Yahoo Tuning Groups Ultimate Backup tuning Re: Using the arithmetic mean to explain consonant-sounding JI scale

--"My question about non octave scale is this, What does one do if one

wants to reinforce a melodic line in a different range. "

Don't get me wrong, octaves can and do repeat themselves in such constructs, just not often.
And if an octave is not available you can use the nearest 5th available to harmonize...though if you have neither you are out of luck.

But you're right, the whole concept of playing in unison IE C5 C6 C8...becomes a lot more difficult in such scales.

>"The idea of 'not' doing something also seems like a troublesome way of thinking to get somewhere. What happens if we all decide to do non-mozart, non-Bach, non electronica, non jazz music?"

I, for one, don't think of it as an either/or situation. A composer could very easily play a harmonized melodic line using many "octave-unisons" in diatonic and then switch into a C D E F# G scale with G being the period. To me, it's not about avoiding the octave but, rather, to the composers discretion, trading the ability to use octave for the ability to pick more diverse types and varieties of chords.

For one, I like using such scales because of what they can do...not at all because they avoid the octave; that is just a coincidence of the formula (1/2)^x + 1 I use to generate the scale: that the period becomes 1.5 instead of 2.
BTW, if you make the equation (1/var)^x+1 = 2 then var would equal 1 and the only note in the scale would be 2!...hence why I don't use 2 as the period in this scale construction method (this certainly has nothing to do with 'trying' to avoid the octave...it's simply a numerical constraint to the formula).

-Michael

🔗Daniel Forro <dan.for@...>

We can't have everything. Yes, maybe we lose octave doubling (anyway it's not much used in contemporary music, anyway it's an overused old junk weapon from the arsenal of classicism/romantism), but we get much more interesting opportunities like heterophony, or quasi parallel mixtures, and much more. I wouldn't call such mixtures "harmony"...

I think there's a lot of composers doing things their own way with lot of NON. Just by using microtones we enter such territories. But probably no extremes are good in both directions. When somebody just copies music of the past, there's no new information and the result will be boring. When somebody does some totally innovative and unusual music (if anything like this is still possible), it will be boring as well, this time because there will be too much information and listener's channels will be soon overfilled. The best music somehow tries to combine and balance old, known principles with new, unknown. Older elements are like a stones in the morass, we can use them as a support and relax with them, new unusual elements bring excitement and necessary contrast.

Daniel Forro

On 20 May 2009, at 4:13 AM, Kraig Grady wrote:
>
> My question about non octave scale is this, What does one do if one
> wants to reinforce a melodic line in a different range. One cannot > do so
> without introducing harmony. The idea of 'not' doing something also
> seems like a troublesome way of thinking to get somewhere. What > happens
> if we all decide to do non-mozart, non-Bach, non electronica, non jazz
> music?

🔗Aaron Krister Johnson <aaron@...>

Daniel,

The octave is the only interval that has a basis for pitch-class identity. Unless your nervous system is somehow unrelated to 'homo sapiens', as a musician and listener, your hear intervals an octave apart as being of the 'same pitch class'...no other interval does this. It seems to me that whether one ignores octave or uses them, octaves do have a very special property in the sonic universe, one that cannot be denied.

In spite of that basic fact, I've been excited by some non-octave music I've heard, and I've been excited to explore it, too. In the end, if it sounds good, and is tastefully done, I guess I like it no matter what it has, or lacks....

But--

I think octave doubling is still valid, and an obvious orchestrational tool *for certain styles*...if it's an old 'junk weapon', so is everything else ever done in the past...so, I respectfully disagree with you on that one! ;)

Best,
AKJ

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
>
> We can't have everything. Yes, maybe we lose octave doubling (anyway
> it's not much used in contemporary music, anyway it's an overused old
> junk weapon from the arsenal of classicism/romantism), but we get
> much more interesting opportunities like heterophony, or quasi
> parallel mixtures, and much more. I wouldn't call such mixtures
> "harmony"...
>
> I think there's a lot of composers doing things their own way with
> lot of NON. Just by using microtones we enter such territories. But
> probably no extremes are good in both directions. When somebody just
> copies music of the past, there's no new information and the result
> will be boring. When somebody does some totally innovative and
> unusual music (if anything like this is still possible), it will be
> boring as well, this time because there will be too much information
> and listener's channels will be soon overfilled. The best music
> somehow tries to combine and balance old, known principles with new,
> unknown. Older elements are like a stones in the morass, we can use
> them as a support and relax with them, new unusual elements bring
> excitement and necessary contrast.
>
> Daniel Forro
>
>
> On 20 May 2009, at 4:13 AM, Kraig Grady wrote:
> >
> > My question about non octave scale is this, What does one do if one
> > wants to reinforce a melodic line in a different range. One cannot
> > do so
> > without introducing harmony. The idea of 'not' doing something also
> > seems like a troublesome way of thinking to get somewhere. What
> > happens
> > if we all decide to do non-mozart, non-Bach, non electronica, non jazz
> > music?
>

🔗Daniel Forro <dan.for@...>

Aaron,

of course I agree octave is an important interval. Using it or not is not quite dependent on octave or non-octave basic material selected for composing - as I showed in some examples, even those chained scales which have different period than octave can include octave intervals and composer can emphasize them or try to hide them. And even in diatonic, pandiatonic or chromatic terrain based on octave we can compose music which emphasizes smaller or bigger intervals in the function of period, or octaves can be even excluded in the extreme case. When talking about all this I mean it in the frame of 12 ET, but it can be easily adapted for microtonal music, where octave can be avoided in more interesting way.

Octave doubling (and unisons) can be still used - for sure in harmony, or when arranging orchestral groups. Melodically it is special effect with very typical sound, which must be used carefully as it can create thick textures which tend to sound muddy. For me personally is more interesting polystratophonic type of orchestral arrangement, based on simultaneous sounding of different contrasting motifs or elements in different layers, and diversity can be even more emphasized by instrumentation and selected range. It's a modern version of traditional polyphony, where one layer is represented by one voice. In polystratophony one layer can be a complex music structure.

I would be the last person who would object against historical (or ethnic) music. I love it, and devoted a lot of time and effort to listening it, analyze it, learn most of those compositional techniques, methods, theories, principles... and also composed lot of music using all that knowledge :-) But it doesn't mean we have to accept everything from the past, there were also cul-de-sacs, mistakes (from later point of view), and even venerable Masters long-bearded or beardless had bad days when they couldn't get out from routine work...

Have a nice day!

Daniel Forro

On 20 May 2009, at 11:52 PM, Aaron Krister Johnson wrote:

>
>
> Daniel,
>
> The octave is the only interval that has a basis for pitch-class > identity. Unless your nervous system is somehow unrelated to 'homo > sapiens', as a musician and listener, your hear intervals an octave > apart as being of the 'same pitch class'...no other interval does > this. It seems to me that whether one ignores octave or uses them, > octaves do have a very special property in the sonic universe, one > that cannot be denied.
>
> In spite of that basic fact, I've been excited by some non-octave > music I've heard, and I've been excited to explore it, too. In the > end, if it sounds good, and is tastefully done, I guess I like it > no matter what it has, or lacks....
>
> But--
>
> I think octave doubling is still valid, and an obvious > orchestrational tool *for certain styles*...if it's an old 'junk > weapon', so is everything else ever done in the past...so, I > respectfully disagree with you on that one! ;)
>

🔗Michael Sheiman <djtrancendance@...>

Aaron>"The octave is the only interval that has a basis for pitch-class identity."

I also realize that...but who says having something with the exact same feel as a certain note below it is necessary? Furthermore, wouldn't it be a more interesting alternative in several cases (for many people, at least) to have the ability to have the pitch-classes consistently rotate for more possible moods instead of "duplicating" the mood/"pitch-class" on each octave?

I, for one, never considered the ability to limit the number of moods possible by "duplicating" the mood on a certain interval as necessarily a good thing. I think Daniel is actually spot on with his idea of chromatic "tone confusion" by avoiding the octave and extending the number of pitch-classes accessible in a single tuning without compromising consonance in any significant way (and often, in fact, enhancing it).

>"I think octave doubling is still valid, and an obvious orchestrational tool *for certain styles*..."
It certainly is...I believe my point, along with Daniels, is that there is no harm in making alternative scales which do not use the octave as a period. Sure, certain music IE Bach and Beethoven, obviously, will need to octave to preserve their original moods and general feel. But for newly composed music...I still do not see harm in scales that do not use the octave, yet maintain most familiar tetra-chords (plus add a few new ones impossible without chromaticism).

-Michael

🔗Michael Sheiman <djtrancendance@...>

>"Yes, maybe we lose octave doubling (anyway it's not much used in contemporary music, anyway it's an overused old

junk weapon from the arsenal of classicism/romanticism)"

Agreed, it is a nice thing to have, but it by no means is (or is nearly) the "backbone" of music. Easy example...the Bohlen Pierce scale has no octave, yet still obtains several very use-able chords and different types of harmonies (just not the usual 2/1 4/1 8/1...type harmonies).

>"When somebody just copies music of the past, there's no new information and the result will be boring.
And, I'm afraid, a lot of pure diatonic mean-tone (including 1/4 comma) and diatonic JI falls into this category. Anything where all notes are only a 3-7 cents off 12TET is likely to sound the same to a huge majority of untrained listeners.

>"When somebody does some totally innovative and unusual music (if anything like this is still possible), it will be boring as well, this time because there will be too much information and listener's channels will be soon overfilled.
In general I agree this is true, but, in some cases, universal theories can break past this. For example, Sethares' theories matching tuning and timbre can explain how/why diatonic systems work...or they can be mixed with other tunings, like 10TET, to make natural-sounding music with unearthly timbres or take "in-harmonic" timbres like bells and use them to construct earthly-sounding matching scales. In cases like these, the result is both overly weird/original and bears relevance to the past.
Actually, the whole (1/variable)^x+1 system, I believe, has similar properties...use it with the 2/1 octave and you get (1/2)^x+1, which generates "normal"/proven JI-style ratios. Meanwhile use PHI or the Silver Ratio in it and you get things like the (1/PHI)^x+1 scale system...which obey a similar mathematic construction formula yet, do to the different generator, make radically different scale results that still manage a good degree of familiarity. Kind of like Sethares tuning/timbre correlation work.

I'm a huge believer in finding different ways to explain JI-type constructs (especially chromatic ones) and then twisting their generators around to create things that are both blatantly new sounding and yet routed in old formulas...depending on how you look at them.

>"The best music somehow tries to combine and balance old, known principles with new, unknown."
I'll again agree for the most part...yet will also admit that what tuning often does (IE trying to inject new as simply adding notes onto old)...often does no work too well and result in things like 5-limit JI scales that sound an awfully lot like diatonic ones yet still require much tougher music theory to learn. Better, IMVHO, to jump to 7-limit or something with more substantial expressive advantages to make it worth learning music new theory for.

Also, personally, I think the whole pattern of insisting that the octave has to be the period for everything and all scale methods must, in the end, fit results higher than 2/1 into the octave by dividing by two is severely hindering the progress of tuning into new territory.
Surely not all good scales can be summarized by a generator^x/2^x type construct (even though mean-tone and MOS scales are built this way)? And surely patterns other than generator^x/2^x exist among the notes in scales like pentatonic and diatonic (of course (1/2)^x + 1 and 1.5 / (1/2)^x + 1) are two...but I'm sure there are several others).

I'm just saying...there's a lot of narrow-mindedness in tuning that seems to cause "tune-niks" to only look for old patterns in old things (IE there 'must' be an octave/unison system or the timbre for the tuning must be the harmonic series)...when they could be looking for new patterns in old things that can be twisted around into new things while maintaining a mysteriously high amount of familiarity.

-Michael

🔗Kraig Grady <kraiggrady@...>

The last century was filled with a 'behaviorist' notion that everything is just a matter of 'conditioning'. Octave are hear to stay. I have used quite a few 'non' octave scales but he result in perception is that what one is really doing is using a master scale with only using certain pitches is a particular range. That such structures are valuable and fertile ground cannot be denied. Perhaps they might be better understood for what they are in the above light. One can easily add octaves to these scales w/o upsetting the notion of the scale that is already in place.

On the other hand it is best not to explain everything in terms of western culture. In the case of the octave they are pretty close to universal ( beating octaves i would propose are still octave) It has done nothing to stand in the way of diversity.

A. Johnson wrote.....
The octave is the only interval that has a basis for pitch-class identity. Unless your nervous system is somehow unrelated to 'homo sapiens', as a musician and listener, your hear intervals an octave apart as being of the 'same pitch class'...no other interval does this. It seems to me that whether one ignores octave or uses them, octaves do have a very special property in the sonic universe, one that cannot be denied.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

🔗Kraig Grady <kraiggrady@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Kraig Grady <kraiggrady@...>

Wilson does not think that the interval of equivalence has to be the octave. No where does he state a preference. In the past because he did not mention other intervals, it was implied he didn't consider them. My views are my own on this subject not his. I have seen him BTW use Viggo Brun's algorithmn with the 5/4 within the 3/2 as just example.

I personally have tried not octave scales and i expressed my problems. The very first cut on my Interiors CD uses such a non octive scale set upon the 22 tones it has. One could say this is cheating in that all the tones are found in the 22. One could take any non octave scale reduce them to one octave and say they are derived from there.

But the problem i have below is that you are dictating what is valuable and expanding the field a priori with example that it has been a furtile ground for many or most.
it is an old way of thinking that trashing the past will lead and is innovation. Sometimes it is . Sometimes going deeper into it produces just as much innovative results.

Where i have found non octave scales most useful has been with solo music if you wish me to add something. I did a violin piece that used one and also modulated with it.

Most of Partch's instruments were non octave instruments within themselves. my tree instrument which involved hanging bars in the eikosany (one of the truly most innovative tunings ever to come out of microtonality) was non octave. My 31 tone marimba was partially non octave cause the 36 tones were set on a 31 tone keyboard. so often a pitch would only be found one place. Working with these i have a pretty good idea of what is useful to me or not.

M>S> wrote
It all begs the question...what if you applied the theories to other intervals and/or occasionally learned something from doing so that could enhance the theories?
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

🔗Michael Sheiman <djtrancendance@...>

Kraig>"I have seen him BTW use Viggo Brun's algorithmn with the 5/4 within the 3/2 as just example."
Interesting...where can I read about this? I had no clues he tried to create a "Moment Of Symmetry" around things other than octaves...

>"But the problem i have below is that you are dictating what is valuable

and expanding the field a priori with example that it has been a furtile

ground for many or most."
Well, consider this. Most of the work I've seen on non-octave scales involves adapting octave-based formulas to work on them, rather than building from the ground up. Not to mention, from what I've read very few have tried...so of course if you take hundreds of thousands (or more) people working on octave based scales and maybe 1000 working on non-octave based one...you're probability of getting a few octave-based ones the work are better.

>"it is an old way of thinking that trashing the past will lead and is

innovation. "
Not necessarily, it means building from the ground up, realizing that doing so may lead either to adopting a lot or very little from the past: it is possible to be open to both. What annoys me is people starting with the past rather than being led to it. It's like trying to take a huge automobile engine and start shaving parts down to make it work as a motorcycle engine with similar power: when you build on a close-minded foundation it's often much harder to customize something to achieve multiple advantages.

>"Where i have found non octave scales most useful has been with solo

music if you wish me to add something. I did a violin piece that used

one and also modulated with it."

Ugh, but that seems to very strongly imply chord-based music is unrealistic with non-octave scales.

An obvious counter-example....play the entire circle of fifths as a chord. Does it related to the past? Certainly. Is it locked into the past and relatively un-customizable? No it can be expanded with notes between the 5ths in many different and not-so-well explored ways. Does it work as a consonant chord? You'd better believe it... Is it octave based? Could be yes or no depending on how you calculate what tones are used in between...or no if you add no tones at all to the circle of 5ths IE 3/2 * 3/2 * 3/2...etc.

I'm not against octave-based tuning...rather, I'm for tuning in general as an art open to the fact the best formulas for creating scale could very well involve without a game plan of either "let's fit the octave" or "let's avoid it". Starting with the plan and having it lead you to a suitable periods, rather than starting with a period and having it lead you to a plan...I believe, will open many more very productive possibilities in the tuning world (and of the type that create chordal, not monophonic, music).

-Michael

--- On Thu, 5/21/09, Kraig Grady <kraiggrady@...> wrote:

From: Kraig Grady <kraiggrady@...>
Subject: [tuning] Re: Using the arithmetic mean to explain consonant-sounding JI scale
To: tuning@yahoogroups.com
Date: Thursday, May 21, 2009, 3:39 PM

Wilson does not think that the interval of equivalence has to be the

octave. No where does he state a preference. In the past because he did

not mention other intervals, it was implied he didn't consider them. My

views are my own on this subject not his. I have seen him BTW use Viggo

Brun's algorithmn with the 5/4 within the 3/2 as just example.

I personally have tried not octave scales and i expressed my problems.

The very first cut on my Interiors CD uses such a non octive scale set

upon the 22 tones it has. One could say this is cheating in that all the

tones are found in the 22. One could take any non octave scale reduce

them to one octave and say they are derived from there.

But the problem i have below is that you are dictating what is valuable

and expanding the field a priori with example that it has been a furtile

ground for many or most.

it is an old way of thinking that trashing the past will lead and is

innovation. Sometimes it is . Sometimes going deeper into it produces

just as much innovative results.

Where i have found non octave scales most useful has been with solo

music if you wish me to add something. I did a violin piece that used

one and also modulated with it.

Most of Partch's instruments were non octave instruments within

themselves. my tree instrument which involved hanging bars in the

eikosany (one of the truly most innovative tunings ever to come out of

microtonality) was non octave. My 31 tone marimba was partially non

octave cause the 36 tones were set on a 31 tone keyboard. so often a

pitch would only be found one place. Working with these i have a pretty

good idea of what is useful to me or not.

M>S> wrote

It all begs the question...what if you applied the theories to other

intervals and/or occasionally learned something from doing so that could

enhance the theories?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_

Mesotonal Music from:

_'''''''_ ^North/Western Hemisphere:

North American Embassy of Anaphoria Island <http://anaphoria. com/>

_'''''''_ ^South/Eastern Hemisphere:

Austronesian Outpost of Anaphoria <http://anaphoriasou th.blogspot. com/>

',',',',',', ',',',',' ,',',',', ',',',',' ,',',',', ',',',',' ,

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> Kraig, you're right: I should have been clearer.
>
> I meant "For example, MOS theory and Sethares critical band theory both >>can<< point toward the diatonic scale" (emphasis on can)
>
> As I mentioned before, Sethares' formula points to diatonic JI assuming normal timbres, on the other hand it points to systems very far from JI to match far different timbre (FM synthesis and in-harmonic bells, for example).
> Same sort of scenario goes for MOS...it can explain the diatonic scale but can also be shifted in other ways to provide systems with intervals that look anything but diatonic.
>
> The only thing that bugs me about both of the systems is they insist on using the octave and not questioning anything else (well, ok, Sethares does at least talk a little about stretched octaves, but not much). And, of course, the moment of symmetry is aimed at symmetry at the 2/1 octave.
>
> It all begs the question...what if you applied the theories to other intervals and/or occasionally learned something from doing so that could enhance the theories?
>
> -Michael
>
Hi Mike,

Sorry for being late. At the beginning of (I think?) this thread you asked something like "why, if at all, do you think the ear looks for multiplicative and not additive symmetry (or do you think it looks both)?". The answer is that waves are "added" via the principle of superposition. So the "sum" of 9Hz and 12Hz, say, equals 3Hz because that is actually the frequency being produced. It is in music where the operation of division takes precedence over all others, and multiplication is just it's inverse operation (As I've said before, it is no mistake that the Gk word "logos" and the Latin "ratio" evolved to mean "logical" and "rational"). 8ve equivalence is also a property of wave addition. So I still don't know where the arithmetic or geometric mean comes in, except perhaps as a novel curiousity.

As for 8ve equivalence, on a recent post you mentioned that your scale (which one I can't remember, probably (1/octave)^x + 1) doesn't produce a MOS at the 8ve but does at the 5th. But 8ve equivalence will still override this. You could always repeat the 'same' scale an 8ve higher or lower, irrespective of how the scale was produced. This could also be applied to one of Bill's "stretched 8ve's" i.e. a stretched 8ve scale an 8ve higher, and so on. In other words, as Kraig seemed to suggest, 8ve equivalence must "go without saying".

Cheers

Rick

🔗Michael Sheiman <djtrancendance@...>

>"which one I can't remember, probably (1/octave)^x + 1) doesn't produce a MOS at the 8ve but does at the 5th"
That's what it does, precisely...and yes you used the right/intended formula.

>"So the "sum" of 9Hz and 12Hz, say, equals 3Hz because that is actually the frequency being produced."
Exactly, that's why I keep on suspecting the harmonic series primarily operates on an additive basis with respect to how it produces consonance. The harmonic series has that distinct pattern in the rate of beating between frequencies that hints where the GCD/"root tone" is...and since it's a difference, it's based in "addition" and not "multiplication".

>"You could always repeat the 'same' scale an 8ve higher or lower, irrespective of how the scale was produced."
>"In other words, as Kraig seemed to suggest, 8ve equivalence must "go without saying".

I get the feeling the only thing special about the 8ve musically is that it preserves the exact tonal color/"class". Which is great if duplicating tonal color over each period is your goal (and I agree it's the best interval, both as physics/signal-processing-theory and musical feel are concerned) to achieve that.

However, I really wonder why, if at all, that should dictate how people must create scales (especially if they, like myself, actually are going for as many tonal colors as possible, rather than "perfectly equal/repeating" ones).
Surely producing keeping repeating tonal colors over a period and consonance are different issues: if they were not, for example, than the circle of 5ths must be terribly dissonant except for (3/2)^12 and (3/2)^1...which is the only combination in the entire circle which really closes in on the 8ve.

-Michael

🔗Kraig Grady <kraiggrady@...>

Erv did not publish these so far but then again he often will not till he fills he has covered enough for it to be more than piece meal.
..where can I read about this? I had no clues he tried to create a "Moment Of Symmetry" around things other than octaves...

Well, consider this. Most of the work I've seen on non-octave scales involves adapting octave-based formulas to work on them, rather than building from the ground up. Not to mention, from what I've read very few have tried...so of course if you take hundreds of thousands (or more) people working on octave based scales and maybe 1000 working on non-octave based one...you're probability of getting a few octave-based ones the work are better.

Consider not using scales at all, or even pitch. why do you even bother with scales then if you feel the past as a weight. cut yourself loose.

Not necessarily, it means building from the ground up, realizing that doing so may lead either to adopting a lot or very little from the past: it is possible to be open to both. What annoys me is people starting with the past rather than being led to it. It's like trying to take a huge automobile engine and start shaving parts down to make it work as a motorcycle engine with similar power: when you build on a close-minded foundation it's often much harder to customize something to achieve multiple advantages.

if say one explores a territory with the means available and doesn't find enough fertile ground , i see no reason to put more effort into it. Once again i do not for the most part find much i cannot do with octave scales

Ugh, but that seems to very strongly imply chord-based music is unrealistic with non-octave scales.
If i don't need octaves why would i need chords. why not throw chords out too. why hold onlto oneelement from the past and throw out others. i would get rid of chords first

An obvious counter-example....play the entire circle of fifths as a chord. Does it related to the past? Certainly. Is it locked into the past and relatively un-customizable? No it can be expanded with notes between the 5ths in many different and not-so-well explored ways. Does it work as a consonant chord? You'd better believe it... depending on the tones as you say below. but why bother with consonants if you care not about octaves?
you concern yourself with scales, consonants, chords which i find more extreme and conservative elements from the past than octaves. only the first do i have much concern with. i have little concern with chords and/or harmonic thinking, this is what leads me back into the past.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

🔗Kraig Grady <kraiggrady@...>

Consider not using scales at all, or even pitch. why do you even bother with scales then if you feel the past as a weight. cut yourself loose.

Ugh, but that seems to very strongly imply chord-based music is unrealistic with non-octave scales.
If i don't need octaves why would i need chords. why not throw chords out too. why hold on to one element from the past and throw out others. i would get rid of chords first

An obvious counter-example....play the entire circle of fifths as a chord. Does it related to the past? Certainly. Is it locked into the past and relatively un-customizable? No it can be expanded with notes between the 5ths in many different and not-so-well explored ways. Does it work as a consonant chord? You'd better believe it... depending on the tones as you say below. but why bother with consonants if you care not about octaves?
you concern yourself with scales, consonants, chords which i find more extreme and conservative elements from the past than octaves. only the first do i have much concern with. i have little concern with chords and/or harmonic thinking, this is what leads me back into the past.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> >"which one I can't remember, probably (1/octave)^x + 1) doesn't produce a MOS at the 8ve but does at the 5th"
> That's what it does, precisely...and yes you used the right/intended formula.
>
> >"So the "sum" of 9Hz and 12Hz, say, equals 3Hz because that is actually the frequency being produced."
> Exactly, that's why I keep on suspecting the harmonic series primarily operates on an additive basis with respect to how it produces consonance. The harmonic series has that distinct pattern in the rate of beating between frequencies that hints where the GCD/"root tone" is...and since it's a difference, it's based in "addition" and not "multiplication".

But Mike, that's why I put inverted commas around "sum" meaning the sum of sine waves. Y(t) = [Acos2pi(9Hz*t)] + [Bcos2pi(12Hz*t)] will produce 3Hz (prove by adding 1/3 seconds x N = 1,2,3,..to the time t. The function Y remains the same for any random choice of t). Here the difference tone is 1/2(12 - 9) = 1.5, an 8ve below the tonic 3. And of course I could have chosen other numbers which have GCD 3 but a completely different difference tone. Why this happens is because 12/9 = 4/3, division and factorisation, not subtraction. Besides, this is not an harmonic series, just two sine waves as yet.
>
> >"You could always repeat the 'same' scale an 8ve higher or lower, irrespective of how the scale was produced."
> >"In other words, as Kraig seemed to suggest, 8ve equivalence must "go without saying".
>
> I get the feeling the only thing special about the 8ve musically is that it preserves the exact tonal color/"class". Which is great if duplicating tonal color over each period is your goal (and I agree it's the best interval, both as physics/signal-processing-theory and musical feel are concerned) to achieve that.

But Mike, don't you think it's quite remarkable that we can play the "same" note on a variety of different instruments which might have nothing else in common? Why can we play "Mary had a little lamb" on an orchestra, kazoo, or whistle it and still recognise the melody? It is a philosophical concept that is sometimes hard to grasp but pitch-classes, like all classes, are not observable. We can never hear the pitch-class of a C note because playing it on one instrument is only one element of the class, not the whole class. Playing it on two instruments is another element, not the class. If we play all elements together, the tonal colour is not the same as playing one at a time. If we write a C note on a score, it remains a pitch-class until we assign an instrument and play it. Therefore, a C note contains all possible tonal colours, and none at all.

So what's preventing us from playing C3 on a piano and C4 an 8ve higher on a sax or the Nepalese Bagpipe Orchestra? It is still the "same" note an 8ve higher. So tone colour and 8ve equivalence are not necessarily related.

-Rick

> However, I really wonder why, if at all, that should dictate how people must create scales (especially if they, like myself, actually are going for as many tonal colors as possible, rather than "perfectly equal/repeating" ones).
> Surely producing keeping repeating tonal colors over a period and consonance are different issues: if they were not, for example, than the circle of 5ths must be terribly dissonant except for (3/2)^12 and (3/2)^1...which is the only combination in the entire circle which really closes in on the 8ve.
>
> -Michael
>