Scalecoding - how it woks for meantone and could for other systems.

In response to Carl's suggestion, here (in text form) is how scalecoding works for all meantone-type tunings

>>
>
> I sure did, and I promise you I'm not the only one. Why not
> explain it on the list?

I will see below:

>
>
> The first problem I ran into is that you seem to assume
> there are recognizable 5ths and 4ths.

In meantone, there is usually little dispute about where the fifth (and hence the fourth) is located - (generally the interval nearest to 700 cents).
In many others, it is also usually clear.
For those tunings which are ambiguous or derived from an entirely "alien" mapping; the same principle could be applied by specifying the reference for the chain.
Chains could be shown from any reference interval, it just happens to be convenient to use the fifth for meantones and many others.

Format of scalecdoing is : x/m1m2..../T
x= The expanse i.e. the number of steps along the chain of fourthsand fifths from the flatmos fourth note to the sharpmost fifth for each scale.
m1m2 etc. = position of missing notes.
T = Tonic.

This is an example of how it works for a meantone-type tuning; and can be applied to any 5 Large 2 small interval = an octave, and produces a unique and unambiguous result
in both directions - from notenames to coding; and coding to notenames.

e.g. scale coding is 9/389/7
Notes in ascending pitch order = C-Db-Eb-F-Gb-A-Bb
Arranged in chain of fifths - left to right = Gb-Db-Ab-Eb-Bb-F-C-G-D-A
Chain is of ten notes i.e. 9 steps (from Gb to A)
Notes in positions 3 (Ab), 8 (G) and 9 (D) are missing
The Tonic is note in position 7 (C)

To analyse a collection of notes.

1. List all the different notes which are used in the piece regardless of octave.

2. Arrange the note names in order of fourths (flats) in one direction and fifths (sharps) in the other, leaving blank spaces where notes are missing. [Sequence ascending in fifths is: Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# F## C## etc.] The fifth may be considered as the dominant, and the fourth as the sub-dominant.

3. Count the total number of steps between the fourthmost (flat) and fifthmost (sharp) note. This is the extent of the string, or chain of fourths/fifths (x).

4. List the missing notes. Identify them by numbering the flatmost as 1 and the following as ascending numbers moving through fifths. Each of the missing notes may be defined as between 2 and x.

5. The megamode may now be defined by the number of steps and the position of the missing notes (m1, m2, etc.). Eg. x=12 m1=2; m2=5; m3=9; and m4=11. Therefore there are four notes missing in the sequence. The extent is (x)=12. So there are thirteen notes of which four (m1 to m4) are missing leaving 13-4 = 9 notes. In this case numbers 1, 3, 4, 6, 7, 8, 9, 10, and 12. 6.

6.The mode is determined by which of the notes is chosen as the start of a sequence of ascending frequencies. This starting note may be identified by stating its position on the chain of fifths. For example, if the notes were six consecutive steps (Eg. F C G D A E B); these pitches could be arranged in seven modes of different ascending pitch orders.

5. The megamode may now be defined by the number of steps and the position of the missing notes (m1, m2, etc.). Eg. x=12 m1=2; m2=5; m3=9; and m4=11. Therefore there are four notes missing in the sequence. The extent is (x)=12. So there are thirteen notes of which four (m1 to m4) are missing leaving 13-4 = 9 notes. In this case numbers 1, 3, 4, 6, 7, 8, 9, 10, and 12. 6.

6.The mode is determined by which of the notes is chosen as the start of a sequence of ascending frequencies. This starting note may be identified by stating its position on the chain of fifths. For example, if the notes were six consecutive steps (Eg. F C G D A E B); these pitches could be arranged in seven modes of different ascending pitch orders.

7. The key of the scale and scale is determined by the tonal center, which may defined as C,D,E,F,G,A, or B with the appropriate sharps or flats. The scale may then be listed in ascending frequency order by note name.

8. A scale or mode may therefore be defined as: Number of steps in chain (x)/position(s) of missing notes (counted from fourths towards fifths)/Position of tonic (counted from fourths towards fifths). Eg. The scale and mode described as 5/25/3 could give the notes F-G-D-E from the chain F-C-G-D-A-E. Using the third note of the chain (G) as the starting note giving a scale of G-D-E-F or the mode of I-V-VI-bVII.

N.B. This system is for Octave Ratio = 2.
5 large + 2 small intervals = one octave.

In the following comments please realise that I am thinking about this from a meantone perspective.

Nevertheless I believe that developments from this type of system can be applied to JI, equal temperaments, and multi-dimensional patterns.

Now I need to start thinking/working.
Within the next few days I intend to post my initial proposals for a similar system which can also embrace other types of tuning systems:
to classify them and devise a coding for the scales that they can produce.

To cover:
Various other quantities of two interval sizes per octave. (e.g. x Large + y small intervals per octave)
Similar systems with more than two interval sizes (e.g. w Gigantic + x Large + y small + z tiny) etc.
JI tunings
Equal temperaments
Non-octave systems
Hybrid systems
other tuning systems that I have yet to seriously think about ...

Charles Lucy - lucy@harmonics.com
------------ Promoting global harmony through LucyTuning -------
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--- In tuning@yahoogroups.com, Charles Lucy <lucy@h...> wrote:
> In meantone, there is usually little dispute about where the fifth
> (and hence the fourth) is located - (generally the interval nearest
> to 700 cents).
> In many others, it is also usually clear.
> For those tunings which are ambiguous or derived from an entirely
> "alien" mapping; the same principle could be applied by specifying
> the reference for the chain.
> Chains could be shown from any reference interval, it just happens to
> be convenient to use the fifth for meantones and many others.
>
> Format of scalecdoing is : x/m1m2..../T
> x= The expanse i.e. the number of steps along the chain of fourthsand
> fifths from the flatmos fourth note to the sharpmost fifth for each
> scale.
> m1m2 etc. = position of missing notes.
> T = Tonic.

Have you much experience working with scales NOT generated by
approximate 3:2's?

I really don't see how this scalecoding of yours is very similar to my
purpose (since you did suggest that we/I consider scalecoding instead
of inventing new terminology). I propose a small set of clear and
logical terms that correspond with universal (among EDOs, anyway)
scale shapes. Scalecoding is a quasi-mathematical formula that gives
you a mode-specific string of numbers that tells you...what? I cannot
see how this would be useful in composition or in comparing tunings.
All it gives you is a string of numbers to identify a mode of a
particular scale that uses a given interval as a generator, in what I
would call a very non-intuitive way. What, do you think, the average
musician would find easier: learning the definitions of a few terms
that refer to a scale shape, or learning this very abstract formula of
yours? Just because your method makes reference to "chains of
fifths/fourths" and works with conventional notation does not
necessarily make it easier.

> This is an example of how it works for a meantone-type tuning; and
> can be applied to any 5 Large 2 small interval = an octave, and
> produces a unique and unambiguous result
> in both directions - from notenames to coding; and coding to
> notenames.

Assuming, of course, that you know what notenames to use! For most of
the tunings I use/am interested in, the note-naming varies depending
on what scale I am using.

>
> e.g. scale coding is 9/389/7
> Notes in ascending pitch order = C-Db-Eb-F-Gb-A-Bb
> Arranged in chain of fifths - left to right = Gb-Db-Ab-Eb-Bb-F-C-G-D-A
> Chain is of ten notes i.e. 9 steps (from Gb to A)
> Notes in positions 3 (Ab), 8 (G) and 9 (D) are missing
> The Tonic is note in position 7 (C)

So let me get this straight: you list the notes in a piece according
to a given note-naming system. Then you arrange them into a chain of
generators, fill in the blanks, and assign ordinality to each number.
Then you make the scalecode which is based on how long the chain of
generators is, which notes are missing from it, and which note is the
"tonic" (presumably if you were using an atonal scale you would leave
that one blank)?

For this to be universal, you would have to also include in the
scalecode: the generating interval, the naming system, and the type of
tuning system (EDO, ED-nonO, circulating, JI etc.). There is nothing
inherent in the code that tells you the shape of the scale without
first actually writing out the scale, unless you memorize all the codes.

>
> Nevertheless I believe that developments from this type of system can
> be applied to JI, equal temperaments, and multi-dimensional patterns.
>
> Now I need to start thinking/working.

Good luck! Considering that most JI scales I have seen on this board
are indeed multidimensional (not using a single generator but rather a
CPS/tonality diamond). I think that once you have accomplished making
this system universal, it will be so complex that it will defeat your
previous goal of "not wanting to overwhelm the non-microtonalist with
too much math". But as I have said, we all must do what works for us.
If scalecoding floats your boat, more power to you! In the meantime,
I will pursue my (very simple) naming of scale families.

Best,

_igs

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:

> Considering that most JI scales I have seen on this board
> are indeed multidimensional (not using a single generator but rather a
> CPS/tonality diamond).

I agree that most JI scales posted here don't use a single generator,
and a few have been CPSs, and a few have been tonality diamonds, but
most have been neither CPSs nor tonality diamonds.

> I agree that most JI scales posted here don't use a single generator,
> and a few have been CPSs, and a few have been tonality diamonds, but
> most have been neither CPSs nor tonality diamonds.

Okay, gross over generalization on my part, as I left out harmonic
series and means that I am not familiar with. But I haven't
encountered (personally, and I admit I am very very far from being an
authority on the subject) hardly any that I recognized as being derived
from one of those three (unless hexanies and eikosanies et al are not
CPS? Forgive me if not, JI isn't really my thing). Nevertheless, my
point was merely to illustrate that very few people generate JI systems
from a single interval, thus requiring a more complex (ergo more
difficult to learn) version of Lucy's scalecoding.

Best,

-Igs

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:
>
> > I agree that most JI scales posted here don't use a single
generator,
> > and a few have been CPSs, and a few have been tonality diamonds,
but
> > most have been neither CPSs nor tonality diamonds.
>
> Okay, gross over generalization on my part, as I left out harmonic
> series and means that I am not familiar with.

Means?

> But I haven't
> encountered (personally, and I admit I am very very far from being
an
> authority on the subject) hardly any that I recognized as being
derived
> from one of those three (unless hexanies and eikosanies et al are
not
> CPS?

Yes, hexanies and eikosanies are perfect examples of CPSs.

> Forgive me if not, JI isn't really my thing).

It seems to me that the majority of JI scales that have been posted
here are Constant Structure scales (or what is much the same thing,
Periodicity Blocks).

> It seems to me that the majority of JI scales that have been posted
> here are Constant Structure scales (or what is much the same thing,
> Periodicity Blocks).

Ah, well go figure then. I have no idea what these are, so of course
I wouldn't really notice them. Would these pose problems in Charles
Lucy's scalecoding system?

-Igs

> In response to Carl's suggestion, here (in text form) is how
> scalecoding works for all meantone-type tunings
//
> Format of scalecdoing is : x/m1m2..../T
> x= The expanse i.e. the number of steps along the chain of
> fourths and fifths from the flatmos fourth note to the
> sharpmost fifth for each scale.
> m1m2 etc. = position of missing notes.
> T = Tonic.
//
> 5. The megamode may now be defined by the number of steps and
> the position of the missing notes (m1, m2, etc.). Eg. x=12
> m1=2; m2=5; m3=9; and m4=11. Therefore there are four notes
> missing in the sequence. The extent is (x)=12. So there are
> thirteen notes of which four (m1 to m4) are missing leaving
> 13-4 = 9 notes. In this case numbers 1, 3, 4, 6, 7, 8, 9, 10,
> and 12.

Now that wasn't hard, was it?

> In the following comments please realise that I am thinking
> about this from a meantone perspective.
>
> Nevertheless I believe that developments from this type of
> system can be applied to JI, equal temperaments, and multi-
> dimensional patterns.
>
> Now I need to start thinking/working.

I look forward to your solution.

-Carl

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:
>
> > It seems to me that the majority of JI scales that have been posted
> > here are Constant Structure scales (or what is much the same thing,
> > Periodicity Blocks).
>
> Ah, well go figure then. I have no idea what these are, so of course
> I wouldn't really notice them. Would these pose problems in Charles
> Lucy's scalecoding system?
>
> -Igs

Yes, but they would pose fewer problems than the Tonality Diamonds or
CPS scales, since they are "epimorphic" or similar to 2D scales of the
sort found in my paper. Take a look at:

http://sonic-arts.org/td/erlich/intropblock1.htm

> Yes, but they would pose fewer problems than the Tonality Diamonds or
> CPS scales, since they are "epimorphic" or similar to 2D scales of the
> sort found in my paper. Take a look at:
>
> http://sonic-arts.org/td/erlich/intropblock1.htm

Ah so! Should have read that much sooner, if for no other reason than
to recognize some of the arcane-looking discussions on this list.
Very interesting concept, these "periodicity blocks". Should I ever
involve myself in JI, I'm sure they will be most useful.

I expect these could be incorporated into scalecoding somehow, but Mr.
Lucy definitely has his work cut out for him!

-Igs