Studies and Observations of 100tet.

This is a call for studies and observations which involve
5,10,20,25,50 and 100tets. All information, links and theoretical
considerations are welcome. Studies and observations are also
welcome for 200tet.

On Aug 23, 2008, at 3:45 PM, robert thomas martin wrote:
> Studies and observations are also welcome for 200tet.

Aaron Hunt uses 205 ET, so perhaps that is related :) Check out http://www.h-pi.com/

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

robert thomas martin wrote:
> This is a call for studies and observations which involve > 5,10,20,25,50 and 100tets. All information, links and theoretical
> considerations are welcome. Studies and observations are also > welcome for 200tet.
> You might know already, but there's a significance to 50-tet: it's virtually the same as 2/7-comma meantone. The fifth is 29 commas or 696 cents (since each step is exactly 24 cents, "comma" would be an appropriate name), the fourth 21/504, the whole tone 8/192. In addition to the chromatic semitone (3/72) and the diatonic semitone (5/120), it has an intermediate semitone, 4 commas or 96 cents; the diesis is half of that, a true quarter tone.

I've used 50-tet myself for meantone. My keyboard has scale tuning to a precision of one cent; each successive fifth is tuned four cents lower than ET than the one before.

Someone already mentioned that 200-edo, a superset of 50, contains both a 696-cent fifth and an almost exact Pythagorean 702-cent fifth. (I didn't catch who said it; I've been having to speedread everything.)

And of course 5-tet is a vague slendro tuning. One of my planned projects is a concerto for six timpani and percussion; the timpani are tuned in 5-equal and need two players. I might do something of the like in 7-tet later on.

~D.

--- In tuning@yahoogroups.com, Danny Wier <dawiertx@...> wrote:
>
> robert thomas martin wrote:
> > This is a call for studies and observations which involve
> > 5,10,20,25,50 and 100tets. All information, links and theoretical
> > considerations are welcome. Studies and observations are also
> > welcome for 200tet.
> >
>
> You might know already, but there's a significance to 50-tet: it's
> virtually the same as 2/7-comma meantone. The fifth is 29 commas or
696
> cents (since each step is exactly 24 cents, "comma" would be an
> appropriate name), the fourth 21/504, the whole tone 8/192. In
addition
> to the chromatic semitone (3/72) and the diatonic semitone (5/120),
it
> has an intermediate semitone, 4 commas or 96 cents; the diesis is
half
> of that, a true quarter tone.
>
> I've used 50-tet myself for meantone. My keyboard has scale tuning
to a
> precision of one cent; each successive fifth is tuned four cents
lower
> than ET than the one before.
>
> Someone already mentioned that 200-edo, a superset of 50, contains
both
> a 696-cent fifth and an almost exact Pythagorean 702-cent fifth. (I
> didn't catch who said it; I've been having to speedread everything.)
>
> And of course 5-tet is a vague slendro tuning. One of my planned
> projects is a concerto for six timpani and percussion; the timpani
are
> tuned in 5-equal and need two players. I might do something of the
like
> in 7-tet later on.
>
> ~D.
>
From Robert. I don't really know very much about this subject
except that I find 100tet very convenient to use when I carry
out my experiments. I am certainly willing to listen and learn
more about it.

--- In tuning@yahoogroups.com, "robert thomas martin"
<robertthomasmartin@...> wrote:
>
> This is a call for studies and observations which involve
> 5,10,20,25,50 and 100tets. All information, links and theoretical
> considerations are welcome. Studies and observations are also
> welcome for 200tet.

Robert,

I'm not sure what your angle is, but it seems you are partial to
numbers based on the counting of the human hand (multiples of 5 and 10).

While these can be interesting tunings, what constitutes and
interesting tuning and temperament really depends on what qualities
one is looking for.

Assuming you're not using just intonation, when looking at EDOs as a
practical starting point, for instance, one might ask the question, if
one is inclined towards the beauty of JI with the practical finitudeof
pitchesthat EDOs offer: "what JI intervals am I going to approximate?"

We know that the scale tree, or mathematically, the 'Stern Brocot'
tree, gives (logaritmic) fractions of the octave which help search the
'EDO space'. As an example, say I want relatively pure fifths. I know
that 7/12 of an octave is a good fifth. The 'parents' of this fraction
in the scale tree are 4/7 and 3/5:

4/7 3/5
7/12

when I search between thee spaces, I can just add numerators and
denominators:

4/7 3/5
7/12
11/19 10/17

another 'generation' gives:

4/7 3/5
7/12
11/19 10/17
15/26 18/31 17/29 13/22

Hence, the popularity, looking at the denominators, of EDOs like
19,31,17, and 22--people don't like to abandon the perfect fifth as a
structural unit of musical cohesiveness, in general....

Now, 50-edo shows up between 11/19 and 18/31 as 29/50, so it's an
important meantone EDO, corresponding with 2/7-comma temperament.

In general, most 'multiple of 5' EDO tunings give a 'Far-Eastern'
flavor because of the relationship they have to 5-edo, with its wide,
but still recognizable and usable fifths. All 'multiple of 5' EDOs
have an 'equal pentatonic' at their disposal, and other resources,
depending on the particular division.

-AKJ.

By the way--one should enable the 'fixed width font' option in your
mail reader or through the Yahoo interface to see the EDO scale trees
I wrote below as I intended them to be seen!

-AKJ.

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
>
> --- In tuning@yahoogroups.com, "robert thomas martin"
> <robertthomasmartin@> wrote:
> >
> > This is a call for studies and observations which involve
> > 5,10,20,25,50 and 100tets. All information, links and theoretical
> > considerations are welcome. Studies and observations are also
> > welcome for 200tet.
>
> Robert,
>
> I'm not sure what your angle is, but it seems you are partial to
> numbers based on the counting of the human hand (multiples of 5 and 10).
>
> While these can be interesting tunings, what constitutes and
> interesting tuning and temperament really depends on what qualities
> one is looking for.
>
> Assuming you're not using just intonation, when looking at EDOs as a
> practical starting point, for instance, one might ask the question, if
> one is inclined towards the beauty of JI with the practical finitudeof
> pitchesthat EDOs offer: "what JI intervals am I going to approximate?"
>
> We know that the scale tree, or mathematically, the 'Stern Brocot'
> tree, gives (logaritmic) fractions of the octave which help search the
> 'EDO space'. As an example, say I want relatively pure fifths. I know
> that 7/12 of an octave is a good fifth. The 'parents' of this fraction
> in the scale tree are 4/7 and 3/5:
>
> 4/7 3/5
> 7/12
>
> when I search between thee spaces, I can just add numerators and
> denominators:
>
> 4/7 3/5
> 7/12
> 11/19 10/17
>
> another 'generation' gives:
>
> 4/7 3/5
> 7/12
> 11/19 10/17
> 15/26 18/31 17/29 13/22
>
> Hence, the popularity, looking at the denominators, of EDOs like
> 19,31,17, and 22--people don't like to abandon the perfect fifth as a
> structural unit of musical cohesiveness, in general....
>
> Now, 50-edo shows up between 11/19 and 18/31 as 29/50, so it's an
> important meantone EDO, corresponding with 2/7-comma temperament.
>
> In general, most 'multiple of 5' EDO tunings give a 'Far-Eastern'
> flavor because of the relationship they have to 5-edo, with its wide,
> but still recognizable and usable fifths. All 'multiple of 5' EDOs
> have an 'equal pentatonic' at their disposal, and other resources,
> depending on the particular division.
>
> -AKJ.
>

I'd go with 200-edo, since you can do so much with it.

Hey, this page might interest you:

http://www.xs4all.nl/~huygensf/doc/measures.html

It's a list of measurement units that have been proposed, and one of them is 360-edo, the Dr�bisch Angle. If the octave is thought of as a circle, then each Angle would be a "degree". I'm a fan of base-60 math, the numeric system used by the Sumerians and Babylonians, and sixty is divisible by two, three, four, five and six. 360-edo contains some useful equal temperaments, including 5, 10, 12, 15, 24, 36 and 72. If you divide the ET whole tone by sixty, you get 360, and dividing the semitone likewise gives you 720, which I'd rather use than 1200 (or we could at least 3600-edo as a measurement; 72 � 50 = 3600, after all).

~D.

robert thomas martin wrote:
> From Robert. I don't really know very much about this subject
> except that I find 100tet very convenient to use when I carry
> out my experiments. I am certainly willing to listen and learn
> more about it.

Sexagesimal arithmetic is interesting, but the size of the base seems
too big a price to pay just to gain a factor of five. A single base
is all that's needed for all logarithmic divisions, but I would favor
one that's of a more manageable magnitude...

144-edo is a logical extention of 12-edo, which can be further
extended to 1728-edo to replace cents.

I noticed in that link you provided that an octave/144 is called a
farab, and an octave/1728 is called a Harmos. I'm glad these units
have been suggested, but the extra names aren't really necessary. We
only need one named unit and prefixed, coherent submultiples.

--- In tuning@yahoogroups.com, Danny Wier <dawiertx@...> wrote:
>
> I'd go with 200-edo, since you can do so much with it.
>
> Hey, this page might interest you:
>
> http://www.xs4all.nl/~huygensf/doc/measures.html
>
> It's a list of measurement units that have been proposed, and one
of
> them is 360-edo, the Dröbisch Angle. If the octave is thought of as
a
> circle, then each Angle would be a "degree". I'm a fan of base-60
math,
> the numeric system used by the Sumerians and Babylonians, and sixty
is
> divisible by two, three, four, five and six. 360-edo contains some
> useful equal temperaments, including 5, 10, 12, 15, 24, 36 and 72.
If
> you divide the ET whole tone by sixty, you get 360, and dividing
the
> semitone likewise gives you 720, which I'd rather use than 1200 (or
we
> could at least 3600-edo as a measurement; 72 × 50 = 3600, after
all).
>
> ~D.
>
> robert thomas martin wrote:
> > From Robert. I don't really know very much about this subject
> > except that I find 100tet very convenient to use when I carry
> > out my experiments. I am certainly willing to listen and learn
> > more about it.
>

robert thomas martin wrote:
> This is a call for studies and observations which involve > 5,10,20,25,50 and 100tets. All information, links and theoretical
> considerations are welcome. Studies and observations are also > welcome for 200tet.

5-ET is much like the Indonesian slendro scale, so I called this piece "Daybreak on Slendro Mountain". I try to modulate around a bit, but it's hard to recognize it as modulation when all keys use the same 5 notes. One thing that's nice about a 5-note scale is that it's easy to adapt to a keyboard (the parts of this were each performed in real time and then MIDI pitch bends were added to retune it).

http://www.io.com/~hmiller/midi/Daybreak.mid

10-tET is not frequently used, but Bill Sethares has some examples on his CDs. He uses retuned sound samples with inharmonic partials adjusted to match the pitches of 10-tET, along with chord progressions based on a circle of thirds instead of fifths.

20-ET is a tuning that I like in spite of its poor approximation to the harmonic series, although I haven't done much with it lately. This example is from 1999.

http://www.io.com/~hmiller/midi/20tet.mid

Dan Stearns does a lot of 20-EDO; you can find some on his zebox page. http://www.zebox.com/daniel_anthony_stearns/

I don't know if much has been done specifically with 25-et, but if you take the 5-et slendro scale, and add another copy of the slendro scale 2 steps down, you have Blackwood's decatonic scale (usually associated with 15-ET). I use a different tuning in this example, but it can be adapted to 25-tet.

http://teamouse.googlepages.com/pilina.mp3

Herman wrote:

> I don't know if much has been done specifically with 25-et,
> but if you take the 5-et slendro scale, and add another copy
> of the slendro scale 2 steps down, you have Blackwood's
> decatonic scale (usually associated with 15-ET).

Excellent review, Herman. One other aspect of 25 that has
been mentioned is its 'no 3s' approximations, where it
starts to look decent up through 2.5.7.11.13.17. In
particular, for 7.11.13 triads, the val <69 85 91] is hard
to beat.

Paul Rapoport's Study In Fives (from Musicworks 61), if you
can locate a copy, is a nice example of 25-ET at work.

-Carl

Brun's Algorithm gives 200 as an MOS of 3/2
--

/^_,',',',_ //^ /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/>

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

Hi Robert,

--- In tuning@yahoogroups.com, "robert thomas martin"
<robertthomasmartin@...> wrote:
>
> This is a call for studies and observations which involve
> 5,10,20,25,50 and 100tets. All information, links and
> theoretical considerations are welcome. Studies and
> observations are also welcome for 200tet.

Here is a list of the percentage error of the
degrees of 100-edo, for the prime-factors from
3 to 41 (41 is merely my arbitrary upper limit; the
lowest primes are generally the most significant ones):

(click "Use Fixed Width Font" under "Options"
if viewing on the stupid Yahoo web interface)

prime .. % error of 100-edo
... 3 .... -50
... 5 .... -19
... 7 .... +26
.. 11 .... +06
.. 13 .... -04
.. 17 .... +25
.. 19 .... +21
.. 23 .... -36
.. 29 .... +20
.. 31 .... -42
.. 37 .... +05
.. 41 .... +24

You can see the prime-factor 3, which is extremely
important in most (but not all) musical practice and
theory, lies almost exactly midway between two of
the 100-edo degrees. I would consider that to be
a major defect of this tuning.

Now of course, whenever you double the number of
divisions of an EDO, any prime-factors with ~50% error
in the smaller EDO, will have hardly an error in the
larger one -- and so it is with 200-edo:

prime .. % error of 200-edo
... 3 .... +01
... 5 .... -39
... 7 .... -47
.. 11 .... +11
.. 13 .... -09
.. 17 .... -49
.. 19 .... +41
.. 23 .... +29
.. 29 .... +40
.. 31 .... +16
.. 37 .... +11
.. 41 .... +49

The errors of 200-edo for prime-factor 5 is large
enough to render it less-than-useful for approximating
or examining tunings which make use of that prime,
and its error for 7 shows that that prime lies nearly
midway between two degrees of 200-edo, and likewise
for prime-factors 17 and 41 (altho those two primes
are not nearly as historically significant as 7).
But it is great for approximating pythagorean tuning
(i.e., based on prime-factor 3).

Of course, 53-edo is already an excellent approximation
of 3-limit tuning, and also gives a very good approximation
of prime-factor 5, so i would consider it far superior to
200-edo for approximating/examining 5-limit tuning -- and
5-limit tuning is quite important, being the ideal
theoretical basis for nearly all Western music between
~1600-1910.

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