Re: What's the name of this scale please?

Temperament schema is always relevant to scales, because a temperament
schema IS a scale.  For example, the 12tet schema yields the 12tet chromatic
scale - and all of its subscales.  However, a logical component arises once
a temperament has been selected and established.  That logical component is
populated by the set of pitches - whatever letter name or number you assign
to them.  That set of pitches is commonly represented by a corresponding set
of letter names - or numbers - which constitute an abstraction of the
specific frequencies involved.  It is at that level of abstraction that
music notation and solfeggio impart meaning.  From that perspective, any
musical entity can be conceptualized as existing in four - at least -
layers.  For the sake of discussion, let us call them here the physical (or
acoustical if you prefer) layer, the logical layer, the visual layer, and
the auditory layer. Other views and terminology undoubtedly exist.

We become aware of music at the auditory layer.  Notated music exists at the
visual layer.  An understanding of pitch relationships and their association
with notation at the visual layer operates at the logical layer.  The
musical sounds themselves exist in the physical acoustical environment. What
I am speaking to deals with the logical layer.

This logical conceptualization is nothing new, and I consider Guido to be
its creator (as far as I know.  You may know of someone earlier, since you
are quite knowledgeable in such matters, and I could be pursuaded that the
diacritic marks of Aristoxenus might qualify).  He developed a logical
system of visual signals that represented he pitches that the musicians who
learned them were to produce. (I consider this to be a defining moment in
the systematic approach to musical notation.  However, history lends itself
to many interpretations.)  One of the upshots is that most formally trained
musicians learn some kind of systematic, symbolic representation of music in
conjunction with learning to play their chosen instruments.  Learning about
the physics of a vibrating string or air column comes much later.

In my experience, when a musician asks the name of a scale, he or she will
generally represent it as notes on a staff or by letter names, just as Mr.
Edwards did at the beginning of this thread.  What they are usually asking
for is some logical way to conceptualize and remember that object.  Now, at
this juncture there are two ways possible to represent such an object, and
they exist at the logical layer of conceptualization.  The one way is to
discuss the intricacies of temperament at the physical layer, including the
physical frequencies, the ratios involved, and a discussion of the nature of
the individual intervals that arise as a result of that specific temperament
schema.  That schema itself is usually represented by some sort of name such
as 12tet, Mean Tone, etc.  Those names encapsulate all of the information
with which they are associated - they are an abstraction.  However, that is
usually far too much information.  The kind of musician who usually is very
interested with scales as of late - meaning for me, the last 20 or 30
years - come from a jazz background, and from that perspective, Masaya
Yamaguchi has written a very intelligent work entitled "The Complete
Thesaurus of Musical Scales" published by Chas. Colin Publishers, NY.
Yamaguchi perceptively understands that the exact physical characteristics
of a temperament schema are not what the reader is interested in, and
presents his work in an abstract numeric notation that represents the
intervallic architecture of the assorted scales.  When a practical jazz
oriented musician asks about a scale, what he or she is interested in is the
logical architecture rather than the exact set of frequencies.  As a
simple-minded example, when asking about a major scale, the musician is not
asking about C-major, D-major, F#-major,  or any-particular-key-major scale.
What that musician is asking for is the logic that underpins the scale along
with a way to classify and remember it.  My only objection to Yamaguchi is
that the work is only complete with respect to a chromatic scale consisting
of 12 elements.  However, that is the area with which most practical,
working musicians - myself included - are constrained, and as a result,
think.  As an aside, Yamaguchi establishes a mapping of his work on the
nomenclature of Alan Forte, and makes a brief comment with that regard in
Appendix I.

My perception of Yamaguchi's method is that he employed the tedious and time
consuming application of the associative law of addition to "flesh out"
Forte's work.  That is fine, and it does exactly what is required within the
domain of a 12 element chromatic scale.  I hasten to add Dr. Schulter, that
none of Yamaguchi concerns itself with the exact pitches that are
represented.  It's objective exists purely at the logical layer of
abstraction.  In a similar manner, the BAD Catalogue - an unpublished work
from the early '80s - exists purely at the logical layer, and is a technique
that anyone can learn.  Since any subscale of any number of elements based
upon any sized chromatic scale can be calculated "ad hoc," it is not
necessary to carry any sort of reference with you at all times.  In point of
fact, any positive decimal integer encapsulates all of the practical,
logical architecture for any scale of any number of elements that is a
subset of any sized universe.  It is actually not even necessary to define
the number of elements for a given scale since that can be gleaned from
inspection of the binary digits.  Secondly, it is not even paramount that
the exact size of the universe from which the scale is derived be indicated,
since the universe itself can be represented by a number - 2047 for a 12
element universe (chromatic scale).  One can determine the minimum size U
(set universe)  necessary by comparing the decimal value of the object scale
with the maximum decimal value of some particular sized universe.  For
example, the minimum size of a universe necessary to derive b.a.d. 3029 -
just a number off the top of my head - will be 13 elements because 2047 <
3029 < 4095, 4095 being the decimal representation of b.a.d.111111111111.
That upper number also defines the exact number of scales possible that are
subset to that particular U, understanding that any U is a subset of itself
by the identity principle.  Another feature is that the binary analogue
abstracts the individual scale elements directly rather than by their
intervallic relationship, which is actually a derived attribute.  This is
useful because only one binary analogue can define any particular decimal
number, and any decimal number can be represented by one and only one binary
analogue.  An upshot of this is that the binary digits are constant,
regardless of the size of a particular U.  For example, the scale that led
to this discussion is represented in decimal format as 729.  The binary
analogue of 729 indicates 7 elements - represented as binary 1's - and will
always have 7 elements in exactly the same relationship to one another,
regardless of the size of the set universe in which 729 exists as a possible
value.  b.a.d. 729 carries the same decimal value whether its analogue is
1011011001, 10110110010, 1011011001000, 10110110010000, or greater.  Notice
that the binary ones exist in exactly the same pattern, and always indicate
exactly the same scale degrees relative to the size of any particular U.
That is not to say that the derived intervallic relationships will be the
same among them relative to Us of different sizes.  That's where the
temperament schema becomes all-important.  b.a.d. 729 is not the same in
12tet as it would be from the temperament you discuss.  In fact, I think
that the way you are approaching that object is as a 7 element temperament
in and of itself, and would be characterized as b.a.d. 65 (Pythagorean), its
analogue being b.a.d. 111111.  You might have already noticed - since I am
satisfied that you an acute and perceptive musician - that the binary
analogue is a rhythm.





"Margo Schulter" <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]
> David Z. Kasler <[EMAIL PROTECTED]> wrote:
> > Sorry about the confusion I created by trying to do a calculation in my
head
> > before having a second cup of coffee.
> > The question you might be asking is:
> > "What does 729 have to do with C C# E F G G# A# C?"
>
> Hello, there, and the curious alternative answer that quickly occurred to
> me -- likely an aside to this thread -- is that if this scale is tuned in
> Pythagorean intonation (based on a chain of pure 3:2 fifths or 4:3
> fourths, respectively around 702 and 498 cents), then a diminished fifth
> such as C#-G has a ratio of 1024:729 (~588 cents), and an augmented fourth
> or tritone such as G-C# a ratio of 729:512 (~612 cents).
>
> In Pythagorean, using an octave of C-C as the reference, we have these
> ratios, with rounded values in cents and deviations from 12-tone equal
> temperament for those who might use this information to set up the tuning
> on a synthesizer or computer software to control a sound card or the like:
>
>
>          C       C#       E    F      G     G#      A#     C
> ratios: 1/1  2187/2048  81/64 4/3    3/2 6561/4096 59049/ 2/1
>                                                    32768
>
> cents:   0      114      408  498    702    816     1020  1200
> 12-et+/- C 0    C#+14    E+8  F-2    G+2    G#+16   Bb+20  C 0
>
> There are some interesting features of this scale like the diminished
> fourth G#-C at 8192:6561 (~384.36 cents), only about 1.95 cents narrow of
> a pure 5:4 major third (~386.31 cents). This contrasts with the more
> active regular Pythagorean major thirds such as C-E and E-G# at 81:64
> (~407.82 cents). Thus C#-F-G# is very close to a pure 4:5:6 (the
> Renaissance ideal), while C-E-G at 64:81:96 is a more active tuning of
> this kind of sonority in a typical 13th-14th century style.
>
> Anyway, in Pythagorean intonation, the number 729 helps define the integer
> ratio of a tritone (equal to three 9:8 whole-tones), and also its octave
> complement of the diminished fifth.
>
> Most appreciatively,
>
> Margo Schulter
> [EMAIL PROTECTED]