60 NOTE MICRO-TUNED SCALE - exhibits interesting phenomenal properties

An writer of alternative music related ideas recently expressed that
a 60 note scale could be both exponential and lograithmic in its
structure and that it would have interesting properties.

I've come across a lot of comments / info on microtuned scales.
Many people discuss the 22, 60, 72 note scale (and higher), but I
have yet to find ANY meaningful details on the 60, 72 note scale.
Paul Erlich does a good job on the 22 note scale, Ive seen an
interesting one based on pi and 22, and the Hindus have their
version of this. No 60 or 72 note scale to be found anywhere.

So;

What is the musical scale for 22 notes, 60 notes, 72 notes where the
scale for each is based as strictly as possible on Pythagorean and
ancient Chinese JUST tuning and where the scale is modal (and
horizontal) , i.e. where ..."There is a strong relationship to the
tonic. When a third is played it always relate to the third degree
as in Western harmonious tradition the third has a relative
position, because it can be the root, the fifth or third of a chord.
- and - "...intervals on which the musical structure is built are
calculated in relation to a permanent tonic. That does not mean that
the relations between other than the tonic are not considered, but
that each note will be established first according to its relation
to the fixed tonic and not, as in the case of cycle of fifths by any
permutations of the basic note. The modal structure can therefore be
compare to the proportional division of the string (straight line)
rather than to the periodic movement of the spiral of fifths. All
the notes obtained in the harmonic system are distinct from those of
the cyclic system, which is based on different data. Though the
notes are theoretically distinct and their sequence follow
completely different rules, in practice they lead to a similar
division of the octave into fifty three intervals.
The scale of proportions is made of a succession of syntonic commas,
81/80, which divide the octave into 53 intervals. Among those, 22
notes was chosen for their specific emotional expressions:"", - see
reference notes included below "The Creation of Musical Scales, part
II. "
I understand further, however, that a 60 note scale can accomodate
both EQUAL LOGARITHMIC AND PYTHAGOREAN JUST TUNING
How would a scale of this type for a sixty note scale appear, from
unison to the end of the scale?
I'd like to be able to study a table for these scales that lists the
scale out, especially for the 60 and 72 note scale in the following
manner;
(snippet from 22 note scale - this entire table included inline in
the below following text, scroll down to review whole table)
Note
degree Interval Interval Name
1 1/1 unison
2 256/243 Pythagorean limma
3 16/15 minor diatonic semitone
4 10/9 minor whole tone
..... up to;
21 15/8 classic major seventh
22 243/128 Pythagorean major seventh
Here, the 22 notes are listed out in seuqence for the whole octave,
from 1 to 22 ('Note degree') in the first column, and then the
interval associated
is shown in the second column. So for a 60 and 72 note scale, what
Im really wanting to find is the same structure, i.e.;
Note
degree Interval Interval Name
1 1/1 unison
2 ?? ??
3 ?? ??
4 ?? ??
.....
59 ?? ??
60 ??? ??
Alternatively, How would a purely logarithmic scale for a 22, 60,
and 72 note scale appear?
============================
REFERENCE NOTES FOLLOW AT END OF EMAIL
=============================

==============
NEXT LINE OF INQUIRY

Below is a list of seven, two note chords. These 'notes' are taken
from a twenty two note scale shown in the full table
shown at the end of this email.
Column (A) = note 1 of the given chord
Column (C) = note 2 of the given chord
I realize that each note shown below is likely somewhat irregular, I
would ask your patience and for the moment ignore the irregularity.
For the first question, all I want to know is what is the specific
interval relationship between (A) and (B) for each chord?
And do you see any odd and or repeating patterns occuring throughout
all of these chords?
If any of the given chords seem to lack harmony, balance, what
additional note expressed as an interval could be placed between (A)
and (C)
if Column (B) to harmonize these better?
In Column (D), could you please show the interval relationship
between (A) and (C), expressed as a fraction?
(A) (B) (C) (D)
chord 1.) 4/2 18/15
chord 2.) 16/1 13/14
chord 3.) 1/15 18/15
chord 4.) 5/15 9/19
chord 5.) 8/9 13/14
chord 6.) 18/22 18/15
chord 7.) 22/10 18/15
=============================
REFERENCE NOTES:

The Creation of Musical Scales, part II.
by Thomas Váczy Hightower.
http://home22.inet.tele.dk/hightower/scales2.htm
The Indian notes relate coarsely to the Western ratios, though the
tuning is very harmonious and create a world of difference. We have
to emphasize that the use of harmony as we know it was and is not
musically practiced.
Here is a crucial point. The Indian music is modal. There is a
strong relationship to the tonic. When a third is played it always
relate to the third degree as in Western harmonious tradition the
third has a relative position, because it can be the root, the fifth
or third of a chord.
Eastern listeners often make remarks such as "Beethoven symphonies
are interesting, but why have all those chords been introduced,
spoiling the charm of the melodies".
The modal music of India is 'horizontal' as the Western
is 'vertical'. The vertical, harmonious system, in which the group
of related sounds is given at once, might be more direct though also
less clear. The accurate discrimination of the different elements
that constitute a chord is not usually possible.
The modal, horizontal system, on the other hand, allows the exact
perception and immediate classification of every note, and therefore
permits a much more accurate, powerful and detailed outlining of
what the music express.
The 22 Shrutis (degrees)
The modal music of India is 'horizontal' as the Western
is 'vertical'. The vertical, harmonious system, in which the group
of related sounds is given at once, might be more direct though also
less clear. The accurate discrimination of the different elements
that constitute a chord is not usually possible.
The modal, horizontal system, on the other hand, allows the exact
perception and immediate classification of every note, and therefore
permits a much more accurate, powerful and detailed outlining of
what the music express.
Musical intervals can be defined in two ways, either by numbers
(string lengths, frequencies) or by their psychological
correspondences, such as feelings and images they necessarily evokes
in our minds. There is no sound without a meaning, so the Indians
consider the emotions that different intervals evoke as exact as
sound ratios. The feeling of the shrutis depends exclusively on
their position in relation to the tonic and indicate the key for the
ragas.
The 22 different keys or degrees encompass what the Indians consider
the most common feelings and reflection of the human mind. They was
aware of the division of the octave into 53 equal parts, the
Pythagorean Comma, and its harmonic equivalent, the comma diesis,
(the syntonic comma, the difference between the major and the minor
tones).
However, they choose the 22nd division of the octave based on the
limit to differentiate the keys as well as psychological and meta
physical reason. The symbolic correspondences of the number 22 and
7, (7 strings and main notes), could also play a part since the
relationship between the circle and the diameter is expressed as the
approximate value of Pi, 22/7.
The modal or Harmonic division of the octave
Indian music is essentially modal, which means that the intervals on
which the musical structure is built are calculated in relation to a
permanent tonic. That does not mean that the relations between other
than the tonic are not considered, but that each note will be
established first according to its relation to the fixed tonic and
not, as in the case of cycle of fifths by any permutations of the
basic note.
The modal structure can therefore be compare to the proportional
division of the string (straight line) rather than to the periodic
movement of the spiral of fifths.
All the notes obtained in the harmonic system are distinct from
those of the cyclic system, which is based on different data. Though
the notes are theoretically distinct and their sequence follow
completely different rules, in practice they lead to a similar
division of the octave into fifty three intervals.
The scale of proportions is made of a succession of syntonic commas,
81/80, which divide the octave into 53 intervals. Among those, 22
notes was chosen for their specific emotional expressions:
Note
degree Interval Value in cents Interval Name Expressive qualities
1 1/1 0 unison marvelous, heroic, furious
2 256/243 90.22504 Pythagorean limma comic
3 16/15 111.7313 minor diatonic semitone love
4 10/9 182.4038 minor whole tone comic, love
5 9/8 203.9100 major whole tone compassion
6 32/27 294.1351 Pythagorean minor third comic, love
7 6/5 315.6414 minor third love
8 5/4 386.3139 major third marvelous, heroic, furious
9 81/64 407.8201 Pythagorean major third comic
10 4/3 498.0452 perfect fourth marvelous, heroic, furious
11 27/20 519.5515 acute fourth comic
12 45/32 590.2239 tritone love
13 729/512 611.7302 Pythagorean tritone comic, love
14 3/2 701.9553 perfect fifth love
15 128/81 792.1803 Pythagorean minor sixth comic, love
16 8/5 813.6866 minor sixth comic
17 5/3 884.3591 major sixth compassion
18 27/16 905.8654 Pythagorean major sixth compassion
19 16/9 996.0905 Pythagorean minor seventh comic
20 9/5 1017.596 just minor seventh comic, love
21 15/8 1088.269 classic major seventh marvelous, heroic, furious
22 243/128 1109.775 Pythagorean major seventh comic, love
================
The Chinese continued the cycle of fifths up to 25,524 notes with a
basic intervals of 0.0021174 savarts. This cycle is very near to
that of precession of the equinoxes, or the Pythagorean great year,
which is of 25,920 solar years. Why the Chinese continued so many
octaves in the cycle of fifths, could have something to do with
their reference tone, Kung.
In practice, for reasons that are symbolic as well as musical, after
the 52nd fifth (53rd note) the Chinese follow the series only for
the next seven degrees, which place themselves above those of the
initial seven-note scale, and they stop the series at the 60th note.
The reason given is that 12 (the number of each cycle) * 5 (the
number of the elements) = 60.
The scale of 60 Lü
The Chinese scale being invariable, constitute in effect a single
mode. Every change in _expression will therefore depend upon
modulation, a change of tonic.
Firstly the choice of gender: fifths whose numbers in the series are
even, are feminine. The odd numbered fifths are masculine.
The choice of tonic are depended of complicated rules and rituals,
which main purpose is to be in accordance with celestial as well as
earthly influx or circumstances. Accordingly, the Chinese has to
choose the right key for the hour of the day and the month. Even
during a performance.
It is a extensive scheme but to get an idea we can say that they
corresponds to political matters, seasons, hour of the day,
elements, color, geographic direction, planets and moon.
This scale of fifths, perfect for transposition because of its
extreme accuracy, also allows the study of astrological
correspondences and of terrestrial influx in their Tone Zodiac.
==========
Thanks for any replies you are comfortable making!
D

Hello D,

> An writer of alternative music related ideas recently expressed
> that a 60 note scale could be both exponential and lograithmic
> in its structure and that it would have interesting properties.

Who was that?

> I've come across a lot of comments / info on microtuned scales.
> Many people discuss the 22, 60, 72 note scale (and higher), but I
> have yet to find ANY meaningful details on the 60, 72 note scale.
> Paul Erlich does a good job on the 22 note scale, Ive seen an
> interesting one based on pi and 22, and the Hindus have their
> version of this. No 60 or 72 note scale to be found anywhere.

72-tone equal temperament has been discussed extensively on
this mailing list. Especially owing to the fact that it is
a tuning of the MIRACLE temperament, which makes the comma
225/224 vanish (among others).

Also, Byzantine music theory divides the octave into 72 parts.

60 is less often discussed, but you can find it here and there.

> What is the musical scale for 22 notes, 60 notes, 72 notes
> where the scale for each is based as strictly as possible on
> Pythagorean and ancient Chinese JUST tuning and where the
> scale is modal (and horizontal) , i.e. where ..."There is
> a strong relationship to the tonic. When a third is played
> it always relate to the third degree as in Western harmonious
> tradition the third has a relative position, because it can
> be the root, the fifth or third of a chord.

Wow, where to begin. 22, 60, and 72 can refer to "equal
temperaments" (equal log divisions of the octave), or to
chains of "perfect fifths" (3:2 ratio), which some would
call "Pythagorean" in nature, or to any scale imaginable,
really. When you say "third", I assume you mean an
approximate 5:4 or 6:5 ratio. 22, 60, and 72-tone equal
temperament all provide better approximations to these
ratios than 12-tone ET does.

>- and - "...intervals on which the musical structure is built
> are calculated in relation to a permanent tonic. That does
> not mean that the relations between other than the tonic are
> not considered, but that each note will be established first
> according to its relation to the fixed tonic and not, as in
> the case of cycle of fifths by any permutations of the basic
> note. The modal structure can therefore be ...

Whoa, you lost me. Who are you quoting?

> I understand further, however, that a 60 note scale can
> accomodate both EQUAL LOGARITHMIC AND PYTHAGOREAN JUST TUNING

60-tET, being a multiple of 12, has the same 3:2 approximations
as you'll find on any piano or guitar -- 700 cents.

It has been known since antiquity that no system of tuning can
provide equal logarithmic divisions of both 2:1 and 3:1.
However, one can have a division of the 3:2 into 35 equal
log parts. This will result in an octave which is stretched
slightly sharp of just.

> Note degree , Interval Interval , Name
> 1 1/1 unison
> 2 256/243 Pythagorean limma
> 3 16/15 minor diatonic semitone
> 4 10/9 minor whole tone
> ..... up to;
> 21 15/8 classic major seventh
> 22 243/128 Pythagorean major seventh
> Here, the 22 notes are listed out in seuqence for the whole
> octave, from 1 to 22 ('Note degree') in the first column,
> and then the interval associated is shown in the second
> column. So for a 60 and 72 note scale, what I'm really
> wanting to find is the same structure, i.e.;
> Note degree , Interval , Interval Name
> 1 1/1 unison
> 2 ?? ??
> 3 ?? ??
> 4 ?? ??
> .....
> 59 ?? ??
> 60 ??? ??

If you download Scala

http://www.xs4all.nl/~huygensf/scala/

and type "pythag" in the box at the bottom, followed by
the following answers to the prompts:

size= 60
formal octave= [return]
fifth degree= [return]
formal fifth= 3/2
count downwards= [return]

and then type "show", I think you'll see what you're
looking for.

> Alternatively, How would a purely logarithmic scale for a 22, 60,
> and 72 note scale appear?

Just type "equal 22", etc. into that box, then "show" and
I think you'll see what you're looking for.

Well, that's a dusey of a first message! I think I'll stop
here for now,

-Carl