Yahoo Tuning Groups Ultimate Backup makemicromusic Sad symmetric septimal minor tune

--- In MakeMicroMusic@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:
> Hi Paul,
>
> Thanks! A useful find :-). This scale is
> very rich in harmonic possibilities isn't it,
> what with the 7/6s as well and the 9/7 and its
> reflection (which you can also pick out in the figure
> fairly easily).

It, and your description below, reminds me a lot of a 'diatonic'
scale in 22-equal formed from a chain of (wide) fifths --
particularly in the aeolian mode, but dorian too. This is a highly
accessible scale, and can be heard in Randy Winchester's music in 22-
equal:

http://web.mit.edu/randy/www/Music/comets.html

Although my excursions in this scale seem to have disappeared from
the internet, Margo flattered me on it once :) The scale has three
approximate 4:6:7:9 chords, which sound great to me, and in JI this
would require extra pitches.

> In fact come to think of it, it is almost like the twelve
> equal diatonic where nearly every
> diad apart from the tritone and semitone (+ tone??)
> is regarded as consonant - but here as pure ratios
> rather than approximations. Only the tiny semitone step

49:48

> isn't a small number ratio +
> similarly, the step from one of the notes of each
> pair to the other note of the other pair of close pitches.

49:32, 49:36, 64:49, 72:49

> The wide interval there is also pretty acceptable for a consonance
> depending on the timbre I expect.

Maybe, but even the 21:16s and 32:21s in the scale don't carry
that "simple ratio sound", I find . . . you'd have to fill in a lot
of the rest of the harmonic series to get the interval to "sing", in
my experience . . . The 22-equal version makes 21;16 and 32;21 the
same as 4;3s and 3;2s, and a lot smoother.

> I'll add a link to his diagram to my page. Also I can call it a
> septimal Dorian, following his example, that will do.

I know some people would expect septimal Dorian to sound more like it
would in Michael Harrison's music,

(there appear to be some free samples here:
http://www.musical-genre.com/michael_harrison.htm )

more like
1/1 9/8 7/6 4/3 3/2 27/16 7/4 or
1/1 9/8 7/6 21/16 3/2 27/16 7/4 or
1/1 8/7 32/27 4/3 32/21 12/7 16/9 or
1/1 8/7 32/27 4/3 3/2 12/7 16/9

The 22-equal septimal Dorian (4 1 4 4 4 1 4) approximates all these
scales, taming the 32:21 wolves they each contain. I find the
corresponding aeolian & phrygian modes affecting as well, and
modulation can be positively spine-chilling!

Anyway, nice tune!!!!

🔗Robert Walker <robertwalker@...>

Hi Kraig,

> I think Al Farabi played allot with dividing the tetrachord into 3
> superparticular ratios, one of which is repeated twice. He also played with
> the rotations also. see page 4 of http://www.anaphoria.com/xen10pur.PDF

Rightio - I see he has it listed there as one of the scales.

I've just looked up in Google to find out about him, rather a nice story here:

http://www.dovesong.com/positive_music/Turkey.asp
:-)

Thanks.

> I find it very useful harmonically . How about the 6-7-8-9 as only one

Yes indeed.

1/1 7/6 4/3 3/2

I've just done an improvisation in the scale exploring some of these chords:
http://www.tunesmithy.netfirms.com/tunes/improvisations.htm#Al_Farabis_very_septimal_Dorian

BTW I played it with the latency of the FM7 set to 400 ms which may be of interest
after the discussion on latency in the main tuning list. Don't seem to have
any problems at all playing that far ahead of time as you can hear :-).
So I think I agree with whoever it was who said that variation
in latency is more important than the amount of it, for me anyway.

Thanks,

Robert

🔗Margo Schulter <mschulter@...>

Dear Robert and everyone,

Thank you for that septimal minor or more specifically Dorian scale,
which I've tried and find gives a special rationale for a JI tuning
based on ratios of 2-3-7 which, unlike other just systems or
approximations I use, makes all the ratios of your scale available,
including those 49:48 steps.

This tuning system has a set of 14 notes, with your scale easily
played on the upper keyboard in my mapping. Here I'll use numbers to
show modifications of a Pythagorean pitch (based on a chain or pure
fifths or fourths) by one or more septimal commas of 64:63 (~27.26
cents):

E1 F#2 G0 A1 B1 C#2 D0 E1
1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1
0 231 267 498 702 933 969 1200

Interestingly, and very pleasantly, the small 49:48 steps (~35.70
cents) sound to me like convincing although "different" semitones,
possibly with the categorical distinctions between large major second
or sixth and small minor third or seventh reinforcing this effect
above a drone.

As has been commented, the special charm of the 49:48 steps
distinguishes this kind of Dorian from another kind which I also love
and indeed favor in just and tempered systems, for example:

D1 E1 F0 G1 A1 B1 C0 D1
1/1 9/8 7/6 4/3 3/2 27/16 7/4 2/1
0 204 267 498 702 906 969 1200

This has the typical 28:27 semitones or thirdtones (since they're
equal to about a third of a 9:8 tone) of Archytas, around 62.96
cents.

I'd like to compose something using Robert's scale, possibly with some
extra tones now and then to facilitate certain progressions, such as
this one involving the great 16:21:24:28 sonority: in my keyboard
tuning discussed above, A1-D0-E1-G0:

G0 F#1
E1 F#1
D0 B1
A1 B1

This calls for the note F#1 to form a pure fifth in the resolving
sonority, in distinction to the "native" F#2 of this scale.

Another remark: Robert, your scale also led me to note this very nice
Dorian scale in 25-tET, which I might call the most "usual" version of
Dorian, and in my keyboard mapping is played like this:

C D Eb F G A Bb C
0 240 288 480 720 912 960 1200

Since 25-tET isn't a regular diatonic tuning (to borrow Easley
Blackwood's term) where you can take five identically sized "whole
tones" and two identically sized "semitones" to make a 2:1 octave,
I'll refer to my "usual quasi-diatonic semitone" of 48 cents. This
step is somewhere between the 49:48 (~36 cents) of Robert's scale and
the 28:27 (~63 cents) of a more familiar (to me) septimal Dorian.

One analysis might explain this scale as simply the 5-tET steps plus
my customary values for the "usual" minor third and major sixth at 288
cents and 912 cents. (In 25-tET, what I call "alternatives thirds" are
also available at 336 cents and 384 cents, and the 240-cent major
second can also serve as a "quasi-third," so that one can get lots of
different nuances).

Again, Robert, thank you for introducing me to such a charming scale:
fascinated as I am with the 48-cent semitones of 25-tET, I'm impressed
by those 36-cent steps which nicely define the difference between a
septimal major second or sixth and a minor third or seventh.

Most appreciatively,

Margo
mschulter@...

🔗Paul Erlich <perlich@...>

Oh, and Aaron, if you were reading it, what I wrote about 22-equal
below also applies to your tuning . . . the chain of (36:7)^(1/4)
s . . . have you tried aeolian and dorian scales with it, or just
pentatonics so far?

--- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In MakeMicroMusic@yahoogroups.com, "Robert Walker"
> <robertwalker@n...> wrote:
> > Hi Paul,
> >
> > Thanks! A useful find :-). This scale is
> > very rich in harmonic possibilities isn't it,
> > what with the 7/6s as well and the 9/7 and its
> > reflection (which you can also pick out in the figure
> > fairly easily).
>
> It, and your description below, reminds me a lot of a 'diatonic'
> scale in 22-equal formed from a chain of (wide) fifths --
> particularly in the aeolian mode, but dorian too. This is a highly
> accessible scale, and can be heard in Randy Winchester's music in
22-
> equal:
>
> http://web.mit.edu/randy/www/Music/comets.html
>
> Although my excursions in this scale seem to have disappeared from
> the internet, Margo flattered me on it once :) The scale has three
> approximate 4:6:7:9 chords, which sound great to me, and in JI this
> would require extra pitches.
>
> > In fact come to think of it, it is almost like the twelve
> > equal diatonic where nearly every
> > diad apart from the tritone and semitone (+ tone??)
> > is regarded as consonant - but here as pure ratios
> > rather than approximations. Only the tiny semitone step
>
> 49:48
>
> > isn't a small number ratio +
> > similarly, the step from one of the notes of each
> > pair to the other note of the other pair of close pitches.
>
> 49:32, 49:36, 64:49, 72:49
>
> > The wide interval there is also pretty acceptable for a consonance
> > depending on the timbre I expect.
>
> Maybe, but even the 21:16s and 32:21s in the scale don't carry
> that "simple ratio sound", I find . . . you'd have to fill in a lot
> of the rest of the harmonic series to get the interval to "sing",
in
> my experience . . . The 22-equal version makes 21;16 and 32;21 the
> same as 4;3s and 3;2s, and a lot smoother.
>
> > I'll add a link to his diagram to my page. Also I can call it a
> > septimal Dorian, following his example, that will do.
>
> I know some people would expect septimal Dorian to sound more like
it
> would in Michael Harrison's music,
>
> (there appear to be some free samples here:
> http://www.musical-genre.com/michael_harrison.htm )
>
> more like
> 1/1 9/8 7/6 4/3 3/2 27/16 7/4 or
> 1/1 9/8 7/6 21/16 3/2 27/16 7/4 or
> 1/1 8/7 32/27 4/3 32/21 12/7 16/9 or
> 1/1 8/7 32/27 4/3 3/2 12/7 16/9
>
> The 22-equal septimal Dorian (4 1 4 4 4 1 4) approximates all these
> scales, taming the 32:21 wolves they each contain. I find the
> corresponding aeolian & phrygian modes affecting as well, and
> modulation can be positively spine-chilling!
>
> Anyway, nice tune!!!!

🔗Robert Walker <robertwalker@...>

Dear Margo, and everyone,

Thanks for your example of a pythagorean scale with
septimal commas to achieve Al Farabi's very septimal
Dorian scale as one of the modes.

I'm interested to hear what you make of it when you
compose in it extended to include your resolution
to a pure fifth from from the 16:21:24:28 sonority.

I'll be doing some more composing in it too,
and will also be interested to compose in the
other Dorian scales in 25 (and also 22) equal, and
also try comparing it with the
1/1 9/8 7/6 4/3 3/2 27/16 7/4 2/1
type septimal Dorian.

Yes it is a most charming scale. I agree that the
semitones are convincing small steps especially
when the harmonic context makes them distinctive
in their role. I have a fondness for small
semitones too.

I had the pleasure of finding this scale for myself
as a new one (to me), though of course it is now
about a thousand years old at least, with Al Farabi in
the tenth century AD as the original discover
(unless someone else preceded him).
I wonder how many people have come across it
in various ways and it explored it in all those
centuries.

I've also been wondering about how you might
extend it to a scale of e.g. 12 notes since
that is convenient on a conventional keyboard
single manual.

BTW I have tried out your double manual style
of playing from time to time. What I found is
that the keyboards I tried are rather deep, so you
can't stretch a single hand from one of the
manuals to the other. But I know there are
keyboards that are less deep than these.

Do you play them two handed with one hand for each manual
(in that case apart from the extra range you get
with two keyboards, and the shorter distance to
move the hands, one could also play on a keyboard divided
left and right with a comma difference in tuning
or whatever one wants between the two halves
- as I did in my improvisation in your
sesquisexta scale).

Or do you have keyboards shallow enough so that
you can play on two at once with a single
hand? Come to think of it, one only needs
one shallow keyboard, for the one that
sits on top.

BTW I have added a new option to FTS
to make any formula you like into
a wave to use e.g. for example
chords in harmonic timbres
- but also they can be made into
sound fonts e.g. for the SB Live! or
for Giga. After I have finished the
upload I'll try making a few fonts
for both of those, and make
them available. (FTS can't play the
waves directly at present, so has to make them
into wave files then play those,
or sound fonts for a wavetable midi synth).

It includes Bessel functions
which make an interesting rounded sawtooth
and I tried x*sin(1/x) and variants
such as x^3*sin(1/x^3) which have
infinitely many waves getting smaller
and smaller as you approach the origin.

It also lets you build the functions up into
a harmonic series and you can quantise the
series to the current scale so I can experiment
with Setherising, I was surprised to find that
sine waves didn't work so well there
as more complex functions - you got beats
within a single note if you use
sine waves for e.g. equal tempered partials
in a harmonic series but when you replaced
them with more complex curves then the beats
went away. So there is clearly a bit of an art
to this. But if I can make a nice Setherised
timbre or two that could be fun to explore.

(any FTS users - this isn't ready yet but
the new upload is probably later today
with some luck, first upload for a number of months
now).

Thanks,

Robet

🔗Paul Erlich <perlich@...>

--- In MakeMicroMusic@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:

> BTW I have added a new option to FTS
> to make any formula you like into
> a wave to use e.g. for example
> chords in harmonic timbres
> - but also they can be made into
> sound fonts e.g. for the SB Live! or
> for Giga. After I have finished the
> upload I'll try making a few fonts
> for both of those, and make
> them available. (FTS can't play the
> waves directly at present, so has to make them
> into wave files then play those,
> or sound fonts for a wavetable midi synth).
>
> It includes Bessel functions
> which make an interesting rounded sawtooth
> and I tried x*sin(1/x) and variants
> such as x^3*sin(1/x^3) which have
> infinitely many waves getting smaller
> and smaller as you approach the origin.
>
> It also lets you build the functions up into
> a harmonic series and you can quantise the
> series to the current scale so I can experiment
> with Setherising, I was surprised to find that
> sine waves didn't work so well there
> as more complex functions - you got beats
> within a single note if you use
> sine waves for e.g. equal tempered partials
> in a harmonic series but when you replaced
> them with more complex curves then the beats
> went away. So there is clearly a bit of an art
> to this.

Hi Robert,

This is one of the most fascinating posts I've seen in a long time.

Would you mind getting a bit deeper on all this on one of the more
analytical lists, such as harmonic_entropy, tuning, or tuning-math?

I'd enjoy that, anyway.

Best,
Paul

🔗Robert Walker <robertwalker@...>

Hi Paul

(continued from MakeMicroMusic, see end of:
/makemicromusic/topicId_6638.html#6700
)

> > BTW I have added a new option to FTS
> > to make any formula you like into
> > a wave to use e.g. for example
> > chords in harmonic timbres
> > - but also they can be made into
> > sound fonts e.g. for the SB Live! or
> > for Giga. After I have finished the
> > upload I'll try making a few fonts
> > for both of those, and make
> > them available. (FTS can't play the
> > waves directly at present, so has to make them
> > into wave files then play those,
> > or sound fonts for a wavetable midi synth).
> >
> > It includes Bessel functions
> > which make an interesting rounded sawtooth
> > and I tried x*sin(1/x) and variants
> > such as x^3*sin(1/x^3) which have
> > infinitely many waves getting smaller
> > and smaller as you approach the origin.
> >
> > It also lets you build the functions up into
> > a harmonic series and you can quantise the
> > series to the current scale so I can experiment
> > with Setherising, I was surprised to find that
> > sine waves didn't work so well there
> > as more complex functions - you got beats
> > within a single note if you use
> > sine waves for e.g. equal tempered partials
> > in a harmonic series but when you replaced
> > them with more complex curves then the beats
> > went away. So there is clearly a bit of an art
> > to this.
>
> This is one of the most fascinating posts I've seen in a long time.
>
> Would you mind getting a bit deeper on all this on one of the more
> analytical lists, such as harmonic_entropy, tuning, or tuning-math?
>
> I'd enjoy that, anyway.

The curves are maybe best explained
with screen shots of them - I'll
do some to follow this up.

The basic mathematical idea I used
is to let the user enter any mathematical formula
to set a curve for either the
quarter, half or complete wave.

If it is intended for a quarter wave, FTS adds a constant
term to ensure that it starts at 0.
Then it does the other three quarters
so that they all join up seamlessly
- continuous lines anyway though they
may not be smooth where the various
quarters meet.

If it is intended for a half curve, then it
does two rotated copies, so again
they meet up and indeed this time
they will meet up smoothly too if the
original curve is smooth at origin.
This time it adds a linear term to
ensure that the start and end points of the
curve are both 0.

If it is a complete wave, then FTS
will add a quadratic term to ensure
that the two endpoints of the curve
are both zero, and that in between
it has as much of the curve above
as below the zero line (so that
the areas above and below are equal).
This won't necessarily always be a smooth
curve where they endpoints meet up
from one wave to the next however.

It then normalises the result so that
whatever your formula, the maximum
height of the wave remains the same.

So you can use any formula for any
of those, e.g. you could use say, x^3
for all three and it would still
work and make continuous
repeating sound waves you can play
and listen to.

You also have a choice of a few
presets too - sine waves,
triangles, a sine triangle, and
repeated sines which make a kind
of a rounded square wave,
and you can morph any of your
curves with a sine wave to
make a curve that is between
the two (e.g. 80 % your curve
and 20 % sine or whatever).

Then it adds together any desred number
of these waves to make
a harmonic series. The series
can be quantised to the current
scale e.g.to 7-et or 12-et or whatever.

Then the Setherised thing is a matter
partly of personal preference I think
- maybe the Setherised sine curve harmonic timbres
are in a sense smoother but they also have more obvious beats
(i.e. for each single note on its own)
while more complex curves end up with
a kind of uniform texture or roughness
so that you don't hear the beats within
that texture so clearly, so it
may sound more harmonious in some sense
if not really smoother.

The x^n*sin(1/x^n) etc family of curves
which I tried out is quite interesting here - when you set
n high you get something more like a series
of fast clicks, so a buzz, rather than a
wave, and the more it is like that then
the less you notice any beats, but the
note gets very kind of rough (an interesting
texture and still with a distinctive
pitch in it but a bit more towards a kind of
coloured noise rather than a note).

Anyway when I have some time after the upload
I'll do a web page about it with graphics
and sound clips, and some sound fonts
and Gigs to download for users of the
SB soundcards or others that can use
SF2 fonts, and for users of GigaSampler
/ GigaStudio. They consist of just
single waves so are tiny and the new

FTS has an option to make wave clips
for all the octaves for a sound font in one go.

However for best pitch accuracy you want
the high pitch waves to be an integral
number of samples. At present it
uses 1/100 of the sample rate
for the 1/1 of the sound font samples as it were,
and then goes up and down in octaves
from that point. So that pitches
them at A = 441, then if you want
A = 440 you can adjust the
result down by 4 cents when you use
the sound font.

If you try A = 440 at 44.1 Khz
that doesn't work well. It is important
to get the numbers right there as
the higher notes have very few samples,
e.g.at A = 441, then a uses
100 samples for a single wave, a' uses 50, and a'' uses
only 25. Clearly something like
49.5 approximated by 50 there would throw
the pitch well out. Best solution would
be if you could use the lowest pitch wave
for everything but in the SF2 fonts
anyway it seems it won't let you raise the
pitch more than about an octave or two.
(will be interested to know if anyone
has any tips about this).

The other solution is to record
a train of e.g. 100 waves
instead of a single wave,
which is an alternative - though the single
wave is more elegant.

Another idea is to set A=440 and
then save the sound clips at
44 Khz which you can do in FTS
and I think the SF2 format anyway
can handle data in that format,
and play it at the appropriate pitch, will see.

More later when I do the web page,
and will be interested to know if
anyone has any tips on the sound fonts.
Also has anyone come across any other program
that lets you play any mathematical equation
you like to enter as a sound wave?

If anyone has any particular questions
be sure to ask too,

Best,

Robert

🔗Robert Walker <robertwalker@...>

🔗Paul Erlich <perlich@...>

🔗Robert Walker <robertwalker@...>

Hi Paul,

> I'm assuming you mean Sethares-style inharmonic timbres, which are
> neither sine curves nor harmonic timbres. Am I understanding you
> correctly?

Yes, I think so. Made by stacking e.g. sine curves, or also
other curves such as rounded saw tooth etc into a harmonic
series of pitches with the pitches quantised to whatever
scale you want e.g. 12-et or 17-et or whatever.

> Yes. Inharmonic timbres will generally have combinational tones which
> beat against the partials and against one another, and according to
> John Chalmers, even virtual pitches can beat against combinational
> tones. I replied to your message on the tuning list, and I'm hoping I
> got your meaning right above, since that's what I assumed in my
> tuning list reply.

Ah right that may be what it is, beating of combinational tones
with each other or the partials. I wondered how it happened.
I was using a harmonic series of quite a few terms, e.g.
5 terms or 13. I wonder if missing out some of the partials
may be a way to reduce those beats then. Well tht will
probably be for another upload after this - I plan to
add options to let the user set the volumes individually
for each partial (and the pitches too if they like).
Right now it just does them with the volumes
decreasing as you go up the series.

Meanwhile what I can do is to add radios to do
only the odd harmonics or only odd prime
harmonics - to reduce the number of partials
so of combination tones - maybe that will help make
the Setherised chords smoother.

Just as a taster, here is a screen shot:

http://www.robertinventor.com/xpn_sin_1oxpn.gif

Shows x^n sin(1/x^n) though can't remember what n was
- it's just a screen shot I made for debugging.

Normally all the ripples would be about the zero line,
but it is curved like this because of the quadratic term
that gets added to make the areas above and below
the zero line the same size, as they are in normal
sound waves.

As you vary the parameters in another window,
(including the n) then the curve
changes instantly in the oscilloscope and you can click
the Play this button to hear it.

I've cc'd this to tuning, and posted another reply
to your post there.

Thanks,

Robert

Sad symmetric septimal minor tune

🔗Robert Walker <robertwalker@...>

🔗mopani@...

🔗Robert Walker <robertwalker@...>

🔗Paul Erlich <perlich@...>

🔗Robert Walker <robertwalker@...>

🔗Paul Erlich <perlich@...>

🔗Robert Walker <robertwalker@...>

🔗Robert Walker <robertwalker@...>

🔗Margo Schulter <mschulter@...>

🔗Paul Erlich <perlich@...>

🔗Robert Walker <robertwalker@...>

🔗Paul Erlich <perlich@...>

🔗Robert Walker <robertwalker@...>

🔗Robert Walker <robertwalker@...>

🔗Paul Erlich <perlich@...>

🔗Robert Walker <robertwalker@...>

🔗Paul Erlich <perlich@...>