Yahoo Tuning Groups Ultimate Backup tuning-math Musical Set Theory in 12-tET

We hold these truths to be self-evident: (Okay, I watched Obama last night)

A. The only scales that can be produced using seven whites (CDEFGAB) ordinary
accidentals, and regular mapping (D->D# means E cannot -> Eb, and D->D# means D
cannot ->Db) and so that all skips (FACEGBD) are either major or minor thirds
are:

1. Major / Minor
2. Melodic Minor
3. Harmonic Minor / Inverse Harmonic Minor.

E#, Fb, Cb, B# are not needed.

B. Same conditions, somewhat more relaxed, allowing augmented and diminished
thirds in the skips (FACEGBD), this picks up the Hungarian scale, for example:

1. All 66 except 0123456, because 01234 pentad cannot be expressed using this
mapping.

2. These 66 septachord/pentachords types encircle ALL hexachord types (pentad up
or septad down)

except these:

012345
012346 and negative
012347 and negative
012348

Interestingly, you can find the 168 scales by going from black to white as
easily as white to black. You get two "regions", the
two-black zone and the three-black zone, for 8 x 21 =
168 combinations which span 65 / 66 scale types (the septachord/pentachords)

168 is an important number also of the Fano plane or PGL(2,7).
More for that on tuning-math.

PGH

Carl Lumma responded on tuning, that I was feeding my own results.
Yes and no. What's amazing to me is that one goes from 5 scales
of 66 to 65 of 66 just by lifting one little restriction on the diatonic nominals. I will upload some of my color-grids relating
to pentad/septads->hexads to the Files section.

It's cool to me that all hexachord Z-relations (they are always
complements) are based on the one tetrachord Z (11), or the 3 pentachord ones, (just 4) and also that the 3 weakly-related 7/5 set complices cluster right in Sector F of my system. The remaining Z-relation is in Sector A and A inverse, and the last one is in the Attic. Of course none of this makes any sense, without it being written up fully.

PGH

🔗Carl Lumma <carl@lumma.org>

PGH wrote:

>Carl Lumma responded on tuning, that I was feeding my own results.
>Yes and no. What's amazing to me is that one goes from 5 scales
>of 66 to 65 of 66 just by lifting one little restriction on the
>diatonic nominals. I will upload some of my color-grids relating
>to pentad/septads->hexads to the Files section.

That's not amazing. There are many transformations that can
do things like this. Rothenberg looks at all scales of all
sizes in 12-ET, sorts by stability and efficiency, and gets
the diatonic and pentatonic scales on top. And while efficiency
and stability make some sense, it's still a bit too convenient.

I don't know what transformation you're using, but can you show:

1. that it also produces the diatonic scales in 31-ET?

2. that people can differentiate stimuli based on it in an
experimental setting?

-Carl

🔗paulhjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> PGH wrote:
>
> >Carl Lumma responded on tuning, that I was feeding my own results.
> >Yes and no. What's amazing to me is that one goes from 5 scales
> >of 66 to 65 of 66 just by lifting one little restriction on the
> >diatonic nominals. I will upload some of my color-grids relating
> >to pentad/septads->hexads to the Files section.
>
> That's not amazing. There are many transformations that can
> do things like this. Rothenberg looks at all scales of all
> sizes in 12-ET, sorts by stability and efficiency, and gets
> the diatonic and pentatonic scales on top. And while efficiency
> and stability make some sense, it's still a bit too convenient.
>
> I don't know what transformation you're using, but can you show:
>
> 1. that it also produces the diatonic scales in 31-ET?
>
> 2. that people can differentiate stimuli based on it in an
> experimental setting?
>
> -Carl
>

The transformation I am using is just every combination of
C D E F G A B with up to 5 sharps and/or flats and the condition
(almost too obvious) that D->D# means D-/->Db and D->D# means
E-/->Eb

1. I've worked some with 31-ET, and Z-relations.(The affine action
in 31-ET isn't so interesting though because it uses every value
of the interval vector 1-30.) This would be fun to try though,
once I find the best diatonic baseline.

2. Don't know, to me personally its meaningful, to get all pentad/septads but one.

This really isn't so much a "tuning" thing as a musical set theory
thing and covering the space. That's what I find amazing. I played with this a little more and I believe I can get all scales up to D4 X S3 X S2 (Which only means up to "M5" and complementation) having only one deformed third per scale.

In terms of tuning, what I am using is:

FACEGBD

And transforming with "normal" sharps and flats.

The M5 transformation then would just hold even pitches fixed and transform odd ones by a tritone. (Which could easily be worked into a scheme of meantone tuning)

I guess what's "amazing" to me is realizing how much there is in
tonal music (even though I learned all this in atonal theory).
Taking jazz harmonies/modal writing etc. You already get 65 of
66 septachords this way, and 31 of 35 hexachord types this way.

If you throw in E#,Fb,Cb,B# you pick up all but one hexachord,
(012345) and you get all pentachords. By complementation you
get all the septachords, indirectly.

PGH

🔗Carl Lumma <carl@lumma.org>

>> I don't know what transformation you're using, but can you show:
>>
>> 1. that it also produces the diatonic scales in 31-ET?
>>
>> 2. that people can differentiate stimuli based on it in an
>> experimental setting?
>>
>> -Carl
>
>The transformation I am using is just every combination of
>C D E F G A B with up to 5 sharps and/or flats and the condition
>(almost too obvious) that D->D# means D-/->Db and D->D# means
>E-/->Eb

This is opaque to me. I thought you were taking 7 from 21, in
ascending order, giving 2187 scales and then imposing the 3rds
restriction.

Which one goes from 65 scales to 5?

>2. Don't know, to me personally its meaningful, to get all
>pentad/septads but one.

Doesn't sound very scientific after implying you can explain
the diatonic scale. Just intonation intervals are identifiable
by naive listeners under laboratory conditions.

>This really isn't so much a "tuning" thing as a musical set theory
>thing and covering the space.

? I thought we were talking about theories that explain music.

>That's what I find amazing. I played
>with this a little more and I believe I can get all scales up to D4 X
>S3 X S2 (Which only means up to "M5" and complementation) having only
>one deformed third per scale.

Again, what do these groups have to do with music?

>In terms of tuning, what I am using is:
>
>FACEGBD

This isn't what we'd call a tuning around here. Can
you explain?

-Carl

🔗paulhjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >> I don't know what transformation you're using, but can you show:
> >>
> >> 1. that it also produces the diatonic scales in 31-ET?
> >>
> >> 2. that people can differentiate stimuli based on it in an
> >> experimental setting?
> >>
> >> -Carl
> >
> >The transformation I am using is just every combination of
> >C D E F G A B with up to 5 sharps and/or flats and the condition
> >(almost too obvious) that D->D# means D-/->Db and D->D# means
> >E-/->Eb
>
> This is opaque to me. I thought you were taking 7 from 21, in
> ascending order, giving 2187 scales and then imposing the 3rds
> restriction.

It's simpler than that. Merely start on F, decide to sharp or not,
go to A, decide to b, natural, #, etc. They are all thirds, but
you can impose restrictions on whether you use diminished thirds/
augmented thirds, etc. It makes more sense with my colorgrids,
showing the 66 pentad/septad types and the hexads they spin out.
(With one per row being a Steiner set, but I'll save that).

> Which one goes from 65 scales to 5?

5 Septads: (Major/Minor Harmonic Scales, Melodic Minor, Major, Minor)
When you have "nice" thirds only. (Major or minor)

65 Septads: (Technically, complements of 65 pentads) when you
allow deformed thirds. This technicality is admittedly a little confusing.

I am still working on the count for one deformed third per chain
(which is really just an open chain, when you don't worry about
it looping back)

> >2. Don't know, to me personally its meaningful, to get all
> >pentad/septads but one.
>
> Doesn't sound very scientific after implying you can explain
> the diatonic scale. Just intonation intervals are identifiable
> by naive listeners under laboratory conditions.

Part of what I am doing is trying to make things as simple as
possible, for example the very simple idea of the circle of
fifths mapping to the chromatic scale by the M5 transform.
Then just choose the tuning you like, its not dictated by the
set theory at all. Of course the reason 12-tET works so well
has to do with infinite series approximations and the fact
that it is "grand central station" for so many linear tunings.

> >This really isn't so much a "tuning" thing as a musical set theory
> >thing and covering the space.
>
> ? I thought we were talking about theories that explain music.

I believe musical set theory and tuning together explain music
even if they are mutually exclusive considerations (sometimes).

> >That's what I find amazing. I played
> >with this a little more and I believe I can get all scales up to D4 X
> >S3 X S2 (Which only means up to "M5" and complementation) having only
> >one deformed third per scale.
>
> Again, what do these groups have to do with music?

Because you get common scales (with some exotic Eastern European
ones). It's just ordinary Scale theory, with the twist of
applying some group theory ideas. Besides the M5 relation
there are other interesting ones. M5 is really nothing more
than the tritone substitution - just ask any jazz double bassist
about that (but I am sure you know it).

> >In terms of tuning, what I am using is:
> >
> >FACEGBD
>
> This isn't what we'd call a tuning around here. Can
> you explain?

Well, the just intonation chain would be a tuning, right?
And then you can map 80->81 with the syntonic comma, and
of course the pythagorean comma fits the circle of fifths into
the octave.

PGH

>
> -Carl
>

🔗Carl Lumma <carl@lumma.org>

Hiya Paul,

>> This is opaque to me. I thought you were taking 7 from 21, in
>> ascending order, giving 2187 scales and then imposing the 3rds
>> restriction.
>
>It's simpler than that. Merely start on F, decide to sharp or not,
>go to A, decide to b, natural, #, etc. They are all thirds, but
>you can impose restrictions on whether you use diminished thirds/
>augmented thirds, etc. It makes more sense with my colorgrids,
>showing the 66 pentad/septad types and the hexads they spin out.
>(With one per row being a Steiner set, but I'll save that).

Er, isn't this 3^7 = 2187? Except you don't allow me to use a
12-ET note twice, which cuts it down somewhat.

>> >2. Don't know, to me personally its meaningful, to get all
>> >pentad/septads but one.
>>
>> Doesn't sound very scientific after implying you can explain
>> the diatonic scale. Just intonation intervals are identifiable
>> by naive listeners under laboratory conditions.
>
>Part of what I am doing is trying to make things as simple as
>possible, for example the very simple idea of the circle of
>fifths mapping to the chromatic scale by the M5 transform.
>Then just choose the tuning you like, its not dictated by the
>set theory at all. Of course the reason 12-tET works so well
>has to do with infinite series approximations and the fact
>that it is "grand central station" for so many linear tunings.

The reason it works so well is that it's has lots of consonances
in very few notes (the grand central station is a consequence
of this). And that's the reason the regular mapping theory says
we use it.

>> Again, what do these groups have to do with music?
>
>Because you get common scales (with some exotic Eastern European
>ones). It's just ordinary Scale theory, with the twist of
>applying some group theory ideas.

That's a pretty big twist, don't you think? I looked up music
Z-relations on wikipedia. It'd be really easy to make audio files
playing short melodies, all of which were z-related except one,
and ask listeners to 'circle the item that doesn't belong'.
Has this been done? If not, why not? And will you volunteer to
do it?

>Besides the M5 relation
>there are other interesting ones. M5 is really nothing more
>than the tritone substitution - just ask any jazz double bassist
>about that (but I am sure you know it).

Is M5 still a tritone substitution in 19-ET? My hunch says no.

-Carl

🔗paulhjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Hiya Paul,
>
> >> This is opaque to me. I thought you were taking 7 from 21, in
> >> ascending order, giving 2187 scales and then imposing the 3rds
> >> restriction.
> >
> >It's simpler than that. Merely start on F, decide to sharp or not,
> >go to A, decide to b, natural, #, etc. They are all thirds, but
> >you can impose restrictions on whether you use diminished thirds/
> >augmented thirds, etc. It makes more sense with my colorgrids,
> >showing the 66 pentad/septad types and the hexads they spin out.
> >(With one per row being a Steiner set, but I'll save that).
>
> Er, isn't this 3^7 = 2187? Except you don't allow me to use a
> 12-ET note twice, which cuts it down somewhat.

Correct. Plus I cut out Fb for example.

> >> >2. Don't know, to me personally its meaningful, to get all
> >> >pentad/septads but one.
> >>
> >> Doesn't sound very scientific after implying you can explain
> >> the diatonic scale. Just intonation intervals are identifiable
> >> by naive listeners under laboratory conditions.
> >
> >Part of what I am doing is trying to make things as simple as
> >possible, for example the very simple idea of the circle of
> >fifths mapping to the chromatic scale by the M5 transform.
> >Then just choose the tuning you like, its not dictated by the
> >set theory at all. Of course the reason 12-tET works so well
> >has to do with infinite series approximations and the fact
> >that it is "grand central station" for so many linear tunings.
>
> The reason it works so well is that it's has lots of consonances
> in very few notes (the grand central station is a consequence
> of this). And that's the reason the regular mapping theory says
> we use it.

Agreed.

> >> Again, what do these groups have to do with music?
> >
> >Because you get common scales (with some exotic Eastern European
> >ones). It's just ordinary Scale theory, with the twist of
> >applying some group theory ideas.
>
> That's a pretty big twist, don't you think? I looked up music
> Z-relations on wikipedia. It'd be really easy to make audio files
> playing short melodies, all of which were z-related except one,
> and ask listeners to 'circle the item that doesn't belong'.
> Has this been done? If not, why not? And will you volunteer to
> do it?

I'm not crazy about the Z-relation. What is meaningful is complementation, (and all hexachords that are Z-related are complements). It's meaningful indirectly, not directly. The other
aspect that is meaningful is how they are built up, from
(0,1,4,6)<->(0,4,6,7) kernel (and 2 pentachord ones). It "explains"
why Z-relations are twisted the way they are. Then you regard the
whole tonal space, and how things deform the texture etc. (of
individual sets, but also of the whole space). It's true
that the Z-relation is probably not acoustically meaningful,
directly, but inverses aren't even that meaningful, as another
poster has pointed out.

> >Besides the M5 relation
> >there are other interesting ones. M5 is really nothing more
> >than the tritone substitution - just ask any jazz double bassist
> >about that (but I am sure you know it).
>
> Is M5 still a tritone substitution in 19-ET? My hunch says no.

Well, here again 19-ET has an 18-cycle chain of M5's (actually
5 works here: (1,5,6,11...) which is 5^n mod 19 multiplicatively.
And of course there is no tritone cuz its an odd temperament.)
>
> -Carl
>

🔗Carl Lumma <carl@lumma.org>

>> >> Again, what do these groups have to do with music?
>> >
>> >Because you get common scales (with some exotic Eastern European
>> >ones). It's just ordinary Scale theory, with the twist of
>> >applying some group theory ideas.
>>
>> That's a pretty big twist, don't you think? I looked up music
>> Z-relations on wikipedia. It'd be really easy to make audio files
>> playing short melodies, all of which were z-related except one,
>> and ask listeners to 'circle the item that doesn't belong'.
>> Has this been done? If not, why not? And will you volunteer to
>> do it?
>
>I'm not crazy about the Z-relation. What is meaningful is
>complementation, (and all hexachords that are Z-related are
>complements). It's meaningful indirectly, not directly. The other
>aspect that is meaningful is how they are built up, from
>(0,1,4,6)<->(0,4,6,7) kernel (and 2 pentachord ones). It "explains"
>why Z-relations are twisted the way they are. Then you regard the
>whole tonal space, and how things deform the texture etc. (of
>individual sets, but also of the whole space). It's true
>that the Z-relation is probably not acoustically meaningful,
>directly, but inverses aren't even that meaningful, as another
>poster has pointed out.

Inverses are recognizable by astute listeners, though they're
borderline. I don't know what a complement is, but I'm going to
assume it's not recognizable by anyone. The fact that this
doesn't seem to bother you (or other music set theorists) is
fairly disturbing. I mean, if you don't care about explaining
music then it's no problem at all. But lots of music set theory
papers do claim to explain music.

And now, I'll briefly digress: Bach used things like retrograde
not because it was an audible transformation, but because it's
challenge to do it and still harmonize the canon. He tells you
he did it (in the canonic riddle) so you're sure to be impressed.
If he were to throw the retrograde on top of the theme without
bothering to harmonize it, that wouldn't have been impressive
at all. With Bach, the transformation was an arbitrary way to
make the canon harder. With serialists, the transformation is
said to be a meaningful way to make harmony obsolete. That's
wrong. It's not a matter of taste. If you can't hear the
transformation, it's not meaningful, period.

>> >Besides the M5 relation
>> >there are other interesting ones. M5 is really nothing more
>> >than the tritone substitution - just ask any jazz double bassist
>> >about that (but I am sure you know it).
>>
>> Is M5 still a tritone substitution in 19-ET? My hunch says no.
>
>Well, here again 19-ET has an 18-cycle chain of M5's (actually
>5 works here: (1,5,6,11...) which is 5^n mod 19 multiplicatively.
>And of course there is no tritone cuz its an odd temperament.)

Of course it has a tritone, and a more accurate one (where 7/5
is the target) than 12-ET. It doesn't temper out 50/49, so you
get a comma shift when doing a tritone substitution, so I guess
you could argue your M5 thing is hip to that. In which case I
suppose I should ask, is M5 a tritone subst. in 22-ET and 26-ET?

-Carl

🔗paulhjelmstad <phjelmstad@msn.com>

It does bother me actually. However, complements are recognizable,
just play a hexachord and then its complement, they have a lot in common. I was able to recognize the complement of the Hungarian
scale in "School for Scandal" by Samuel Barber the other day also
(Even though this is 7 / 5 complementation). The reason being (same
as Z-relation): Complementary hexachords have the same intervallic structure, or in physics terms "The same energy charges on the necklace" to bring in necklace theory. Figuring out where all
these dyads (intervals) land on the necklace is another matter.
Then you have to consider the 3-set interval vector, 4-set etc.

Most of what I am doing revolves around the interval vectors
of musical sets, and looking at the slots. For example, there
are exactly 24 pentachords up to D4 X S3.(Slot 1, 5 swappable).

> And now, I'll briefly digress: Bach used things like retrograde
> not because it was an audible transformation, but because it's
> challenge to do it and still harmonize the canon. He tells you
> he did it (in the canonic riddle) so you're sure to be impressed.
> If he were to throw the retrograde on top of the theme without
> bothering to harmonize it, that wouldn't have been impressive
> at all. With Bach, the transformation was an arbitrary way to
> make the canon harder. With serialists, the transformation is
> said to be a meaningful way to make harmony obsolete. That's
> wrong. It's not a matter of taste. If you can't hear the
> transformation, it's not meaningful, period.

Fourier analysis / Spectrum analysis probably casts better light
on some of these issues --- but for me,

Well, its like learning modern music. After playing Bartok for
hours it starts to sound normal, even ordinary. So much of
voice leading is based on certain symmetries of motion. (Common-tone
theorems point to this).

> >> >Besides the M5 relation
> >> >there are other interesting ones. M5 is really nothing more
> >> >than the tritone substitution - just ask any jazz double bassist
> >> >about that (but I am sure you know it).
> >>
> >> Is M5 still a tritone substitution in 19-ET? My hunch says no.
> >
> >Well, here again 19-ET has an 18-cycle chain of M5's (actually
> >5 works here: (1,5,6,11...) which is 5^n mod 19 multiplicatively.
> >And of course there is no tritone cuz its an odd temperament.)
>
> Of course it has a tritone, and a more accurate one (where 7/5
> is the target) than 12-ET. It doesn't temper out 50/49, so you
> get a comma shift when doing a tritone substitution, so I guess
> you could argue your M5 thing is hip to that. In which case I
> suppose I should ask, is M5 a tritone subst. in 22-ET and 26-ET?

Depends on how you define the tritone. I can't remember if there
are any flat-line interval distributions in 19-ET, but I am sure
there are plenty of scrambles. I would say 22-ET has the more
meaningful tritone sub, since it's tritone is 2^1/2.

Tritones are meaningful in the same was as complements: They
are both involutions (f(f(x))=x. And these involutions are related
in my theorizing when I-Ching diagrams are used. Interestingly,
it's the tritone(s) that move in a complementation of a set.

It would take a neurophysiologist or something to test how meaningful this is. But if harmony affects brain-chemistry, I think it is
significant.

PGH

> -Carl
>

🔗Steven Grainger <srgrainger@yahoo.com.au>

Dear Maths Gods,
I have gotten interested in Just Intonation and Archytas system of working out scales based on epimoric ratios and harmonic means.

I didn't finish junior high school so I have had to learn about multiplying and dividing ratios/fractions. but I am not sure what exactly is being multiplied when we are dealing with string lengths and sound waves.

+++++++

For example: in 3:2 x 5:4 = 15/8,

I can understand in 3:2, the '3' could refer to a string 3 times the length of a fundamental unit, and the '2' refer to a string 2 times the fundamental. And that the difference between these two strings produces an interval of 3:2.

But what I can't quite visualize is what is happening, what basic operation is occuring to these strings, when we multiple one fraction by another.

My groping thinking goes: we have these two strings in proportion of 3:2, but what is happening when we multiple these strings by '5:4'? Does that mean: we take 15 of the '3' length strings and 4 of the '2' length strings - so that we then have two very long strings but in ratio of 15:8. Yet we would then have two very long strings both lower than the fundamental string length.

I like to have a good concrete model of what the numbers are representing,. Any help would be appreciated in showing me the errors and incompleteness of my thinking or providing me with more far-reaching ways of understanding what is happening, as I am trying to get a deeper feel of the exquiste relationships between intervals based on harmonic ratios.

Yours prayerfully
Steve

________________________________
From: paulhjelmstad <phjelmstad@msn.com>
To: tuning-math@yahoogroups.com
Sent: Fri, 29 January, 2010 9:08:47 AM
Subject: [tuning-math] Musical Set Theory in 12-tET

We hold these truths to be self-evident: (Okay, I watched Obama last night)

1. Major / Minor
2. Melodic Minor
3. Harmonic Minor / Inverse Harmonic Minor.

E#, Fb, Cb, B# are not needed.

B. Same conditions, somewhat more relaxed, allowing augmented and diminished
thirds in the skips (FACEGBD), this picks up the Hungarian scale, for example:

1. All 66 except 0123456, because 01234 pentad cannot be expressed using this
mapping.

2. These 66 septachord/pentacho rds types encircle ALL hexachord types (pentad up
or septad down)

except these:

012345
012346 and negative
012347 and negative
012348

168 is an important number also of the Fano plane or PGL(2,7).
More for that on tuning-math.

PGH

__________________________________________________________________________________
Yahoo!7: Catch-up on your favourite Channel 7 TV shows easily, legally, and for free at PLUS7. www.tv.yahoo.com.au/plus7

🔗Carl Lumma <carl@lumma.org>

Hiya Steve,

> For example: in 3:2 x 5:4 = 15/8,
[snip]
> My groping thinking goes: we have these two strings in
> proportion of 3:2, but what is happening when we multiple
> these strings by '5:4'? Does that mean: we take 15 of the
> '3' length strings and 4 of the '2' length strings - so
> that we then have two very long strings but in ratio of
> 15:8. Yet we would then have two very long strings both
> lower than the fundamental string length.

Sounds like one valid way to imagine it. Except I think
you meant we take 3 strings of length 5 and tie them
together, and 2 strings of length 4 and tie them together.
Now we have two strings of lengths 15 and 8. They're very
long, but the interval between them is indeed a type of
major 7th.

> I like to have a good concrete model of what the numbers
> are representing.

It can be a good thing to think about, but I wouldn't get
hung up too much on the whys of basic operations like this.
The important thing is that pitch is heard as the logarithm
of frequency, so to add two intervals together, you would
add their logarithms, which is equivalent to multiplying
their ratios.

-Carl

🔗Steven Grainger <srgrainger@yahoo.com.au>

Re:
The important thing is that pitch is heard as the logarithm
of frequency, so to add two intervals together, you would
add their logarithms, which is equivalent to multiplying
their ratios.

I am not sure how 'pitch is heard as the logarith of a frequency'.

Would you mind giving me an example of adding the logorhythms of the frequencies together as a way of multiplying ratios.

Is it like 81/64 = (3x2) to the power of 4.

The wikipedia article about logorithms was a bit beyond me, whereas as I can almost understand the multiplying of fractions and the concrete model they relate to.

Perhaps I should do a basic maths course but I am really only interested in understanding the relationships in the harmonic scales.so as to see more musical possibilites. It seems this forum is for people to share higher level understanding rather than the beginner level I am at.

Thanks
Steve

________________________________
From: Carl Lumma <carl@lumma.org>
To: tuning-math@yahoogroups.com
Sent: Sat, 30 January, 2010 11:12:18 AM
Subject: [tuning-math] Re: Dear maths Gods

Hiya Steve,

> I like to have a good concrete model of what the numbers
> are representing.

-Carl

🔗Carl Lumma <carl@lumma.org>

Hi Steve,

>Re:
>The important thing is that pitch is heard as the logarithm
>of frequency, so to add two intervals together, you would
>add their logarithms, which is equivalent to multiplying
>their ratios.
>
>I am not sure how 'pitch is heard as the logarith of a frequency'.
>
>Would you mind giving me an example of adding the logorhythms of the frequencies together as a way of multiplying ratios.
>
>Is it like 81/64 = (3x2) to the power of 4.

Forget music for a second and just think about logarithms.
If you have two numbers x and y, and you multiply them
together, you get the same result as if you'd added their
logarithms instead. That is,

x * y = log(x) + log(y)

When I said 'pitch is log frequency' I just meant that two
musical intervals are heard as being the same size if they
have the same frequency ratio, not the same frequency
difference. For example, the interval 300Hz : 500 Hz is
heard to be the same size as 600Hz : 1000Hz, even though
there's a gap of 200 Hz in the first case and 400 Hz in the
latter case. If you take their logarithms instead, the gaps
will be the same. That is,

log(500) - log(300) = log(1000) - log(600)

Try it with a calculator!

>Perhaps I should do a basic maths course but I am really only interested in understanding the relationships in the harmonic scales.so as to see more musical possibilites. It seems this forum is for people to share higher level understanding rather than the beginner level I am at.

Yeah, this forum is more advanced. The regular tuning list
/tuning
would be more appropriate.

But harmonic scales are very simple (and good!). It sounds
like you already understand them pretty well.

-Carl

🔗Graham Breed <gbreed@gmail.com>

On 1 February 2010 01:28, Carl Lumma <carl@lumma.org> wrote:

> x * y = log(x) + log(y)

In case you confuse the poor chap with the shortcut, let's state the
true identity.

log(x * y) = log(x) + log(y)

That's something you can easily check with a calculator.

The derivation of 81/64 is as follows:

Start with a perfect fifth, called F where F=log(3/2). Take four
fifths, so 4*F. Lower it by two octaves, giving 4*F - 2*log(2).

Using natural logarithms, that gives

4*ln(3/2) - 2*ln(2) = .2355660714

and ln(81/64) = .2355660713

The two numbers are the same to within the calculator's precision.

Note, it's convenient to take the logarithms to base 2, so that an
octave is one unit

F = log(3/2) / log(2)

log(81/64)/log(2) = 4*F - 2

Another useful thing, if you want to know how many steps 81/64 is on a
12 note scale, you can try

12*log(81/64)/log(2) = 4.078...

So approximately 4 steps. This is why logarithms are convenient. You
only need to remember the numbers for basic intervals like an octave,
fifth, and third, and you can easily calculate interval sizes.

Graham

🔗Steven Grainger <srgrainger@yahoo.com.au>

Thanks Carl and Graham,
I think I will try the other other tuning forum you mentioned as I only have elementary level maths and do not even understand what a logarithm is, letting only adding them, but I really appreciate your answers and has helped me clarify what my next learning steps may be.

harmonically yours
Steve

________________________________
From: Graham Breed <gbreed@gmail.com>
To: tuning-math@yahoogroups.com
Sent: Mon, 1 February, 2010 1:41:40 PM
Subject: Re: [tuning-math] logorithm of a frequency

On 1 February 2010 01:28, Carl Lumma <carl@lumma.org> wrote:

> x * y = log(x) + log(y)

In case you confuse the poor chap with the shortcut, let's state the
true identity.

log(x * y) = log(x) + log(y)

That's something you can easily check with a calculator.

The derivation of 81/64 is as follows:

Start with a perfect fifth, called F where F=log(3/2). Take four
fifths, so 4*F. Lower it by two octaves, giving 4*F - 2*log(2).

Using natural logarithms, that gives

4*ln(3/2) - 2*ln(2) = .2355660714

and ln(81/64) = .2355660713

The two numbers are the same to within the calculator's precision.

Note, it's convenient to take the logarithms to base 2, so that an
octave is one unit

F = log(3/2) / log(2)

log(81/64)/log( 2) = 4*F - 2

Another useful thing, if you want to know how many steps 81/64 is on a
12 note scale, you can try

12*log(81/64) /log(2) = 4.078...

Graham

🔗Carl Lumma <carl@lumma.org>

🔗Steven Grainger <srgrainger@yahoo.com.au>

Dear All,
Re: Epimoric Ratios

I would appreciate any ideas, suggestions, praise songs or forumulas about tuning based on epimoric ratios and their possible relationships to binaural beating.

Thanks
Steve

________________________________
From: Graham Breed <gbreed@gmail.com>
To: tuning-math@yahoogroups.com
Sent: Mon, 1 February, 2010 1:41:40 PM
Subject: Re: [tuning-math] logorithm of a frequency

On 1 February 2010 01:28, Carl Lumma <carl@lumma.org> wrote:

> x * y = log(x) + log(y)

In case you confuse the poor chap with the shortcut, let's state the
true identity.

log(x * y) = log(x) + log(y)

That's something you can easily check with a calculator.

The derivation of 81/64 is as follows:

Start with a perfect fifth, called F where F=log(3/2). Take four
fifths, so 4*F. Lower it by two octaves, giving 4*F - 2*log(2).

Using natural logarithms, that gives

4*ln(3/2) - 2*ln(2) = .2355660714

and ln(81/64) = .2355660713

The two numbers are the same to within the calculator's precision.

Note, it's convenient to take the logarithms to base 2, so that an
octave is one unit

F = log(3/2) / log(2)

log(81/64)/log( 2) = 4*F - 2

Another useful thing, if you want to know how many steps 81/64 is on a
12 note scale, you can try

12*log(81/64) /log(2) = 4.078...

Graham

🔗paulhjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >> >> Again, what do these groups have to do with music?
> >> >
> >> >Because you get common scales (with some exotic Eastern European
> >> >ones). It's just ordinary Scale theory, with the twist of
> >> >applying some group theory ideas.
> >>
> >> That's a pretty big twist, don't you think? I looked up music
> >> Z-relations on wikipedia. It'd be really easy to make audio files
> >> playing short melodies, all of which were z-related except one,
> >> and ask listeners to 'circle the item that doesn't belong'.
> >> Has this been done? If not, why not? And will you volunteer to
> >> do it?
> >
> >I'm not crazy about the Z-relation. What is meaningful is
> >complementation, (and all hexachords that are Z-related are
> >complements). It's meaningful indirectly, not directly. The other
> >aspect that is meaningful is how they are built up, from
> >(0,1,4,6)<->(0,4,6,7) kernel (and 2 pentachord ones). It "explains"
> >why Z-relations are twisted the way they are. Then you regard the
> >whole tonal space, and how things deform the texture etc. (of
> >individual sets, but also of the whole space). It's true
> >that the Z-relation is probably not acoustically meaningful,
> >directly, but inverses aren't even that meaningful, as another
> >poster has pointed out.
>
> Inverses are recognizable by astute listeners, though they're
> borderline. I don't know what a complement is, but I'm going to
> assume it's not recognizable by anyone. The fact that this
> doesn't seem to bother you (or other music set theorists) is
> fairly disturbing. I mean, if you don't care about explaining
> music then it's no problem at all. But lots of music set theory
> papers do claim to explain music.
>
> And now, I'll briefly digress: Bach used things like retrograde
> not because it was an audible transformation, but because it's
> challenge to do it and still harmonize the canon. He tells you
> he did it (in the canonic riddle) so you're sure to be impressed.
> If he were to throw the retrograde on top of the theme without
> bothering to harmonize it, that wouldn't have been impressive
> at all. With Bach, the transformation was an arbitrary way to
> make the canon harder. With serialists, the transformation is
> said to be a meaningful way to make harmony obsolete. That's
> wrong. It's not a matter of taste. If you can't hear the
> transformation, it's not meaningful, period.
>
> >> >Besides the M5 relation
> >> >there are other interesting ones. M5 is really nothing more
> >> >than the tritone substitution - just ask any jazz double bassist
> >> >about that (but I am sure you know it).
> >>
> >> Is M5 still a tritone substitution in 19-ET? My hunch says no.
> >
> >Well, here again 19-ET has an 18-cycle chain of M5's (actually
> >5 works here: (1,5,6,11...) which is 5^n mod 19 multiplicatively.
> >And of course there is no tritone cuz its an odd temperament.)
>
> Of course it has a tritone, and a more accurate one (where 7/5
> is the target) than 12-ET. It doesn't temper out 50/49, so you
> get a comma shift when doing a tritone substitution, so I guess
> you could argue your M5 thing is hip to that. In which case I
> suppose I should ask, is M5 a tritone subst. in 22-ET and 26-ET?
>
> -Carl

Well, I was thinking a little more about "Phase Two" (My attempt
to integrate my set theory model with the regular mapping paradigm).
And I thought of two things that are meaningful.

The M5 relationship merely reflects a set across a horizontal axis.
(Also, it transposes odd steps by a tritone and keeps even fixed).
So that is definitely meaningful. This boils my 10 pentads/septad scales down to only 6, and these 6 should have a meaningful pattern - but I will get back to that -

For example, consider either C4 X C3 (based on 2^1/4 X 2^1/3) or
3^m X 5^n. For simplicity I won't choose anything in the 3^2 column.

Since the vertical axis doesn't shift, we can consider for example
1, 3/2, 4/3, 5/4, 6/5, 8/5 hexachord. This maps to 1, 16/15, 15/16,
5/4, 5/6, 8/5...where the tritone sub is fairly obvious. Then
we just need to ask the question is 3/2 -> 16/15 meaningful? I think
it could be worked into the geometries of hexanies, tetrads, etc.

The second case is complementation. I'll choose a Z-related
set (all of them have the tetrachord Z-relation and/or pentachord
Z-relations as kernels) What happens, is the "kernel" part
has the M5 relation, mentioned above, or is the same, or perhaps
inverse. All of which make sense in terms of regular mappings (fractions). For example, {0,4,5,6,9,10} Z {1,2,3,7,8,11} has
{0,4,9,10} Z {1,7,8,11} "inside", these are M5 related, easily mapped
as above --- and the remaining tones {4,5} -> (2,3) are a simple
transposition, even though as fractions they might be different
(actually, here they are both 16/15 based on 4/3:5/4 and 6/5:9/8)

Of course these are the relationships that enter into the interval
vectors, which are the same in complementary and/or Z-related sets.

There are a few Z relations that are worse, (only 4 set types) using
pentachord-Z's on one side but the same principles apply.

Let me know what you think! Thanks PGH

🔗Carl Lumma <carl@lumma.org>

🔗paulhjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >Let me know what you think! Thanks PGH
>
> I'm very glad you're thinking about this, but I'm not the person
> to ask to evaluate it. Hopefully, folks like Hudson, Jon Wild,
> John Chalmers, or others will join in. Is Stephen Soderberg around
> these days?
>
> -Carl
>

Thanks. It's funny, although the two sides (musical set theory and regular mapping paradigm) are like big chamber of a cave, not much connects them. And what does is somewhat stale and flat. However, I did boil my fractions down to these six, so, perhaps,just in terms of pure math, "Is this Anything?"

6/5, 9/5, 4/3
9/8, 9/5, 4/3
8/5, 6/5, 4/3
8/5, 6/5, 5/4
16/15, 5/4, 15/8
16/15, 5/3, 5/4

(These six fractions, along with the tritone (1/1 and 45/32) create
six pentads, which along with their M5 brothers, give 10 which
canvas 70 hexads with one or two tritones (40 + 30). The remaining
10 are pretty easy to remap. (There are 80 hexad types total,
which reduce to 35 under D12 X S2). I am kinda working backwards,
in the respect that when considering M12 I take 66*2=132 pentads to
generate 924 hexads, (seven supersets per pentad) instead of taking
132 Steiner hexads and working down to 792 pentads. But working
backwards can shed light on so much, of course.

Not that anyone asked but a little more theory. M12 permutes
each pentad to another one (this is called quintuple-transitivity)
and is in fact sharply transitive. It's pretty slick math,
M12 has order 95040 and each "g" will permute the pentads such
that you go between 132 Steiner blocks * 720 combinations per
block. (Think of going one block, two blocks, etc and then having
720 combinations). Technically, you are considering 6 pentads
per block of six and then taking 5! combinations so 6! = 6 * 5!.

PGH

🔗paulhjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "paulhjelmstad" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> >
> > >Let me know what you think! Thanks PGH
> >
> > I'm very glad you're thinking about this, but I'm not the person
> > to ask to evaluate it. Hopefully, folks like Hudson, Jon Wild,
> > John Chalmers, or others will join in. Is Stephen Soderberg around
> > these days?
> >
> > -Carl
> >
>
> Thanks. It's funny, although the two sides (musical set theory and regular mapping paradigm) are like big chamber of a cave, not much connects them. And what does is somewhat stale and flat. However, I did boil my fractions down to these six, so, perhaps,just in terms of pure math, "Is this Anything?"
>
> 6/5, 9/5, 4/3
> 9/8, 9/5, 4/3
> 8/5, 6/5, 4/3
> 8/5, 6/5, 5/4
> 16/15, 5/4, 15/8
> 16/15, 5/3, 5/4
>
> (These six fractions, along with the tritone (1/1 and 45/32) create
> six pentads, which along with their M5 brothers, give 10 which
> canvas 70 hexads with one or two tritones (40 + 30). The remaining
> 10 are pretty easy to remap. (There are 80 hexad types total,
> which reduce to 35 under D12 X S2). I am kinda working backwards,
> in the respect that when considering M12 I take 66*2=132 pentads to
> generate 924 hexads, (seven supersets per pentad) instead of taking
> 132 Steiner hexads and working down to 792 pentads. But working
> backwards can shed light on so much, of course.
>
> Not that anyone asked but a little more theory. M12 permutes
> each pentad to another one (this is called quintuple-transitivity)
> and is in fact sharply transitive. It's pretty slick math,
> M12 has order 95040 and each "g" will permute the pentads such
> that you go between 132 Steiner blocks * 720 combinations per
> block. (Think of going one block, two blocks, etc and then having
> 720 combinations). Technically, you are considering 6 pentads
> per block of six and then taking 5! combinations so 6! = 6 * 5!.
>
> PGH

I found a few interesting things this weekend, but I need a write
a short paper and upload it for everything to make sense. I've
found that complements of hexachords (Z-related ones) fall in place
in a certain exact way in my pentad->hexad grid diagrams (Excel spreadsheets, also need to post them) but more readily, I can state that:

"Every hexachord with 0 to 2 tritones, up to the M5 relation, can be found from a pentachord that is part of a diatonic (major), melodic minor, harmonic minor or harmonic major scale"

The remaining chromatic ones can be mapped by merely holding even
pitches fixed and moving the odd ones a tritone (pretty much
chromatic type collections).

Also, outside of the tritone, one only needs this trios (in different
positions) 024, 025, 027, 045, 047, 048 all of which are diatonic
except the last one. But the resultant pentad is always as stated
above.

This leaves only 4 hexachord types, with 2 tritones:

1. the whole tone scale 0.2.4.6.8.10
2. 0.2.3.6.7.9 and it mirror inverse
3. 0.1.2.6.7.8

PGH

🔗paulhjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "paulhjelmstad" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "paulhjelmstad" <phjelmstad@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> > >
> > > >Let me know what you think! Thanks PGH
> > >
> > > I'm very glad you're thinking about this, but I'm not the person
> > > to ask to evaluate it. Hopefully, folks like Hudson, Jon Wild,
> > > John Chalmers, or others will join in. Is Stephen Soderberg around
> > > these days?
> > >
> > > -Carl
> > >
> >
> > Thanks. It's funny, although the two sides (musical set theory and regular mapping paradigm) are like big chamber of a cave, not much connects them. And what does is somewhat stale and flat. However, I did boil my fractions down to these six, so, perhaps,just in terms of pure math, "Is this Anything?"
> >
> > 6/5, 9/5, 4/3
> > 9/8, 9/5, 4/3
> > 8/5, 6/5, 4/3
> > 8/5, 6/5, 5/4
> > 16/15, 5/4, 15/8
> > 16/15, 5/3, 5/4
> >
> > (These six fractions, along with the tritone (1/1 and 45/32) create
> > six pentads, which along with their M5 brothers, give 10 which
> > canvas 70 hexads with one or two tritones (40 + 30). The remaining
> > 10 are pretty easy to remap. (There are 80 hexad types total,
> > which reduce to 35 under D12 X S2). I am kinda working backwards,
> > in the respect that when considering M12 I take 66*2=132 pentads to
> > generate 924 hexads, (seven supersets per pentad) instead of taking
> > 132 Steiner hexads and working down to 792 pentads. But working
> > backwards can shed light on so much, of course.
> >
> > Not that anyone asked but a little more theory. M12 permutes
> > each pentad to another one (this is called quintuple-transitivity)
> > and is in fact sharply transitive. It's pretty slick math,
> > M12 has order 95040 and each "g" will permute the pentads such
> > that you go between 132 Steiner blocks * 720 combinations per
> > block. (Think of going one block, two blocks, etc and then having
> > 720 combinations). Technically, you are considering 6 pentads
> > per block of six and then taking 5! combinations so 6! = 6 * 5!.
> >
> > PGH
>
> I found a few interesting things this weekend, but I need a write
> a short paper and upload it for everything to make sense. I've
> found that complements of hexachords (Z-related ones) fall in place
> in a certain exact way in my pentad->hexad grid diagrams (Excel spreadsheets, also need to post them) but more readily, I can state that:
>
> "Every hexachord with 0 to 2 tritones, up to the M5 relation, can be found from a pentachord that is part of a diatonic (major), melodic minor, harmonic minor or harmonic major scale"
>
> The remaining chromatic ones can be mapped by merely holding even
> pitches fixed and moving the odd ones a tritone (pretty much
> chromatic type collections).
>
> Also, outside of the tritone, one only needs this trios (in different
> positions) 024, 025, 027, 045, 047, 048 all of which are diatonic
> except the last one. But the resultant pentad is always as stated
> above.
>
> This leaves only 4 hexachord types, with 2 tritones:
>
> 1. the whole tone scale 0.2.4.6.8.10
> 2. 0.2.3.6.7.9 and it mirror inverse
> 3. 0.1.2.6.7.8
>
> PGH

Ugh sorry I meant

This leaves only 4 hexachord types, with 3 tritones:

1. the whole tone scale 0.2.4.6.8.10
2. 0.1.3.6.7.9 and it mirror inverse
3. 0.1.2.6.7.8