The Book (from MMM)

--- In MakeMicroMusic@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> * Circle of fifths
> * Modulation
> * God laughs: the Pythagorean comma

There is no circle of fifths in JI theory. One thing there is, which you don't mention, are a variety of ways of constructing JI scales in a systematic way--Fokker blocks, combination product sets, Euler generi, dwarves, etc.

> ** Regular temperaments [are "well-temperaments" included in this?]

They'd better be--those are of central importance.

> -----
> * Scale as architecture, mode as mood color
> ** Sometime you use all notes, e.g., traditional diatonics and pentatonics
> ** Most of the time you don't use all possible notes in a scale, e.g.,
> traditional "major / minor" modes as part of chromatic scale
> ** Traditional Greek modes, of which major & minor are only two
> ** "Standard" 12-EDO modes derived from but now independent of the
> Greek (same notes, different keys; same key, different notes)

Why rake over endlessly already familiar material? Even something so simple as modes/scales of 31edo is going to be completely new to most people not on these lists.

> *** 12-EDO whole tone / 6-EDO
> *** Messaien and other modes

12edo 12edo 12edo, on and on. It's pretty well-known by now.

> *** Non-octave scales (e.g., Bohlen-Pierce, Carlos Alpha / Beta / Gamma)

To the extent those actually are. Non-octave isn't important, what's important is that these are examples of subgroup scales/temperaments, and any discussion should start from consideration of subgroups, and note their generally close relationship with larger groups which contain them.

> ** traditional non-triad chords

Why should we care overmuch about tradition in these parts? If we do care, I suggest adopting a meantone rather than a 12edo view for much of the discussion.

> *** 7ths: dominant, major, minor
> *** sus4 / sus2
> *** other jazz / classical extended chords
> * adapting these ideas to non-twelve harmonies

Or simply presenting more general ideas, such as chords which can only be defined in terms of temperament, ASSes, etc.

Some of the points Gene makes seem to be related to The Book's
purpose. Is it a reference for people who know something about
xentonality and want to go deeper? Or is it a book that leads
non-xentonal musicians to xentonality?

If we're trying to reach people who are new to the idea of
xentonality, and even new to the idea that the 12-EDO scale isn't the
"natural" or "only" scale there is, it seems likely that anchoring new
content in terms of 12-EDO -- whether by showing similarities or
differences or both -- may help them relate.

Thus:

Me:
>> * Circle of fifths
>> * Modulation
>> * God laughs: the Pythagorean comma

Gene:
> There is no circle of fifths in JI theory.

JI fifths don't circulate, but everyone knows about the Circle of
Fifths. One of the things that made me realize that my understanding
of standard 12-EDO tuning was inadequate was coming to terms with the
Pythagorean comma.

That led me to the tuning list, where everyone talks about tempering
out *other* commas. Because by then I understood that tempering out
the Pythagorean comma gave a circle of fifths, and I knew that was
desirable, I was able to see that tempering other commas might be
useful for other reasons. (I still don't know what all the reasons
might be, but I figure I'll learn some as I lurk.)

So that's why I suggest that specific structure for the book: octaves,
fifths, and fourths show how the familiar 12-EDO maps almost perfectly
to the harmonic series. 5-limit ratios build out the diatonic scale
(without mentioning how badly "out of tune" they are in 12-EDO).
Stacking intervals seems like a normal and reasonable way to build out
a scale, and that leads to the circle of fifths.

Which fails to circulate, contrary to expectation.

And that leads directly to commas and temperament.

This may not be the way you want to think about commas now, but as a
didactic approach I thought it could be effective.

> One thing there is, which you don't mention, are a variety of ways of constructing
> JI scales in a systematic way--Fokker blocks, combination product sets, Euler generi,
> dwarves, etc.

I have a lot to learn. I tried reading about dwarves on the
Xenharmonic wiki and didn't get it fully -- I wasn't even sure whether
it applies to JI specifically or if it is used to generate selections
of notes from other scales, that's how ignorant I am. And I don't know
anything substantial about the others, either.

This kind of thing might make it reasonable to break some things into
"basic" and "advanced" sections: basic JI theory giving what I've
given, leading to the circle of fifths (that isn't), leading to
commas, leading to basic temperament theory. Later, an advanced JI
section could deal with the other things you mention, and it seems
likely that an advanced temperament section could be useful, too.

>> ** Regular temperaments [are "well-temperaments" included in this?]
>
> They'd better be--those are of central importance.

I wasn't sure which was a subset of the other. I'll get my terms right
eventually, I promise. :)

>> * Scale as architecture, mode as mood color
>> [Snip]
>> ** "Standard" 12-EDO modes derived from but now independent of the
>> Greek (same notes, different keys; same key, different notes)
>
> Why rake over endlessly already familiar material? Even something so simple as
> modes/scales of 31edo is going to be completely new to most people not on these
> lists.

I'm not really a musician, so maybe everyone knows about modes, but
back when I was reading guitar magazines on a regular basis it seemed
like not everyone did. In my limited experience, people think in terms
of "scales": E major, A natural minor, Blues scale in G. That leads to
the idea that there's a canonical set of notes for a scale -- but
that's precisely the mindset we want to break people out of.

The traditional modes are different ways of subsetting the same 12-EDO
scale. Now, with non-12-EDO scales, I have to decide what subsets to
use. How do I do that? Having a grip on modes seems important here.
Getting a mode of 31-EDO or Blackwood seems natural if I know that the
world already does that with 12-EDO.

>> *** 12-EDO whole tone / 6-EDO
>> *** Messaien and other modes
>
> 12edo 12edo 12edo, on and on. It's pretty well-known by now.

Except to the people you're trying to reach.

I never really liked messing around with Debussy's "artificial"
whole-tone scale. I couldn't control it, and I had grown up accustomed
to perfect fifths as a dominant harmonic factor. I get the feeling I
would treat it differently if I thought about it microtonally as an
EDO instead.

Similarly with Messaien, though I don't know his music as well.

In other words, these composers are interesting because they're
treating the ordinary 12-EDO scale as an EDO rather than as an
approximation to JI. ("See, newbie? It even works to some extent on
your beloved piano.")

>> *** Non-octave scales (e.g., Bohlen-Pierce, Carlos Alpha / Beta / Gamma)
>
> To the extent those actually are. Non-octave isn't important, what's important is
> that these are examples of subgroup scales/temperaments, and any discussion
> should start from consideration of subgroups, and note their generally close
> relationship with larger groups which contain them.

Okay. I don't know what this means, so my ability to respond is
limited. Among the hodgepodge of things I read, I see a lot of people
talking about them as non-octave scales, but I'd be happy to know that
there are better ways of thinking about them.

>> ** traditional non-triad chords
>
> Why should we care overmuch about tradition in these parts? If we do care, I
> suggest adopting a meantone rather than a 12edo view for much of the discussion.

If the goal is to introduce people to something new, you have to see
it from their perspective, which means relative to their tradition.
Most people don't know meantone, but everyone knows 12-EDO.

Once you have them in your clutches, you can twist their perspective
in other interesting ways. :)

>> *** 7ths: dominant, major, minor
>> *** sus4 / sus2
>> *** other jazz / classical extended chords
>> * adapting these ideas to non-twelve harmonies
>
> Or simply presenting more general ideas, such as chords which can only be defined in
> terms of temperament, ASSes, etc.

Sure. Ideally, I think you'd want to show people how these things
relate to extended chords that they already know.

Regards,
Jake

On 2/8/11, genewardsmith <genewardsmith@...> wrote:
>
>
> --- In MakeMicroMusic@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
>> * Circle of fifths
>> * Modulation
>> * God laughs: the Pythagorean comma
>
> There is no circle of fifths in JI theory. One thing there is, which you
> don't mention, are a variety of ways of constructing JI scales in a
> systematic way--Fokker blocks, combination product sets, Euler generi,
> dwarves, etc.
>
>> ** Regular temperaments [are "well-temperaments" included in this?]
>
> They'd better be--those are of central importance.
>
>
>> -----
>> * Scale as architecture, mode as mood color
>> ** Sometime you use all notes, e.g., traditional diatonics and pentatonics
>> ** Most of the time you don't use all possible notes in a scale, e.g.,
>> traditional "major / minor" modes as part of chromatic scale
>> ** Traditional Greek modes, of which major & minor are only two
>> ** "Standard" 12-EDO modes derived from but now independent of the
>> Greek (same notes, different keys; same key, different notes)
>
> Why rake over endlessly already familiar material? Even something so simple
> as modes/scales of 31edo is going to be completely new to most people not on
> these lists.
>
>
>> *** 12-EDO whole tone / 6-EDO
>> *** Messaien and other modes
>
> 12edo 12edo 12edo, on and on. It's pretty well-known by now.
>
>
>> *** Non-octave scales (e.g., Bohlen-Pierce, Carlos Alpha / Beta / Gamma)
>
> To the extent those actually are. Non-octave isn't important, what's
> important is that these are examples of subgroup scales/temperaments, and
> any discussion should start from consideration of subgroups, and note their
> generally close relationship with larger groups which contain them.
>
>> ** traditional non-triad chords
>
> Why should we care overmuch about tradition in these parts? If we do care, I
> suggest adopting a meantone rather than a 12edo view for much of the
> discussion.
>
>> *** 7ths: dominant, major, minor
>> *** sus4 / sus2
>> *** other jazz / classical extended chords
>> * adapting these ideas to non-twelve harmonies
>
> Or simply presenting more general ideas, such as chords which can only be
> defined in terms of temperament, ASSes, etc.
>
>
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>
>

Understanding tuning-space is central to understanding commas, I think. The Pythagorean comma is easy to understand if you first imagine a 2-dimensional tuning space where the x-axis is powers of 3/1 and the y-axis is powers of 2/1. It's obvious from this conception that because 2 & 3 are co-prime, no amount of movement along the axes will do anything but lead to larger and larger ratios--any power of 3 will be co-prime with a power of 2. No power of 3 will ever equal a power of 2, so you'll never be able to subtract a power of 2 from a power of 3 and get 0, but you *can* get arbitrarily close to 0. A comma in this space is really just a point where a power of 3 comes close to a power of 2, such that when put into a ratio, the ratio is close to 1/1.

Now, imagine that we just "say" a given comma EQUALS 1/1. We have basically assigned values x and y such that 2^x=3^y, meaning that some power of 2 *does equal* some power of 3. What this effectively does is "collapse" our two-dimensional space into a one-dimensional space, because there is now an equivalence relationship between powers of 2's and powers of 3's, meaning we can now get to some powers of 2 using only powers of 3. This means subtracting 3^y from 2^x will equal 0, in other words getting us back to our root note. At this point, because we have collapsed a 2-dimensional space to a 1-dimensional one, we have a "Rank-1" temperament, aka an EDO. In the case of the Pythagorean comma, tempering it out gives 12-EDO.

So in the 3-limit, we can actually call ANY 3-limit ratio a "comma" and temper it out to get an EDO. If we temper out 9/8 (which is the difference between 3^2 and 2^3, i.e. 9/1 and 8/1), we get 2-EDO, for example.

Now, the interesting thing is that we can also take that 2-dimensional space and, instead of tempering, just draw a "boundary" at the comma. I mean, 3-limit JI effectively contains an infinite number of intervals, right? But what if we just draw a boundary using the Pythagorean comma, and make a "box" in the 2-D space--we're saying "we're only going to use notes in this box". Well, now we have a 12-note scale, but this scale will have two step-sizes instead of one. It is, in fact, what we would call a "distributionally-even" (DE) scale, wherein the small steps are as evenly-distributed between the large steps as possible. If we draw our box using the comma 3^7/2^11 (2187/2048), we get the Pythagorean diatonic scale, which is the result of stacking seven 3/1's (or 3/2's) and reducing to fit in 1 octave. Again, two step sizes and DE.

Now, what about the 5-limit? Here, we have 3 primes: 2, 3, and 5, so our tuning space is 3-D, rather than 2-D. Tempering out a single comma (whether it's a (2,3) comma, (2,5) comma, or (3,5) comma doesn't matter) does NOT get us to an EDO. It gets us to a 2-D tuning-space, where we can either temper out another comma to get an EDO or we can draw a boundary using another comma to get a DE scale. We can also, in this 3-D space, use two commas to define a block, and this is where the JI diatonic scale comes from. The JI diatonic scale has three step-sizes: 16/15, 10/9, and 9/8, and anyone who tells you that this is the scale that music theory is founded on, they have no idea what they're talking about. This scale does not translate to the circle/spiral of fifths at all, since it's inherently 3-dimensional.

This is what took me the longest time to figure out. To get to "Meantone Temperament" in from 5-limit JI, you start by collapsing the 5-limit so that it can be described by a "warped 3-limit", by setting 81/80 (aka 3^4/(5*(2^4))) equal to 1/1. At this point, I'm not sure but I *think* you have exactly 1/4-comma meantone. THEN, you can temper out any number of 3-limit commas to get to a variety of EDOs, and any of them will technically be "versions of Meantone temperament". However, they will not all be psychoacoustically "valid" versions of Meantone, because depending on the size of the Pythagorean comma you temper out to get to your EDO, you might warp the (already warped) 3-limit so much that four 3/2's minus two 2/1's no longer gets you to something that actually sounds like a 5/4. So to get an EDO that does a good job of approximating Meantone, you need to temper out a small 3-limit comma, i.e. one that is close to 1/1. You can also think of this 3-limit comma as just "modifying the size" of 81/80, since both 81/80 and your 3-limit comma of choice will be used to alter the size of your 3/1 (and by octave equivalence, your 3/2 as well).

HTH! Oh, and if I got something wrong in this explanation, I hope someone will correct me.

-Igs

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> Some of the points Gene makes seem to be related to The Book's
> purpose. Is it a reference for people who know something about
> xentonality and want to go deeper? Or is it a book that leads
> non-xentonal musicians to xentonality?
>
> If we're trying to reach people who are new to the idea of
> xentonality, and even new to the idea that the 12-EDO scale isn't the
> "natural" or "only" scale there is, it seems likely that anchoring new
> content in terms of 12-EDO -- whether by showing similarities or
> differences or both -- may help them relate.
>
> Thus:
>
> Me:
> >> * Circle of fifths
> >> * Modulation
> >> * God laughs: the Pythagorean comma
>
> Gene:
> > There is no circle of fifths in JI theory.
>
> JI fifths don't circulate, but everyone knows about the Circle of
> Fifths. One of the things that made me realize that my understanding
> of standard 12-EDO tuning was inadequate was coming to terms with the
> Pythagorean comma.
>
> That led me to the tuning list, where everyone talks about tempering
> out *other* commas. Because by then I understood that tempering out
> the Pythagorean comma gave a circle of fifths, and I knew that was
> desirable, I was able to see that tempering other commas might be
> useful for other reasons. (I still don't know what all the reasons
> might be, but I figure I'll learn some as I lurk.)
>
> So that's why I suggest that specific structure for the book: octaves,
> fifths, and fourths show how the familiar 12-EDO maps almost perfectly
> to the harmonic series. 5-limit ratios build out the diatonic scale
> (without mentioning how badly "out of tune" they are in 12-EDO).
> Stacking intervals seems like a normal and reasonable way to build out
> a scale, and that leads to the circle of fifths.
>
> Which fails to circulate, contrary to expectation.
>
> And that leads directly to commas and temperament.
>
> This may not be the way you want to think about commas now, but as a
> didactic approach I thought it could be effective.
>
> > One thing there is, which you don't mention, are a variety of ways of constructing
> > JI scales in a systematic way--Fokker blocks, combination product sets, Euler generi,
> > dwarves, etc.
>
> I have a lot to learn. I tried reading about dwarves on the
> Xenharmonic wiki and didn't get it fully -- I wasn't even sure whether
> it applies to JI specifically or if it is used to generate selections
> of notes from other scales, that's how ignorant I am. And I don't know
> anything substantial about the others, either.
>
> This kind of thing might make it reasonable to break some things into
> "basic" and "advanced" sections: basic JI theory giving what I've
> given, leading to the circle of fifths (that isn't), leading to
> commas, leading to basic temperament theory. Later, an advanced JI
> section could deal with the other things you mention, and it seems
> likely that an advanced temperament section could be useful, too.
>
> >> ** Regular temperaments [are "well-temperaments" included in this?]
> >
> > They'd better be--those are of central importance.
>
> I wasn't sure which was a subset of the other. I'll get my terms right
> eventually, I promise. :)
>
> >> * Scale as architecture, mode as mood color
> >> [Snip]
> >> ** "Standard" 12-EDO modes derived from but now independent of the
> >> Greek (same notes, different keys; same key, different notes)
> >
> > Why rake over endlessly already familiar material? Even something so simple as
> > modes/scales of 31edo is going to be completely new to most people not on these
> > lists.
>
> I'm not really a musician, so maybe everyone knows about modes, but
> back when I was reading guitar magazines on a regular basis it seemed
> like not everyone did. In my limited experience, people think in terms
> of "scales": E major, A natural minor, Blues scale in G. That leads to
> the idea that there's a canonical set of notes for a scale -- but
> that's precisely the mindset we want to break people out of.
>
> The traditional modes are different ways of subsetting the same 12-EDO
> scale. Now, with non-12-EDO scales, I have to decide what subsets to
> use. How do I do that? Having a grip on modes seems important here.
> Getting a mode of 31-EDO or Blackwood seems natural if I know that the
> world already does that with 12-EDO.
>
> >> *** 12-EDO whole tone / 6-EDO
> >> *** Messaien and other modes
> >
> > 12edo 12edo 12edo, on and on. It's pretty well-known by now.
>
> Except to the people you're trying to reach.
>
> I never really liked messing around with Debussy's "artificial"
> whole-tone scale. I couldn't control it, and I had grown up accustomed
> to perfect fifths as a dominant harmonic factor. I get the feeling I
> would treat it differently if I thought about it microtonally as an
> EDO instead.
>
> Similarly with Messaien, though I don't know his music as well.
>
> In other words, these composers are interesting because they're
> treating the ordinary 12-EDO scale as an EDO rather than as an
> approximation to JI. ("See, newbie? It even works to some extent on
> your beloved piano.")
>
> >> *** Non-octave scales (e.g., Bohlen-Pierce, Carlos Alpha / Beta / Gamma)
> >
> > To the extent those actually are. Non-octave isn't important, what's important is
> > that these are examples of subgroup scales/temperaments, and any discussion
> > should start from consideration of subgroups, and note their generally close
> > relationship with larger groups which contain them.
>
> Okay. I don't know what this means, so my ability to respond is
> limited. Among the hodgepodge of things I read, I see a lot of people
> talking about them as non-octave scales, but I'd be happy to know that
> there are better ways of thinking about them.
>
> >> ** traditional non-triad chords
> >
> > Why should we care overmuch about tradition in these parts? If we do care, I
> > suggest adopting a meantone rather than a 12edo view for much of the discussion.
>
> If the goal is to introduce people to something new, you have to see
> it from their perspective, which means relative to their tradition.
> Most people don't know meantone, but everyone knows 12-EDO.
>
> Once you have them in your clutches, you can twist their perspective
> in other interesting ways. :)
>
> >> *** 7ths: dominant, major, minor
> >> *** sus4 / sus2
> >> *** other jazz / classical extended chords
> >> * adapting these ideas to non-twelve harmonies
> >
> > Or simply presenting more general ideas, such as chords which can only be defined in
> > terms of temperament, ASSes, etc.
>
> Sure. Ideally, I think you'd want to show people how these things
> relate to extended chords that they already know.
>
> Regards,
> Jake
>
> On 2/8/11, genewardsmith <genewardsmith@...> wrote:
> >
> >
> > --- In MakeMicroMusic@yahoogroups.com, Jake Freivald <jdfreivald@> wrote:
> >
> >> * Circle of fifths
> >> * Modulation
> >> * God laughs: the Pythagorean comma
> >
> > There is no circle of fifths in JI theory. One thing there is, which you
> > don't mention, are a variety of ways of constructing JI scales in a
> > systematic way--Fokker blocks, combination product sets, Euler generi,
> > dwarves, etc.
> >
> >> ** Regular temperaments [are "well-temperaments" included in this?]
> >
> > They'd better be--those are of central importance.
> >
> >
> >> -----
> >> * Scale as architecture, mode as mood color
> >> ** Sometime you use all notes, e.g., traditional diatonics and pentatonics
> >> ** Most of the time you don't use all possible notes in a scale, e.g.,
> >> traditional "major / minor" modes as part of chromatic scale
> >> ** Traditional Greek modes, of which major & minor are only two
> >> ** "Standard" 12-EDO modes derived from but now independent of the
> >> Greek (same notes, different keys; same key, different notes)
> >
> > Why rake over endlessly already familiar material? Even something so simple
> > as modes/scales of 31edo is going to be completely new to most people not on
> > these lists.
> >
> >
> >> *** 12-EDO whole tone / 6-EDO
> >> *** Messaien and other modes
> >
> > 12edo 12edo 12edo, on and on. It's pretty well-known by now.
> >
> >
> >> *** Non-octave scales (e.g., Bohlen-Pierce, Carlos Alpha / Beta / Gamma)
> >
> > To the extent those actually are. Non-octave isn't important, what's
> > important is that these are examples of subgroup scales/temperaments, and
> > any discussion should start from consideration of subgroups, and note their
> > generally close relationship with larger groups which contain them.
> >
> >> ** traditional non-triad chords
> >
> > Why should we care overmuch about tradition in these parts? If we do care, I
> > suggest adopting a meantone rather than a 12edo view for much of the
> > discussion.
> >
> >> *** 7ths: dominant, major, minor
> >> *** sus4 / sus2
> >> *** other jazz / classical extended chords
> >> * adapting these ideas to non-twelve harmonies
> >
> > Or simply presenting more general ideas, such as chords which can only be
> > defined in terms of temperament, ASSes, etc.
> >
> >
> >
> >
> >
> > ------------------------------------
> >
> > You can configure your subscription by sending an empty email to one
> > of these addresses (from the address at which you receive the list):
> > tuning-subscribe@yahoogroups.com - join the tuning group.
> > tuning-unsubscribe@yahoogroups.com - leave the group.
> > tuning-nomail@yahoogroups.com - turn off mail from the group.
> > tuning-digest@yahoogroups.com - set group to send daily digests.
> > tuning-normal@yahoogroups.com - set group to send individual emails.
> > tuning-help@yahoogroups.com - receive general help information.
> > Yahoo! Groups Links
> >
> >
> >
> >
>

> The Book ... scales ... temperaments ... Circle of Fifths ...

Very interesting considerations in this short exchange!

Of course, I had similar basic ideas when I put together my Unequal Temperaments book in 1978, and rewrote it 30 years later (both to favourable reviews and comments by scholars and musicians). That was quite a mammoth task: I managed to include the full theory of intervals, scales and temperament, plus every important temperament treated in detail, a historical overview, tuning and intonation instructions, multiple divisions and other complementary topics.

However, my book is lacking for members of this forum: it is meant for music up to c.1900! I would welcome that somebody VERY conversant with MODERN SCALES AND TUNINGS undertakes this new mammoth scrutinising the main modern trends, proposal and music. (My UT could be used as background to avoid reinventing the wheel about the basic acoustics of the musical scale.)

Best wishes

Claudio
http://temper.braybaroque.ie

(Note: the Enharmonic Keyboards treatise by Prof. Barbieri is specifically meant for early or classical music with more than 12 pitches per octave, but it also covers up to c1900).

Igs,

Clearest explanation I've ever heard. thanks. And now I have some idea
what Paul has been posting to facebook as well.

Chris

On Wed, Feb 9, 2011 at 3:16 PM, cityoftheasleep <igliashon@...> wrote:
>
>
>
> Understanding tuning-space is central to understanding commas, I think. The Pythagorean comma is easy to understand if you first imagine a 2-dimensional tuning space where the x-axis is powers of 3/1 and the y-axis is powers of 2/1. It's obvious from this conception that because 2 & 3 are co-prime, no amount of movement along the axes will do anything but lead to larger and larger ratios--any power of 3 will be co-prime with a power of 2. No power of 3 will ever equal a power of 2, so you'll never be able to subtract a power of 2 from a power of 3 and get 0, but you *can* get arbitrarily close to 0. A comma in this space is really just a point where a power of 3 comes close to a power of 2, such that when put into a ratio, the ratio is close to 1/1.
>

On Wed, Feb 9, 2011 at 3:16 PM, cityoftheasleep <igliashon@...> wrote:
>
> This is what took me the longest time to figure out. To get to "Meantone Temperament" in from 5-limit JI, you start by collapsing the 5-limit so that it can be described by a "warped 3-limit", by setting 81/80 (aka 3^4/(5*(2^4))) equal to 1/1. At this point, I'm not sure but I *think* you have exactly 1/4-comma meantone.

You don't have 1/4-comma meantone, you just end up with a
tuning-inspecific meantone mapping.

> THEN, you can temper out any number of 3-limit commas to get to a variety of EDOs, and any of them will technically be "versions of Meantone temperament". However, they will not all be psychoacoustically "valid" versions of Meantone, because depending on the size of the Pythagorean comma you temper out to get to your EDO, you might warp the (already warped) 3-limit so much that four 3/2's minus two 2/1's no longer gets you to something that actually sounds like a 5/4.

You can also describe it as a second 5-limit comma, e.g. 12-equal is
81/80 and 128/125 vanishing. You don't need only one 5,3 comma and
then one 3,2 comma. I think you can probably translate any comma into
a purely 3-limit one though. The rest of your explanation was pretty
spot on as far as I can tell.

> So to get an EDO that does a good job of approximating Meantone, you need to temper out a small 3-limit comma, i.e. one that is close to 1/1. You can also think of this 3-limit comma as just "modifying the size" of 81/80, since both 81/80 and your 3-limit comma of choice will be used to alter the size of your 3/1 (and by octave equivalence, your 3/2 as well).

I also like to think of it in terms of the simplest intervals that are
equated. So magic means that a bunch of major thirds gets you to a
fifth, okay. But it also happens to mean that 25/24 and 128/125 end up
being equated to the same interval. And porcupine means that 250/243
vanishes, whatever that means. Well, it happens to mean that 10/9 and
27/25 end up being equated to the same interval, and hence that 25/24
and 81/80 end up being equated to the same interval. I have a quick
algorithm to figure out the simplest intervals that are equated for a
certain comma being tempered out, if anyone's interested. I know it
works for the 5-limit and should hold for higher-dimensional
temperaments (I think).

-Mike

Hi Igs,

I read your post three times and it clarified for me a lot of things I had read before on the List but never understood. You said:

<Now, what about the 5-limit? Here, we have 3 primes: 2, 3, and 5, so our tuning space is 3-D, rather than 2-D.>

What about 29 limit? (2, 3, 5, 7, 11, 13, 17, 19, 23, 29) How would you work with a 10 dimensional space? Has it ever been done? (29/6 is a good harmony interval in my system.)

Your elucidation makes sense but it's seems to me to be at odds with my own approach. I know a little bit about commas and (if you can believe it) I rediscovered meantone temperament in the late 90's by using the fourth root of 80 instead of 3 in the circle of fifths.

I should point out that my meantone had seventeen notes where (if D is the starting note and you go up 8 notes and down 8 notes from D) A# was slightly lower than Bb, C# was slightly lower than Db and so on. But I think regular meantone is a subset of this.

My approach has nothing at all to do with commas. I choose a tonic (1/1) and I select eleven more notes (between 1/1 and 2/1) so that (i) all the chosen notes go well with the tonic, 1/1 and (ii) the greatest possible number of *good* harmony intervals (either containing the tonic or otherwise and an octave or less wide) occur, over a one octave range. I had to strike a balance between (i) and (ii) to get to best of both worlds. All the intervals chosen are just.

Then I take my just scale and temper it to get a scale with the maximum number of good harmony intervals (an octave or less wide) and each note in the tempered version is less than 256/255 (6.776 cents) away from the original note. (In fact, in the end, every note in the tempered scale was less than 5.4 cents away from the notes in the original just scale.)

So, commas don't figure in my chosen method for scale building although they are obviously valid for other methods. Which method is better?

John.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Understanding tuning-space is central to understanding commas, I think. The Pythagorean comma is easy to understand if you first imagine a 2-dimensional tuning space where the x-axis is powers of 3/1 and the y-axis is powers of 2/1. It's obvious from this conception that because 2 & 3 are co-prime, no amount of movement along the axes will do anything but lead to larger and larger ratios--any power of 3 will be co-prime with a power of 2. No power of 3 will ever equal a power of 2, so you'll never be able to subtract a power of 2 from a power of 3 and get 0, but you *can* get arbitrarily close to 0. A comma in this space is really just a point where a power of 3 comes close to a power of 2, such that when put into a ratio, the ratio is close to 1/1.
>
> Now, imagine that we just "say" a given comma EQUALS 1/1. We have basically assigned values x and y such that 2^x=3^y, meaning that some power of 2 *does equal* some power of 3. What this effectively does is "collapse" our two-dimensional space into a one-dimensional space, because there is now an equivalence relationship between powers of 2's and powers of 3's, meaning we can now get to some powers of 2 using only powers of 3. This means subtracting 3^y from 2^x will equal 0, in other words getting us back to our root note. At this point, because we have collapsed a 2-dimensional space to a 1-dimensional one, we have a "Rank-1" temperament, aka an EDO. In the case of the Pythagorean comma, tempering it out gives 12-EDO.
>
> So in the 3-limit, we can actually call ANY 3-limit ratio a "comma" and temper it out to get an EDO. If we temper out 9/8 (which is the difference between 3^2 and 2^3, i.e. 9/1 and 8/1), we get 2-EDO, for example.
>
> Now, the interesting thing is that we can also take that 2-dimensional space and, instead of tempering, just draw a "boundary" at the comma. I mean, 3-limit JI effectively contains an infinite number of intervals, right? But what if we just draw a boundary using the Pythagorean comma, and make a "box" in the 2-D space--we're saying "we're only going to use notes in this box". Well, now we have a 12-note scale, but this scale will have two step-sizes instead of one. It is, in fact, what we would call a "distributionally-even" (DE) scale, wherein the small steps are as evenly-distributed between the large steps as possible. If we draw our box using the comma 3^7/2^11 (2187/2048), we get the Pythagorean diatonic scale, which is the result of stacking seven 3/1's (or 3/2's) and reducing to fit in 1 octave. Again, two step sizes and DE.
>
> Now, what about the 5-limit? Here, we have 3 primes: 2, 3, and 5, so our tuning space is 3-D, rather than 2-D. Tempering out a single comma (whether it's a (2,3) comma, (2,5) comma, or (3,5) comma doesn't matter) does NOT get us to an EDO. It gets us to a 2-D tuning-space, where we can either temper out another comma to get an EDO or we can draw a boundary using another comma to get a DE scale. We can also, in this 3-D space, use two commas to define a block, and this is where the JI diatonic scale comes from. The JI diatonic scale has three step-sizes: 16/15, 10/9, and 9/8, and anyone who tells you that this is the scale that music theory is founded on, they have no idea what they're talking about. This scale does not translate to the circle/spiral of fifths at all, since it's inherently 3-dimensional.
>
> This is what took me the longest time to figure out. To get to "Meantone Temperament" in from 5-limit JI, you start by collapsing the 5-limit so that it can be described by a "warped 3-limit", by setting 81/80 (aka 3^4/(5*(2^4))) equal to 1/1. At this point, I'm not sure but I *think* you have exactly 1/4-comma meantone. THEN, you can temper out any number of 3-limit commas to get to a variety of EDOs, and any of them will technically be "versions of Meantone temperament". However, they will not all be psychoacoustically "valid" versions of Meantone, because depending on the size of the Pythagorean comma you temper out to get to your EDO, you might warp the (already warped) 3-limit so much that four 3/2's minus two 2/1's no longer gets you to something that actually sounds like a 5/4. So to get an EDO that does a good job of approximating Meantone, you need to temper out a small 3-limit comma, i.e. one that is close to 1/1. You can also think of this 3-limit comma as just "modifying the size" of 81/80, since both 81/80 and your 3-limit comma of choice will be used to alter the size of your 3/1 (and by octave equivalence, your 3/2 as well).
>
> HTH! Oh, and if I got something wrong in this explanation, I hope someone will correct me.
>
> -Igs
>
> --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@> wrote:
> >
> > Some of the points Gene makes seem to be related to The Book's
> > purpose. Is it a reference for people who know something about
> > xentonality and want to go deeper? Or is it a book that leads
> > non-xentonal musicians to xentonality?
> >
> > If we're trying to reach people who are new to the idea of
> > xentonality, and even new to the idea that the 12-EDO scale isn't the
> > "natural" or "only" scale there is, it seems likely that anchoring new
> > content in terms of 12-EDO -- whether by showing similarities or
> > differences or both -- may help them relate.
> >
> > Thus:
> >
> > Me:
> > >> * Circle of fifths
> > >> * Modulation
> > >> * God laughs: the Pythagorean comma
> >
> > Gene:
> > > There is no circle of fifths in JI theory.
> >
> > JI fifths don't circulate, but everyone knows about the Circle of
> > Fifths. One of the things that made me realize that my understanding
> > of standard 12-EDO tuning was inadequate was coming to terms with the
> > Pythagorean comma.
> >
> > That led me to the tuning list, where everyone talks about tempering
> > out *other* commas. Because by then I understood that tempering out
> > the Pythagorean comma gave a circle of fifths, and I knew that was
> > desirable, I was able to see that tempering other commas might be
> > useful for other reasons. (I still don't know what all the reasons
> > might be, but I figure I'll learn some as I lurk.)
> >
> > So that's why I suggest that specific structure for the book: octaves,
> > fifths, and fourths show how the familiar 12-EDO maps almost perfectly
> > to the harmonic series. 5-limit ratios build out the diatonic scale
> > (without mentioning how badly "out of tune" they are in 12-EDO).
> > Stacking intervals seems like a normal and reasonable way to build out
> > a scale, and that leads to the circle of fifths.
> >
> > Which fails to circulate, contrary to expectation.
> >
> > And that leads directly to commas and temperament.
> >
> > This may not be the way you want to think about commas now, but as a
> > didactic approach I thought it could be effective.
> >
> > > One thing there is, which you don't mention, are a variety of ways of constructing
> > > JI scales in a systematic way--Fokker blocks, combination product sets, Euler generi,
> > > dwarves, etc.
> >
> > I have a lot to learn. I tried reading about dwarves on the
> > Xenharmonic wiki and didn't get it fully -- I wasn't even sure whether
> > it applies to JI specifically or if it is used to generate selections
> > of notes from other scales, that's how ignorant I am. And I don't know
> > anything substantial about the others, either.
> >
> > This kind of thing might make it reasonable to break some things into
> > "basic" and "advanced" sections: basic JI theory giving what I've
> > given, leading to the circle of fifths (that isn't), leading to
> > commas, leading to basic temperament theory. Later, an advanced JI
> > section could deal with the other things you mention, and it seems
> > likely that an advanced temperament section could be useful, too.
> >
> > >> ** Regular temperaments [are "well-temperaments" included in this?]
> > >
> > > They'd better be--those are of central importance.
> >
> > I wasn't sure which was a subset of the other. I'll get my terms right
> > eventually, I promise. :)
> >
> > >> * Scale as architecture, mode as mood color
> > >> [Snip]
> > >> ** "Standard" 12-EDO modes derived from but now independent of the
> > >> Greek (same notes, different keys; same key, different notes)
> > >
> > > Why rake over endlessly already familiar material? Even something so simple as
> > > modes/scales of 31edo is going to be completely new to most people not on these
> > > lists.
> >
> > I'm not really a musician, so maybe everyone knows about modes, but
> > back when I was reading guitar magazines on a regular basis it seemed
> > like not everyone did. In my limited experience, people think in terms
> > of "scales": E major, A natural minor, Blues scale in G. That leads to
> > the idea that there's a canonical set of notes for a scale -- but
> > that's precisely the mindset we want to break people out of.
> >
> > The traditional modes are different ways of subsetting the same 12-EDO
> > scale. Now, with non-12-EDO scales, I have to decide what subsets to
> > use. How do I do that? Having a grip on modes seems important here.
> > Getting a mode of 31-EDO or Blackwood seems natural if I know that the
> > world already does that with 12-EDO.
> >
> > >> *** 12-EDO whole tone / 6-EDO
> > >> *** Messaien and other modes
> > >
> > > 12edo 12edo 12edo, on and on. It's pretty well-known by now.
> >
> > Except to the people you're trying to reach.
> >
> > I never really liked messing around with Debussy's "artificial"
> > whole-tone scale. I couldn't control it, and I had grown up accustomed
> > to perfect fifths as a dominant harmonic factor. I get the feeling I
> > would treat it differently if I thought about it microtonally as an
> > EDO instead.
> >
> > Similarly with Messaien, though I don't know his music as well.
> >
> > In other words, these composers are interesting because they're
> > treating the ordinary 12-EDO scale as an EDO rather than as an
> > approximation to JI. ("See, newbie? It even works to some extent on
> > your beloved piano.")
> >
> > >> *** Non-octave scales (e.g., Bohlen-Pierce, Carlos Alpha / Beta / Gamma)
> > >
> > > To the extent those actually are. Non-octave isn't important, what's important is
> > > that these are examples of subgroup scales/temperaments, and any discussion
> > > should start from consideration of subgroups, and note their generally close
> > > relationship with larger groups which contain them.
> >
> > Okay. I don't know what this means, so my ability to respond is
> > limited. Among the hodgepodge of things I read, I see a lot of people
> > talking about them as non-octave scales, but I'd be happy to know that
> > there are better ways of thinking about them.
> >
> > >> ** traditional non-triad chords
> > >
> > > Why should we care overmuch about tradition in these parts? If we do care, I
> > > suggest adopting a meantone rather than a 12edo view for much of the discussion.
> >
> > If the goal is to introduce people to something new, you have to see
> > it from their perspective, which means relative to their tradition.
> > Most people don't know meantone, but everyone knows 12-EDO.
> >
> > Once you have them in your clutches, you can twist their perspective
> > in other interesting ways. :)
> >
> > >> *** 7ths: dominant, major, minor
> > >> *** sus4 / sus2
> > >> *** other jazz / classical extended chords
> > >> * adapting these ideas to non-twelve harmonies
> > >
> > > Or simply presenting more general ideas, such as chords which can only be defined in
> > > terms of temperament, ASSes, etc.
> >
> > Sure. Ideally, I think you'd want to show people how these things
> > relate to extended chords that they already know.
> >
> > Regards,
> > Jake
> >
> > On 2/8/11, genewardsmith <genewardsmith@> wrote:
> > >
> > >
> > > --- In MakeMicroMusic@yahoogroups.com, Jake Freivald <jdfreivald@> wrote:
> > >
> > >> * Circle of fifths
> > >> * Modulation
> > >> * God laughs: the Pythagorean comma
> > >
> > > There is no circle of fifths in JI theory. One thing there is, which you
> > > don't mention, are a variety of ways of constructing JI scales in a
> > > systematic way--Fokker blocks, combination product sets, Euler generi,
> > > dwarves, etc.
> > >
> > >> ** Regular temperaments [are "well-temperaments" included in this?]
> > >
> > > They'd better be--those are of central importance.
> > >
> > >
> > >> -----
> > >> * Scale as architecture, mode as mood color
> > >> ** Sometime you use all notes, e.g., traditional diatonics and pentatonics
> > >> ** Most of the time you don't use all possible notes in a scale, e.g.,
> > >> traditional "major / minor" modes as part of chromatic scale
> > >> ** Traditional Greek modes, of which major & minor are only two
> > >> ** "Standard" 12-EDO modes derived from but now independent of the
> > >> Greek (same notes, different keys; same key, different notes)
> > >
> > > Why rake over endlessly already familiar material? Even something so simple
> > > as modes/scales of 31edo is going to be completely new to most people not on
> > > these lists.
> > >
> > >
> > >> *** 12-EDO whole tone / 6-EDO
> > >> *** Messaien and other modes
> > >
> > > 12edo 12edo 12edo, on and on. It's pretty well-known by now.
> > >
> > >
> > >> *** Non-octave scales (e.g., Bohlen-Pierce, Carlos Alpha / Beta / Gamma)
> > >
> > > To the extent those actually are. Non-octave isn't important, what's
> > > important is that these are examples of subgroup scales/temperaments, and
> > > any discussion should start from consideration of subgroups, and note their
> > > generally close relationship with larger groups which contain them.
> > >
> > >> ** traditional non-triad chords
> > >
> > > Why should we care overmuch about tradition in these parts? If we do care, I
> > > suggest adopting a meantone rather than a 12edo view for much of the
> > > discussion.
> > >
> > >> *** 7ths: dominant, major, minor
> > >> *** sus4 / sus2
> > >> *** other jazz / classical extended chords
> > >> * adapting these ideas to non-twelve harmonies
> > >
> > > Or simply presenting more general ideas, such as chords which can only be
> > > defined in terms of temperament, ASSes, etc.
> > >
> > >
> > >
> > >
> > >
> > > ------------------------------------
> > >
> > > You can configure your subscription by sending an empty email to one
> > > of these addresses (from the address at which you receive the list):
> > > tuning-subscribe@yahoogroups.com - join the tuning group.
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> > > tuning-help@yahoogroups.com - receive general help information.
> > > Yahoo! Groups Links
> > >
> > >
> > >
> > >
> >
>

> Now, imagine that we just "say" a given comma EQUALS 1/1. We have basically assigned values x and y such that 2^x=3^y, meaning that some power of 2 *does equal* some power of 3. What this effectively does is "collapse" our two-dimensional space into a one-dimensional space, because there is now an equivalence relationship between powers of 2's and powers of 3's, meaning we can now get to some powers of 2 using only powers of 3. This means subtracting 3^y from 2^x will equal 0, in other words getting us back to our root note. At this point, because we have collapsed a 2-dimensional space to a 1-dimensional one, we have a "Rank-1" temperament, aka an EDO. In the case of the Pythagorean comma, tempering it out gives 12-EDO.
>
> So in the 3-limit, we can actually call ANY 3-limit ratio a "comma" and temper it out to get an EDO. If we temper out 9/8 (which is the difference between 3^2 and 2^3, i.e. 9/1 and 8/1), we get 2-EDO, for example.

I think it would be good to point out the two general motivations for choosing commas: accuracy and simplicity. Accuracy is how close to 1 your ratio that you're setting equal to one actually is, and simplicity is how few powers of your primes you have to go through to get to that comma.

The pythagorean comma is a good compromise between the two because when you set 3^12=2^19, you are setting 524,288=531,441. Dividing both sides by 524,288 shows that you are setting 1.01364326... equal to 1. That's relatively close to being true, so tempering that comma is retains reasonable accuracy, and we didn't have to go very far in powers of 2 and three the get there, so it's reasonably simple.

Tempering 9/8=1/1 (which you do by setting 3^2=2^3) gives you 1.125=1. Not near as accurate as the pythagorean comma (1.0136...=1), but it is admittedly MUCH simpler. You go through very few powers of 2 and 3 to get to this comma.

Conversely, if you go much much further in the powers of 3 and 2, you can set 3^53=2^84 tempering a comma so that 1.00209031=1. This comma gives you extremely good accuracy compared to the pythagorean comma, but it is also far more *complex*, taking far more powers of 2 and 3 to reach.

This battle between accuracy and simplicity has perceptual implications. Commas that are too complex (regardless of their accuracy) tend to lead to musical structures that are hard to deal with compositionally and in performance, and they're hard to follow for a listener. Commas that are too inaccurate (regardless of simplicity) will lead to musical structures with very out of tune harmonies that will probably make you cringe.

As per your taste, you may lean towards exploring temperaments with inaccurate commas that have unique or simple structures, or you may decide to work with some of the overly complex temperaments to take advantage of some very precise harmonies. Just as a general rule though, people seem to gravitate towards compromises between the two.

John

I'm not Igs, but I'll respond to the obvious math questions anyway.

On Wed, Feb 9, 2011 at 5:28 PM, john777music <jfos777@...> wrote:
>
> What about 29 limit? (2, 3, 5, 7, 11, 13, 17, 19, 23, 29) How would you work with a 10 dimensional space? Has it ever been done? (29/6 is a good harmony interval in my system.)

I don't think that anyone's bothered to work with the 29-limit. I
could be wrong. On tuning-math they've done pretty exhaustive searches
for consistent ET's up to all kinds of ridiculous limits, in search of
the One True Universal ET, usually something really high-numbered to
replace 1200-ET or something like that. At least that's what I got
from it.

> Your elucidation makes sense but it's seems to me to be at odds with my own approach. I know a little bit about commas and (if you can believe it) I rediscovered meantone temperament in the late 90's by using the fourth root of 80 instead of 3 in the circle of fifths.

That would be specifically quarter-comma meantone.

> I should point out that my meantone had seventeen notes where (if D is the starting note and you go up 8 notes and down 8 notes from D) A# was slightly lower than Bb, C# was slightly lower than Db and so on. But I think regular meantone is a subset of this.

Regular meantone doesn't specify how many notes are in the scale -
just how the different intervals are mapped relative to one another.
We tend to talk in "MOS's" around here. MOS stands for "Moment of
Symmetry," and refers to a scale in which there are only two step
sizes (and only two sizes of third, two sizes of fourth, etc). So
meantone, for any decent tuning of the generator, will have MOS's at
1, 2, 5, 7, 12, 19, etc notes. Your 17-note scale could be said to be
a subset of the 19-note one, which ends up yielding some useful
properties if higher-order meantones are desired. Or it could be said
to be a superset of the 12-tone one, with 5 extra notes, if you want.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> You don't have 1/4-comma meantone, you just end up with a
> tuning-inspecific meantone mapping.

Okay. I don't know why I thought that...I guess it seemed that at 1/4 comma, four 3/2's adds up to an *exact* 5/4, since the mapping is based on equating four 3/2's with 5/4. But yes, you're right.

> You can also describe it as a second 5-limit comma, e.g. 12-equal is
> 81/80 and 128/125 vanishing. You don't need only one 5,3 comma and
> then one 3,2 comma. I think you can probably translate any comma into
> a purely 3-limit one though. The rest of your explanation was pretty
> spot on as far as I can tell.

Now that part seems confusing to me. If you temper out a 5,3 comma, doesn't that mean you've basically reduced the entire prime-5 dimension to terms of prime-3? How do you temper out 128/125 from the 2-dimensional 81/80-tempered 3-limit Meantone lattice?

> I have a quick
> algorithm to figure out the simplest intervals that are equated for a
> certain comma being tempered out, if anyone's interested. I know it
> works for the 5-limit and should hold for higher-dimensional
> temperaments (I think).

Now THAT is cool. I am interested!

-Igs

On Wed, Feb 9, 2011 at 5:38 PM, John Moriarty <JlMoriart@...> wrote:
> > So in the 3-limit, we can actually call ANY 3-limit ratio a "comma" and temper it out to get an EDO. If we temper out 9/8 (which is the difference between 3^2 and 2^3, i.e. 9/1 and 8/1), we get 2-EDO, for example.
>
> I think it would be good to point out the two general motivations for choosing commas: accuracy and simplicity. Accuracy is how close to 1 your ratio that you're setting equal to one actually is, and simplicity is how few powers of your primes you have to go through to get to that comma.

I used to think this way, but nowadays I'd add one more motivation,
which is somewhat at odds with the other two: setting up interesting
puns. Blackwood is a great example. It's hardly based off of an ideal
comma to temper - 256/243 - which is about 90 cents large and more
complex than the nearby 250/243, which is 49 cents and defines
porcupine temperament.

These are both great tunings with interesting puns, but I find
Blackwood's especially interesting because of the drastically
shortened circle of fifths that arises from it, which leads to some
really interesting chord progressions. Whitewood does something
similar but builds everything off of a 7-tet scale rather than a 5-tet
scale.

This goal is, like I said, somewhat at odds with the other ones
mentioned: it's really easy to make arbitrarily interesting puns by
just equating random intervals together. Doing so tends to produce a
lot of tuning error, so a bit of tact is needed here.

-Mike

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> What about 29 limit? (2, 3, 5, 7, 11, 13, 17, 19, 23, 29) How would you work with a 10
> dimensional space? Has it ever been done? (29/6 is a good harmony interval in my
> system.)

Well, to get to a rank-2 temperament, you'd need to temper out 8 29-limit commas, although if you're not after 11, 13, 17, 19, and 23, you don't need to include them. There's no reason you couldn't do a 2.3.5.7.29 subgroup. Although if you only want 29/6 and not 29/1 (or 29/4, 29/8, 29/16 etc.) you can do a 2.3.5.7.29/6 or a 2.3.5.7.29/3 subgroup. The beauty of tuning space is that you can construct it however you want.

However, if you're working with octaves, you're not going to be able to avoid getting 29/12 and 29/24 if you have 29/3 or 29/6. And 29/24 isn't a very strong identity, I believe it's firmly in the dyadic entropy well of 6/5. Thus I'm pretty sure if you try to combine 5-limit consonances with ratios of 29, the two are going to fight each other during optimization and probably end up splitting the difference and having 29/24 mapped to the same interval as 6/5.

> My approach has nothing at all to do with commas.

From your description, it sounds like your approach is very similar to that taken by Michael S. (aka djtrancendance). If I understand it correctly, your approach has little to do with commas because it has little to do with limits. You have worked out a set of ratios that you consider desirable, and you have worked out an error threshold that you consider to be acceptable, as well as a maximum number of notes. I'll have to read your book to get much deeper into the nuts and bolts.

Looking at your Blue JI scale, it looks to me to be a 4-dimensional scale of mostly 5-limit intervals but with two notes that poke into the dimension of prime-7, as Kraig handily diagrammed here with the dimension of prime-2 omitted since octaves are equivalent:

http://anaphoria.com/OsullivanBLUESjiscale.gif

As a JI scale, I'd say it's better than strict 5-limit chromatic (which has a 16/15 and a 45/32 instead of 15/14 and 7/5), but still has all the same problems all JI systems do: limited modulation ability due to harsh discords in distant keys.

Now, your temperament is essentially what is considered a "well-temperament", whereby intervals are tempered less in some keys and more in others. The end result is something which really isn't all that different than 1/4-comma meantone or the Meantone[12] scale in 31-EDO. The difference between it and your JI system is not very much, either.

You ask which method is better--well, that depends on your goal. A regular temperament spreads the error out as evenly as possible, and a TOP temperament (which allows some tempering of the octave) is a temperament where the tuning of the period (i.e. the octave) and the generator (i.e. the fifth) are both tempered to minimize the absolute psychoacoustic damage over the whole range. You should read Paul Erlich's paper called "A Middle Path", it explains everything I just did as well as TOP optimization:

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

In your temperament, the error is not very evenly spread out, so that in some keys the intervals are very in tune, but in others bad intervals come up. In this sense, your tuning optimizes for the highest *quality* of consonance. However, as I'm sure you discovered, if you have a fixed scale size of 12 notes, improving the consonance of some intervals decreases it for others, meaning the *quantity* of dissonances goes up with the quality of consonances. In an equal scale, the quality of the consonances and the quantity of dissonances have balanced out, so that no key is any more or less dissonant than any other, and every key is tuned acceptably. This is why people liked 12-tET and why 12-tET eventually eclipsed Meantone. So really, it's not true that you've improved on 12-tET--you've simply reinstated Meantone and given it your own personal well-tempered twist (you might also want to look into the history of well-tempering, specifically the work of Werckmeister). Thus, your tuning does not compete with 12-tET, because it does not improve on the features that initially made 12-tET popular. You are certainly not alone in advocating a return to tunings near to JI, but if you *really* want to "sell" your temperament, it would be good to show a comparison between it, 1/4-comma meantone, 1/3-comma meantone, and a few of the more popular well-temperaments that preceded 12-tET. It would also be good to do a tally of the total number of consonant chords in each temperament, and compare those results to the total number in 12-tET.

In my personal opinion, 19-tET is a better solution overall because it produces a significant increase in consonance for only a marginal increase in complexity, while retaining all the benefits of an equal temperament and "secretly" importing a large variety of additional non-Meantone temperaments of both the 5 and 7-limit. It thus fills the need for more "in-tune" harmonies as well as the need for "new" harmonies (like subminor and supermajor 3rds), "new" scales, microtonal pitch inflections (due to its small single-step interval of around 63 cents), and backwards-compatibility with existing music and music theory and the ability to modulate to all 19 keys. As a 19-tET guitar player, I can attest that it is no more difficult than 12-tET and a good deal easier than some irregularly-fretted systems I've tried.

-Igs

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> What about 29 limit? (2, 3, 5, 7, 11, 13, 17, 19, 23, 29) How would you work with a 10
> dimensional space? Has it ever been done? (29/6 is a good harmony interval in my
> system.)

Well, to get to a rank-2 temperament, you'd need to temper out 8 29-limit commas, although if you're not after 11, 13, 17, 19, and 23, you don't need to include them. There's no reason you couldn't do a 2.3.5.7.29 subgroup. Although if you only want 29/6 and not 29/1 (or 29/4, 29/8, 29/16 etc.) you can do a 2.3.5.7.29/6 or a 2.3.5.7.29/3 subgroup. The beauty of tuning space is that you can construct it however you want.

However, if you're working with octaves, you're not going to be able to avoid getting 29/12 and 29/24 if you have 29/3 or 29/6. And 29/24 isn't a very strong identity, I believe it's firmly in the dyadic entropy well of 6/5. Thus I'm pretty sure if you try to combine 5-limit consonances with ratios of 29, the two are going to fight each other during optimization and probably end up splitting the difference and having 29/24 mapped to the same interval as 6/5.

> My approach has nothing at all to do with commas.

From your description, it sounds like your approach is very similar to that taken by Michael S. (aka djtrancendance). If I understand it correctly, your approach has little to do with commas because it has little to do with limits. You have worked out a set of ratios that you consider desirable, and you have worked out an error threshold that you consider to be acceptable, as well as a maximum number of notes. I'll have to read your book to get much deeper into the nuts and bolts.

Looking at your Blue JI scale, it looks to me to be a 4-dimensional scale of mostly 5-limit intervals but with two notes that poke into the dimension of prime-7, as Kraig handily diagrammed here with the dimension of prime-2 omitted since octaves are equivalent:

http://anaphoria.com/OsullivanBLUESjiscale.gif

As a JI scale, I'd say it's better than strict 5-limit chromatic (which has a 16/15 and a 45/32 instead of 15/14 and 7/5), but still has all the same problems all JI systems do: limited modulation ability due to harsh discords in distant keys.

Now, your temperament is essentially what is considered a "well-temperament", whereby intervals are tempered less in some keys and more in others. The end result is something which really isn't all that different than 1/4-comma meantone or the Meantone[12] scale in 31-EDO. The difference between it and your JI system is not very much, either.

You ask which method is better--well, that depends on your goal. A regular temperament spreads the error out as evenly as possible, and a TOP temperament (which allows some tempering of the octave) is a temperament where the tuning of the period (i.e. the octave) and the generator (i.e. the fifth) are both tempered to minimize the absolute psychoacoustic damage over the whole range. You should read Paul Erlich's paper called "A Middle Path", it explains everything I just did as well as TOP optimization:

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

In your temperament, the error is not very evenly spread out, so that in some keys the intervals are very in tune, but in others bad intervals come up. In this sense, your tuning optimizes for the highest *quality* of consonance. However, as I'm sure you discovered, if you have a fixed scale size of 12 notes, improving the consonance of some intervals decreases it for others, meaning the *quantity* of dissonances goes up with the quality of consonances. In an equal scale, the quality of the consonances and the quantity of dissonances have balanced out, so that no key is any more or less dissonant than any other, and every key is tuned acceptably. This is why people liked 12-tET and why 12-tET eventually eclipsed Meantone. So really, it's not true that you've improved on 12-tET--you've simply reinstated Meantone and given it your own personal well-tempered twist (you might also want to look into the history of well-tempering, specifically the work of Werckmeister). Thus, your tuning does not compete with 12-tET, because it does not improve on the features that initially made 12-tET popular. You are certainly not alone in advocating a return to tunings near to JI, but if you *really* want to "sell" your temperament, it would be good to show a comparison between it, 1/4-comma meantone, 1/3-comma meantone, and a few of the more popular well-temperaments that preceded 12-tET. It would also be good to do a tally of the total number of consonant chords in each temperament, and compare those results to the total number in 12-tET.

In my personal opinion, 19-tET is a better solution overall because it produces a significant increase in consonance for only a marginal increase in complexity, while retaining all the benefits of an equal temperament and "secretly" importing a large variety of additional non-Meantone temperaments of both the 5 and 7-limit. It thus fills the need for more "in-tune" harmonies as well as the need for "new" harmonies (like subminor and supermajor 3rds), "new" scales, microtonal pitch inflections (due to its small single-step interval of around 63 cents), and backwards-compatibility with existing music and music theory and the ability to modulate to all 19 keys. As a 19-tET guitar player, I can attest that it is no more difficult than 12-tET and a good deal easier than some irregularly-fretted systems I've tried.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> This is what took me the longest time to figure out. To get to "Meantone Temperament" in from 5-limit JI, you start by collapsing the 5-limit so that it can be described by a "warped 3-limit", by setting 81/80 (aka 3^4/(5*(2^4))) equal to 1/1. At this point, I'm not sure but I *think* you have exactly 1/4-comma meantone.

Nope; I'm afraid this is just any old meantone. To get 1/4 comma, or any specific 5-limit meantone tuning, you need to add two independent intervals that, rather than being tempered out, are left fixed: the "eigenmonzos". In the case of 1/4 comma meantone, these we can take as 2 and 5, or equivalently 2 and 5/4, or if you allow "fractional monzos", 2 and 5^(1/4), the period and generator.

The only exception to this is where you have an edo tuning, which means it is actually rank one. The you take two commas and one eigenmonzo. For instance to get 31et, you can temper out 81/80 and 393216/390625 (the Wuerschmidt comma.) For the eigenmonzo, 2.

Thank you, Gene. That's a very important piece of the puzzle you just gave me. The picture is now even more in focus.

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
>
> > This is what took me the longest time to figure out. To get to "Meantone Temperament" in from 5-limit JI, you start by collapsing the 5-limit so that it can be described by a "warped 3-limit", by setting 81/80 (aka 3^4/(5*(2^4))) equal to 1/1. At this point, I'm not sure but I *think* you have exactly 1/4-comma meantone.
>
> Nope; I'm afraid this is just any old meantone. To get 1/4 comma, or any specific 5-limit meantone tuning, you need to add two independent intervals that, rather than being tempered out, are left fixed: the "eigenmonzos". In the case of 1/4 comma meantone, these we can take as 2 and 5, or equivalently 2 and 5/4, or if you allow "fractional monzos", 2 and 5^(1/4), the period and generator.
>
> The only exception to this is where you have an edo tuning, which means it is actually rank one. The you take two commas and one eigenmonzo. For instance to get 31et, you can temper out 81/80 and 393216/390625 (the Wuerschmidt comma.) For the eigenmonzo, 2.
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Now that part seems confusing to me. If you temper out a 5,3 comma, doesn't that mean you've basically reduced the entire prime-5 dimension to terms of prime-3? How do you temper out 128/125 from the 2-dimensional 81/80-tempered 3-limit Meantone lattice?

You could do it in stages as this suggests: 128/125, if 5 ~ (3/2)^4, becomes 128/(3/2)^12, which is the reciprocal of the Pythagorean comma. But the math is easier to do it all at once.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Looking at your Blue JI scale, it looks to me to be a 4-dimensional scale of mostly 5-limit intervals but with two notes that poke into the dimension of prime-7, as Kraig handily diagrammed here with the dimension of prime-2 omitted since octaves are equivalent:
>
> http://anaphoria.com/OsullivanBLUESjiscale.gif
>
> As a JI scale, I'd say it's better than strict 5-limit chromatic (which has a 16/15 and a 45/32 instead of 15/14 and 7/5), but still has all the same problems all JI systems do: limited modulation ability due to harsh discords in distant keys.
>
> Now, your temperament is essentially what is considered a "well-temperament", whereby intervals are tempered less in some keys and more in others. The end result is something which really isn't all that different than 1/4-comma meantone or the Meantone[12] scale in 31-EDO.

This strikes me as pretty misleading. The Blue JI scale has a wolf fifth, but also two fifths flat by a comma, and two others flat by 225/224. Tempering in meantone makes all the fifths but the wolf the same, giving Meantone[12]. However, only tempering out the septimal kleisma (225/224) gives you a 12-note scale in marvel temperament you can also arrive at from various other starting points, eg the Duodene, and it's much closer in sound and concept to Blue JI. It comes up so much it probably deserves a name of its own, such as the Marveldene.

The difference between it and your JI system is not very much, either.

I don't agree.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Thank you, Gene. That's a very important piece of the puzzle you just gave me. The picture is now even more in focus.

Thanks! I oversimplified a little; maybe I should do that more often.

"john777music" <jfos777@...> wrote:

> What about 29 limit? (2, 3, 5, 7, 11, 13, 17, 19, 23, 29)
> How would you work with a 10 dimensional space? Has it
> ever been done? (29/6 is a good harmony interval in my
> system.)

It's possible to run searches in the 29-limit, and here's
an example:

http://x31eq.com/cgi-bin/pregular.cgi?limit=29&error=5.0

You end up with either very complex or poorly tuned
scales. It's difficult to get so many intervals in tune.
Also, temperaments with a few good intervals tend to have
them canceled out by bad intervals, so nothing really
stands out. The first result from that search is simply
29-equal in the 29-limit with an extra chain for the
mapping of the prime 29.

I don't think anybody uses full 29-limit temperaments
because you run into diminishing returns for the
complexity. Subgroups might be interesting.

> My approach has nothing at all to do with commas. I
> choose a tonic (1/1) and I select eleven more notes
> (between 1/1 and 2/1) so that (i) all the chosen notes go
> well with the tonic, 1/1 and (ii) the greatest possible
> number of *good* harmony intervals (either containing the
> tonic or otherwise and an octave or less wide) occur,
> over a one octave range. I had to strike a balance
> between (i) and (ii) to get to best of both worlds. All
> the intervals chosen are just.

Regular temperaments all imply commas being tempered out
(unison vectors). If you found Meantone, you must have been
considering regular temperaments once. The algebra works
out such that you can either use the mapping or the unison
vectors to define the temperament class (usually). The
mapping's the more direct way but unison vectors encode the
harmonic tricks you can perform.

Graham

Mike Battaglia <battaglia01@...> wrote:

> I also like to think of it in terms of the simplest
> intervals that are equated. So magic means that a bunch
> of major thirds gets you to a fifth, okay. But it also
> happens to mean that 25/24 and 128/125 end up being
> equated to the same interval. <snip>

I cover the magic comma without its ratio here:

http://x31eq.com/magic/tripod.pdf

It's my model for a regular temperament paper with minimal
mathematics. It is still quite mathematical. You have to
wade through:

- Lattices

- Numbers identifying overtones

- Numbers for "feet"

- The "nine" in "nine-limit"

- Numbers for scale degrees

- Counting notes in a scale

- Arithmetic of intervals

There are other complexities involved. There are new, or
obscure, technical terms. I say "Didymic" instead of
"5-limit" which avoids a number but doesn't simplify
anything. I introduce terms for the basic Magic
intervals. There are two new variants of staff notation
which are unavoidable given that they're what I was writing
about.

I hope musically literate readers will be able to follow
it. I haven't had any feedback as to what they have
problems with. Let's assume that musically illiterate
readers won't be interested in notation in the first
place. Teaching them about magic temperament is a
different problem.

Note that even musicians without a scientific background
might prefer the ratios to be there in all their glory.
Kyle Gann for example:

http://www.kylegann.com/JIreasons.html

"6. I have a tremendous natural talent for fractions and
logarithms. It would be a shame to waste it. I warn my
students that if they don't have a good head for fractions
and logarithms they should leave just intonation alone.
It's not for everyone. "

Graham

Jake Freivald <jdfreivald@...> wrote:
> Some of the points Gene makes seem to be related to The
> Book's purpose. Is it a reference for people who know
> something about xentonality and want to go deeper? Or is
> it a book that leads non-xentonal musicians to
> xentonality?

It's unlikely The Book could ever exist, because the closer
it comes to reality the more people will realize it wasn't
the book they were thinking of. Maybe a book might come
about if somebody can be bothered to write it.

There are a load of different people who might be
interested. Non-musicians can learn microtonal music from
scratch. Scientists with an interest in music can learn
the arithmetic of tuning. Intuitive musicians can be given
something to play with. Serious composers can learn the
theoretical underpinnings of different tuning systems.
Existing microtonalists can be indoctrinated into the
regular mapping paradigm. All these viewpoints would lead
to different books -- or, more realistically, web pages by
people with some time and inclination but not enough to
write books.

You can take the contributors to the UnTwelve competition
as a cross-section of the microtonal world:

http://www.untwelve.org/2010_competition_notes.html

Most have degrees in music. Many specialize in electronic
music. (Electronics help to get microtonal pieces heard,
of course.) Most show no awareness of regular mappings.
You could treat them as an audience to be reached, or as
wayfarers who have chosen their own paths and can't be
persuaded to change now. But, at least, they give some
idea of the potential audience.

The interviews with the winners include the question of how
they came to be interested in microtonality. The first
prize winner Monroe Golden:

http://www.untwelve.org/interviews/golden.html

Read Genesis of a Music (a great book on microtonality) as
an undergraduate (of music, eventually composition). Made
an electronic piece using Genesis-43. Ended up studying
with Ben Johnston after being pushed towards his music.

Soressa Gardner:

http://www.untwelve.org/interviews/gardner.html

Played in a gamelan and learned microtonality as part of
electronic music, itself part of her degree.

Igliashon Jones:

http://www.untwelve.org/interviews/jones.html

Known in these parts, of course. Wanted to do something
different, tried more notes.

Graham

Igs,

Phenomenal post, and thank you very much for taking the time with it. Also many thanks to the follow-on commenters. This really helps me understand what a lot of people are talking *about,* and hopefully what they're *actually saying* will become clearer, too. :)

Naturally, I'm still trying to absorb it and all of the commentary on it. I was struck by this, though, and was wondering if you or anyone else would comment:

> The JI diatonic scale has three step-sizes: 16/15, 10/9, and 9/8,
> and anyone who tells you that this is the scale that music theory
> is founded on, they have no idea what they're talking about.
> This scale does not translate to the circle/spiral of fifths at
> all, since it's inherently 3-dimensional.

Once upon a time, mere weeks ago, I thought I understood the way Equal Temperament came to be: The Pythagorean concept of stacking fifths led to a scale that gave JI approximations to desirable ratios; people (e.g., Didymus) found ways to alter the tuning to give better sounds (hence the "Didymus comma", a.k.a. the Syntonic comma, because Didymus discovered that a Pythagorean third reduced by 81/80 gives a 5/4 third); altering the stacks of fifths in other ways led to alternative temperaments (e.g., meantones); then, people realized that if you flattened the fifths by just two cents, you could do the pythagorean stacking, every key would be the same, and you'd get a circle of fifths. That story, if true, defines a relationship (historical rather than mathematical) between JI and the circle of fifths.

Are you saying that 12-EDO really didn't have this evolutionary relationship to JI? Did the people who developed 12-EDO (12-TET, really) think in terms of three-dimensional prime spaces?

Thanks,
Jake

On Thu, Feb 10, 2011 at 1:20 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> > This is what took me the longest time to figure out. To get to "Meantone Temperament" in from 5-limit JI, you start by collapsing the 5-limit so that it can be described by a "warped 3-limit", by setting 81/80 (aka 3^4/(5*(2^4))) equal to 1/1. At this point, I'm not sure but I *think* you have exactly 1/4-comma meantone.
>
> Nope; I'm afraid this is just any old meantone. To get 1/4 comma, or any specific 5-limit meantone tuning, you need to add two independent intervals that, rather than being tempered out, are left fixed: the "eigenmonzos". In the case of 1/4 comma meantone, these we can take as 2 and 5, or equivalently 2 and 5/4, or if you allow "fractional monzos", 2 and 5^(1/4), the period and generator.

How would you get golden meantone then - you'd have to have real
monzos, rather than fractional ones?

-Mike

> Mike wrote :
> I don't think that anyone's bothered to work with the 29-limit.

I did, with many satisfactions.
During 10 years I held a psychoacoustic regular research group and
one of the subjects we worked on was to validate the musical
interactions of each prime with precedent ones ; we did this through
an exploration of basic "Dudon scales" up to the 37th harmonic limit
and even a few over, each one with a photosonic disk using the
coherent subharmonics chords for each note.
A Dudon scale (name given by Gene after my works) for a given comma n/
d is a scale containing all the divisors of both n & d, transposed in
the same octave :
it considers a comma as a coïncidence between primes and gathers all
harmonic paths to arrive to it.
ex. Dudon(385/384) = [1 35 77 5 11 3-385 55 7]
(you can also have scales compiling several commas together, ex. Dudon(320/319, 145/144, 88/87) = [1 9-145 319-5 11-87 3 29])
For each coïncidence notes were given by the participants according
to their musical resonance and I did in the end a synthesis of the
results.
Though not a very rigorous scientific experimentation, it allowed
among other things a selection of what were considered as best
musical coïncidences by the participants and gave a certain
indication of each new prime limit's specific harmonic interest.
To resume these results, unequal drops of musical interest were
observed for each new prime :
Big drop with 7 and 11 compared to 3 and 5, but rather regain of
interest with 13,
relative drop with 17, but new interest again with 19,
relative drop with 23 and 29 but still significantly musical interest,
more important drop with 31 & 37 and after.
Commas tested in 23-limit were 46/45, 161/160, 208/207 897/896,
115/114, 69/68, 460/459, 736/735,
and in 29-limit, 88/87, 175/174, 320/319, 145/144, 117/116, 116/115,
58/57, 609/608, 841/840.
Summing all selected coïncidences by global limits, 5-limit, 13-
limit, 19-limit and 29-limit appeared as the most promising boundaries.
Of course, even if there are no limitations, subgroups involving 23
or 29 (what I did here) are easier to play with. I found that 23 has
good relations with 5, 7 and 17, and 29 with 3, 5 and 13.
- - - - - -
Jacques

On Thu, Feb 10, 2011 at 6:27 AM, Graham Breed <gbreed@...> wrote:
>
> There are a load of different people who might be
> interested. Non-musicians can learn microtonal music from
> scratch. Scientists with an interest in music can learn
> the arithmetic of tuning. Intuitive musicians can be given
> something to play with. Serious composers can learn the
> theoretical underpinnings of different tuning systems.
> Existing microtonalists can be indoctrinated into the
> regular mapping paradigm. All these viewpoints would lead
> to different books -- or, more realistically, web pages by
> people with some time and inclination but not enough to
> write books.

There's also a huge section of the jazz community comprised of people
that feel they're on a spiritual search for meaning in music, are
tired of reinventing the wheel, are frustrated that they too can't be
harmonic pioneers like Coltrane and Miles and so on were, and are
generally very attracted to new and interesting and different theories
about "how music works." Offshoots of this mentality are often seen
amongst modern shredder guitar players and comp majors and such. If
someone could manage to reach these people, I think there's a very
large silent market there.

-Mike

On Thu, Feb 10, 2011 at 11:55 AM, Jacques Dudon <fotosonix@...> wrote:
>
> Mike wrote  :
> I don't think that anyone's bothered to work with the 29-limit.
>
> I did, with many satisfactions.
> During 10 years I held a psychoacoustic regular research group and one of the subjects we worked on was to validate the musical interactions of each prime with  precedent ones ; we did this through an exploration of basic "Dudon scales" up to the 37th harmonic limit and even a few over, each one with a photosonic disk using the coherent subharmonics chords for each note.
> A Dudon scale (name given by Gene after my works) for a given comma n/d is a scale containing all the divisors of both n & d, transposed in the same octave :
> it considers a comma as a coïncidence between primes and gathers all harmonic paths to arrive to it.
> ex. Dudon(385/384) = [1   35   77   5   11  3-385   55   7]

This is based off of the equal-beating/recurrence relation stuff?

-Mike

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
> Are you saying that 12-EDO really didn't have this evolutionary relationship to JI? Did the
> people who developed 12-EDO (12-TET, really) think in terms of three-dimensional prime > spaces?

LOL, no, your understanding is correct, Jake. The "regular temperament paradigm" is a relatively recent innovation, LOOOOOONG after the advent of Meantone. I should note though that 12-EDO is actually much older than we take it to be, having been discovered in China as an essentially 3-limit temperament (which it is very good for). But no, the conception of multi-dimensional tuning spaces had absolutely nothing to do with the historical procession from JI to Meantone to 12-tET. However, I suspect that that is because people were so fixated on a particular small subset of harmonies that they never bothered to explore alternatives. The idea of generating a musical scale by chaining some interval other than a fifth, and arriving at a temperament from some other comma, is quite rare (if not nonexistent) in the history of musical tunings. So no one ever bothered to try to generalize the theory of temperament to a higher level of mathematical abstraction until very recently. But the results of these recent innovations have been quite remarkable. There are some fascinating alternative temperaments now at our disposal, and although none match both the simplicity and the accuracy of Meantone for dealing with 5-limit harmonies, there is much to be said of exploring equally-simple, equally-accurate, or different prime-limit (or subgroup) temperaments.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> How would you get golden meantone then - you'd have to have real
> monzos, rather than fractional ones?

This is why I said I was oversimplifying. To get to golden meantone, you would indeed need "real monzos", or what I'd call points in interval space, since they no longer correspond 1-1 to intervals. But still, there will be fractional monzos arbitrarily close, and so intervals, including rational ones, arbitrarily close to being left unchanged by the tempering. Using these for eigenmozos, you can get tunings arbitrarily close to golden meantone.

See how much easier it all becomes when you oversimplify?

On Thu, Feb 10, 2011 at 1:33 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > How would you get golden meantone then - you'd have to have real
> > monzos, rather than fractional ones?
>
> This is why I said I was oversimplifying. To get to golden meantone, you would indeed need "real monzos", or what I'd call points in interval space, since they no longer correspond 1-1 to intervals. But still, there will be fractional monzos arbitrarily close, and so intervals, including rational ones, arbitrarily close to being left unchanged by the tempering. Using these for eigenmozos, you can get tunings arbitrarily close to golden meantone.
>
> See how much easier it all becomes when you oversimplify?

Ha! Sure does. Oversimplifying, with asterisks for the full picture at
the bottom, would work wonders on the xenharmonic wiki.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> This is based off of the equal-beating/recurrence relation stuff?

Doesn't need to be. Here's Dudon(676/675):

135/128 9/8 75/64 5/4 675/512 169/128 45/32 3/2 25/16 13/8 27/16 225/128 15/8 2

Here it is transposed:

16/15 10/9 52/45 6/5 5/4 4/3 64/45 3/2 8/5 5/3 16/9 15/8 169/90 2

Here it is in 940et:

! islandude.scl
Dudon(676/675) in 940et
13
!
111.06383
182.55319
248.93617
315.31915
386.80851
497.87234
608.93617
702.12766
813.19149
884.68085
995.74468
1088.93617
1200.00000

Note that after tempering, it has only 13 notes, not 14. It's got only one barbados triad, but there's plenty of other stuff. The step sizes are not too irregular, but it's not proper: fifths in two different interval classes, for instance.

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> Once upon a time, mere weeks ago, I thought I understood the way Equal Temperament came to be: The Pythagorean concept of stacking fifths led to a scale that gave JI approximations to desirable ratios; people (e.g., Didymus) found ways to alter the tuning to give better sounds (hence the "Didymus comma", a.k.a. the Syntonic comma, because Didymus discovered that a Pythagorean third reduced by 81/80 gives a 5/4 third); altering the stacks of fifths in other ways led to alternative temperaments (e.g., meantones); then, people realized that if you flattened the fifths by just two cents, you could do the pythagorean stacking, every key would be the same, and you'd get a circle of fifths. That story, if true, defines a relationship (historical rather than mathematical) between JI and the circle of fifths.

What actually happened was not too dissimilar. The stacking of fifths lead to the use of Pythagorean thirds and fifths, which were tempered to purer forms in the Renaissance. A lot of this was adaptively, no doubt, as there was a great deal of a capella, but various rather flattish meantones, favoring thirds, such as 1/3 and 2/7 comma were proposed and used, and 1/4 comma became quite the thing. Then the fifth crept up sharper values, and circulating temperaments were introduced, until finally 12edo was accepted as the theoretical standard.

Jake>"What actually happened was not too dissimilar. The stacking of fifths
lead to the use of Pythagorean thirds and fifths, which were tempered to
purer forms in the Renaissance. A lot of this was adaptively, no doubt,
as there was a great deal of a capella, but various rather flattish
meantones, favoring thirds, such as 1/3 and 2/7 comma were proposed and
used, and 1/4 comma became quite the thing. Then the fifth crept up
sharper values, and circulating temperaments were introduced, until
finally 12edo was accepted as the theoretical standard."

    Big hulking question...so why was 12EDO considered over 31EDO (IE virtually the same as quarter-comma-meantone)?

  Both enable transposition, the diatonic scale in 31EDO is purer than in 12...  The only thing I can think of is 12 gets the job done in less notes, thus making building instruments around it and playing them (to an extent) easier...any other reasons?

--- On Thu, 2/10/11, genewardsmith <genewardsmith@...> wrote:

From: genewardsmith <genewardsmith@...>
Subject: [tuning] Temperament Basics for Jake (was Re: The Book (from MMM))
To: tuning@yahoogroups.com
Date: Thursday, February 10, 2011, 11:29 AM

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> Once upon a time, mere weeks ago, I thought I understood the way Equal Temperament came to be: The Pythagorean concept of stacking fifths led to a scale that gave JI approximations to desirable ratios; people (e.g., Didymus) found ways to alter the tuning to give better sounds (hence the "Didymus comma", a.k.a. the Syntonic comma, because Didymus discovered that a Pythagorean third reduced by 81/80 gives a 5/4 third); altering the stacks of fifths in other ways led to alternative temperaments (e.g., meantones); then, people realized that if you flattened the fifths by just two cents, you could do the pythagorean stacking, every key would be the same, and you'd get a circle of fifths. That story, if true, defines a relationship (historical rather than mathematical) between JI and the circle of fifths.

What actually happened was not too dissimilar. The stacking of fifths lead to the use of Pythagorean thirds and fifths, which were tempered to purer forms in the Renaissance. A lot of this was adaptively, no doubt, as there was a great deal of a capella, but various rather flattish meantones, favoring thirds, such as 1/3 and 2/7 comma were proposed and used, and 1/4 comma became quite the thing. Then the fifth crept up sharper values, and circulating temperaments were introduced, until finally 12edo was accepted as the theoretical standard.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Feb 10, 2011 at 11:55 AM, Jacques Dudon <fotosonix@...> wrote:
> >
> > Mike wrote  :
> > > I don't think that anyone's bothered to work with the 29-limit.
> >
> > I did, with many satisfactions.
> > During 10 years I held a psychoacoustic regular research group and one of the subjects we worked on was to validate the musical interactions of each prime with  precedent ones ; we did this through an exploration of basic "Dudon scales" up to the 37th harmonic limit and even a few over, each one with a photosonic disk using the coherent subharmonics chords for each note.
> > A Dudon scale (name given by Gene after my works) for a given comma n/d is a scale containing all the divisors of both n & d, transposed in the same octave :
> > it considers a comma as a coïncidence between primes and gathers all harmonic paths to arrive to it.
> > ex. Dudon(385/384) = [1   35   77   5   11  3-385   55   7]
>
> This is based off of the equal-beating/recurrence relation stuff?
>
> -Mike

It was not the purpose. Many JI scales can produce all kinds of synchronous beatings of course, and this one does (the harmonic beatings of 64/55 and 77/64 here for example are equal), but synchronous and equal beating properties usually refer to temperaments (whether rational or not). By definition, these (JI) scales are not temperaments. They actually show the way to temperaments, once you temper the concerned primes in order to dissolve the commas. Then yes, this can be done in many equal-beating and recurrent ways.
- - -
Jacques

On Thu, Feb 10, 2011 at 1:53 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > This is based off of the equal-beating/recurrence relation stuff?
>
> Doesn't need to be. Here's Dudon(676/675):
>
> 135/128 9/8 75/64 5/4 675/512 169/128 45/32 3/2 25/16 13/8 27/16 225/128 15/8 2

Huh. What exactly is this doing? Is this just a way to rationalize
different temperaments in general, or does this have any certain
properties?

-Mike

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

>     Big hulking question...so why was 12EDO considered over 31EDO (IE virtually the same as quarter-comma-meantone)?

Because people were using keyboard instruments with twelve notes.

On Thu, Feb 10, 2011 at 2:34 PM, Michael <djtrancendance@...> wrote:
>
> Jake>"What actually happened was not too dissimilar. The stacking of fifths lead to the use of Pythagorean thirds and fifths, which were tempered to purer forms in the Renaissance. A lot of this was adaptively, no doubt, as there was a great deal of a capella, but various rather flattish meantones, favoring thirds, such as 1/3 and 2/7 comma were proposed and used, and 1/4 comma became quite the thing. Then the fifth crept up sharper values, and circulating temperaments were introduced, until finally 12edo was accepted as the theoretical standard."
>
>     Big hulking question...so why was 12EDO considered over 31EDO (IE virtually the same as quarter-comma-meantone)?

Probably because those ~9/7's still kind of sound like 5/4's, and
those wolf 3/2's still kind of sound like 3/2's. Once people decided
at some point to start coming up with well-temperaments that already
indicated they heard 128/125 as a unison vector, it was only a matter
of time.

I've often wondered why we didn't spring for 19-EDO though. Probably
because we were used to working on 12-note keyboards, is my only
explanation. The music history buffs on here know a lot more about it
than I do.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Huh. What exactly is this doing? Is this just a way to rationalize
> different temperaments in general, or does this have any certain
> properties?

I'll be interested in what Jacques has to say, but I can tell you it has some properties.

(1) One thing it does is to find simple intervals between which there is the comma. For instance, Dudon(81/80) is 9/8-5/4-81/64-3/2-27/16-2, and between 5/4 and 81/64 there appears 81/80. Transposing this scale to its various modes, we find 10/9-9/8-4/3-3/2-16/9-2, and thus we see 81/80 appears between 10/9 and 9/8.

(2) Tempering out the comma gives a scale tempered from the subgroup defined by the prime divisors of the numerator or denominator of the comma which is small but which actually does make use of the tempering. For instance, from Dudon(81/80) above we get Meantone[5].

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > This is based off of the equal-beating/recurrence relation stuff?
>
> Doesn't need to be. Here's Dudon(676/675):
>
> 135/128 9/8 75/64 5/4 675/512 169/128 45/32 3/2 25/16 13/8 27/16 225/128 15/8 2
>
> Here it is transposed:
>
> 16/15 10/9 52/45 6/5 5/4 4/3 64/45 3/2 8/5 5/3 16/9 15/8 169/90 2

Not that bad, I have seen worse Dudon scales ! ;)
Except that the rare 13-limit intervals are lost in a sea of 5-limit. It needs some judicious extra 13-limit comma in here...

Now, the next step can be to find the "Dudon scales resolutions" of any good JI scale (ignoring the commas). Probably a child play, for programmers...

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> I'll be interested in what Jacques has to say, but I can tell you it has some properties.

When I investigated those, it had several purposes. First it was mainly didactic, as you very well just pointed in your example of Dudon(81/80) in the sense that it lights up a relation between some of our most common traditional scales (here the anhemitonic pentatonic scale) and some archetypal coïncidences between harmonics (such as between 3 and 5 here with the syntonic comma). So the idea was to see if I could find other examples, and I did find many indeed. Then it was tempting to compare the musicality of all possible commas through this very basic traduction, and to compare musicalities produced by different limits, etc. Also while doing this I started quickly to extend this idea to groups of two or three commas together, and to use it as a tool for scale creation.
Another purpose was more directly related with a specific photosonic disk technique that I call omission, where I replace one black arc every one given number of them, by a blank space. This generates a subharmonic drone in polyphony. And commas are simply the kind of notes that will generate the more rich polyphonies.

> (1) One thing it does is to find simple intervals between which there is the comma. For instance, Dudon(81/80) is 9/8-5/4-81/64-3/2-27/16-2, and between 5/4 and 81/64 there appears 81/80. Transposing this scale to its various modes, we find 10/9-9/8-4/3-3/2-16/9-2, and thus we see 81/80 appears between 10/9 and 9/8.
>
> (2) Tempering out the comma gives a scale tempered from the subgroup defined by the prime divisors of the numerator or denominator of the comma which is small but which actually does make use of the tempering. For instance, from Dudon(81/80) above we get Meantone[5].
>

--- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@...> wrote:
>

> Not that bad, I have seen worse Dudon scales ! ;)

Sometimes great, and sometimes messed up. And alternative to try sometimes is the minimal Euler genus in the subgroup of primes of the comma containing the comma as a step.

Igs wrote:

> No power of 3 will ever equal a power of 2, so you'll never be able
> to subtract a power of 2 from a power of 3 and get 0 [snip]
> A comma in this space is really just a point where a power of 3
> comes close to a power of 2, such that when put into a ratio,
> the ratio is close to 1/1.
>
> Now, imagine that we just "say" a given comma EQUALS 1/1.

We can say a monkey is a motorboat, but it ain't so. :) What you're really saying here, I think, is that in this "tuning space" I can substitute a different number for the prime I'd like.

What follows is probably rudimentary for a lot of you, but here I go.

If I want equal temperament, I can subtract a small amount, beta=0.003385845, from 3 and get 2.996614155. Call 3-minus-beta="Three".

Now the Pythagorean comma (3^12)/(2^19) gets replaced by (Three^12)/(2^19) = 1.

2/1 = 2, or 1200 cents, and Three/2 = 1.49831, or 700 cents. Shazam, I have traditional equal temperament. When you "temper a comma", that's the kind of thing you're talking about.

With traditional 12-tET, only the Three gets modified. You could also do the same procedure with 2 (replacing it with 2-alpha=Two), and any other primes I'd like to. I was going to ask why people only reduce with the Three and never talk about messing with the Two -- you could, after all, reduce the sharpness of the 12-tET major third by flattening the octave (i.e., reducing the Two). But after spending too much time in Excel I realized what should have been intuitive: Even with a 10-cent decrease in the octave (i.e., an octave = 1190 cents, or Two = 0.012), tempering out the Pythagorean comma (Three^12)/(Two^19) = 1 yields little substantial benefit. A fifth is very flat at 694 cents, a major third is still way too sharp at 397 cents.

This does make me note the difference between 19-tET and 19-EDO: You could temper the octave to make a 19-tET that's a little bit "better" than 19-EDO. (By "better", of course, I mean "something that I like more". I tend to be bothered much more by flat notes than by sharp ones.) By increasing the octave a little bit, you can get errors in the fifth and major third that seem significantly lower (although the fourth becomes much sharper).

Just Intonation
316 -- m3
386 -- M3
498 -- P4
702 -- P5
884 -- M6
969 -- Harmonic 7

19-tET, Octave = 1200 cents / 19-EDO
316 -- m3 (exact)
379 -- M3 (flat, 7 cents)
505 -- 4 (sharp, 7 cents)
695 -- 5 (flat, 7 cents)
884 -- M6 (exact)
947 -- H7 (flat, 22 cents)

19-tET, Octave = 1208 cents
318 -- m3 (sharp, 2 cents)
381 -- M3 (flat, 5 cents)
509 -- 4 (sharp, 11 cents)
699 -- 5 (flat, 3 cents)
890 -- M6 (sharp, 6 cents)
954 -- H7 (flat 16 cents)

I don't think I'd want to distort the octave any more than that, but maybe I could. I'd have to play it by ear.

So, this leads to a more general question: How do people choose which primes to temper, and how much, when the "temper out the 128/125" or whatever? Do we just always presume octave equivalence unless otherwise stated, such that, in the case of 128/125, you only have the option of tempering the five? And what if there are other primes?

I realize I've been asking a lot of questions, and I appreciate everyone's willingness to talk.

Thanks,
Jake

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
I was going to ask why people only reduce with the
> Three and never talk about messing with the Two -- you could, after all,
> reduce the sharpness of the 12-tET major third by flattening the octave
> (i.e., reducing the Two).

That may be true of most venues but it certainly is not true of the Yahoo lists. Tempering 2 is a major industry around here.

Gene said:

> That [people don't temper Two] may be true of most venues but it
> certainly is not true of the Yahoo lists. Tempering 2 is a major
> industry around here.

Okay, maybe I just haven't understood it when people talk about it. But when I hear people say, "Temper out 81/80 and you'll get XYZ scale", it sounds like (perhaps because I don't have enough experience) there's only one way to temper that comma. In reality, it looks like there's an infinite number of ways. How do you all know how you're distorting which primes to temper the comma?

Thanks,
Jake

On Fri, Feb 11, 2011 at 9:14 AM, Jake Freivald <jdfreivald@...> wrote:
>
> Igs wrote:
>
> > No power of 3 will ever equal a power of 2, so you'll never be able
> > to subtract a power of 2 from a power of 3 and get 0 [snip]
> > A comma in this space is really just a point where a power of 3
> > comes close to a power of 2, such that when put into a ratio,
> > the ratio is close to 1/1.
> >
> > Now, imagine that we just "say" a given comma EQUALS 1/1.
>
> We can say a monkey is a motorboat, but it ain't so. :) What you're
> really saying here, I think, is that in this "tuning space" I can
> substitute a different number for the prime I'd like.

The point isn't just to "say" it, the point is to detune the intervals
so that it actually works out that way. You tune something that
approximates 3/2 so that a bunch of them gets you to something that
approximates 5/1.

It isn't about substituting numbers for primes, it's about how those
numbers will be perceived. 700 cents will be perceived as though it's
a 3/2 interval. There's no point in pretending that it's a different
thing.

In the case of certain tunings, like 17-tet, more than one obvious
choice exists on how to map 5. So you can pick one, or you can say
it's a no-5's temperament. That's when things start to break down.

The theory is very abstract and is based on theoretical "maps" of
intervals - you manipulate the maps and see how different theoretical
intervals relate to one another within the map. This has nothing to do
with any specific tuning, but the idea is that once you settle on a
tuning (like say 15-tet), you then pick a way to "map" each prime
within that tuning. The rest then falls into place.

-Mike

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>How do you all know how you're distorting which primes to temper the comma?

That's apparently what "eigenmonzos" are--intervals which remain untempered. So in 1/4-comma meantone, the 2 and the 5 are eigenmonzos (or something, I may not have this concept 100% yet). Keeping the 2 and the 5 "pure" means that the 3 has to be tempered an exact amount: 1/4 of the syntonic comma. On the other hand, you could make 2 and 6/5 your eigenmonzos, which means 1/3 of the syntonic comma has to be tempered out of the 3.

Another way is to just have two commas (in the 5-limit or any 3-dimensional subgroup). If you temper out any two 5-limit commas, you get one (and only one) tuning (assuming 2 is an eigenmonzo).

Another way is "Tenney OPtimal" tuning, or "TOP" tuning, which is where you distribute the fractions of the comma across all of the primes to minimize the overall damage in proportion to the simplicity of the intervals being tempered.

-Igs

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> Gene said:
>
> > That [people don't temper Two] may be true of most venues but it
> > certainly is not true of the Yahoo lists. Tempering 2 is a major
> > industry around here.
>
> Okay, maybe I just haven't understood it when people talk about it. But when I hear people say, "Temper out 81/80 and you'll get XYZ scale", it sounds like (perhaps because I don't have enough experience) there's only one way to temper that comma. In reality, it looks like there's an infinite number of ways. How do you all know how you're distorting which primes to temper the comma?

Maybe this will help:

http://xenharmonic.wikispaces.com/Abstract+regular+temperament

Tunings which retune the octave as well as the rest of the primes include TOP, Tenney-Euclidean, and Frobenius, and in the case of equal temperaments, Zeta. If you run Graham's temperament finder

http://x31eq.com/temper/

you'll get lots of tempered octaves, and there are Scala functions to do this also.

On 12 February 2011 05:30, Jake Freivald <jdfreivald@...> wrote:

> Okay, maybe I just haven't understood it when people talk about it. But when I hear people say, "Temper out 81/80 and you'll get XYZ scale", it sounds like (perhaps because I don't have enough experience) there's only one way to temper that comma. In reality, it looks like there's an infinite number of ways. How do you all know how you're distorting which primes to temper the comma?

It's worse than that -- there are infinity squared tunings because
there are two generators! Most of the time it doesn't matter which
one you choose as long as you choose a sensible one.

There's an infinity of routes that can take you from Paris to Rome as
well. (All roads lead to Rome.) You probably won't have to specify
yours when you submit your expenses claim unless you sneaked in an
appointment at Moscow or something strange like that.

Graham

On 12 February 2011 06:19, cityoftheasleep <igliashon@...> wrote:

> Another way is "Tenney OPtimal" tuning, or "TOP" tuning, which is where you distribute the fractions of the comma across all of the primes to minimize the overall damage in proportion to the simplicity of the intervals being tempered.

The original TOP (which can also be Tempered Octaves, Please!)
naturally defines two eigenmonzos. More generally, R for a rank R
temperament.

A Kees-max optimization, which is a pure octaves equivalent of TOP,
also gives eigenmonzos. One is fixed at 2:1. For example, the
optimal Kees-max meantone is the quarter comma tuning where 5:4 is
pure.

TE error uses irrational weights which means the eigenmonzos of the
optimal tuning don't map to ratios. There are variants involving a
rational weighting matrix, and then you get true eigenmonzos. They're
generally so complex that I don't see a need to care about them, but
there they are.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> The original TOP (which can also be Tempered Octaves, Please!)
> naturally defines two eigenmonzos. More generally, R for a rank R
> temperament.

Eh? TOP has the same problem TE does: irrational weights leading to only approximate eigenmonzos. However, both least squares and minimax lead to exact eigenmonzos. 1/4 comma meantone is the best known example; it's the minimax tuning for both 5-limit and 7-limit meantone.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Jake Freivald <jdfreivald@...> wrote:
> > Some of the points Gene makes seem to be related to The
> > Book's purpose. Is it a reference for people who know
> > something about xentonality and want to go deeper? Or is
> > it a book that leads non-xentonal musicians to
> > xentonality?
>
> It's unlikely The Book could ever exist, because the closer
> it comes to reality the more people will realize it wasn't
> the book they were thinking of. Maybe a book might come
> about if somebody can be bothered to write it.
>
> There are a load of different people who might be
> interested. Non-musicians can learn microtonal music from
> scratch. Scientists with an interest in music can learn
> the arithmetic of tuning. Intuitive musicians can be given
> something to play with. Serious composers can learn the
> theoretical underpinnings of different tuning systems.
> Existing microtonalists can be indoctrinated into the
> regular mapping paradigm. All these viewpoints would lead
> to different books -- or, more realistically, web pages by
> people with some time and inclination but not enough to
> write books.

Over 45 years ago, when I first got into microtonality, I observed that there was no book that would serve as a good introduction to microtonality, covering all of the major aspects: reasons for seeking alternatives to 12-equal; the acoustical basis for consonance & dissonance; a brief history of tuning; an unbiased comparison of JI vs. temperament (equal & unequal, regular & irregular) vs. equal division; approaches to tonality, harmony, and constructing scales; microtonal notation; microtonal instruments (new techniques vs. building new or modifying existing instruments; new keyboard designs); etc.

I'm sorry to say that this is still the case. There are many more resources available now than when I started, but IMO none of these is, by itself, comprehensive enough to be suitable as a textbook for a course in microtonality and alternative tunings.

About 13 years ago I decided to begin writing a book that would fill this void. I sought to make it as interesting as possible, beginning with the story of how I got into microtonality in the first place. In the course of my investigations I encountered "heroes" (such as Helmholtz, Partch and Fokker) and "villains" (naysayers such as Paul Hindemith and sources of misinformation masquerading as "authorities" who have written articles for music reference books) who often disgreed with one another. Independent study enabled me to sort through the contradictions and half-truths in order to draw my own conclusions, and before long I was exploring new territory (such as grouping EDO's into temperament classes according to generating intervals) and discovering useful concepts (such as odd-limit consistency in EDO's). Having gone beyond what I had learned from my reading, I became a primary source of information, one who could speak from first-hand experience, qualified to write The Book.

After writing about ten chapters I realized that I didn't have a satistifactory notation (or notations) for all of the tunings I wanted to cover, so I decided to suspend work on the book until I had adequately addressed that problem. Around that time I was contacted by several members of this tuning list and invited to participate. It was here that I was introduced to Dave Keenan, and together we spent several years working on a system of notation that will handle tunings much more complicated and diverse than I had ever anticipated.

Nine years later I have not yet gotten back to working on The Book, which is now sorely in need of an overhaul, since so much has taken place in the interim. Whereas I originally conceived this as paper-and-ink hard copy, I'm now thinking of something on CD-ROM, with numerous links to sound files to illustrate brief musical examples discussed in the text. I'm thinking that this would initially be made available (at no cost) only to members of the tuning lists as a work in progress, with suggestions and constructive criticism welcomed!

--- In MakeMicroMusic@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> Jake Freivald wrote:
> > I would love to see a book on tuning theory. I understand the need for
> > a book that is for players, but I'd want something that helped me
> > understand the history and theory, too. ...
>
> This looks like a fine, ambitious project. But the outline titles
> suggest the necessity to pursue the matter for dozens of pages per
> subject along with ample demonstrations.
>
> Oz.

I believe that The Book be not only about tuning theory but also some practical aspects (such as tips for building or modifying instruments, woodwind fingering charts for microtones, and useful software).

The first section of about 7 or 8 chapters would cover the basics, and after that one or more sections with chapters on supplementary material dealing with topics of special interest (for more limited audiences), e.g., for those who are more theoretically and/or mathematically inclined. I would want to include sound files of entire compositions (in a variety of styles, from a variety of contributors) that demonstrate or exemplify techniques and tunings discussed in the text. I would also like to include articles written by others on these lists as additional resources that go into greater detail on selected topics that may be of special interest.

As Dr. Oz observed, this is a very ambitious project, and I don't expect to be able to do it alone. When I undertook my notation project, I had no idea that it would take about ten times as long as I expected and that the result would be at least ten times as versatile as I had imagined -- and it would not have been possible without having Dave Keenan as collaborator, not to mention those who contributed helpful suggestions and encouragement. If anyone else would like to help me with this project, I would be most grateful. There are probably some out there who could contribute a chapter or two on topics in which you have more knowledge and expertise than I do.

I should mention that I was not expecting to resume work on my book for another 3 years, at which time I plan to retire from my full-time job. But since there has been so much discussion about the need for The Book, I'm thinking that perhaps this may be the time to review my chapter outline and start reworking those ten chapters.

Your thoughts, anyone?

--George

On Sun, Feb 13, 2011 at 2:09 AM, gdsecor <gdsecor@...> wrote:
>
> Nine years later I have not yet gotten back to working on The Book, which is now sorely in need of an overhaul, since so much has taken place in the interim. Whereas I originally conceived this as paper-and-ink hard copy, I'm now thinking of something on CD-ROM, with numerous links to sound files to illustrate brief musical examples discussed in the text. I'm thinking that this would initially be made available (at no cost) only to members of the tuning lists as a work in progress, with suggestions and constructive criticism welcomed!

That would be fantastic. As you know I much admired your paper on
17-TET. I think you have a knack for talking about this stuff,
actually.

-Mike

On 12 February 2011 19:51, genewardsmith <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> The original TOP (which can also be Tempered Octaves, Please!)
>> naturally defines two eigenmonzos.  More generally, R for a rank R
>> temperament.
>
> Eh? TOP has the same problem TE does: irrational weights leading to only approximate eigenmonzos. However, both least squares and minimax lead to exact eigenmonzos. 1/4 comma meantone is the best known example; it's the minimax tuning for both 5-limit and 7-limit meantone.

What examples do you have where TOP gives irrational eigenmonzos?
Isn't 1/4 comma meantone the optimal Kees-max tuning like I said?
There are at least some special cases where any minimax will lead to
some interval being tempered pure.

Least squares will give rational eigenmonzos with rational weights, like I said.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What examples do you have where TOP gives irrational eigenmonzos?

What about meantone? In tuning space, that's <1+e 1-e 1+e| where e must satisfy the linear relationship tempering out 81/80. That turns out to mean e = (-4+4log2(3)-log2(5))/(4+4log2(3)+log(5)). This is an element of the field Q(log2(3), log2(5)) where log2(3) and log2(5) are transcendental and algebraically unrelated, so that unweighting <1+e 1-e 1+e| still leaves you with transcendental quantities. For number theoretic reasons, raising 2 to the power of these quantities leads to more transcendental numbers.

> Isn't 1/4 comma meantone the optimal Kees-max tuning like I said?

It is.

On 14 February 2011 22:00, genewardsmith <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> What examples do you have where TOP gives irrational eigenmonzos?
>
> What about meantone? In tuning space, that's <1+e 1-e 1+e| where e must satisfy the linear relationship tempering out 81/80. That turns out to mean e = (-4+4log2(3)-log2(5))/(4+4log2(3)+log(5)). This is an element of the field Q(log2(3), log2(5)) where log2(3) and log2(5) are transcendental and algebraically unrelated, so that unweighting <1+e 1-e 1+e| still leaves you with transcendental quantities. For number theoretic reasons, raising 2 to the power of these quantities leads to more transcendental numbers.

Okay. Is it the special case where an eigenmonzo comes out as a prime
that it stays rational? (Just getting clear on the basics :-)

Graham