Using the arithmetic mean to explain consonant-sounding JI scales...

Continuing where I left off...
Using the arithmetic series formula (which naturally creates JI scales in a very simple way)
ratio = (1/octave)^x
..gives....
1.5 where x = 1 (3/2)
1.25 where x = 2 (5/4)
1.125 where x = 3 (9/8)
1.0625 where x = 4 (17/16)
Note: all the above are straight from the harmonic series and the 17/16 is not in common with diatonic JI which is: 1/1
9/8
5/4
4/3
3/2
5/3
15/8
2/1
...and, if we take 1.5 / ((1/octave)^x): AKA the same thing in reverse we get
1.411764 where x = 1 (24/17)

Here we get 24/17, which is not in any variation of diatonic JI that I know about.
So the final scale is
1
1.0625
1.125
1.25
1.411764
1.5 (period)

So what's missing? Note that the 4/3 typically used as F in the C diatonic scale is completely gone. And, so is the note C6, which conflicts with B (the nearest tone to the usual B is 1.5 * 1.411764 = 2.117476 above it and 1.25 * 1.5 = 1.875 below it).

I believe this is very important as in the pentatonic scale virtually any combination of notes can make a consonant chord while in the diatonic scale no combination of E5 and F5 or B 5 and C5 can make a consonant chord (one flaw with diatonic JI that really annoys me as it limits the number of chords I can make with a fixed/non-adaptive scale)..

So, hopefully, such a scale construction can finally allow us to us near-semitone type notes together in consonant chords within the same octave.

Any thoughts on the possibilities for this (and/or why you don't think it will work)?

-Michael

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> I believe this is very important as in the pentatonic scale virtually any combination of notes can make a consonant chord while in the diatonic scale no combination of E5 and F5 or B 5 and C5 can make a consonant chord (one flaw with diatonic JI that really annoys me as it limits the number of chords I can make with a fixed/non-adaptive scale)..
>

I have always thought that the inverted major 7th chords like BCEG and ACEF are far more consonant than the wolf-minor DFA in JI. Note that these are both just major 7ths (8:10:12:15) in the diatonic JI. Are these not considered consonant?

Billy

William> "I have always thought that the inverted major 7th chords like BCEG and ACEF are far more consonant than the wolf-minor DFA in JI."

From what I see, DFA is particularly nasty because the root note has 8 as the denominator (9/8) while the two following notes have 3 as theirs (4/3 and 5/3) and appear to make the mind confused over which notes (the root vs. the next two) constitutes the base tone since they both point toward different tones.

This is part of why I see the fact the arithmetic mean scale omits having F within the same 3/2 period as D or A, for example, is so important.

Come to think of it BCEG (in 5-limit JI diatonic) also has x/8 as the common denominator and is thus a straight series of partials from the harmonic series...so it is obvious that would be consonant. I don't think ACEF work as well though...though A C and F have a common denominator of 3 E (5/4) does not seem to.
Although at least in this case (unlike DFA) most of the notes in the chord aim toward a single common denominator/u-tonal relationship...and it sounds consonant to me though not as consonant as BCEF.

A weird coincidence in the arithmetic mean scale is the chord
E F# G A where every single tone shares a common denominator except F#. However, F# has a value of 24/17 which is just a tiny bit away from 23/16 which would give it a common denominator.

To make things more obvious how this scale works, here's a subset of the scale that fits almost perfectly into 12TET that includes C D E F# G repeated where the period/repeating-interval is G AKA 3/2.

So a close equivalent to the arithmetic mean scale in 12TET is:
>>
C4 D4 E4 F#4 G4 (g is the period) A4 B4 C#5 D5 (d = period) E5 F#5 G#5 A5 (a = period) B5 C#6 D#6 E6 (e = period) F#6 G#6 A#6 B6 (b = period) C#7 D#7 F7 F#7 (f# = period) G#7 A#7 C8 C#8 (c# = period)
<<

Notice how the scale follows the circle of 5ths (since the 5th is the period in this scale system) and consistently avoids having more than one note per triad that has a different common denominator than the other two notes (per triad)...thus making even the worst-case chord scenarios at least in step with your BCEF example chord, so far as I can tell.

-Michael

I have used exactly this scale 26 years ago in one of my works, those
time I did some research about such modal chains smaller than octave.
This one based on Lydian tetrachord is interesting from compositional
point of view, as there's no octave to the first note of every cycle
which can be intentionally emphasized to get an interesting
"confusing" effect. For me personally Lydian scale was always
important, as lot of Moravian music is based on it, it's the most
North/Western occurance of this shepherd's flute scale in Europe,
where this type of Eastern, Balcan, Carpathian culture ends. (Moravia
is Eastern half of Czech Republic, with very rich and old culture
going to Grand Moravian Empire, and connected to Byzantion, whereas
Northern Moravia = Silesia is culturally oriented to Poland. Bohemia
is oriented to German culture. For example Alois Hába, Leoš
Janáček were composers of Moravian origin.)

Just small repair: there should be E#7 in B period, and H#8 in F#
period.

Daniel Forró

On 18 May 2009, at 12:20 PM, djtrancendance@... wrote:
> To make things more obvious how this scale works, here's a subset
> of the scale that fits almost perfectly into 12TET that includes C
> D E F# G repeated where the period/repeating-interval is G AKA 3/2.
>
> So a close equivalent to the arithmetic mean scale in 12TET is:
> >>
> C4 D4 E4 F#4 G4 (g is the period) A4 B4 C#5 D5 (d = period) E5 F#5
> G#5 A5 (a = period) B5 C#6 D#6 E6 (e = period) F#6 G#6 A#6 B6 (b =
> period) C#7 D#7 F7 F#7 (f# = period) G#7 A#7 C8 C#8 (c# = period)
> <<
>
> Notice how the scale follows the circle of 5ths (since the 5th is
> the period in this scale system) and consistently avoids having
> more than one note per triad that has a different common
> denominator than the other two notes (per triad)...thus making even
> the worst-case chord scenarios at least in step with your BCEF
> example chord, so far as I can tell.
>
> -Michael
>

Daniel,

   So the whole use of the results of the formula (1/octave)^x + 1 to generate a JI-complaint scale have already been discovered (apparently by you over 20 years ago).  Cool. 

   A few questions:

A) What other scales did you find in your experimentation?
B) What other innovations have been made in research of modal chains not related to the octave (links would be much appreciated)?

C) You said "Just small repair: there should be E#7 in B period, and H#8 in F# period." 
   Pardon my newbie-ism concerning this notation but
       I) What's the difference between an E# and an F (I have a vague idea what you're talking about)?
       II) What on earth is an H# (in this case, I
truly have no clue what you're talking about)?

-Michael

Michael,

I didn't use microtones for these modal chains, it was done in the
frame of 12 ET. So I didn't need your hi math formula :-) And I don't think I was the first composer using this principle, despite the fact
I'm not aware about any work of 20th century music using similar
system. Anyway it's nothing special, just a way how to organize basic
material for composition. Then I have worked with it in a modal,
serial or intervallic way. Such scales are only a fraction of my
compositional language.

A) Concerning scales made from such chains, there's an unlimited
number of them. For me it was enough to find how to create them. When
I will need some, it can be easily done. Any interval can be used
which allows those "shifts" thru standard octave terrain, be it pure
or tempered, or microtonal. In the case of 12 ET the smallest
interval should be fourth, then scales can be constructed from
tetrachords (another field of my interest).
I was mainly interested in those scales which avoid octave interval
(for example tetrachordal structure 131 - C Db E F Gb A Bb Cb D Eb Fb
G Ab... - first octave step C is missing). Anyway when octave is
there, it's often a different "grade" in the chain period, and it can
be used intentionally to confuse listeners and hide real periodicity
blocks.

Fourth-ave and fifth-ave are the most usable as they go through all
circle and the resulting scales have big range. That's good for
piano, organ, orchestral music. Interval 10 halftones allows to
create scale with 5 octaves span. Interval 9 creates three octaves
scale (root notes of 4 periods create diminished seventh chord), 8
goes to 2 octaves (root notes of 3 periods create augmented chord).
Of course only in the case that last tone of one period is the first
tone of the next period. There's always possibility to have bridge
interval.

I like a combination of modal and serial principles, so I focused
mainly on scales which use all 12 tones. And it's like to find a
small treasure when some mathematically beautiful scale is
discovered, combining modal chains with 12tone serialism, avoiding
normal octave in its internal structure, and additionally having some
more features like symmetry, and musically interesting subsets as
well. For example this one:

C-Db-Eb-Gb-Ab-A-B-D-E-F-G-Bb-(C)

Period is 8 halftones, no bridge interval, so "root" notes of periods
are C, Ab, E, and after two standard octaves whole scale is repeated.
And period intervallic structure is 1232, three times repeated. We
can find pentatonic cells 232, major/minor chords 353, quartal chord
555, all 12 dominant seventh chords, diminished chords, minor
tetrachords, tritone relations, symmetric chords like 35153, 35353,
35653, 52125, 53235, 53535, 53735, 55355, 2533352, 2353532...

Principle can be easily used also for scales with periodicity block
bigger than octave. For me personally these scales are more
interesting, especially when they are constructed in such a way they
avoid standard octave step and have different tones in higher range
(behind octave). It's more easy to create 12tone series and get a
synthesis of modal and serial principles. I like to make it even more
interesting and work with intervallic method in composition.
One of possible scales with period 14 halftones, "serpent-like"
intervallic structure 14131211, no bridge interval between period,
with span of 7 standard octaves:
C-Db-F-Gb-A-Bb-C-Db-D-Eb-G-Ab-B-C-D-Eb-E-F-A-Bb-Db-D-E-F-F#-G-B-C-D#-
E-F#-G-G#-A-C#-D-F-F#-G#-A-A#-B-D#-E-F-G#-A-B-(C)

As you can see this scale has often starting C in higher octaves
which can be again emphasized in composition.
To make this scale more interesting, every second period can have
inverted structure 11213141.

When I think more far in this direction, it's possible to make scales
using different pattern in every period. For example chained scale
constructed from tetrachords - every tetrachord can have different
internal structure. As I have 20 different tetrachords in my
tetrachordal system, there's a lot of combinations possible, and when
bridge interval between tetrachords comes into the game, even more.
To make it more beautiful it's again possible to work with
intervallic structure of tetrachords, and order them symmetrically
for example.

B) I have no idea if somebody worked with similar scales.

C) On standard piano keyboard there's no difference between E# and F,
but in music theory is. To ignore enharmonic changes looks, hm, how
to say politely....
You have used diatonic pattern 2-2-2-1, major-major-major-minor
second, so you have to keep this intervallic structure in transpositions as well. Starting on B it must be B-C#-D#-E#-F#. By
using F you destroy intervallic structure, because D#-F is diminished
third, and F-F# augmented prime (chromatic halftone, not diatonic).
[If you want to use F, you must enharmonically change the other notes
as well to get result Cb-Db-Eb-F-Gb, which is also OK.]
Therefore starting on F# it should be F#-G#-A#-B#-C#, on C# - C#-D#-
E#-Fx-G# etc.

H# is just my mistake, as my native language (plus German and more
others) differentiates between B and H, B is English Bb, H is B.
Therefore I could compose some works on acronym B-A-C-H, or H-A-B-A,
Schumann could use a German name of Czech city Aš (A-S-C-H) and
Gustav Holst started Uranus in Planets with his own signature (G-(u)S-
(t)A-(v)-H-(olst)). Not possible with English note names :-) When I'm
tired, I sometimes do this mistake...

Daniel Forró

On 19 May 2009, at 2:26 AM, djtrancendance@... wrote:
> Daniel,
>
> So the whole use of the results of the formula (1/octave)^x + 1
> to generate a JI-complaint scale have already been discovered
> (apparently by you over 20 years ago). Cool.
>
> A few questions:
>
> A) What other scales did you find in your experimentation?
> B) What other innovations have been made in research of modal
> chains not related to the octave (links would be much appreciated)?
>
> C) You said "Just small repair: there should be E#7 in B period,
> and H#8 in F# period."
> Pardon my newbie-ism concerning this notation but
> I) What's the difference between an E# and an F (I have a
> vague idea what you're talking about)?
> II) What on earth is an H# (in this case, I
> truly have no clue what you're talking about)?
>
> -Michael
>
>

Daniel> "I didn't use microtones for these modal chains, it was done
in the frame of 12 ET. So I didn't need your hi math formula :-)"

I just find it a bit profound that all the tones in your scale are within a few cents of tones generated by a formula that works the same way as the one used for the latest PHI scale I favor, that's all.
And, to be honest, I don't consider evaluating (1/octave)^x + 1 that much high math, not much different in difficulty than the 2^(x/12) used to generate 12TET. :-D And, at least in my mind, high math isn't the point anyhow, the point is finding universally applicable formulas that make scales easily understandable to the mind in several different situations.

> "And I don't think I was the first composer using this principle,"
Which is why I was wondering about other research on the subject: so much has been done far as special modes and 12TET I'm sure a good few have run into something similar when playing with the spiral of 5ths.

>"Anyway it's nothing special, just a way how to organize basic
material for composition."
Nothing special so far as not really being or sounding micro-tonal. But the reason it interests me is that it fits the same (1/var)^x+1 formula as the Silver and Golden ratio scales I found to be the best I've heard. It makes me wonder if the human mind in general prefers scales generated using that formula and if that sty;e formula has more possible forms and/or wider applications. To be sure, I like the sound of the scale we both ran into a lot more than the usual major/minor scale construct and it appears to contain a wider range of chords.

> "Fourth-ave and fifth-ave are the most usable as they go
through all circle and the resulting scales have big range."
Right...now the question to me becomes how to summarize a 4th-ave in (1/var)^x + 1 form...or if that's possible at all.

> "I like a combination of modal and serial principles, so I
> focused mainly on scales which use all 12 tones."
Which is great, because it seems so often people give up on chromaticism in scales (as opposed to tunings) as being too hard to achieve and/or too dissonant.

> "When I think more far in this direction, it's possible to
make scales using different pattern in every period. "
That's pretty sweet...though it goes beyond what I know how to do IE how to you ensure a note several semi-tones (say, 25) up works with one much further down in harmony (or can you at all)?

> "B) I have no idea if somebody worked with similar scales."
Well, it's reassuring to know I'm not the only one with this "I came across this but am not sure who else did" problem. :-)

> "using F you destroy intervallic structure, because D#-F is
diminished third, and F-F# augmented prime (chromatic halftone, not
diatonic). "
So, if I have this right, this stems from the fact a diatonic semitone is larger than a chromatic one, and the difference between the two is enough to skew the observed interval into sounding like a different interval?

-Michael

On 19 May 2009, at 11:29 PM, Michael Sheiman wrote:
> I just find it a bit profound that all the tones in your scale are
> within a few cents of tones generated by a formula that works the
> same way as the one used for the latest PHI scale I favor, that's all.
>
Such coincidence is possible, many times it happened two or more
people came to similar results through different ways.

> And, to be honest, I don't consider evaluating (1/octave)^x + 1
> that much high math, not much different in difficulty than the 2^(x/
> 12) used to generate 12TET. :-D
>
That was only joke from my side...

> And, at least in my mind, high math isn't the point anyhow, the
> point is finding universally applicable formulas that make scales
> easily understandable to the mind in several different situations.
>
I don't believe much in universal theories of all, universal means
also uniform. I prefer diversity.

> > "And I don't think I was the first composer using this principle,"
> Which is why I was wondering about other research on the subject:
> so much has been done far as special modes and 12TET I'm sure a > good few have run into something similar when playing with the
> spiral of 5ths.
>
>
There's so much composers in 20th century, and certainly lot of them
tried to work with the old material in some new way, so I think also
somebody had to use similar methods. It would be tough job to go
through all composer's descriptions of their methods (if ever they
published their secrets), or do analysis...

> >"Anyway it's nothing special, just a way how to organize basic
> material for composition."
> Nothing special so far as not really being or sounding micro-tonal.
> But the reason it interests me is that it fits the same (1/var)^x+1
> formula as the Silver and Golden ratio scales I found to be the
> best I've heard. It makes me wonder if the human mind in general
> prefers scales generated using that formula and if that sty;e
> formula has more possible forms and/or wider applications. To be
> sure, I like the sound of the scale we both ran into a lot more
> than the usual major/minor scale construct and it appears to
> contain a wider range of chords.
>

Somehow I don't see any connection or common elements between the
principle I have described, and your formula, or with those expensive
metals ratios...

Concerning chords and harmony, such artificially created scales can
really offer more in both directions. I suppose we both consider
modal harmony which uses only scale tones, not other notes, and in
the case of chained scales where period is larger than octave and in
higher range there are different tones than in lower part, it is not
allowed to do octave transpositions of notes as this will destroy
scale structure).
> > "Fourth-ave and fifth-ave are the most usable as they go
> through all circle and the resulting scales have big range."
> Right...now the question to me becomes how to summarize a 4th-ave
> in (1/var)^x + 1 form...or if that's possible at all.
>
Don't understand what you mean. If chaining is possible with fifths,
fourths are just an inversion. Fifth circle starts again after 7
octaves, fourth circle after 5.

> > "I like a combination of modal and serial principles, so I
> > focused mainly on scales which use all 12 tones."
> Which is great, because it seems so often people give up on
> chromaticism in scales (as opposed to tunings) as being too hard to
> achieve and/or too dissonant.
>

Why do you think so? I think there's still a lot of composers trying
to continue with 12tone music, serialism, multiserialism... despite
all later waves and tendencies like aleatorics, timbre music,
minimalism, new simplicity, world music, microtonality...

I don't think 12tone music must be dissonant, everything depends on
intervallic structure of the tone series, used harmony and the other
principles. Chromaticism of all kind is my life obsession, and thanks
to my study of some interesting extremely chromatic elements in works
of composers like Solage, Gesualdo, Bach, Mozart, Rejcha or Skriabin
I have found 30 years ago my own methods of 12tone serialism which
sound quite consonant.

> > "When I think more far in this direction, it's possible to
> make scales using different pattern in every period. "
> That's pretty sweet...though it goes beyond what I know how to do> IE how to you ensure a note several semi-tones (say, 25) up works
> with one much further down in harmony (or can you at all)?
>
I don't understand what you mean...

> > "B) I have no idea if somebody worked with similar scales."
> Well, it's reassuring to know I'm not the only one with this "I
> came across this but am not sure who else did" problem. :-)
>
>
Problem is easily eliminated if you don't consider it to be
problem :-) Call me ignorant, but I don't take care if anybody found
it before me, and also I never say it's my invention and I was the
first one. This is not important to me. I just use it.

> > "using F you destroy intervallic structure, because D#-F is
> diminished third, and F-F# augmented prime (chromatic halftone, not
> diatonic). "
> So, if I have this right, this stems from the fact a diatonic
> semitone is larger than a chromatic one, and the difference between
> the two is enough to skew the observed interval into sounding like
> a different interval?
>
Not so complex, it's much more easy and it has nothing to do with a
sound. It's only one of basic rules of music theory, part of
enharmony. On the piano D#-F sound same as Eb-F, D#-E#, Eb-E#, Eb-Gbb
or D#-Gbb, but size of interval is different (with one exception in
these examples):
D#-F and Eb-Gbb are diminished thirds (derived from basic shapes in
the standard notation D-F and E-G, that means without accidentals)
Eb-F and D#-E# are major seconds (derived from E-F and D-E)
Eb-E# is double augmented prime (derived from E-E)
D#-Gbb is triple diminished fourth (derived from D-G)
That's all.

Daniel Forró

Daniel>"That was only joke from my side..."
   I figured on that, hence the :-D emoticon. :-)

>"I don't believe much in universal theories of all, universal means

also uniform. I prefer diversity."
  Hmm...the way I think of it is of one theory that can be used in many different ways.  I don't believe, however, that just because a theory has many applications that it has to be a dominant one or come at the expense of other theories.  For example, MOS theory and Sethares critical band theory both point toward the diatonic scale...however they can be "re-routed" using other variables to do bizarrely diverse things (such as make 10TET relatively consonant by finding matching timbres, in the case of Sethares).  In a twisted way, finding what makes things "the same" can often be turned around to help discover how the polar opposite might be made to work well.

>"Somehow I don't see any connection or common elements between the principle I have described, and your formula, or with those expensive metals ratios..."
    Taking (1/2)^x + 1 and 1.5 - ((1/2)^1 + 1) gives the scale 1 1.0625 1.125 1.25 1.411764 1.5...basically c,c#,d,e,f#,g with g as the period...which coincides with your scale (only yours seems to be missing the c# which equals about 1.0625...not a major difference).

>"Concerning chords and harmony, such artificially created scales can

really offer more in both directions. "
   If I'm reading you well...I agree: there are so many fascinating options for scales that form so many more chords and types of harmony if you agree to let go of the octave restriction...even within pure 12TET.

>"Don't understand what you mean. If chaining is possible with fifths,

fourths are just an inversion. Fifth circle starts again after 7

octaves, fourth circle after 5."
    Now I see what you mean: for example going a 4th down from the octave and a 5th up from the octave before it arrive at the same tone. 

>"Call me ignorant, but I don't take care if anybody found

it before me, "
  Well neither do I, unless people blatantly yell at my for "trying to steal someone else's work" when I didn't intend to.  Which, unfortuately, seems to happen on this list all the time.  Truth is I don't have 12 hours a day to look up the works of every single brilliant tuning expert...and I'm beginning to come to the conclusion that if people want to blame me for stealing credit, that's their problem not mine.  I just enjoy shifting through equations, testing theories by ear, and trying to make beautiful music.  :-)

>"I don't think 12tone music must be dissonant, everything depends on

intervallic structure of the tone series, used harmony and the other

principles. "
  Of course not...and looking over your own work gives hope that we've by no means exhausted the combinations of chords and harmonies available with chromaticism.  I know I've been harsh on 12TET in the past but, to be honest, I was focussed on the limitations of the usual 7-tone diatonic scales and imitations of it (such as diatonic mean-tone) and not full-blown chromaticism.  Nowadays I'm getting more and more interested in chromaticism as an ideal way to provide new yet easily accessible scales for the usual harmonic-series-type timbres.

>"I don't understand what you mean..."
    You discussed circles of 3rds, 5ths and other intervals other than the octave.  That's what I was talking about.

-Michael