II starts from the second K and
Piagui III from the first P.
TABLE X - FIRST SEQUENCE OR PIAGUI I
NOTE SF CELL No. RELATIVE FREQUENCY
FREQUENCY (Hz)
C -- 1 1 261.6255 *
C# K 52 1.06066017178 = K 277.4958
D K 104 1.125 = K2
294.3287 **
Eb P 153 1.189207115 = K2P
311.127 *
E K 205 1.26134462288 = K3P 330

F K 257 1.33785800438 = K4P 350.0178
F# P 306 1.41421356237 = K4P2 369.9944 *
G K 358 1.5 = K5P2 392.4383 **
Ab K 410 1.59099025767 = K6P2 416.2437
A P 459 1.68179283051 = K6P3
440 *
Bb K 511 1.7838106725 = K7P3 466.6905
B K 563 1.89201693432 = K8P3 495
2C P 612 2 = K8P4 523.2511 *

* Tempered and Piagui I frequencies
** Pythagorean frequencies
SF = Semitone Factor

Eight tones of any of the three new octaves differ from the tempered
ones; the other four have identical values (C, Eb, F#, A). Eight of
the Piagui I frequencies are higher than their corresponding values
in the tempered octave. Four of the Piagui II are higher while the
other four are lower and eight Piagui III pitches are even lower.
Each sequence begins with keynote C making successive groups of KKP,
KPK and PKK. It follows that if keynote is changed to C# whose
frequency is 277.4958 Hz, the chromatic scale is made up by sequence
KPK. Likewise, when it is changed to D whose frequency is 294.3287
Hz, the PKK sequence works in this new chromatic scale. Thus,
chromatic scales of the three sequences work simultaneously.
TABLE XI - SECOND SEQUENCE OR PIAGUI II
NOTE SF CELL No. RELATIVE FREQUENCY
FREQUENCY (Hz)
C 1 = 1 261.6255 *
C# K 52 1.06066017178 = K 277.4958
D P 101 1.12119522034 = KP
293.3333
Eb K 153 1.189207115 = K2P
311.127 *
E K 205 1.26134462288 = K3P 330
F P 254 1.333333... = K3P2 348.834 **
F# K 306 1.41421356237 = K4P2 369.9944 *
G K 358 1.5 = K5P2 392.4383 **
Ab P 407 1.58560948667 = K5P3 414.836
A K 459 1.68179283051 = K6P3 440 *
Bb K 511 1.7838106725 = K7P3 466.6905
B P 560 1.88561808316 = K7P4 493.3259
2C K 612 2 = K8P4 523.2511 *

* Tempered and Piagui II frequencies
** Pythagorean frequencies
SF = Semitone Factor

The final Piagui sequence that could be applied in the manufacture
of musical instruments would depend on the opinion of musicians and
analysts, while, the three sequences will be considered in the
analyses. However, the relative D and G frequencies of Piagui I
coincide with 9/8 and 3/2 transcendent frequencies of Pythagoras
scale, a subjective motive when considering the Piagui I scale, as
the basic solution should it satisfy chord evaluations.
TABLE XII - THIRD SEQUENCE OR PIAGUI III
NOTE SF CELL No. RELATIVE FREQUENCY
FREQUENCY (Hz)
C 1 1 261.6255 *
C# P 49 1.05707299111 = P 276.5573
D K 101 1.12119522034 = KP
293.3333
Eb K 153 1.189207115 = K2P 311.127 *
E P 202 1.25707872211 = K2P2 328.8839
F K 254 1.333333... = K3P2 348.8341 **
F# K 306 1.41421356237 = K4P2 369.9944 *
G P 355 1.49492696045 = K4P3 391.1111...
Ab K 407 1.58560948667 = K5P3 414.836
A K 459 1.68179283051 = K6P3 440 *
Bb P 508 1.777777... = K6P4 465.1121
B K 560 1.88561808316 = K7P4 493.3259
2C K 612 2 = K8P4 523.2511 *

* Tempered and Piagui III frequencies
** Pythagorean frequency
SF = Semitone Factor

Three variants of the Piagui scale were unexpected. Instead of only
twelve tones of the octave and according to Table XIII, twenty tones
are provided by the three types of octaves. Their frequency values
(Hz) depend on the KKP, KPK or PKK sequence. It was hoped that
there might be bases to set aside two sequences, and it was decided
that this matter should be discussed as soon as it was proved that
at least one of the options provided better chords. Later on, we
shall see that the three option features enable us to make perfect
chords whether some frequencies in the three types of octaves differ
or not.
TABLE XIII - WORKING FREQUENCIES OF PIAGUI SCALES
NOTE RELATIVE FREQUENCY CELL FREQUENCY SCALE
C = 1 1 261.6255 Hz I – II – III
C# P = 1.0570729911 49 276.5573 III
C# K = 1.06066017178 52 277.4958 I –
II
D KP = 1.12119522036 101 293.3333... II – III
D K2 = 1.125 104 294.3287 I
Eb K2P = 1.189207115 153 311.127 I – II – III
E K2P2 = 1.25707872212 202 328.8839 III
E K3P = 1.26134462288 205 330 I – II
F K3P2 = 1.333333... 254 348.8341 II – III
F K4P = 1.33785800438 257 350.0178 I
F# K4P2 = 1.41421356237 306 369.9944 I – II – III
G K4P3 = 1.49492696047 355 391.1111... III
G K5P2 = 1.5 358 392.4383 I – II
Ab K5P3 = 1.58560948668 407 414.8359 II – III
Ab K6P2 = 1.59099025767 410 416.2437 I
A K6P3 = 1.68179283053 459 440 I – II – III
Bb K6P4 = 1.777777... 508 465.1121 III
Bb K7P3 = 1.7838106725 511 466.6904 I – II
B K7P4 = 1.88561808317 560 493.3258 II – III
B K8P3 = 1.89201693434 563 495 I
2C K8P4 = 2 612 523.2511 I – II – III

As the object of the analysis, is to obtain only twelve tone
frequencies per octave instead of the above twenty ones, the chord
evaluations are expected to give more information regarding harmony,
in order to clear up this matter.
Note that the stressed data are common figures of Piagui scales,
they coincide with the Tempered values.

TABLE XIV - THE PIAGUI AND TEMPERED FREQUENCIES (Hz)
PIAGUI I PIAGUI II PIAGUI III TEMPERED
(KKP) (KPK) (PKK)
C 261.6255 261.6255 261.6255 261.6255
C# 277.4958 277.4958 276.5573 277.1826
D 294.3287 293.3333 293.3333 293.6648
Eb 311.127 311.127 311.127 311.127
E 330 330 328.8839 329.6276
F 350.0178 348.8341 348.8341 349.2282
F# 369.9944 369.9944 369.9944 369.9944
G 392.4383 392.4383 391.1111 391.9954
Ab 416.2437 414.836 414.836 415.3047
A 440 440 440 440
Bb 466.6905 466.6905 465.1121 466.1637
B 495 493.3259 493.3259 493.8833
2C 523.2511 523.2511 523.2511 523.2511
IV. 3 MOST CONSONANT INTERVALS AND THE NEW INTONATION
Historians and analysts affirm that the most consonant intervals are
the perfect fifth (ratio 3/2) and the perfect fourth (ratio 4/3).
Following is a more detailed explanation of these elementary
concepts.
When the Pythagoras sonometer string is seen as having five equal
lengths (5/5), and its slider is placed at 2/5 of its whole length,
the frequency produced by this shorter length is [(3/5) / (2/5)] =
(3/2) = 1.5 times the frequency obtained from the longer one. By
plucking both lengths simultaneously or consecutively, the frequency
ratio is evaluated by our brain that detects the perfect fifth. The
longer part 3/5 gives the basic frequency represented by the unit
figure of Pythagoras note C while the shorter one 2/5 corresponds to
note G.
Similarly, when the string is seen as having seven equal lengths
(7/7) and the slider is placed at 3/7 of its whole length, the
frequency produced by this shorter length is [(4/7) / (3/7)] = (4/3)
= 1.33333... times the frequency obtained from the longer one. In
this case, the perfect fourth is detected by plucking both lengths
simultaneously or consecutively. The longer part 4/7 is now the
basic length giving the unit figure of note C while the shorter one
3/7 corresponds to note F.
In this connection, it is interesting to note that Bartholomew, in
his "Acoustics of Music", when referring to perfect fifths and
fourths, states that "Nearly all primitive scales include these
intervals as primary division points of the octave".
The scientific basis of consonance and concordance and the needed
rules of harmony are to be found in the tone frequency relationships
of data given in Tables X, XI and XII.
Perfect consonant intervals are set out in Table X:
(G/C) = (Ab /C#) = (Bb /Eb) = (B/E) = K5P2 = (3/2) = 1.5 = Perfect
fifth
(F# / C#)=(G/D)=(A/E)=(Bb /F) = K3P2 =(4/3) =1.33333... = Perfect
fourth
Identical relations are obtained from Table XI of Piagui II:
(G/C) = (A/D) = (Bb /Eb) = (2C/F) = K5P2 = (3/2) = 1.5 = Perfect
fifth
(F/C)=(F# / C#)=(Ab /Eb) = (A/E) = (B/F#) = (2C/G) = K3P2 =
1.333333... = Perfect fourth.
The same perfect intervals are found in Table XII of Piagui III.
In the following diagrams, each tone contained in four Piagui
octaves is marked by a point on a square perimeter. All tones are
linked by perfect fifths and perfect fourths in a cyclical way to
continue on more octaves. The origins of these characteristics are
based on commas M, J and U that led to the Natural Progression of
Musical Cells from which the K and P semitone factors were deduced.
These are the basic features that will create the best chord
expressions as will be discussed in Chapters VII and VIII.

PERFECT FIFTHS AND FOURTHS IN PIAGUI SCALES

Perfect fifth
Perfect fourth

Probably, man's common sense expected perfect links in the octaves
over the centuries. The precise and suitable values of K and P
produce these remarkable results.
The combined sounds produced by adding C + G or C + F, where G and F
are the perfect fifth and perfect fourth respectively with respect
to note C, are detected not only by our brain but also through
computer-printing responses. Thus, the mentioned perfect intervals
are the basic contributions, classifying the Piagui scales as the
best way to perfecting harmony. This technical procedure is applied
on triad evaluations where three tone frequencies are added as
discussed in Chapter VIII.

Chapter V
MUSICAL INSTRUMENTS RULED
BY THE PIAGUI SCALE
Since research began, the purpose was to determine a set of eleven
relative frequencies with respect to C = 1 to generate the new
octave and apply the new set to musical instruments instead of the
tempered scale. This object was achieved from the mathematical
point of view. However, chord evaluations will analyze and provide
more information on this matter.
V. 1 ELECTRONIC ORGAN
The tempered tones of electronic organs are obtained by using a
stable master oscillator to feed frequency dividers to get stable
top octave frequencies, the lower ones being derived from successive
divisions by two. For example, the old integrated circuit ECG 2043
contained the proper dividers according to the master oscillator
frequency. Together, they produced the top octave, whose tone
frequencies have shown considerable discrepancies regarding
theoretical tempered ones.
The manufacturers' criteria until almost the end of the last century
were 100% hardware. Electronic organ keyboards use software at the
present time. A few companies in the world use their own technology
to supply the world market. Considering that keyboard manufacturers
do not frequently change their designs and software, the Piagui
keyboard could be manufactured by using past technology.

Table XV shows the ideal fundamental frequencies of the new
keyboards. It does not consider the imperfection of man's ear,
which demands a slight increase in the higher tone frequencies and a
slight decrease in the lower ones. From about key No. 16 up to No.
68 ear efficiency is normal. The upper and lower octaves are
listened slightly out of tune, except for those organs that could be
designed and manufactured with additional electronic means to
compensate for human imperfection.
Eight frequencies of any Piagui octave differ from the tempered
ones. Cello and double bass players will be familiar with the right
emissions of the new tone frequencies as soon as the Piagui tones
and harmonies are appreciated.

TABLE XV
TONE FREQUENCIES (Hz) OF THE PIAGUI KEYBOARDS
No. NOTE PIAGUI I PIAGUI II PIAGUI III
1 ¼ A 27.5 27.5 27.5
2 ¼ Bb x K = 29.1681 x K = 29.1681 x P = 29.0695
3 ¼ B x K = 30.9375 x P = 30.8329 x K = 30.8329
4 ½ C x P = 32.7032 x K = 32.7032 x K = 32.7032
5 ½ C# x K = 34.687 x K = 34.687 x P = 34.5696
6 ½ D x K = 36.7911 x P = 36.6666 x K = 36.6666
7 ½ Eb x P = 38.8909 x K = 38.8909 x K = 38.8909
8 ½ E 41.25 41.25 41.1105
9 ½ F 43.7522 43.6043 43.6043
10 ½ F# 46.2493 46.2493 46.2493
11 ½ G 49.0548 49.0548 48.8888
12 ½ Ab 52.0304 51.8545 51.8545
13 ½ A 55 55 55
14 ½ Bb 58.3363 58.3363 58.139
15 ½ B 61.875 61.6657 61.6657

No. NOTE PIAGUI I PIAGUI II PIAGUI III
16 C 65.4064 65.4064 65.4064
17 C# 69.3739 69.3739 69.1393
18 D 73.5822 73.3333 73.3333
19 Eb 77.7817 77.7817 77.7817
20 E 82.5 82.5 82.221
21 F 87.5045 87.2085 87.2085
22 F# 92.4986 92.4986 92.4986
23 G 98.1096 98.1096 97.7777
24 Ab 104.0609 103.709 103.709
25 A 110 110 110
26 Bb 116.6726 116.6726 116.278
27 B 123.75 123.3315 123.3315
28 2C 130.8128 130.8128 130.8128
29 2C# 138.7479 138.7479 138.2786
30 2D 147.1644 146.6666 146.6666
31 2Eb 155.5635 155.5635 155.5635
32 2E 165 165 164.442
33 2F 175.0089 174.417 174.417
34 2F# 184.9972 184.9972 184.9972
35 2G 196.2192 196.2192 195.5555
36 2Ab 208.1219 207.418 207.418
37 2A 220 220 220
38 2Bb 233.3452 233.3452 232.556
39 2B 247.5 246.6629 246.6629
40 4C 261.6255 261.6255 261.6255
41 4C# 277.4958 277.4958 276.5573
42 4D 294.3288 293.3333 293.3333
43 4Eb 311.127 311.127 311.127
44 4E 330 330 328.8839
45 4F 350.0178 348.8341 348.8341

No. NOTE PIAGUI I PIAGUI II PIAGUI III

46 4F# 369.9944 369.9944 369.9944
47 4G 392.4383 392.4383 391.1111
48 4Ab 416.2437 414.836 414.836
49 4A 440 440 440
50 4Bb 466.6905 466.6905 465.1121
51 4B 495 493.3259 493.3259
52 8C 523.2511 523.2511 523.2511
53 8C# 554.9916 554.9916 553.1146
54 8D 588.6575 586.6666 586.6666
55 8Eb 622.254 622.254 622.254
56 8E 660 660 657.7679
57 8F 700.0357 697.6682 697.6682
58 8F# 739.9888 739.9888 739.9888
59 8G 784.8767 784.8767 782.2222
60 8Ab 832.4874 829.672 829.672
61 8A 880 880 880
62 8Bb 933.381 933.381 930.2242
63 8B 990 986.6518 986.6518
64 16 C 1046.5023 1046.5023 1046.5023
65 16 C# 1109.9833 1109.9833 1106.2293
66 16 D 1177.315 1173.3333 1173.3333
67 16 Eb 1244.5079 1244.5079 1244.5079
68 16 E 1320 1320 1315.5357
69 16 F 1400.0714 1395.3363 1395.3363
70 16 F# 1479.9777 1479.9777 1479.9777
71 16 G 1569.7534 1569.7534 1564.4444
72 16 Ab 1664.9749 1659.3439 1659.3439
73 16 A 1760 1760 1760
74 16 Bb 1866.7619 1866.7619 1860.4485
75 16 B 1980 1973.3036 1973.3036

No. NOTE PIAGUI I PIAGUI II PIAGUI III
76 32 C 2093.0045 2093.0045 2093.0045
77 32 C# 2219.9665 2219.9665 2212.4585
78 32 D 2354.63 2346.6666 2346.6666
79 32 Eb 2489.0159 2489.0159 2489.0159
80 32 E 2640 2640 2631.0714
81 32 F 2800.1428 2790.6727 2790.6727
82 32 F# 2959.9554 2959.9554 2959.9554
83 32 G 3139.5068 3139.5068 3128.8888
84 32 Ab 3329.9498 3318.6878 3318.6878
85 32 A 3520 3520 3520
86 32 Bb 3733.5238 3733.5238 3720.8969
87 32 B 3960 3946.6072 3946.6072
88 32 C 4186.009 4186.009 4186.009

K = (9/8)1/2 = 1.06066017178
P = [(8/9) 21/4] = 1.05707299111

Important is the maximum permissible discrepancy of Piagui tone
frequencies regarding theoretical values. The desirable solution
would be to achieve the figures given in Table XV. However, since
these frequencies demand an accurate frequency output by the master
oscillator as well as exact divisors to provide a flawless top
octave, it is impossible to attain such ideal keyboard frequencies.
Quartz oscillators within ± 25 parts per million basic accuracy can
be used to bring tone frequencies closer to the ideal, so the best
set of figures will depend on the quartz oscillator frequency and
the frequency dividers (divisors).
The divisors, which are integers, impose some frequency errors,
depending on the design of the top octave stage.
Let us determine the tentative figures of maximum and minimum
fluctuations permissible in Piagui I tone frequencies.
As the least distinguishable consonance is M = 1.00112915, the
frequency discrepancies must be sufficiently lower than M, in order
to produce the expected tone frequencies.
If any pitch is M times higher than ideal pitch, its error
percentage is given by:
[(1.00112915 – 1) 100] b  + 0.113%
Similarly, when it is M times lower, the error is – 0.113%
Since an error of 0% is unattainable, a tentative maximum figure
could be about ± (0.113 / 5) b  ± 0.02%.
The ideal frequency of note G in Piagui I intonation is 392.4383 Hz
and 0.02% of this value is 0.08 Hz. Therefore, the minimum
permissible value would be (392.4383 – 0.08) = 392.3583 and its
maximum permissible frequency would be (392.4383 + 0.08) = 392.5183
Hz.
The following diagram shows a tentative design of the Piagui I Top
Octave. The accuracy of the frequency output of the quartz
oscillator is ± 25 parts per million. That is to say maximum error
of ± 0.0025%.
Note that frequency ratios 3/2 and 4/3 corresponding to perfect
fifths and perfect fourths respectively have been used in the Top
Octave design. Ratios (B/E) = (Bb /Eb) = (Ab /C#) = (3/2) are
perfect fifths and (2C/G) = (Bb /F), perfect fourths.
The ratio (Perfect fifth / Perfect fourth) = (9/8) was also used.
If musicians and analysts accept one of the Piagui intonations, the
electronic industry could provide better technical criteria to set
the top octaves.

Mario Pizarro
ELECTRONIC ENGINEER
(PIAGUISCALE)

A 7/6 could very well be a "tone." What is a tone? Are we saying that
a tone is something more or less around the 200 cents range? In that
case, scales like A B C D E F G# A would be supposedly "improper"
because of the F-G# augmented second.

I have no problem with trying to "reverse-engineer" the diatonic
numbering systems we have now, which are rooted in meantone tradition,
and expand them to apply to all of JI. 7/6 is just 7/6, and we can
call it an "augmented tone" or a "subminor third" or a "diminished
fourth" if we want. These are just names, but that interval obviously
exists in that scale, so who cares what to call it?

-Mike

On Tue, Jun 3, 2008 at 8:54 AM, Graham Breed <gbreed@...> wrote:
> Mike Battaglia wrote:
>> Graham: I think he means that having G-A# be a septimal subminor third
>> somehow disqualifies it as a tone. He's saying that for the major
>> scale, you have ttsttts so that the "t"s represent something in the
>> category of a "tone," be it a major or minor tone, and the "s"'s
>> represent something in the category of a semitone.
>
> I don't know what he means and as he isn't generally clear
> about what kind of "diatonic" he's talking about I don't
> like to second guess him. But I least I agree that a 7:6
> isn't a tone if that's the point.
>
> So fine, if you want something like a tone when the diatonic
> semitone's 15:14 then you can use 28:25. This will still
> give minor thirds of 6:5. I don't know why you'd want to do
> that but what you'd get looks like a diatonic.
>
> For a more obviously useful diatonic, with 9-limit chords,
> you can replace the 6:5 minor thirds with 7:6. This
> naturally leads to a "tone" of 8:7 because 7/6 * 8/7 = 4/3.
> But that implies a semitone of 28:27, not 15:14. So 28:27
> is the septimal semitone that's most likely to be used in a
> diatonic, and hence a "diatonic semitone". Maybe that's his
> main point.
>
>> And ham: Again, I don't see why the problem can't just be solved by
>> calling 7/6 an "augmented" second, which it is. That's the point of
>> "augmented" and "diminished" intervals.
>
> That would make 15:14 a minor second. But has Ham accepted
> the equivalence between "minor second" and "diatonic semitone"?
>
> Graham
>
>