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Regarding Nature based Music Phenomenon and Lucy 'Pi' Tuning

🔗Dan Amateur <xamateur_dan@...>

10/3/2006 8:20:47 PM

In the course of recent dialog with Charles Lucy,
I have found myself fascinated with his work and
related subjects. I share the below with the group,
and look forward to any comments, feedback.....

At the end of the email, articles on the following are
also provided;

Natural Musical Scale Derived From Vesica Pisces
Octave as a Sine Wave and 1.2 AC waveforms
43.1 Spherical Music

To Charles,

I have gone through all the text, research papers on
your site and have studied them at length. Am still
digesting the information and have been thinking about
everything you have stated therein. It seems like a
very great and highly important work you have embarked
on and it seems to me that you are not just offering
something beautiful to mankind in the way of musical
alternatives but have made great efforts to put a
momentum in place that should have every chance of
having positive implications in many areas of human
consciousness and the applied and theoretical
sciences. I trust that one day you will be given the
recognition and reward due to true pioneers of merit.

Within this email, I have some questions, observations
and articles of information which I hope are of
interest to you - as they seem to directly and
indirectly confirm many of your concepts in various
ways, and perhaps you are already aware of these info
bytes.

The info articles are at the end of this email, and I
would very much like to hear your thoughts on these.

Below , first, some simple questions about deriving a
simple pi scale based on input you have offered, using
your formula I have tried to create a rudimentary pi
scale.

Following, are notes I have made about the germaine
principles found on the Harrison research papers, I
have preceded inline questions with >>, if possible
would very much like to hear your feedback on these.

"The solution that I use is very simple:
multiple sharps and flats A to G derived from
A4=440Hz, and the Large interval (i.e. between the
fourth and fifth) is: the ratio of the two pi root of
2. i.e. 2^(1/(2*pi)) or the radian angle in geometric
terms = 360/2pi degrees."

Using the Large interval; 1.11663288

432 X 1.11663288

Based on 432 Hertz

1 261.0399936 C
2 269.3237016 C #
3 291.4858399 D f
4 300.7357006 D
5 325.4826729 E f
6 335.8113715 E
7 363.4446544 F #
8 374.9780189 G f
9 418.7127852 A f
10 432 A
11 467.5484633 B f
12 482.3854042 B

Based on 440 Hertz

440 X 1.11663288

1 C 265.8740675
2 C # 274.3111776
3 D 296.8837258
4 D # 306.3048802
5 E 331.5101298
6 E # 342.0301006
7 G f 370.175111
8 G f 381.9220563
9 A f 426.4667257
10 A 440
11 B f 476.2067682
12 B 491.3184672

===============
Real Intervals (Steps) of Natural Melody, providing
real consonances or chords of natural harmony

OCTAVE RATIO, CIRCLE CIRCUMFERENCE; REPRESENT BY
LOGARITHM OF 2 [viz. .30103]

(.30103 x 2 = .60206 + .09582 = .69788)

space or quantity of two octaves and a sharp 3rd be
taken

Circle Circumference to diameter = 3.1416 is to 1

Octave = Logarithm of 2 = .30103

Two Octaves = (.30103 x 2 = .60206 )
Sharp 3rd = 0.09582

The Circles Circumference to its diameter is as Two
Octaves + Sharp 3rd to the Octave?

3.1416 to 1: So, as 3.1416 is to 1, so is .30103 to
.09582

Circles Circumference to Diameter is as the Octave
(.30103) to Sharp 3rd (.09582)
===========

Octave contains five larger notes and two lesser notes
(but no actual major or minor notes)

Half the diameter = the larger note = .04791

There are four 5ths, each = .17447 .17447 x 4 = .69788
>>>Somehow, Harrison says this equals two octaves and
a sharp 3rd, but I don't see the connection, how is
this so?

Each of the four 5ths can be generated by subtracting
5*radius from the circumference, the remainder = a
spatial quantity equivalent to two equal lesser notes
which = .06148, divide this by 2 = .03074

Subtract the lesser note (.03074) from the greater
note (.04791)
.04791 - .03074 = .01717
This is not the same as minor half note, but may be
called a tone major and or a tone minor

A fifth must contain three of the larger notes and one
of the lesser
Fifth = .04791+.04791+.04791+.03074 = .17447

There are no ratios exisitng in this musical scale,
except that of the octave itself.

Chords are best described logarithmically

We can derive the ratio of any chord by subtracting
the logarithm of the lesser number from that of the
greater
This approach may cause differences from conventional
ratios

The resulting sounds will tend to 'sound better' and
this will be observed via the latitude of the flatness
or sharpness of the resultant

For example; (Note: the 5th will sound better than
the 6th)

Interval Latitude
----------------------------------------------------------------------------------------------
5ths .00162 Flat
4th (complement of the 5th to the octave) .00162
Sharp
3rd Sharp .00109 Flat
6th Sharp .00053
----------
Four 4ths and a sharp 3rd = Two octaves
----------

Robert Smiths Observations;

In a monochord, the major 3rd to the fifth is as as
the diameter of a circle, to its circumference.
The octave = five meantones and two limmas, making it
a little bigger than six such tones, or three major
3rds.
>>> When he says 'six such' tones, does he imply six
'meantones' ?

The circumference of the circle is a little larger
than three of its diameters.

Time of a single vibration = 8 31/12/1748
>>> Any idea what this actually means, is?

Two musical strings with the same thickness, density
and tension, differing in length only, the times of
their vibrations are proportional to their lengths.
>>>What happens if one or more of the other factors
differ; thickness,density, tension?
=============

Came across the following, which seems to lend support
to your Lucy Tuning based on pi and the circle, your
thoughts on this?

Octave as a Sine Wave and 1.2 AC waveforms

When an alternator produces AC voltage, the voltage
switches polarity over time, but does so in a very
particular manner. When graphed over time, the "wave"
traced by this voltage of alternating polarity from an
alternator takes on a distinct shape, known as a sine
wave:
+- Time Graph of AC voltage over time (the sine wave)
In the voltage plot from an electromechanical
alternator, the change from one polarity to the other
is a smooth one, the voltage level changing most
rapidly at the zero ("crossover") point and most
slowly at its peak. If we were to graph the
trigonometric function of "sine" over a horizontal
range of 0 to 360 degrees, we would �nd the exact same
pattern:

1.2. AC WAVEFORMS 7

Angle Sine(angle)
in degrees

0 0.0000 -- zero
15 0.2588
30 0.5000
45 0.7071
60 0.8660
75 0.9659
90 1.0000 -- positive peak
105 0.9659
120 0.8660
135 0.7071
150 0.5000
165 0.2588
180 0.0000 -- zero
195 -0.2588
210 -0.5000
225 -0.7071
240 -0.8660
255 -0.9659
270 -1.0000 -- negative peak
285 -0.9659
300 -0.8660
315 -0.7071
330 -0.5000
345 -0.2588
360 0.0000 -- zero
----------------------------------
43.1 Spherical Music

..."in the spiral of perfect fifths, there the major
and minor scales are the reverse of each other when
the arithmetic vibration frequency ratios are put into
geometrical form in a spiral diagram".

The 3-4-5 triangle can be written thus:
Representing Single and Double Squares.

3^2 = 9
4^2 = 16
5^2 = 25

3 x 96 = 288 cps (3rd harmonic of 96 cps).
4 x 96 = 384 cps (4th harmonic of 96 cps).
5 x 96 = 480 cps (5th harmonic of 96 cps).

"The 12 Lynus Tonal System or the cycle of fifths of
China and the 22 natural intonation Scruti scales of
India, are arithmetical progressions... the musical
scale systems of different peoples can be
mathematically oriented into a spiral complex when
translated into digits 1, 2, 3, 4 and 5. This means,
as already noted, that a "spherical music" may
eventually express the evolution toward a world-wide
acceptance of a common tonal system, perhaps the
Mercator, equal-tempered 53 tones-to-the-octave scale
within the overall mathematical perimeter of the
circle of fifths"...

(Correspondences, File, "30 Unique Divisions").

..."the binary aspect of the octave intervals of the
tonal spiral requires a Boolean algebra".
Dr. Andrew Pikler, International Congress on
Acoustics, Tokyo, Aug. 1968, Generation and Plotting
of Musical Tone System with the Digital Computer.

..."possibility of a "double helix" in the domain of
the musical tone system...the "sharps" and "flats'
furnish helical formations, each separately split with
the Pythagorean comma"...
Dr. Andrew Pikler.

43.2 "1, 2, 3 and 5 "

"Harold's concept of giving a measure number to
opposite forces (yin-yang) in order to describe them,
seems to relate to music, that is:

If a note in a minor key is at once the same sound and
a different sound, it must be more or less of one
sound. Therefore...if...a minor key...represents
negative, and a major key represents positive...
Harold's theory assigns 2 as the measure number of the
minor, 3 as the measure of the major, and 5 as the
measure of wholeness or the 1 sound that can be played
in either the major or minor key- but cannot be played
in both at the same instant of time".

If the foregoing is true, we ourselves cannot make the
1 sound, with the measure number 5, until we strike a
chord:

3 (+ major) and 2 (- minor).

Harold's description of the nuclear force of the
hydrogen atom can be read as 3(+) / 2(-). It would
say, "that a chord can be struck using a formula that
could read:

2(-) / 3(+), or 3(+) / 2(-),

these 2 would not sound alike, but either would convey
the sound of wholeness.
================================
Natural Musical Scale Derived From Vesica Pisces

In the process of going from One to Two, a Circle
with a diameter length has a single vibration, but
when another Circle is added interesecting it in its
center, a Vesica Pisces is formed. This is the area
that it shares which looks like the female generative
organ, and from where in sacred geometry all things
come from. Part of the reproductive creative process
How natures multiplication and reproduction came about
from One to Many Vibrations that we see and feel and
hear.

If there was a string across the diameter of our
first circle, that could be our base frequency of
1,The second circle makes the Vesica Pisces, and
extends the distance to 1.5 times the first. Therefore
any frequency times 1.5 gives a pitch that is in tune
with the original pitch or vibration.
And then if we continue on with the construction
of a new Vesica Pisces by the mating of two circles,
it makes a new larger circle of 2.25 times larger than
our original circle. And so if we half this diamter,
into a range between 1 and 2, it is 1.125. Therefore
any original frequency sounds in tune with that
frequency times 1.125. And lo and behold we can go on
making twelve notes between 1 and its octave of 2.
1.125 being the B note on the piano.
Therefore using this same process we get a table,
as seen on the left, with each of its corresponding
notes that are all harmoniously in tune with each
other, and now you know why.

Circle Diameter Ratio Note Hertz
1 = 1 1 A 440
2 = 1.5 1.5 E 659
3 = 2.25 1.125 B 494
4 = 3.375 1.69 F# 743
5 = 5.062 1.265 C# 554
6 = 7.539 1.88 G# 827
7 = 11.39 1.42 D# 622
8 = 17.08 1.06 A# 466
9 = 25.62 1.59 F 698
10=38.44 1.19 C 523
11=57.66 1.79 G 787
12=86.49 1.35 D 582

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🔗misterbobro <misterbobro@...>

10/4/2006 12:40:03 AM

Synchronicity, very interesting. I discovered "Lucy Tuning" a few
days ago by typing what seemed to be key words and phrases taken
from my hands-on experiments with a fretless guitar and the idea,
whether it's sci-fi or not, that the number 22 in the Indian Sruti
system originates in the 22/7 rational approximation of Pi. The
beginning of the process is described in real-time, warts and all,
in the "Shake Yer Sruti" thread at the Alternate Tuning group.

Got the "sharp 3d" right away, very convenient because I'm working
on refining a close-to equal 17-tone tuning which of course has high
3ds. Now with the computer I'm continuing, but ignoring what I
consider to the meaningless division of a circumfrence into 360
degrees. One experiment generated a tuning averaging out about 2
cents different than one of Margo Schulter's tunings in the Scala
archive and about 4 cents from one of my own which I did basically
by ear.

Been reading Margo Schulter's archived posts on the groups because
my "what sounds best to me"-based approach to tuning coincides
amazingly with work she's already done over the years and I'll now
be able to save a lot of reinventing the wheel, so to speak. Most
importantly, her articles on the real-life USE of tunings are a
treasure; this is no small or idle matter for someone playing live
and recording outside the 12-tET dictatorship.

Using a circle/spiral of fifths derived from spherical/circular
relationships is just one possibility, there are plenty of others.

By the way, this is all, actually, Just Intonation, like it or not:
it's just that it's coming not from an ideal string, but from an
ideal POINT.

It's important to remember that Just Intonation isn't about numbers
as such, it's about relationships between partials. The third partial
may lie at 2.98001/2 on a real-life instrument, you could call
it "Aunt Mathilda" or "3/2", it doesn't matter as long as you don't
confuse an idealized NAME (number) with reality. The work of William
Sethares demonstrates this in a spectacular way.

-Cameron Bobro

--- In MakeMicroMusic@yahoogroups.com, Dan Amateur
<xamateur_dan@...> wrote:
>
>
> In the course of recent dialog with Charles Lucy,
> I have found myself fascinated with his work and
> related subjects. I share the below with the group,
> and look forward to any comments, feedback.....
>
> At the end of the email, articles on the following are
> also provided;
>
> Natural Musical Scale Derived From Vesica Pisces
> Octave as a Sine Wave and 1.2 AC waveforms
> 43.1 Spherical Music
>
>
>
> To Charles,
>
> I have gone through all the text, research papers on
> your site and have studied them at length. Am still
> digesting the information and have been thinking about
> everything you have stated therein. It seems like a
> very great and highly important work you have embarked
> on and it seems to me that you are not just offering
> something beautiful to mankind in the way of musical
> alternatives but have made great efforts to put a
> momentum in place that should have every chance of
> having positive implications in many areas of human
> consciousness and the applied and theoretical
> sciences. I trust that one day you will be given the
> recognition and reward due to true pioneers of merit.
>
> Within this email, I have some questions, observations
> and articles of information which I hope are of
> interest to you - as they seem to directly and
> indirectly confirm many of your concepts in various
> ways, and perhaps you are already aware of these info
> bytes.
>
> The info articles are at the end of this email, and I
> would very much like to hear your thoughts on these.
>
> Below , first, some simple questions about deriving a
> simple pi scale based on input you have offered, using
> your formula I have tried to create a rudimentary pi
> scale.
>
> Following, are notes I have made about the germaine
> principles found on the Harrison research papers, I
> have preceded inline questions with >>, if possible
> would very much like to hear your feedback on these.
>
>
> "The solution that I use is very simple:
> multiple sharps and flats A to G derived from
> A4=440Hz, and the Large interval (i.e. between the
> fourth and fifth) is: the ratio of the two pi root of
> 2. i.e. 2^(1/(2*pi)) or the radian angle in geometric
> terms = 360/2pi degrees."
>
>
>
> Using the Large interval; 1.11663288
>
> 432 X 1.11663288
>
> Based on 432 Hertz
>
> 1 261.0399936 C
> 2 269.3237016 C #
> 3 291.4858399 D f
> 4 300.7357006 D
> 5 325.4826729 E f
> 6 335.8113715 E
> 7 363.4446544 F #
> 8 374.9780189 G f
> 9 418.7127852 A f
> 10 432 A
> 11 467.5484633 B f
> 12 482.3854042 B
>
>
> Based on 440 Hertz
>
> 440 X 1.11663288
>
> 1 C 265.8740675
> 2 C # 274.3111776
> 3 D 296.8837258
> 4 D # 306.3048802
> 5 E 331.5101298
> 6 E # 342.0301006
> 7 G f 370.175111
> 8 G f 381.9220563
> 9 A f 426.4667257
> 10 A 440
> 11 B f 476.2067682
> 12 B 491.3184672
>
> ===============
> Real Intervals (Steps) of Natural Melody, providing
> real consonances or chords of natural harmony
>
> OCTAVE RATIO, CIRCLE CIRCUMFERENCE; REPRESENT BY
> LOGARITHM OF 2 [viz. .30103]
>
> (.30103 x 2 = .60206 + .09582 = .69788)
>
> space or quantity of two octaves and a sharp 3rd be
> taken
>
> Circle Circumference to diameter = 3.1416 is to 1
>
> Octave = Logarithm of 2 = .30103
>
> Two Octaves = (.30103 x 2 = .60206 )
> Sharp 3rd = 0.09582
>
>
> The Circles Circumference to its diameter is as Two
> Octaves + Sharp 3rd to the Octave?
>
> 3.1416 to 1: So, as 3.1416 is to 1, so is .30103 to
> .09582
>
> Circles Circumference to Diameter is as the Octave
> (.30103) to Sharp 3rd (.09582)
> ===========
>
> Octave contains five larger notes and two lesser notes
> (but no actual major or minor notes)
>
> Half the diameter = the larger note = .04791
>
> There are four 5ths, each = .17447 .17447 x 4 = .69788
> >>>Somehow, Harrison says this equals two octaves and
> a sharp 3rd, but I don't see the connection, how is
> this so?
>
> Each of the four 5ths can be generated by subtracting
> 5*radius from the circumference, the remainder = a
> spatial quantity equivalent to two equal lesser notes
> which = .06148, divide this by 2 = .03074
>
> Subtract the lesser note (.03074) from the greater
> note (.04791)
> .04791 - .03074 = .01717
> This is not the same as minor half note, but may be
> called a tone major and or a tone minor
>
> A fifth must contain three of the larger notes and one
> of the lesser
> Fifth = .04791+.04791+.04791+.03074 = .17447
>
> There are no ratios exisitng in this musical scale,
> except that of the octave itself.
>
> Chords are best described logarithmically
>
> We can derive the ratio of any chord by subtracting
> the logarithm of the lesser number from that of the
> greater
> This approach may cause differences from conventional
> ratios
>
> The resulting sounds will tend to 'sound better' and
> this will be observed via the latitude of the flatness
> or sharpness of the resultant
>
> For example; (Note: the 5th will sound better than
> the 6th)
>
> Interval Latitude
> -------------------------------------------------------------------
---------------------------
> 5ths .00162 Flat
> 4th (complement of the 5th to the octave) .00162
> Sharp
> 3rd Sharp .00109 Flat
> 6th Sharp .00053
> ----------
> Four 4ths and a sharp 3rd = Two octaves
> ----------
>
> Robert Smiths Observations;
>
> In a monochord, the major 3rd to the fifth is as as
> the diameter of a circle, to its circumference.
> The octave = five meantones and two limmas, making it
> a little bigger than six such tones, or three major
> 3rds.
> >>> When he says 'six such' tones, does he imply six
> 'meantones' ?
>
> The circumference of the circle is a little larger
> than three of its diameters.
>
> Time of a single vibration = 8 31/12/1748
> >>> Any idea what this actually means, is?
>
> Two musical strings with the same thickness, density
> and tension, differing in length only, the times of
> their vibrations are proportional to their lengths.
> >>>What happens if one or more of the other factors
> differ; thickness,density, tension?
> =============
>
> Came across the following, which seems to lend support
> to your Lucy Tuning based on pi and the circle, your
> thoughts on this?
>
> Octave as a Sine Wave and 1.2 AC waveforms
>
> When an alternator produces AC voltage, the voltage
> switches polarity over time, but does so in a very
> particular manner. When graphed over time, the "wave"
> traced by this voltage of alternating polarity from an
> alternator takes on a distinct shape, known as a sine
> wave:
> +- Time Graph of AC voltage over time (the sine wave)
> In the voltage plot from an electromechanical
> alternator, the change from one polarity to the other
> is a smooth one, the voltage level changing most
> rapidly at the zero ("crossover") point and most
> slowly at its peak. If we were to graph the
> trigonometric function of "sine" over a horizontal
> range of 0 to 360 degrees, we would ¯nd the exact same
> pattern:
>
> 1.2. AC WAVEFORMS 7
>
>
> Angle Sine(angle)
> in degrees
>
> 0 0.0000 -- zero
> 15 0.2588
> 30 0.5000
> 45 0.7071
> 60 0.8660
> 75 0.9659
> 90 1.0000 -- positive peak
> 105 0.9659
> 120 0.8660
> 135 0.7071
> 150 0.5000
> 165 0.2588
> 180 0.0000 -- zero
> 195 -0.2588
> 210 -0.5000
> 225 -0.7071
> 240 -0.8660
> 255 -0.9659
> 270 -1.0000 -- negative peak
> 285 -0.9659
> 300 -0.8660
> 315 -0.7071
> 330 -0.5000
> 345 -0.2588
> 360 0.0000 -- zero
> ----------------------------------
> 43.1 Spherical Music
>
> ..."in the spiral of perfect fifths, there the major
> and minor scales are the reverse of each other when
> the arithmetic vibration frequency ratios are put into
> geometrical form in a spiral diagram".
>
> The 3-4-5 triangle can be written thus:
> Representing Single and Double Squares.
>
> 3^2 = 9
> 4^2 = 16
> 5^2 = 25
>
> 3 x 96 = 288 cps (3rd harmonic of 96 cps).
> 4 x 96 = 384 cps (4th harmonic of 96 cps).
> 5 x 96 = 480 cps (5th harmonic of 96 cps).
>
>
> "The 12 Lynus Tonal System or the cycle of fifths of
> China and the 22 natural intonation Scruti scales of
> India, are arithmetical progressions... the musical
> scale systems of different peoples can be
> mathematically oriented into a spiral complex when
> translated into digits 1, 2, 3, 4 and 5. This means,
> as already noted, that a "spherical music" may
> eventually express the evolution toward a world-wide
> acceptance of a common tonal system, perhaps the
> Mercator, equal-tempered 53 tones-to-the-octave scale
> within the overall mathematical perimeter of the
> circle of fifths"...
>
> (Correspondences, File, "30 Unique Divisions").
>
> ..."the binary aspect of the octave intervals of the
> tonal spiral requires a Boolean algebra".
> Dr. Andrew Pikler, International Congress on
> Acoustics, Tokyo, Aug. 1968, Generation and Plotting
> of Musical Tone System with the Digital Computer.
>
> ..."possibility of a "double helix" in the domain of
> the musical tone system...the "sharps" and "flats'
> furnish helical formations, each separately split with
> the Pythagorean comma"...
> Dr. Andrew Pikler.
>
>
>
> 43.2 "1, 2, 3 and 5 "
>
> "Harold's concept of giving a measure number to
> opposite forces (yin-yang) in order to describe them,
> seems to relate to music, that is:
>
> If a note in a minor key is at once the same sound and
> a different sound, it must be more or less of one
> sound. Therefore...if...a minor key...represents
> negative, and a major key represents positive...
> Harold's theory assigns 2 as the measure number of the
> minor, 3 as the measure of the major, and 5 as the
> measure of wholeness or the 1 sound that can be played
> in either the major or minor key- but cannot be played
> in both at the same instant of time".
>
> If the foregoing is true, we ourselves cannot make the
> 1 sound, with the measure number 5, until we strike a
> chord:
>
> 3 (+ major) and 2 (- minor).
>
> Harold's description of the nuclear force of the
> hydrogen atom can be read as 3(+) / 2(-). It would
> say, "that a chord can be struck using a formula that
> could read:
>
> 2(-) / 3(+), or 3(+) / 2(-),
>
> these 2 would not sound alike, but either would convey
> the sound of wholeness.
> ================================
> Natural Musical Scale Derived From Vesica Pisces
>
>
> In the process of going from One to Two, a Circle
> with a diameter length has a single vibration, but
> when another Circle is added interesecting it in its
> center, a Vesica Pisces is formed. This is the area
> that it shares which looks like the female generative
> organ, and from where in sacred geometry all things
> come from. Part of the reproductive creative process
> How natures multiplication and reproduction came about
> from One to Many Vibrations that we see and feel and
> hear.
>
> If there was a string across the diameter of our
> first circle, that could be our base frequency of
> 1,The second circle makes the Vesica Pisces, and
> extends the distance to 1.5 times the first. Therefore
> any frequency times 1.5 gives a pitch that is in tune
> with the original pitch or vibration.
> And then if we continue on with the construction
> of a new Vesica Pisces by the mating of two circles,
> it makes a new larger circle of 2.25 times larger than
> our original circle. And so if we half this diamter,
> into a range between 1 and 2, it is 1.125. Therefore
> any original frequency sounds in tune with that
> frequency times 1.125. And lo and behold we can go on
> making twelve notes between 1 and its octave of 2.
> 1.125 being the B note on the piano.
> Therefore using this same process we get a table,
> as seen on the left, with each of its corresponding
> notes that are all harmoniously in tune with each
> other, and now you know why.
>
>
> Circle Diameter Ratio Note Hertz
> 1 = 1 1 A 440
> 2 = 1.5 1.5 E 659
> 3 = 2.25 1.125 B 494
> 4 = 3.375 1.69 F# 743
> 5 = 5.062 1.265 C# 554
> 6 = 7.539 1.88 G# 827
> 7 = 11.39 1.42 D# 622
> 8 = 17.08 1.06 A# 466
> 9 = 25.62 1.59 F 698
> 10=38.44 1.19 C 523
> 11=57.66 1.79 G 787
> 12=86.49 1.35 D 582
>
>
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🔗Andreas Sparschuh <a_sparschuh@...>

3/25/2008 1:52:30 PM

--- In MakeMicroMusic@yahoogroups.com, Dan Amateur <xamateur_dan@...>
wrote:
>
>
> Circle Diameter Ratio Note Hertz
> 1 = 1 1 A 440
> 2 = 1.5 1.5 E 659
> 3 = 2.25 1.125 B 494
> 4 = 3.375 1.69 F# 743
> 5 = 5.062 1.265 C# 554
> 6 = 7.539 1.88 G# 827
> 7 = 11.39 1.42 D# 622
> 8 = 17.08 1.06 A# 466
> 9 = 25.62 1.59 F 698
> 10=38.44 1.19 C 523
> 11=57.66 1.79 G 787
> 12=86.49 1.35 D 582
>
>
that absolute pitches can also be interpreted as "3x+-1"
Collatz-sequence too:

A: 440Hz 220
E: (3A=660>) 659
B: (3E=1977>) 1976 988 494 247
F#: (3B=741 370.5<) 371.5 743 (>742 371 (>370 185=555/3))
C#: (3*185=555>) 554 277
G#:= 3C# = 3*277 = 827 (<828 414 207)
D#: (3*207=622>) 621 622 311
A#: (3D#=933>) 932 466 233
F: (3A#=699>) 698 349
C: (3F=1047>) 1046 523 (>524 262 131)
G: (3*131=393 786 <) 787 (> 786 388 194 97)
D: 582 291 = 3*97 (>292 146 73)
A: (3*73=219<) 220cps

that's chromatically in ascending order:

!xamateur_dan12.scl
!
dan_amateur's 12 absolute pitches
!
554/523 ! C#
582/523 ! D
622/523 ! D#
659/523 ! E
698/523 ! F
743/523 ! F#
787/523 ! G
827/523 ! G#
880/523 ! A
932/523 ! Bb
988/523 ! B
2/1
!

have a lot of fun with that
A.S.