Yahoo Tuning Groups Ultimate Backup tuning Shortcut Questions

Okay, so I've got quite a few questions I've been trying to sort out with the help of the xenwiki and just lurking on the list and such, but in the interest of better understanding certain recent discussions, I've decided just to ask:

1) What is the regular temperament paradigm? (Apologies to Mike Battaglia; I realize you asked basically the same thing, perhaps rhetorically, but I'll bet you have a much better grasp on it than I do.) ;-)

2) What is the regular mapping paradigm? (Is it the same thing?)

3) What exactly is harmonic entropy, and is this what is meant by "HE"?

4) A while back on the xenwiki, I found a somewhat helpful description of what a comma pump does, but could someone please give me a few well-commented examples?

5) I think I have a rough idea, maybe, how the ratios for intervals can be used to build consonant chords; however, I've gotten the impression the harmonic series can also be used in a similar way. Is this actually the case, and if so, what's the procedure?

6) What does "omnitetrachordal" mean?

7) '13-EDO as a Father temperament or 11-EDO as a Hanson temperament ... / Hell, for all my blathering about 13-EDO really being "Uncle", I tend to treat it as "A-Team", the 2.9.21 subgroup that I guess I discovered' -- can anyone explain what any of that means? (I don't mean that to mock or anything; I really want to understand.)

8) "If I'm not going to use 12-TET, what can I use instead and how can I make it sound good?" (Thanks for putting it in a nutshell, Igs.) ;-)

I've got more, but with any luck some of that will keep me busy for a while. :-)

🔗Mike Battaglia <battaglia01@...>

On Thu, Jun 9, 2011 at 11:00 PM, Jay Random <cortaigne@...> wrote:
>
> Okay, so I've got quite a few questions I've been trying to sort out with the help of the xenwiki and just lurking on the list and such, but in the interest of better understanding certain recent discussions, I've decided just to ask:
>
> 1) What is the regular temperament paradigm? (Apologies to Mike Battaglia; I realize you asked basically the same thing, perhaps rhetorically, but I'll bet you have a much better grasp on it than I do.) ;-)
>
> 2) What is the regular mapping paradigm? (Is it the same thing?)

The regular temperament or regular mapping paradigm, in its simplest
form, is what you're doing when you look at 12-equal and you say that
7 steps is a mistuned 3/2, and that 4 steps is a mistuned 5/4. In this
case, what you're actually doing is coming up with a schema relating
the equal temperament to a mistuned form of JI. The general way to go
about doing this is to first "map" the primes (like 2/1, 3/1, 5/1,
etc) and then work out how "composite" intervals (like 5/4) would be
mapped by combining the mapped primes.

> 3) What exactly is harmonic entropy, and is this what is meant by "HE"?

HE is Harmonic Entropy, yes. HE is a preliminary attempt to merge
music theory and information theory together; it's a model of
consonance proposed by Paul Erlich that attempts to measure how
confused your brain is when it hears an interval. When you hear
50001/40000, for example, it's probably going to get sucked into the
perception of 5/4. But when you hear 11/9, on the other hand, there
are going to be a lot of intervals competing for the "perception" of
that interval. The postulate here is that more confusion = more
dissonance.

The current hot topic in music theory around here is to what extent
learning can affect the accuracy of this model.

> 4) A while back on the xenwiki, I found a somewhat helpful description of what a comma pump does, but could someone please give me a few well-commented examples?

Meantone is the temperament eliminating 81/80. This means that it
equates 81/64 and 5/4, 32/27 and 6/5, and 9/8 and 10/9. It also means,
by definition, that it equates 81/80 and 1/1. Comma pumps work off of
this principle. A comma pump is a chord progression that, if you were
to tune it in JI with the most consonant root movements possible at
every turn, wouldn't actually end you back where you started, but
rather some comma up. For example -

A good example in meantone would be Cmaj - Am - Dm - Gmaj - Cmaj. If
you were to do this in JI, you'd go down 6/5, up 4/3, up 4/3, and down
3/2. If you multiply that all out, that means that the net effect of
that chord progression would be to move you down by 81/80 in JI.
However, in meantone, you end up back where you started, because 81/80
is tempered to be equal to 1/1.

The main point is that this if you tried to actually tune this in
5-limit JI, you'd end up some 73 octaves below where you started by
the end of the song.

http://www.youtube.com/watch?v=AapxXRlsdwA

Or you'd have to move by Pythagorean intervals or have comma shifts in
the middle of the chord progression, which is fine, but you'd probably
like it better if you just meantone-tempered the whole thing.

Lastly, comma pumps may be a huge part of how tonality works - they
enable you to leave the tonic along direction x, and then come back to
it from direction y. Here's a porcupine comma pump (excuse my
butchering of the explanation of this on camera, it was like 6 AM :))

http://www.youtube.com/watch?v=XSfnyr1MhXE

Some other attempts to work the porcupine comma pump into tonal
harmony can be found here:

http://soundcloud.com/mikebattagliamusic

> 5) I think I have a rough idea, maybe, how the ratios for intervals can be used to build consonant chords; however, I've gotten the impression the harmonic series can also be used in a similar way. Is this actually the case, and if so, what's the procedure?

What do you mean by this? What is the difference between a ratio from
the harmonic series and one that's not from the harmonic series?

> 6) What does "omnitetrachordal" mean?

An omnitetrachordal scale is one that displays sub-periodicity around
the 4/3; e.g. for any pattern of steps inside of some 4/3 in the
scale, that pattern is replicated in another 4/3-chunk going either up
or down from it. There are a few additional restrictions on various
strong and weak versions of this depending on who you talk to, but
that's the basic gist of it.

> 7) '13-EDO as a Father temperament or 11-EDO as a Hanson temperament ... / Hell, for all my blathering about 13-EDO really being "Uncle", I tend to treat it as "A-Team", the 2.9.21 subgroup that I guess I discovered' -- can anyone explain what any of that means? (I don't mean that to mock or anything; I really want to understand.)
>
> 8) "If I'm not going to use 12-TET, what can I use instead and how can I make it sound good?" (Thanks for putting it in a nutshell, Igs.) ;-)

It depends on your personality. Some people here really like moving
more and more towards the higher-accuracy, JI/microtemperament
direction (Carl, Gene), others like moving more and more towards the
direction of having more interesting puns and comma pumps (Petr), and
others like exploring high-error temperaments to see how they shake up
the limits of your perception (Igs), and others like me are still
finding their voice and trying to explore as much as possible. There
really is no "magic bullet." If you prefer a gradual evolution of your
paradigm, you might like to start with 19-equal first, since it also
supports meantone but offers additional possibilities as well. If you
want to shock your nervous system into something new, you might find
that you vibe really well with 15-equal or 16-equal. If what you want
is to explore higher-limit harmony, 22-equal might be the tuning for
you. If you want a combination of higher-limit harmonies as well as
higher accuracy, and don't mind having more notes, 31-equal, 41-equal,
46-equal might be the tuning for you. If you have no practical
limitations at all and don't care how many notes you want to use, but
you just want a tuning that's easy to "think in," you just can't beat
72-equal, period.

-Mike

🔗domeofatonement <domeofatonement@...>

🔗Jake Freivald <jdfreivald@...>

Also, if you're looking for JI consonances, you might check this out on Wikipedia:

http://en.wikipedia.org/wiki/Tonality_diamond

"The tonality diamond is often regarded as comprising the set of consonances <http://en.wikipedia.org/wiki/Consonance_and_dissonance> of the n-limit. "

"[Harry] Partch arranged the elements of the tonality diamond in the shape of a rhombus <http://en.wikipedia.org/wiki/Rhombus>, and subdivided into (n+1)^2 /4 smaller rhombuses. Along the upper left side of the rhombus are placed the odd numbers from 1 to n, each reduced to the octave (divided by the minimum power of 2 such that 1 \le r < 2). These intervals are then arranged in ascending order. Along the lower left side are placed the corresponding reciprocals, 1 to 1/n, also reduced to the octave (here, /multiplied/ by the minimum power of 2 such that 1 \le r < 2). These are placed in descending order. At all other locations are placed the product of the diagonally upper- and lower-left intervals, reduced to the octave. This gives all the elements of the tonality diamond, with some repetition. Diagonals sloping in one direction form Otonalities <http://en.wikipedia.org/wiki/Otonality_and_Utonality> and the diagonals in the other direction form Utonalities. One of Partch's instruments, the diamond marimba <http://en.wikipedia.org/wiki/Diamond_marimba>, is arranged according to the tonality diamond."

[edit
<http://en.wikipedia.org/w/index.php?title=Tonality_diamond&action=editÂ§ion=2>]
7-limit tonality diamond

7/4 <http://en.wikipedia.org/wiki/Harmonic_seventh>

3/2 <http://en.wikipedia.org/wiki/Perfect_fifth>
7/5 <http://en.wikipedia.org/wiki/Septimal_tritone>

5/4 <http://en.wikipedia.org/wiki/Major_third>
6/5 <http://en.wikipedia.org/wiki/Just_minor_third>
7/6 <http://en.wikipedia.org/wiki/Septimal_minor_third>
1/1 <http://en.wikipedia.org/wiki/Unison>
1/1
1/1
1/1

8/5 <http://en.wikipedia.org/wiki/Just_minor_sixth>
5/3 <http://en.wikipedia.org/wiki/Major_sixth>
12/7 <http://en.wikipedia.org/wiki/Septimal_major_sixth>

4/3 <http://en.wikipedia.org/wiki/Perfect_fourth>
10/7 <http://en.wikipedia.org/wiki/Septimal_tritone>

8/7 <http://en.wikipedia.org/wiki/Septimal_major_second>

I'm not sure whether the tonality diamond above will come out correctly over email. The top row is 7/4, in the middle. The second row is 3/2, 7/5. The third row is 5/4, 6/5, 7/6. And so on.

Regards,
Jake

🔗genewardsmith <genewardsmith@...>

🔗jlmoriart <JlMoriart@...>

The frequency ratios between the pitches in the harmonic series ARE those same whole number ratios that we talk about being in "just intonation" to build chords.

It is the harmonic series (1/1, 2/1, 3/1, 4/1, 5/1) that relates the pitches of overtones in most standard music making apparati like vibrating strings and air columns. When used to relate the pitches of sine waves, the harmonic series of overtones creates the most unified timbre.

When using multiple voices that contain that harmonic series in their sine wave overtones, it is at combinations of those frequency ratios the sine wave overtones will line up together and will not "beat", or clash and go WAH-WAH-WAH.

For example, if two trumpets play two tones, and one is a frequency 5/4 (1.25) times the other, then the fifth partial of the lower one will line up with the fourth partial of the higher one, and those two sines will not beat against each other, and the sound will blend more. This is one of the models for what makes consonance consonance.

-John

🔗Graham Breed <gbreed@...>

"Jay Random" <cortaigne@...> wrote:

> 2) What is the regular mapping paradigm? (Is it the same
> thing?)

The Regular Mapping Paradigm is a web page here:

http://x31eq.com/paradigm.html

I chose "mapping" over "temperament" because there are
cases where the mapping is relevant even if you aren't
using a regular temperament. You can use just intonation
in a regular-mapping way if you think of complex ratios as
standing in for simpler ratios. Erv Wilson's "Scales of Mt
Meru" are rational constructions that can be tied to regular
mappings. Circular temperaments can be thought of as
detunings of equal temperaments, and may be inspired by
higher rank mappings.

Most importantly, a scheme for adaptive temperament can be
specified in the light of the relevant mappings. This is a
model for how a flexible pitch performance will work. You
can think of a score as a set of pitch relationships that
the performers try to preserve (according to taste) and
melodic constraints that arise out of those harmonic
relationships. The mapping makes all this explicit.
Conventional staff notation encodes the meantone mapping,
and is sometimes taken to describe the mapping for
12-equal.

If the regular mapping paradigm is really a paradigm, it
should be possible to specify all musical pitch in terms of
it. Because the ideal intervals can be whatever you like,
it defaults to a "fuzzy pitch paradigm" where each pitch is
specified with an allowable range of error around its ideal
tuning and the mapping disappears behind the sofa. This
paradigm is more expressive than the conventional approach,
where a fixed scale is specified (usually in terms of cents
or ratios) and it's assumed that the music was written in
that scale and the performers should be following that scale
with no guide as to how far they can deviate from it. The
mappings come in because sometimes you describe the scales
with a simpler framework than the ideal intervals you think
you're approximating.

Depending on the definition, a regular temperament is
either a fixed tuning with a mapping from just intonation,
or a set of tunings with a few free parameters. As many
observed tunings aren't regular, a regular temperament
paradigm can't have universal application -- unless you
expand the concept of "temperament" to a point where it
becomes meaningless.

Regular mappings are more general because the resulting
tuning doesn't have to be regular. But, still, the regular
mapping paradigm is most useful when the tuning is close to
regular and the ideal tuning follows some meaningful
framework (usually small-integer ratios). Where regular
mappings aren't useful I don't know of any alternative
paradigm that does better.

> 5) I think I have a rough idea, maybe, how the ratios for
> intervals can be used to build consonant chords; however,
> I've gotten the impression the harmonic series can also
> be used in a similar way. Is this actually the case, and
> if so, what's the procedure?

You take frequency ratios relative to the root and multiply
through by the lowest common denominator. So, a 5-limit
major triad has intervals

5:4 6:5

The relative to the root, the chord becomes

1/1:1/1*5/4:1/1*5/4*6/5

1/1:5/4:3/2

The lowest common denominator is 4, and multiplying through
by that gives

4:5:6

A minor triad has the same intervals in a different order:

6:5 5:4

That works out as

1/1:6/5:3/2

The lowest common denominator is 10, giving a harmonic
series fragment of

10:12:15

For more complex harmonies, a lot of us find that following
the harmonic series is a better way of predicting
consonance than looking at the simplicity of the
constituent intervals in a chord.

> 6) What does "omnitetrachordal" mean?

It means a scale contains the same pattern of intervals,
covering a fourth, repeated twice in the scale. C major is
an example: C-D-E-F is "tone tone semitone" and so is
G-A-B-C.

Graham