IRC Channel, visual representations & iPhone apps

Hello everyone!

I just got introduced to this group by Dante

firstly, can I point out that there is a really friendly IRC channel
#music-theory on irc.freenode.net which would welcome related discussion. (
http://webchat.freenode.net/)

secondly, may I introduce what I'm doing? I am currently developing
iPhone applications to teach musical fluency. my motivation is that I am
striving for this fluency myself.

so far I have released one app:
http://itunes.apple.com/gb/app/chordwheel-pro/id406836326?mt=8

and my second is going to be a perfect pitch training game, but this has
led me into deep and new waters; ie looking at tuning systems notably just
intonation, and going backwards even further looking at pure ratios, and
the fundamental mathematical nature of music.

Already I have a problem. I have got it working by positioning the 12
equal ratio semitone pitch classes at the 12 points of a clock face, and
have a needle point in the direction currently being sung. but this is
really unsuitable. I have no desire to sing in this tuning system. It sounds
all wrong to me. it is not natural for my voice. I need to represent
these tones using ratios. Fair enough, I can use just intonation. but now
I have a problem. let's say the tone at 12 o'clock is always the keynote,
and the remaining 11 get positioned using just intonation. The attached
picture gives an idea of this. but what if I want to change key? let's
say I modulate from C to D. if I simply use the previous 'D', and
recalculate everything around that, that is surely the best option. But
this would mean that after some adventurous modulating I might never get
back to the original C. and maybe that wouldn't matter?!

I can't see any other viable alternative... what are my options?

- - - - -

on a related note, I started looking at new representations for music last
summer, putting up research at www.toneme.org, but my understanding has
come a long way since writing those pages. while I still think there is
merit in the system I propose, my focus has shifted...

I have been trying to come up with a musical system of representation that
is as aesthetically pleasing to the eye and logic centre as is to the
ears the music it represents.

it seems that some compromise had to be reached to represent the
infinitude of harmonious sounds that generate from a root pitch using a
finite number of physical notes/keys/bells etc.

and appears that contemporary musical theory is based not on the
underlying mathematics of harmony but upon this representation.

this seems analagous to a flatlander observing the shadow of a
three-dimensional object, and attempting to model around the shadow.

or trying to explain planetary motion from a geocentric standpoint

so I'm currently trying to get my head around what is at the core.

this is my most recent understanding:
http://wehearandplay.com/forum/viewtopic.php?p=4990#4990

the third post...

so I am looking at possible systems of representation that can illustrate
this graphically.

I'm guessing this path has been walked before, and I don't really want to
waste my remaining life energy reinventing the wheel. can anyone see what
I'm trying to get at and give me some pointers?

Also, I am very open to new concepts to code up on iPhone/iPad, so any
creative input is very much appreciated! if there is something you would
like to see out there, I may be able to get it done!

Pi

--- In tuning@yahoogroups.com, Pi <sunfish7@...> wrote:

Fair enough, I can use just intonation. but now
> I have a problem. let's say the tone at 12 o'clock is always the keynote,
> and the remaining 11 get positioned using just intonation. The attached
> picture gives an idea of this. but what if I want to change key? let's
> say I modulate from C to D. if I simply use the previous 'D', and
> recalculate everything around that, that is surely the best option. But
> this would mean that after some adventurous modulating I might never get
> back to the original C. and maybe that wouldn't matter?!

This is the problem of the comma pump. The best-known example is from the diatonic scale: if you go up a fourth to F, down a minor third to D, up a fourth to G, and then down to C, you won't get back to the C you started at if all the intervals are just. In fact, you'll get to (4/3)*(5/6)*(4/3)*(2/3) = 80/81. The small interval of 80/81 is known as the syntonic comma, or comma of Didymus. One solution to this is to temper: you can flatten the fifth somewhat--ideally more than the two cents it is flattened in 12 equal--and make the tone from C to D the same size as the one from D to E. If you make the third from C to E a just 5/4, then C up a ninth to D' can be divided in two to get a flattened G. This is called 1/4 comma meantone tuning, and is very close to what you get by dividing the octave into 31 equal parts. This and related systems were in general use for hundreds of years.

--- In tuning@yahoogroups.com, Pi <sunfish7@...> wrote:
> and my second is going to be a perfect pitch training game, but this has
> led me into deep and new waters; ie looking at tuning systems notably just
> intonation, and going backwards even further looking at pure ratios, and
> the fundamental mathematical nature of music.

There is no "fundamental mathematical nature of music". Or rather, there is no "one" fundamental mathematical nature of music, but a plethora of equally-valid ones. Many succumb to the siren-song of thinking it's "all about ratios and prime numbers" because of its rational simplicity and historical popularity, and indeed much great music has resulted from this approach. But a brief examination of the irregularities inherent in human hearing (as well as the irregularities of musical instruments and other sound-producing devices), it becomes quite obvious and clear that neither do our ears nor sounds themselves obey the strict regularity of JI frequency ratios. Where the relevance of ratios begins (and ends) is in their use of the construction of abstract musical systems and structures, which can then be only imperfectly realized in practice (except, perhaps in the realm of synthesis, but even that is questionable). The same can be said for all other mathematical characterizations of musical intervals--especially including temperaments.

Music, as a physical and/or psychological phenomenon, deals only with probabilities. The best that can be said of any exact frequency relationship is that it might describe an *actual* relationship between sound frequencies for a statistically-significant portion of the time. All this means, of course, is that we (as musicians) have absolute freedom to choose (or ignore) any of the multitudes of mathematically-described musical systems, according to our personal preferences; we can all rest assured that there is no one "right, pure, true, and fundamental way to tune".

> Already I have a problem. I have got it working by positioning the 12
> equal ratio semitone pitch classes at the 12 points of a clock face, and
> have a needle point in the direction currently being sung. but this is
> really unsuitable. I have no desire to sing in this tuning system. It sounds
> all wrong to me. it is not natural for my voice. I need to represent
> these tones using ratios. Fair enough, I can use just intonation. but now
> I have a problem. let's say the tone at 12 o'clock is always the keynote,
> and the remaining 11 get positioned using just intonation. The attached
> picture gives an idea of this. but what if I want to change key? let's
> say I modulate from C to D. if I simply use the previous 'D', and
> recalculate everything around that, that is surely the best option. But
> this would mean that after some adventurous modulating I might never get
> back to the original C. and maybe that wouldn't matter?!
>
> I can't see any other viable alternative... what are my options?

Your viable alternatives are to temper, to choose different ratios in some keys so that the comma-gap does not appear between the two C's, or to not "recalculate" when you modulate, keeping a system of 12 fixed pitches based on one key (thus some keys will mostly have your target ratios, and others will have "wolf intervals" that are less-desirable but unavoidable in fixed JI). Personally, I prefer the first option, as I find neither psychological nor aesthetic advantage in dealing with integer ratios, but many others DO find advantage in dealing with ratios, so your mileage may vary.

-Igliashon Jones

P.S. Welcome to the Tuning List!

Igs>"Music, as a physical and/or psychological phenomenon, deals only with
probabilities. The best that can be said of any exact frequency relationship is
that it might describe an *actual* relationship between sound frequencies for a
statistically-significant portion of the time."

Exactly! And every time you "optimize" more for one theory...you lose
optimization for another theory. The best I can say is...be open minded and
listen to all theories, try making some of your own...and see what you end up
liking: never let anyone tell you they have the a right way that always works
which causes you to ignore other alternatives.

Just this week I've became experimenting obsessively with "septimal minor
sixths" IE 14/9 as alternative fifths and the super-major and minor triads that
go with them. Mathematics and theories like Harmonic Entropy tell me they
shouldn't work very well...but my ears say otherwise. Just...try everything:
there's no harm in exploring.

Far as your problem
A) Like Igliashon, I think tempering (IE using TET temperaments) is a viable
option. The problem is some intervals will become hugely "wolf/sour" while
others won't.
B) Perhaps better, IMVHO, is irregular tempering. Personally I try to get all
dyads within less than 8 cents of the ratios I want, so the variation is 0-8
cents instead of the 0-13+ cent errors (or even 20+ cent "commatic" errors)
common in temperaments.

Plus, with irregular temperaments, you can often make your scale into a FIXED
system of 12 or less notes without having any notes drift (unlike using a huge
temperament like 53TET+ and selecting 12 notes at a time...)

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

> Just this week I've became experimenting obsessively with "septimal minor
> sixths" IE 14/9 as alternative fifths and the super-major and minor triads that
> go with them. Mathematics and theories like Harmonic Entropy tell me they
> shouldn't work very well...but my ears say otherwise. Just...try everything:
> there's no harm in exploring.

I hope you aren't confusing someone who is trying to learn the basics with this stuff. No one else seems to think 14/9 sounds like a fifth, and you yourself became confused on the issue of where the actual with was located in a chord which could be described as the second inversion of a supermajor triad. Since it is acting like a sort of sixth in these triads, why persist in calling it a fifth?

> Far as your problem
> A) Like Igliashon, I think tempering (IE using TET temperaments) is a viable
> option. The problem is some intervals will become hugely "wolf/sour" while
> others won't.

Not necessarily. If you are willing to use all the intervals in your equal temperament, not at all, unless it is sour in every key as some seem to prefer.

Gene>"Since it is acting like a sort of sixth in these triads, why persist in
calling it a fifth?"

Ok so, technically (if you read my last message) I agree that it IS simply
part of an inversion and that the 1/1 7/6 14/9 chord is an inversion of a chord
with a 14/9 (last octave) 1/1 7/6 (next octave) setup...in which case the 14/9
is indeed 3/2 away from 7/6.

I think you were very unclear, however...you said the chord 1/1 7/6 14/9 had
a "normal fifth". It doesn't. You did say "octave equivalent", but I missed
what you meant by that IE "octave equivalent...what interval?".

However, the chord it was an inversion of, 14/9 (last octave) 1/1 7/6 (next
octave) DOES have a normal fifth between the 14/9 when the 7/6 is moved up to
the "equivalent?!" next octave
. You didn't specify directly (IE in numerical form) what the inversion you
were talking about was...nor whether the "normal fifth" was in the inversion or
the original chord. You gave me two sentence explanation and I was left to
guess.

Also I said "fifth" AFTER when I posted the chord (and said Scala called it a
"septimal minor 6th").
But, composition-wise...I figure I can choose to use it as a sort of widened
fifth...after all a minor 6th from common practice music is 8/5 and the perfect
5th is 3/2...and both are an almost equally large distance away from 16/9.

Yes I was wrong (IE missing the "hint" you gave me about octave equivalence
and the inversion), but yes, you were very vague in explaining. I also don't
see why I 100% must treat the chord as an inversion when I make a scale or write
music.
Sometimes I feel I'm being prodded to either state composition methods
according strictly to Scala's dyadic classification scheme (which often seems to
agree not-so-much with other theories IE negliably different distance in
common-practice theory from a fifth and sixth) or not say a word...seems kind
of Puritanistic.

To the tuning list,

Despite I have a lot of musical subjects to learn from many of you let me inform you that from some time ago I am playing with three sets of endless tone and subtone frequencies where the classic 2C = 2 is out of the scene. Since equal tones along the progression are rejected, 2^n do not take part of any of the three modes excepting when n = 0 that gives the initial tone C = 1.

Man introduced the concept of the octave. The natural frequency progression of sound has nothing to do with octaves if we walk in the music field. I suspect that a conflict appears when schools of singers or musical instrument manufacturers open the doors to the aesthetic responses of 2^n (n is a round number) and simultaneously over millenia musicians and mathematicians impose tone segments which are called octaves. Consonance and virtual music existed since life exist, who is the scientist that is able to convince us that octave is the right parameter to study and explain the wide phenomenons of music.

Thanks
Mario
Lima, January 01, 2011
----- Original Message -----
From: Michael
To: tuning@yahoogroups.com
Sent: Saturday, January 01, 2011 7:46 PM
Subject: Re: [tuning] Re: IRC Channel, visual representations & iPhone apps

Igs>"Music, as a physical and/or psychological phenomenon, deals only with probabilities. The best that can be said of any exact frequency relationship is that it might describe an *actual* relationship between sound frequencies for a statistically-significant portion of the time."

Exactly! And every time you "optimize" more for one theory...you lose optimization for another theory. The best I can say is...be open minded and listen to all theories, try making some of your own...and see what you end up liking: never let anyone tell you they have the a right way that always works which causes you to ignore other alternatives.

Just this week I've became experimenting obsessively with "septimal minor sixths" IE 14/9 as alternative fifths and the super-major and minor triads that go with them. Mathematics and theories like Harmonic Entropy tell me they shouldn't work very well...but my ears say otherwise. Just...try everything: there's no harm in exploring.

Far as your problem
A) Like Igliashon, I think tempering (IE using TET temperaments) is a viable option. The problem is some intervals will become hugely "wolf/sour" while others won't.
B) Perhaps better, IMVHO, is irregular tempering. Personally I try to get all dyads within less than 8 cents of the ratios I want, so the variation is 0-8 cents instead of the 0-13+ cent errors (or even 20+ cent "commatic" errors) common in temperaments.

Plus, with irregular temperaments, you can often make your scale into a FIXED system of 12 or less notes without having any notes drift (unlike using a huge temperament like 53TET+ and selecting 12 notes at a time...)

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>
> This is the problem of the comma pump. The best-known example is from the
> diatonic scale: if you go up a fourth to F, down a minor third to D, up a
> fourth to G, and then down to C, you won't get back to the C you started at
> if all the intervals are just. In fact, you'll get to
> (4/3)*(5/6)*(4/3)*(2/3) = 80/81. The small interval of 80/81 is known as the
> syntonic comma, or comma of Didymus. One solution to this is to temper: you
> can flatten the fifth somewhat--ideally more than the two cents it is
> flattened in 12 equal--and make the tone from C to D the same size as the
> one from D to E. If you make the third from C to E a just 5/4, then C up a
> ninth to D' can be divided in two to get a flattened G. This is called 1/4
> comma meantone tuning, and is very close to what you get by dividing the
> octave into 31 equal parts. This and related systems were in general use for
> hundreds of years.

great, a hands-on example!

this raises several questions for me.

I can see that this is a problem for finite key (discrete pitch? there is a
better name for this?) instruments. I can see that it is also a problem
for notation. but is it actually a musical problem? if we are singing
barbershop, or playing violins and trombones, maybe no correction is
needed?

So going C -> G -> D -> F -> C

maybe musically we have the option of returning back to the established
tonal centre (the original C), or shifting tonal centre. while our new
note is at a ratio of 81/80 to the original C, it has a more comfortable
ratio with G,D,F that were passed through in order to arrive at it.

it may be that devices such as the iPad with touchscreen capability could
become the first instruments to allow dynamic recreation of related tones,
ie it is conceivable that whatever note you hit, the entire screen
rearranges around that point showing you your current closest matches, and
maybe highlighting the recent path you have taken to get to the
now. although really polyphony is the only thing that touchscreen would have
over conventional computer interface... Has anyone experimented with this?

I am still struggling with a lot of concepts

one thing I notice is that starting with a frequency which I represent by 1

I construct an octave, which I can write as 1/2

splits up into 2/3 and 3/4 (2/3 x 3/4 = 1/2) ie P5 + P4

each of these we can split up

look at 4/5 x 5/6 = 2/3 ie M3 + m3

now the terminology seems to break down, for example - there is no term
for the next interval that I know of. but we can still keep splitting ad
infinitum

6/7 x 7/8 = 3/4

etc.

8/9 x 9/10 = 4/5 ...
etc etc

now what of these smaller intervals? I have never heard any mention of
any musical system or research that examines these intervals.

another phenomenon I notice is the spectral composition of chords: I
linked to a post I made on this subject, giving an example of playing a C4
simultaneously with G4, and how this implies a fundamental frequency at
C3, with a rich pattern repeating through its harmonics

http://wehearandplay.com/forum/viewtopic.php?p=4994#4994

I can see this readily extending to any chord or progression, in a fractal
like way, so that maybe an entire piece of music could be considered to be
dancing through the harmonics of its root tone. also it should be
possible to suggest fundamentals well below the range of human hearing; I
wonder if the incredibly complex harmonies of Rachmaninov are doing this,
and that this music is inaccessible to many people; maybe it requires
substantial familiarity to train the brain to be able to recreate such
fundamentals. a friend was pointing out to me how our brains are naturally
tuned to recreating a complete picture given just a small sample of data,
how a handful of brush strokes can imply a gondolier or a tiger. so it
seems to me that music is dancing around some underlying harmony, and our
voices minds eyes and instruments can somehow resonate with this dance,
evoke this dance... yet somehow the dance is beyond the representation.

Sorry, that is a real meander...

the next thing I am wondering is about the wisdom of my initial design,
which was to work modulo the octave; ie C4 and C5 would both shoot the
needle to the 12 o'clock position. now I haven't quite got my head around
this, but it seems as though there is nothing unique in the concept of
octave. It simply uses the number 2. we could construct a similar concept
using the number 3. and we would detect a 'threeness' consonance among
such sounds.

this is the fifth, yes? mixing twoness with threeness...

so if we also throw in the number 5, we get our major third and minor
third, and if I'm not mistaken, most just intonation systems use 2,3,5 and
no higher

there must be some research into laying out these tones in some way that
illustrates the musical connection between them. for example, all 2^k *
note_i are somehow in the same octaves-category, pitchclass_2. all 3^k
notes are in the same pitchclass_3 category. these categories can
combine, just the same way prime numbers can combine to give integers.

so maybe such a representation would represent tones by their prime
composition

but then we have the same age-old problem we have had between the prime
numbers and the integers. The presence of the prime numbers in the
integers seems to be a logic unto its own.

I would have thought that if you start with a tone, and Mark around at all
of the low ratios, so that you have a visual indication, may be the
distance represents the actual distance in pitch, and the angle represents
the composition of primes. and you just start filling it in with the
lower numbers..... what sort of mathematical shape would give a decent
representation? Maybe each constituent prime would come with its own colour,
so you could at a glance get some measure...

I am very very curious about this: looking at it as a completely abstract
problem, how can we represent on paper what we perceive as beautiful music?
does it really all disappear down the rabbit hole?

Welcome to the tuning list!

On Sun, Jan 2, 2011 at 12:07 AM, Pi <sunfish7@gmail.com> wrote:
>
> I can see that this is a problem for finite key (discrete pitch?  there is a better name for this?) instruments.   I can see that it is also a problem for notation.   but is it actually a musical problem?  if we are singing barbershop,  or playing violins and trombones, maybe no correction is needed?

That would be something more akin to what we call "Adaptive JI" -
where the chords are just, but the root movements are tempered so as
to allow comma pumps like this. There are also different "puns" that
you can make in other tunings as well, all of which have their own
flavor; blackwood[10] has become a favorite of mine. It's two chains
of 5-et a major third apart, so that modulation by five fifths brings
you back to where you started (as opposed to 12 fifths like we're used
to).

>  maybe musically we have the option of returning back to the established tonal centre (the original C),  or  shifting tonal centre.    while our new note is at a ratio of 81/80  to the original C,  it has a more comfortable ratio with G,D,F that were passed  through in order to arrive at it.

Yes, but if you temper the root movements, you may never notice, while
still having perfectly just chords. Or you could explore modulation by
81/80 for its own musical effect, which can be quite beautiful. For
example, in 5-limit JI, the following chord progression modulates up
by 81/80 each time:

||: Cmaj9 | Dmaj9 | Dm9 :||

Voice it and arpeggiate like this: ||: C-E-G-B-D | D-F#-A-C#-E | D-F-A-C-E :||

But squeeze it into one octave, so that the B-D is movement down by a
major sixth rather than up by a minor third. It drifts every time, but
I think it's beautiful.

>  it may be that devices such as the iPad with touchscreen capability could become the first instruments to allow dynamic recreation of related tones, ie  it is conceivable that whatever note you hit, the entire screen rearranges around that point showing you your  current closest matches,  and maybe highlighting the recent path you have taken to get to the now. although really polyphony is the only thing that touchscreen would have over conventional computer interface...  Has anyone experimented with this?

I don't think that something like this currently exists. In general,
this community is starving for new tools to write music with, so feel
free to make it happen.

>  now the terminology seems to break down,  for example - there is no term for the next interval that I know of.  but we can still keep splitting ad infinitum
>
> 6/7 x 7/8 = 3/4

7/6 is often called a "subminor third," and 8/7 is often called a
"supermajor second." I've seen some people abbreviate them sm3 and
SM2, but I prefer to make subminor just a lowercase "s" and supermajor
an uppercase s: 7/6 = s3, 9/7 = S3, 8/7 = S2, etc.

> 8/9 x 9/10 = 4/5 ...

9/8 is usually called a "major whole tone," and 10/9 is usually called
a "minor whole tone." I have never liked these names because they're
both used in both the major and minor 5-limit scales, but that's the
name for them that I know.

> etc etc
>  now what of these smaller intervals?   I have never heard any mention of any musical system or research that examines these intervals.

That would be because most of the research involving microtonal
intervals like this has taken place outside of academia. A lot of it
has taken place on this mailing list. You should read Carl Lumma's
"too condensed tuning-math outline," it's a great introduction to all
of this stuff: http://www.lumma.org/music/theory/tctmo/

Paul Erlich's "A Middle Path" is also a great paper to read to get
started: eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

>  another phenomenon I notice is the spectral composition of chords:   I linked to a post I made on this subject,  giving an example of playing a C4 simultaneously with G4,  and how this implies a  fundamental frequency at C3,  with a rich pattern repeating through its harmonics
> http://wehearandplay.com/forum/viewtopic.php?p=4994#4994
>  I can see this readily extending to any chord or progression,  in a fractal like way,  so that  maybe an entire piece of music could be considered to be dancing through the harmonics of its root  tone.

I think that this is part of it, but that part of music is also the
jumping around of root tones, as well as causing an ambiguity between
which tone really is the "root" for a certain interval.

>   there must be some research into laying out these tones in some way that illustrates the musical connection between them.    for example, all 2^k * note_i  are somehow in the same octaves-category, pitchclass_2.    all 3^k  notes are in the same pitchclass_3  category.    these categories can combine,  just the same way prime numbers can combine to give integers.

If there's one thing that there's a hell of a lot of research on, it's
that. Start here: http://xenharmonic.wikispaces.com/

>  I would have thought that if you start with a tone,  and Mark around at all of the low ratios, so that you have a visual indication,  may be the distance represents the actual distance in pitch,  and the angle represents the composition of primes.   and you just start filling  it in with the lower numbers.....  what sort of mathematical shape would give a decent representation? Maybe each constituent prime would come with its own colour,  so you could at a glance get some measure...

I've often thought about representations such as these. There are a
lot of good ones but I haven't gotten too far into them... yet. Feel
free to come up with your own diagrams and post them here.

>  I am very very curious about this: looking at it as a completely abstract problem, how can we represent on paper what we perceive as beautiful music?    does it really all disappear down the rabbit hole?

No, but I think that there's more to it than math. You spoke earlier
about understanding music in terms of mathematics; what you should be
thinking more about is understanding music in terms of
psychoacoustics. There are real limits to our perception of different
integer ratios - 501/400, for example, is probably going to be heard
as 5/4, even though technically it's not. These limits can be used to
create their own musical effects. Look up "harmonic entropy" for more
research on this.

On top of the psychoacoustic layer is the cognitive layer, where
things like cultural factors and learned scale patterns and personal
preferences come into it. How much of the perception of music is
psychoacoustic and how much is cognitive is an open question. There
are lots of opinions on it.

-Mike

>
> Already I have a problem. I have got it working by positioning the 12
> equal ratio semitone pitch classes at the 12 points of a clock face, and
> have a needle point in the direction currently being sung. but this is
> really unsuitable. I have no desire to sing in this tuning system. It sounds
> all wrong to me. it is not natural for my voice.

That is the constant complaint that choir directors have been hearing from singers for the last century or two...

You might want to try dividing the octave into 19 equal intervals instead. In my experience this is the system that singers actually sing, when left to their own devices.

This system also happens to very closely correspond to our traditional notation, with its 7 naturals, 7 sharps and 7 flats. But in 19-tone equal, E#=Fb and also B#=Cb which reduces the 21 notes to 19. The 19-tone equal temperament divides the octave into 1/3 tones instead of semitones. The diatonic semitone, e.g. E-F becomes a 2/3 tone, while the chromatic semitone, e.g. Eb-E is a 1/3 tone in this system.

As others have explained to you, it is not practical to use just fifths and fourths (i.e. Pythagorean Intonation) to tune the other consonances, as there are commatic discrepancies from just.

The division of the octave into 19 equal intervals closely corresponds to a tonal web of just major sixths and just minor thirds. This implies fifths flat by only 1/3 comma and fourths sharp by 1/3 comma. This is preferable to Pythagorean thirds and sixths, off by a whole comma.

The only other possibility of any real importance is the division of the octave into 31 equal intervals, which closely corresponds to a web of just major thirds and just minor sixths. But 31 is far too many intervals in the octave to be practical, either for singers or instruments. Its 1/5 tone is much too narrow to be sung accurately; if it is (more or less), the result is similar to the aimless divagations of the tone-deaf.

I could of course treat this matter in almost infinitely greater depth, and give a dozen or more reasons why the 19-tone equal temperament is to be preferred to all the others, including involved speculations on the binary nature of human hearing, but for your purposes that is all you really need to know, perhaps. Be careful or this subject can absorb you past all hope!

I will however note in passing that many rock melodies cannot be represented in the 12-tone equal temperament at all, for the singers are singing 1/3 tones. 1/4 tones are however never habitually used by singers, not even Arab or Indian singers; they are too narrow for the human mind to reliably remember or reproduce. For all practical purposes, the choice is between 1/2 and 1/3 tones and their multiples. Crocker notes by the way that the neumes of ancient and mediaeval Western music were microtonal; only quite late did Western music become diatonic. In many ways jazz and rock music are a popular revolt against being confined to the diatonic. But even the diatonic is much easier to sing in its 19-tone equal version.

This board seems to be dominated by advocates of the untempered ratios. There are severe problems with such an idea, beginning with the most obvious: most musical timbres omit most partials, so that the supposed 'naturalness' of these ratios is very doubtful. It is much safer to stick to the six consonances of the senario, in my opinion. Of course this is not to say that the thirteen dissonant intervals of the octave are of no importance.

You would benefit by studying a text on mathematics and music, if you have not already done so.

Gregg Gibson

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

> You might want to try dividing the octave into 19 equal intervals instead. In my experience this is the system that singers actually sing, when left to their own devices.

It seems most unlikely that singers sing in any equal temperament, and from what I understand measurement shows that they don't.

> The division of the octave into 19 equal intervals closely corresponds to a tonal web of just major sixths and just minor thirds. This implies fifths flat by only 1/3 comma and fourths sharp by 1/3 comma. This is preferable to Pythagorean thirds and sixths, off by a whole comma.

"Only" 1/3 comma? How do singers consistently hit a fifth seven cents flat? Are they performing with a 19-equal keyboard?

> The only other possibility of any real importance is the division of the octave into 31 equal intervals, which closely corresponds to a web of just major thirds and just minor sixths. But 31 is far too many intervals in the octave to be practical, either for singers or instruments.

Given that instruments can and have done that, and that the 1/4 comma tuning, very close to 31, was in very widespead use for a long time, this claim is easily refuted.

> Its 1/5 tone is much too narrow to be sung accurately; if it is (more or less), the result is similar to the aimless divagations of the tone-deaf.

You don't need to attempt the full range of possibilities like Vicentino. He was very much the exception.

> I will however note in passing that many rock melodies cannot be represented in the 12-tone equal temperament at all, for the singers are singing 1/3 tones.

Easy to say, but can you provide a single example or supporting evidence?

> 1/4 tones are however never habitually used by singers, not even Arab or Indian singers; they are too narrow for the human mind to reliably remember or reproduce.

Except that measurements seem to show they do use intervals that small.

> For all practical purposes, the choice is between 1/2 and 1/3 tones and their multiples. Crocker notes by the way that the neumes of ancient and mediaeval Western music were microtonal; only quite late did Western music become diatonic. In many ways jazz and rock music are a popular revolt against being confined to the diatonic. But even the diatonic is much easier to sing in its 19-tone equal version.

This is ass-backwards. The reason 19 is of interest is in good part *because* you can use it for the diatonic scale and extensions, and hence lots of familiar music can be played in it. It can also do other, less familiar things, but meantone and the diatonic scale is its big selling point to start out with.

> This board seems to be dominated by advocates of the untempered ratios.

This is hilarious. How long have you been reading here? JI advocates complain that it is dominated by advocates of temperament, and with better reason than your complaint.

> There are severe problems with such an idea, beginning with the most obvious: most musical timbres omit most partials, so that the supposed 'naturalness' of these ratios is very doubtful.

Most musical timbres most emphatically do NOT omit most partials, starting with the one you start from--the human voice.

On Sat, Jan 22, 2011 at 11:19 PM, gregggibson <gregggibson@...> wrote:
>
> >
> > Already I have a problem. I have got it working by positioning the 12
> > equal ratio semitone pitch classes at the 12 points of a clock face, and
> > have a needle point in the direction currently being sung. but this is
> > really unsuitable. I have no desire to sing in this tuning system. It sounds
> > all wrong to me. it is not natural for my voice.
>
> That is the constant complaint that choir directors have been hearing from singers for the last century or two...
>
> You might want to try dividing the octave into 19 equal intervals instead. In my experience this is the system that singers actually sing, when left to their own devices.

They're probably doing something closer to adaptive-JI while
conceptualizing things around 12-tet. I doubt they naturally tend to
sing fifths that are that flat.

> The division of the octave into 19 equal intervals closely corresponds to a tonal web of just major sixths and just minor thirds. This implies fifths flat by only 1/3 comma and fourths sharp by 1/3 comma. This is preferable to Pythagorean thirds and sixths, off by a whole comma.

There's also tunings that don't eliminate 81/80. What's wrong with those?

> The only other possibility of any real importance is the division of the octave into 31 equal intervals, which closely corresponds to a web of just major thirds and just minor sixths. But 31 is far too many intervals in the octave to be practical, either for singers or instruments. Its 1/5 tone is much too narrow to be sung accurately; if it is (more or less), the result is similar to the aimless divagations of the tone-deaf.

What's wrong with 22? There are lots of useful divisions of the octave
if you're willing to get away from meantone.

> I could of course treat this matter in almost infinitely greater depth, and give a dozen or more reasons why the 19-tone equal temperament is to be preferred to all the others, including involved speculations on the binary nature of human hearing, but for your purposes that is all you really need to know, perhaps.

19-tet has some useful properties, but it fails spectacularly in the 7-limit.

> I will however note in passing that many rock melodies cannot be represented in the 12-tone equal temperament at all, for the singers are singing 1/3 tones. 1/4 tones are however never habitually used by singers, not even Arab or Indian singers; they are too narrow for the human mind to reliably remember or reproduce. For all practical purposes, the choice is between 1/2 and 1/3 tones and their multiples. Crocker notes by the way that the neumes of ancient and mediaeval Western music were microtonal; only quite late did Western music become diatonic. In many ways jazz and rock music are a popular revolt against being confined to the diatonic. But even the diatonic is much easier to sing in its 19-tone equal version.

I'm not sure that 19-tone power chords would be really good for rock.

> This board seems to be dominated by advocates of the untempered ratios. There are severe problems with such an idea, beginning with the most obvious: most musical timbres omit most partials, so that the supposed 'naturalness' of these ratios is very doubtful. It is much safer to stick to the six consonances of the senario, in my opinion. Of course this is not to say that the thirteen dissonant intervals of the octave are of no importance.

Uh, where are you getting this? In all seriousness, this list is where
modern temperament theory has basically come from.

Most musical timbres omit most partials?

-Mike

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

> There's also tunings that don't eliminate 81/80. What's wrong with those?

I suspect that schismatic temperaments such as 53 would be easier to train a choir to sing in than 19. The problem is finding music for it.

> What's wrong with 22? There are lots of useful divisions of the octave
> if you're willing to get away from meantone.

How readily would choirs take to 22? The fifth is as sharp as the 19et fifth is flat, and that would be harder to get people used to.

> 19-tet has some useful properties, but it fails spectacularly in the 7-limit.

Not as spectacularly as 12.

> I'm not sure that 19-tone power chords would be really good for rock.

And so 22 would be no better. What is the range of tunings acceptable for power chords, I wonder?

> Uh, where are you getting this? In all seriousness, this list is where
> modern temperament theory has basically come from.

That neglects the profound evil which is (or at least was) tuning-math.

On Sun, Jan 23, 2011 at 5:21 AM, genewardsmith
<genewardsmith@...> wrote:
>
> > What's wrong with 22? There are lots of useful divisions of the octave
> > if you're willing to get away from meantone.
>
> How readily would choirs take to 22? The fifth is as sharp as the 19et fifth is flat, and that would be harder to get people used to.

Well first off, I said that in response to Gregg's assertation that
there existed no useful division of the octave besides 12, 19 and 31.
But since you bring the point up, I don't think that choirs would have
to sing the fifths sharp. They'd probably sing them closer to just, as
do 12-tet choirs with major thirds. I do, however, think that it would
be easy enough to learn to conceptualize 22-tet internally, as far as
mapping out what's going on is concerned. If someone does that I
wouldn't be surprised if they habitually sung the fifths a little
sharp.

I'm not exactly sure what singers will spontaneously do as far as
12-equal is concerned, but I'd assume they sing the major thirds
somewhere between 12-tet and just.

Either way, this is all besides the point: choirs could learn to sing
in 22-tet or anything else. Even if they habitually sang the fifths
flatter and closer to just, they'd still be able to figure out how to
modulate around the superpyth cycle if they wanted and if you trained
them.

> > 19-tet has some useful properties, but it fails spectacularly in the 7-limit.
>
> Not as spectacularly as 12.

They both fail spectacularly enough for me to not really consider them
for 7-limit harmony at all.

> > I'm not sure that 19-tone power chords would be really good for rock.
>
> And so 22 would be no better. What is the range of tunings acceptable for power chords, I wonder?

I don't know, but for maximum power you're going to be running that
chord through a distortion pedal. Distortion means that you take the
frequency spectrum and convolve it with itself a lot. This means there
will be lots and lots of beating everywhere.

I rendered this up with sines and arctan distortion and 19 and 22 are
actually both pretty decent. You can't tell one from the other with
enough distortion. I thought 19 would be a lot worse. Here it is:
http://www.mikebattagliamusic.com/music/powerchords.wav

Starts with 19, switches to 22 but you can't hear the switch because
they sound the same, and then 12 at the end.

> > Uh, where are you getting this? In all seriousness, this list is where
> > modern temperament theory has basically come from.
>
> That neglects the profound evil which is (or at least was) tuning-math.

http://images2.memegenerator.net/ImageMacro/5381357/i-see-what-you-did-there.jpg?imageSize=Medium&generatorName=Joseph-Ducreux

-Mike

> > This board seems to be dominated by advocates of the untempered
> > ratios.
>
> This is hilarious. How long have you been reading here?

You are replying to Greg Gibson, the world's foremost expert
on 19-ET. He's been around these parts since 1997.

-Carl

Gene wrote:

> I suspect that schismatic temperaments such as 53 would be
> easier to train a choir to sing in than 19. The problem is
> finding music for it.

It's important when talking about real ensembles to
distinguish between vertical and melodic intonation.
In the case of 53, the 5-limit errors are smaller than
the vertical accuracy of any choir. So in one dubious
sense, some choirs already do sing in 53.

It is incredibly difficult to sing accurately out of
tune in a choral setting. More measurements need to be
done. At the moment, there are measurements establishing
vertical 9-limit JI in a capella vocal groups, and none
to my knowledge supporting tempered vertical intonation.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Gene wrote:
>
> > I suspect that schismatic temperaments such as 53 would be
> > easier to train a choir to sing in than 19. The problem is
> > finding music for it.
>
> It's important when talking about real ensembles to
> distinguish between vertical and melodic intonation.

I wasn't talking about vertical intonation, which as you point out would not be relevant, but of using something like Schismatic[17] or 53et on a keyboard instrument to go with a suitable score and an appropriately trained choir. I think it's possible that would present less of a problem than 19.

MikeB>"Well first off, I said that in response to Gregg's assertation that there
existed no useful division of the octave besides 12, 19 and 31."

31TET, IMVHO, is pretty a awesome "does most things" tuning. It does great
at traditional meantone...plus nails the 7 and 11-limit. Stinks at 9-limit,
though...and, as I recall, misses 13 and 15-limit for those who dare to go that
far.

Personally, 19TET does too many things too much like 12TET...grabs a little
bit of the 7-limit, but that's it.

My two main strikes against tunings like 12,19,31,53 ("ideal" meantone
tunings)...is they all seem to hint at the idea that
A1) Fifths and thirds are king and all other intervals are less valid
A2) Thus it follow basic major and minor triads are supposedly the most valuable
If all I wanted was 5-limit and tons of triads...I'd be plenty happy with
12TET and likely not bother with microtonality. Saying only good meantone-like
tunings are valuable, to me, is fairly akin to saying microtonality beyond what
has already been done in history with tweaking meantones is not valuable.
Boo....

>"Either way, this is all besides the point: choirs could learn to sing in 22-tet
>or anything else. Even if they habitually sang the fifths flatter and closer to
>just..."

This all seems to fall under the assumption that notes have to be 5-limit to
be easily "tuned" by musicians. Ok, so that may be the easiest way to
tune...but I don't for a second believe that means people can't taught to sing
other intervals...I just think they might (gasp) need to learn other ways than
by testing for "periodicity buzz". Another option, in some cases, is the whole
"drat" technique you/Mike have been discussing that creates an alternative type
of periodicity buzz for odd types of intervals.

On Sun, Jan 23, 2011 at 5:51 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"Well first off, I said that in response to Gregg's assertation that there existed no useful division of the octave besides 12, 19 and 31."
>
>     31TET, IMVHO, is pretty a awesome "does most things" tuning.  It does great at traditional meantone...plus nails the 7 and 11-limit.  Stinks at 9-limit, though...and, as I recall, misses 13 and 15-limit for those who dare to go that far.

I dunno, I don't mind 31-tet too much in the 13-limit. And major 7
chords sound okay to me.

>     Personally, 19TET does too many things too much like 12TET...grabs a little bit of the 7-limit, but that's it.

It has some other interesting features too, like that it supports the
island/Parizekmic/etc temperament we've been talking about, but yeah,
for 7-limit it's not the greatest.

>     My two main strikes against tunings like 12,19,31,53 ("ideal" meantone tunings)...is they all seem to hint at the idea that

?? 53 isn't a meantone...

> A1) Fifths and thirds are king and all other intervals are less valid
> A2) Thus it follow basic major and minor triads are supposedly the most valuable
>     If all I wanted was 5-limit and tons of triads...I'd be plenty happy with 12TET and likely not bother with microtonality.  Saying only good meantone-like tunings are valuable, to me, is fairly akin to saying microtonality beyond what has already been done in history with tweaking meantones is not valuable.  Boo....

I think the real value here is finding 5-limit systems that aren't
what we're used to, like Blackwood and Porcupine. The fact that 22
supports porcupine is one of its biggest strengths. I'd also like to
find some comparable 7-limit systems to Pajara sometime, which contain
full 7-limit tetrads with reasonable complexity.

>    This all seems to fall under the assumption that notes have to be 5-limit to be easily "tuned" by musicians.  Ok, so that may be the easiest way to tune...but I don't for a second believe that means people can't taught to sing other intervals...I just think they might (gasp) need to learn other ways than by testing for "periodicity buzz".

We were talking about the sharp fifths, not the higher-limit intervals...

> Another option, in some cases, is the whole "drat" technique you/Mike have been discussing that creates an alternative type of periodicity buzz for odd types of intervals.

Here's an interesting thing to do: Go into 19-equal, use the GM "Reed
Organ" sound, and find random higher-limit chords that occur in the
tuning, but really accurately. I was poking around with it last night
(should have written all of this down!) and I'd find random chords
that buzz. The augmented second in this tuning buzzes pretty well, as
does the augmented sixth. By finding good rational approximations to
this, you end up getting random chords that buzz that correspond to
like 39-limit chords and weird stuff like that. This still sound like
mistuned versions of lower limit chords, as a model like harmonic
entropy would dictate, but the fact that they buzz is a nice pleasant
sound. I'll post some examples later when I'm not swamped for time.

I also think that this might be too restrictive, it obviously doesn't
matter if all of the partials beat in sync with one another if by the
time you get to like the 8th partial it's too low to hear. It also
probably doesn't matter if there's strict syncing between the brat and
the drat if the brat is really slow. It also might not matter if
there's strict syncing between the brat and drat at all.

I think what does matter is making sure that different important
partials avoid the evil 70 cents critical band maximum of dissonance.
So a good way to model this might be to take Plomp and Llevelt's curve
and convolve it in the Mellin sense with a decaying impulse train.
I'll leave you with that cluster$#*& of DSP jargon and now I have to
go.

I also want to go back to reviewing Sethares' work here.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > And so 22 would be no better. What is the range of tunings acceptable for power
> > chords, I wonder?
>
> I don't know, but for maximum power you're going to be running that
> chord through a distortion pedal. Distortion means that you take the
> frequency spectrum and convolve it with itself a lot. This means there
> will be lots and lots of beating everywhere.

Dude, I play power chords in 20-EDO (i.e. 5-EDO) all the dang time. You'll see when my UnTwelve entry goes live, there's a whole doom-metal section in 5-EDO with gallons of distortion and it's FINE. 7-EDO works just fine too. For the record though, I'd say they're both about the threshold of it, 16-EDO and 18-EDO "power chords" just don't cut it. But 19 and 22 are both a cinch. Didn't anyone listen to my old 22-EDO metal stuff? If anyone can tell those power-chords are off-Just, I'll be amazed.

-Igs

MikeB>"I think the real value here is finding 5-limit systems that aren't what
we're used to, like Blackwood and Porcupine."
Fair enough. Granted though, doesn't that mean going beyond ideas like that
major and minor "common practice" triads are the most important chords (IE the
very reason people in the past have gravitated toward 12,19,31TET)?

>"Here's an interesting thing to do: Go into 19-equal, use the GM "Reed Organ"
>sound, and find random higher-limit chords that occur in the tuning, but really
>accurately. "

There seems to be some sort of alternative periodic buzz that allows easy
by-ear tuning/identification of higher limits, right?

>"I think what does matter is making sure that different important partials avoid
>the evil 70 cents critical band maximum of dissonance."

Funny, you say 70, Igs says 50, and I could swear it was more like 90 IE from
-> http://sethares.engr.wisc.edu/images/image3.gif. Can anyone here give a
definitive answer to where the maximum critical band dissonance point is?

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

> I'd also like to
> find some comparable 7-limit systems to Pajara sometime, which contain
> full 7-limit tetrads with reasonable complexity.

The painfully obvious answer to that one is meantone. Other possibilities are orwell and myna. Add a touch more complexity, and you can have miracle, magic, garibaldi, sensi and valentine. If even meantone (graham complexity 10) or myna (same thing) are too complex for you, you can try augene (9), injera (8), semaphore (8), negri (7), keemun (6), blacksmith (5), diminished (4) in place of pajara (6). But really, meantone or myna are not all that complex.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> Personally, 19TET does too many things too much like 12TET...grabs a little
> bit of the 7-limit, but that's it.

LOL, I used to think that too, and that's why it took me over 7 years to really give 19 a serious look. 19-EDO does a whole lot of things very differently than 12-tET. For starters, you've got about a bazillion more MOS scales in 19, most of which involve very different structures and harmonic resources. For instance, there's the scale you get by dividing the ~4/3 in half, which incidentally tempers out 49/48 (the difference between two 7/6's and a 4/3) and makes a nice 9-note MOS scale of 3 3 1 3 1 3 1 3 1 and has 7 consonant triads with two sizes of 3rd each, or the scale you get by dividing the major 3rd into 3 parts, giving 2 2 2 2 3 2 2 2 2, or the scale you get by dividing the 5/3 in half, giving 2 2 3 2 2 3 2 3, or the f'ed-up 7-note version of the whole-tone scale it does, 3 3 3 3 3 3 1. 19 does some cool 13-limit stuff pretty decently, like 13/7, 13/9, and 13/10, and it's the smallest EDO that approximates harmonics 1-31 within at least 30 cents. It's not great in the 7-limit but it at least gets in the ball-park. It fascinates me how, unlike in 12-EDO, the 5-limit and 7-limit are separated out in 19, but the 11- and 13-limit are wrapped up with both the 5- and 7-limit harmonies (particularly how 8/5 and 13/8 are approximated by the same interval of about 821 cents, and also that 7/5=11/8=18/13)...yet if you use the right harmonic configurations, you can totally distinguish the different prime identities. It's also not significantly worse than 22 in the 7-limit, despite what you might think, and to my ears actually sounds more "septimal" than 22 (even if less in-tune).

Even the 5-limit diatonic sounds a whole lot different than 12-tET to my ears, much more of a departure than either the 31-EDO diatonic or the 17-EDO diatonic (probably because of the wide ~14/13 semitone).

There's lot's to like about 19 that has little to do with meantone and 12-tET. You just gotta know how to look.

-Igs

On Sun, Jan 23, 2011 at 7:47 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"I think the real value here is finding 5-limit systems that aren't what we're used to, like Blackwood and Porcupine."
>     Fair enough.  Granted though, doesn't that mean going beyond ideas like that major and minor "common practice" triads are the most important chords (IE the very reason people in the past have gravitated toward 12,19,31TET)?

None of those support porcupine or blackwood, though. And you can have
your own compositional ideas on what chords are important or useful to
use. There is obviously lots of value in going past the 5-limit, which
was why I was so interested in 2.3.13/10 subgroups a little while ago.
But I think that as far as someone who's just getting used to
microtonality is concerned, alternative 5-limit tunings are a great
place to start. Specifically I think the way to go is to load up
Blackwood (the 20-tet or 35-tet versions are good, maybe you prefer
15), put on a decent timbre (something smooth, like the "Ocarina"
patch if you're limited to GM), load up a major9#11 chord in the chord
generator, and right click around the Chromatic Clavier for a while.
Make it so that if you right click on a major root, you get a maj9#11
chord, and if you right click on a minor root, you get a m11 chord.
Then go explore some limma puns and watch their heads explode.

> >"Here's an interesting thing to do: Go into 19-equal, use the GM "Reed Organ" sound, and find random higher-limit chords that occur in the tuning, but really accurately. "
>
>   There seems to be some sort of alternative periodic buzz that allows easy by-ear tuning/identification of higher limits, right?

No, it is periodicity buzz. The irrational versions turn out to only
buzz evenly when sines are involved. I think that it could be used for
ear training purposes, but I'm not sure how easy it would be or how
specific you could get with the identification. Whether or not you can
construct an irrational version that still sounds pleasant, due to
some property of the buzz that I haven't mapped out yet, is an
interesting question. There are a few interesting observations here,
to reiterate

1) There's probably a critical threshold for the drat at which the
whole thing stops being heard as "synced up" at all. I doubt we hear a
301/200 polyrhythm as anything but a rushing 3/2 polyrhythm. We need
to review the literature on the perception of rhythm to formalize this
more precisely.
2) It may not be the end of the world if the brat and the drat aren't
perfectly in sync, or if there are two rhythms going on in parallel,
so long as they don't conflict with one another in such a way that is
irritating. The literature on the perception of rhythm would also be a
good place to look, but I don't have much time now.
3) If you end up with lots of partials that are in the red zone of
awful beating, it sounds awful.
4) The volume of each harmonic gets lower as the harmonic gets higher,
so it probably doesn't matter what's going on at partial 9 and 10 and
what not.

> >"I think what does matter is making sure that different important partials avoid the evil 70 cents critical band maximum of dissonance."
>
>   Funny, you say 70, Igs says 50, and I could swear it was more like 90 IE from -> http://sethares.engr.wisc.edu/images/image3.gif.   Can anyone here give a definitive answer to where the maximum critical band dissonance point is?

I don't know. Whatever it is, we'll tweak it to that. I'll set up
different curves for different values and we'll see what works best.
70 was probably too high, after thinking about it. I think 50 is
probably closer.

-Mike

On Sun, Jan 23, 2011 at 7:52 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I'd also like to
> > find some comparable 7-limit systems to Pajara sometime, which contain
> > full 7-limit tetrads with reasonable complexity.
>
> The painfully obvious answer to that one is meantone. Other possibilities are orwell and myna. Add a touch more complexity, and you can have miracle, magic, garibaldi, sensi and valentine. If even meantone (graham complexity 10) or myna (same thing) are too complex for you, you can try augene (9), injera (8), semaphore (8), negri (7), keemun (6), blacksmith (5), diminished (4) in place of pajara (6). But really, meantone or myna are not all that complex.

I guess I was shooting for 7-limit tetrads in a diatonic-sized scale.
Orwell doesn't handle that too well. I'll check out myna and the other
ones - I remember liking injera the first time I messed with it.

-Mike

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

> I guess I was shooting for 7-limit tetrads in a diatonic-sized scale.
> Orwell doesn't handle that too well. I'll check out myna and the other
> ones - I remember liking injera the first time I messed with it.

Myna has a 7-note MOS, but that has no tetrads. The 11 note MOS does better, but it's with the 15 note MOS that myna really starts to cook. One advantage it holds over meantone is that meantone doesn't have a MOS between 12 and 19.

On Sun, Jan 23, 2011 at 7:54 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> > Personally, 19TET does too many things too much like 12TET...grabs a little
> > bit of the 7-limit, but that's it.
>
> LOL, I used to think that too, and that's why it took me over 7 years to really give 19 a serious look. 19-EDO does a whole lot of things very differently than 12-tET. For starters, you've got about a bazillion more MOS scales in 19, most of which involve very different structures and harmonic resources. For instance, there's the scale you get by dividing the ~4/3 in half, which incidentally tempers out 49/48 (the difference between two 7/6's and a 4/3) and makes a nice 9-note MOS scale of 3 3 1 3 1 3 1 3 1 and has 7 consonant triads with two sizes of 3rd each

This is one of the my favorite scales in the world. I like it better
in 24-tet, but the 19-tet one probably has better melodic properties.

> Or the scale you get by dividing the major 3rd into 3 parts, giving 2 2 2 2 3 2 2 2 2, or the scale you get by dividing the 5/3 in half, giving 2 2 3 2 2 3 2 3, or the f'ed-up 7-note version of the whole-tone scale it does, 3 3 3 3 3 3 1.

Or start with the 5/4 and get magic[7], 5 1 5 1 5 1 1, which is like
an expanded version of 12-tet's augmented scale.

-Mike

On 24 January 2011 03:23, Mike Battaglia <battaglia01@...> wrote:

> I think the real value here is finding 5-limit systems that aren't
> what we're used to, like Blackwood and Porcupine. The fact that 22
> supports porcupine is one of its biggest strengths. I'd also like to
> find some comparable 7-limit systems to Pajara sometime, which contain
> full 7-limit tetrads with reasonable complexity.

Looking at 7-limit systems with an arbitrarily defined average error
of around 7 cents would be one way to do that:

http://x31eq.com/cgi-bin/pregular.cgi?limit=7&error=7

Graham

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Or start with the 5/4 and get magic[7], 5 1 5 1 5 1 1, which is like
> an expanded version of 12-tet's augmented scale.
>

Ugh. I make a point of avoiding fair improper scales (# of s's > # of L's, but L:s > 2:1), I just can't stand how melodically and progressionally-awkward they are.

-Igs

Igs>"Ugh. I make a point of avoiding fair improper scales (# of s's > # of
L's, but L:s > 2:1), I just can't stand how melodically and
progressionally-awkward they are."

I get a similar impression...although I've also noticed there are many
options for strictly proper scales that don't fit into the usual MOS scale
subsets of TET tunings bin.

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

> Ugh. I make a point of avoiding fair improper scales (# of s's > # of L's, but L:s > 2:1), I just can't stand how melodically and progressionally-awkward they are.

Har. Check out Myna[7] then.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I think the real value here is finding 5-limit systems that aren't
> what we're used to, like Blackwood and Porcupine. The fact that 22
> supports porcupine is one of its biggest strengths.

I'm all for alternative 5-limit temperaments, except that Porcupine is just *heinous* in the 7- or 8-note MOS. Oh, the chords are fine, but they fit together all wrong and melody is just atrocious. You've basically got to use the 15-note MOS to get any decent music out of it. I've never heard a piece of music written in Porcupine[7] or [8] that I've liked, I tend to find even 11-EDO preferable!

On the other hand, have you tried Negri[9] or [10]? It's not as simple as Porcupine, but I think it's pretty sweet.

> I'd also like to find some comparable 7-limit systems to Pajara sometime, which contain
> full 7-limit tetrads with reasonable complexity.

Uh...Blackwood? Proper Blackwood[10] in 25-EDO has very low error in both prime 5 and 7, so as long as the funky 3/2 doesn't bother you, it's great!

Honestly though, if you want a full 2.3.5.7 system with any real level of accuracy (i.e. <8 cents) in all primes, there's no good simple solution. If you want a significant number of tetrads (i.e. tetrads on more than 2/3 of the possible roots), you're looking at 12 to 15-note scales, bare minimum.

-Igs

Igs wrote:

> Honestly though, if you want a full 2.3.5.7 system with any
> real level of accuracy (i.e. <8 cents) in all primes, there's
> no good simple solution. If you want a significant number of
> tetrads (i.e. tetrads on more than 2/3 of the possible roots),
> you're looking at 12 to 15-note scales, bare minimum.

The best that can be done is pajara[10], with 4/5 of the
roots having a full tetrad. 10 cents error on 7, < 6 cents
on the other primes. -C.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> The best that can be done is pajara[10], with 4/5 of the
> roots having a full tetrad. 10 cents error on 7, < 6 cents
> on the other primes. -C.
>

I was incredulous at first, thinking "22-EDO isn't that accurate!" But then I remembered pajara's a temperament and not a scale, and that it can be tuned much more accurately than 22-EDO. Duh. Man, there needs to be more music written in (optimal) pajara[10].

Out of curiousity, what's the next-best?

-Igs

Igs wrote:

> > The best that can be done is pajara[10], with 4/5 of the
> > roots having a full tetrad. 10 cents error on 7, < 6 cents
> > on the other primes. -C.
>
> I was incredulous at first, thinking "22-EDO isn't that accurate!"
> But then I remembered pajara's a temperament and not a scale, and
> that it can be tuned much more accurately than 22-EDO. Duh. Man,
> there needs to be more music written in (optimal) pajara[10].
>
> Out of curiousity, what's the next-best?

Meantone is technically better than pajara but it can't
meet the 2/3 of roots in < 12 tones requirement you set.

Keep in mind your error requirement is about primes, which
are large intervals. The optimal pajara I mentioned still
has a 984-cent 7:4.

Negri is technically worse than pajara, its 7:4 is worse,
and it can't begin to meet your complexity requirement.

Keemun is probably next-best. But Keemun[11] still only
gives five complete tetrads (less than 1/2 of the roots).
Its 7:4 is also worse at 953 cents.

Pajara is really it when it comes to 'doing in the 7-limit
what meantone does in the 5-limit'.

-Carl

On Mon, Jan 24, 2011 at 12:20 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > Or start with the 5/4 and get magic[7], 5 1 5 1 5 1 1, which is like
> > an expanded version of 12-tet's augmented scale.
> >
>
> Ugh. I make a point of avoiding fair improper scales (# of s's > # of L's, but L:s > 2:1), I just can't stand how melodically and progressionally-awkward they are.
>
> -Igs

? Do you not like the augmented scale? This sounds like the same thing
to me, with an extra note thrown in there.

I've heard composers do interesting things with it - Graham wrote a
piece in magic a while ago that was pretty tight.

-Mike

Just curious, where can I find a Scala file of Pajara[10]? I'm very curious
to see how this compares to Gene's posted Minerva tuning, which seems chalk-full
of good 7-limit intervals...

On Mon, Jan 24, 2011 at 10:14 PM, Michael <djtrancendance@...> wrote:
>
>     Just curious, where can I find a Scala file of Pajara[10]?  I'm very curious to see how this compares to Gene's posted Minerva tuning, which seems chalk-full of good 7-limit intervals...

If you want the 22-tet version, just load up 22-equal and then go to
"Mode," where you'll see the Pajara modes all listed under the
"decatonic" moniker.

The Pajara[10] MOS that I like the most is the symmetrical static
major mode. Load up this 9-limit chord: C E\ Bb D G - and then
transpose it up and down decatonically throughout the scale. You end
up with the following chords:

C9
C#\7#9
Daug7#9
Em6/9
Fm6/9
F#\9
G7#9
Bbm6/9
B\m6/9
C9

This scale also works in 12-equal, which supports Pajara, but
obviously doesn't support the 7-limit as well. That's right, new
proper 12-equal scales! Astound your friends, amaze your audience!
Blah.

The Pajara scales that have received the most attention aren't the MOS
ones, but rather the near-MOS scales. These scales are related to
Pajara in the same way that the melodic minor scale is related to
natural minor - although natural minor is an MOS, melodic minor isn't,
because only some of the interval classes have two specific interval
sizes. However, the same unison vectors are tempered out anyway - if
you look at the PB for one of these near-MOS scales, it looks like the
normal PB but with a chunk taken out and pasted on the other side. So
you get jigsaw-shaped pieces tiling the lattice instead of
parallelograms, or whatever the higher-dimensional equivalent of these
is.

The near-MOS Pajara scales that everyone likes to use are the
"pentachordal" scales, so named because I forget why. Something to do
with splitting up the scale into 2 pentachords, instead of 2
tetrachords like the diatonic scale, but I'm not sure how it works
out. Anyway, start with standard pentachordal major and play around.

The fifth mode of standard pentachordal major works really well as a
"tonal" version of the blues scale, especially in 12-tet where 81/80
vanishes as well. It isn't CS when used in this fashion, though.

-Mike

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Just curious, where can I find a Scala file of Pajara[10]?
> I'm very curious to see how this compares to Gene's posted Minerva
> tuning, which seems chalk-full of good 7-limit intervals...

I'd recommend Graham's temperament finder.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> ? Do you not like the augmented scale? This sounds like the same thing
> to me, with an extra note thrown in there.

No, I can't say that I like the augmented scale much. Oh, I've heard good things done with it, but it's not my cup of tea.

> I've heard composers do interesting things with it - Graham wrote a
> piece in magic a while ago that was pretty tight.

Well, MOS-wise Magic works basically like a really improper version of Dicot/hemififths/mohajiraish/Beatles/whatever. I'm sure it's workable in inspired hands, but I've never been inspired by it.

-Igs

None of your objections are valid. The last in particular is entirely without basis, and leads me to question your knowledge of musical acoustics.

I unfortunately do not have the time to reply to you in depth. This would require in part that I present some of my unpublished discoveries on this subject, especially concerning the binary theory of human hearing, symmetry and the differential limen, which I have found to be constant throughout the vocal register, and much narrower than in the literature. Also, this is of course not an appropriate forum to present such discoveries. Finally, your somewhat impolite and unscholarly tone gives me pause. I will however, in order to be helpful to others, respond to your first objection with a simple experiment that I like to use with subjects.

Ask a singer to sing three ascending major thirds, with the tone intercalated for greater ease in intonation. For example:

C D E E F# G# G# A# B#

You can, if you like, prompt the singer using whatever temperament you like, provided it has consonant major thirds in the chain of its fifths. It will make no difference in the results. I have used 12- 19- 26- 31- 43- 45- 50- and 55-tone equal for this purpose. I have also used 22- and 53-tone equal values, though these temperaments either exaggerate the comma (22-tone equal) or preserve it (53-tone equal) which results in highly dissonant fifths or thirds at some point in the tonal fabric. This of course is the reason for temperament in the first place, historically considered.

If the singer is singing just major thirds, the B# will fall short of the octave C by a diesis, about 1/5 tone; If 12-tone equal major thirds, B will equal C exactly; if Pythagorean major thirds, B will be sharper than C by a comma; finally, if the singer is using 19-tone equal major thirds, or what is audibly the same thing, major thirds diminished by 1/3 comma, B will be a diesis plus a comma flatter than C, or about 1/3 tone flatter. You can also ask them to complete the octave by singing C. Many singers are amazed that they always come out about 1/3 tone flat of C.

What happens? In virtually all cases, singers end about 1/3 tone flatter than C. When they do not, they are invariably wildly off by a tone or more, presumably they have not understood that they have been asked to sing major thirds. Musical training seems to make no difference.

This is very strong prima facie evidence that the human brain does indeed think in 19-tone equal, and in 1/3 tones, which are of course just narrower than the melodic limen of about 55-60 cents, as cited in the literature, and confirmed by my own experiments. This theory of the primacy of the 19-tone equal is not mine; there are two Italian professors who have presented evidence in favor of this, although I do not have their names to hand at the moment. I have however greatly extended and deepened this theory.

I have many interests and duties other than musical theory; I am just finishing my translation of Kafka's Stories at the moment, but will publish my musico-theoretical results in due course. Until then, I wish this board all the best.

Thanks to Carl Lumma for his kind words. It is true that I am one of the growing number of authorities on this subject, but only in the sense that Newton cited... super umeris gigantium...

Gregg Gibson

On Wed, Jan 26, 2011 at 11:35 PM, gregggibson <gregggibson@...> wrote:
>
> None of your objections are valid. The last in particular is entirely without basis, and leads me to question your knowledge of musical acoustics.

Who are you responding to? Me or Gene? Nonetheless, how are we
supposed to react? You're telling us that no divisions of the octave
other than 19 and 31 have any use. That's quite a bold statement to
make - what is wrong with 22-tet? Compared to 22, 19 is atrocious in
the 11-limit, and there are lots of interesting 5-limit systems out
there other than meantone.

> This is very strong prima facie evidence that the human brain does indeed think in 19-tone equal, and in 1/3 tones, which are of course just narrower than the melodic limen of about 55-60 cents, as cited in the literature, and confirmed by my own experiments. This theory of the primacy of the 19-tone equal is not mine; there are two Italian professors who have presented evidence in favor of this, although I do not have their names to hand at the moment. I have however greatly extended and deepened this theory.

Since when is the JND supposed to be 55 cents? 5 cents is more like it.

> Thanks to Carl Lumma for his kind words. It is true that I am one of the growing number of authorities on this subject, but only in the sense that Newton cited... super umeris gigantium...

Much respect to you for your work. Paul Erlich is one of the world's
leading authorities on 22-equal, and I've read his paper on it dozens
of times and internalized a lot of it. I don't think anyone ever
claimed that 19-tet wasn't a good tuning for meantone, but I jump off
the train when sweeping meantone-centric generalizations about how
it's the only useful tuning after 12 are made.

-Mike

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

> I don't think anyone ever
> claimed that 19-tet wasn't a good tuning for meantone, but I jump off
> the train when sweeping meantone-centric generalizations about how
> it's the only useful tuning after 12 are made.

Anyway, we've already got a candidate for the only good meantone, and that's Lucy Tuning. That's 3/10 comma, not 1/3 comma.

On Thu, Jan 27, 2011 at 12:06 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I don't think anyone ever
> > claimed that 19-tet wasn't a good tuning for meantone, but I jump off
> > the train when sweeping meantone-centric generalizations about how
> > it's the only useful tuning after 12 are made.
>
> Anyway, we've already got a candidate for the only good meantone, and that's Lucy Tuning. That's 3/10 comma, not 1/3 comma.

I forgot all about that. 19 supports meantone[7], but not
LucyTuning[7]. What use is it anyway?

-Mike

Gregg>"This is very strong prima facie evidence that the human brain does indeed
think in 19-tone equal, and in 1/3 tones, which are of course just narrower than
the melodic limen of about 55-60 cents, as cited in the literature, and
confirmed by my own experiments."

If I got what you are saying correctly...I think you are right that around
25/24 is about where the human ear starts to think of two notes as being far
enough apart to be melodically "notice-ably different". In that case, I think
19TET has about hits the spot dead-on melodically.

However, I'm less convinced of it 19TET far as representing most to all
relevant dyad basis of chords the mind can think in IE the 11/9 and 11/6
(neutral third and seventh used in Blues and African and Middle Eastern music)
are simply not there and neither is the Middle Eastern 22/15 diminished fifth.

On Thu, Jan 27, 2011 at 12:19 AM, Michael <djtrancendance@...> wrote:
>
> Gregg>"This is very strong prima facie evidence that the human brain does indeed think in 19-tone equal, and in 1/3 tones, which are of course just narrower than the melodic limen of about 55-60 cents, as cited in the literature, and confirmed by my own experiments."
>
>     If I got what you are saying correctly...I think you are right that around 25/24 is about where the human ear starts to think of two notes as being far enough apart to be melodically "notice-ably different".  In that case, I think 19TET has about hits the spot dead-on melodically.

19-TET does have some killer melodic properties. I didn't realize this
until Carl pointed this out to me a while ago, because the diatonic
semitone in 19-TET is pretty sharp. It's long been said that melodic
resolution by a leading tone works better if it's sharpened to be
around the magical 70 cent interval, which is, as George Secor has
pointed out, one of the reasons why 17-TET has such fantastic melodic
properties. Whereas 17-TET maps this interval to the diatonic
semitone, 19-TET maps it to the chromatic semitone, and so raising the
leading tone by a diesis for V->I makes for really good melodic
properties.

-Mike

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gregg>"This is very strong prima facie evidence that the human brain does indeed
> think in 19-tone equal, and in 1/3 tones, which are of course just narrower than
> the melodic limen of about 55-60 cents, as cited in the literature, and
> confirmed by my own experiments."
>
> If I got what you are saying correctly...I think you are right that around
> 25/24 is about where the human ear starts to think of two notes as being far
> enough apart to be melodically "notice-ably different". In that case, I think
> 19TET has about hits the spot dead-on melodically.
>
> However, I'm less convinced of it 19TET far as representing most to all
> relevant dyad basis of chords the mind can think in IE the 11/9 and 11/6
> (neutral third and seventh used in Blues and African and Middle Eastern music)
> are simply not there and neither is the Middle Eastern 22/15 diminished fifth.
>

Correction: I should of course have said that the 1/3 tone of the 19-tone equal (ideally about 63,3 cents with an octave stretch) is just WIDER than the melodic limen of 55-60 cents. Strange how such errors creep in.

The idea that there is a 'neutral' third is what is called a ghost-concept, arising from the habit of thinking in cents. Always remember that the cent is a purely artificial creation. I have even heard a musician claim that the fact that there are 1200 cents 'proves' that it is most natural to divide the octave into 12 equal semitones. He was quite serious...

Only under Western influence have the Arabs for example adopted the idea of 24-tone equal temperament and the quarter tone. But their singers have repeatedly been found to use the 1/3 tone as their narrowest melodic interval.

But the ratios are also a form of ghost-concept, at least insofar as they are not actually tunable by beats in a given timbre, and insofar as there exists no other mental device to permit their memorization. The identification of these devices has occupied me recently, but appears to be confined to the senario. Even when the musical timbre does permit the accurate tuning of a ratio such as 11:9, it is still melodically swallowed up by the nearest multiple of the 1/3 tone, in this case the minor third.

Gregg Gibson

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
>
> 19-TET does have some killer melodic properties. I didn't realize this
> until Carl pointed this out to me a while ago, because the diatonic
> semitone in 19-TET is pretty sharp. It's long been said that melodic
> resolution by a leading tone works better if it's sharpened to be
> around the magical 70 cent interval, which is, as George Secor has
> pointed out, one of the reasons why 17-TET has such fantastic melodic
> properties. Whereas 17-TET maps this interval to the diatonic
> semitone, 19-TET maps it to the chromatic semitone, and so raising the
> leading tone by a diesis for V->I makes for really good melodic
> properties.
>

Hmm, when I played around with 19edo, I did not like the 19edo semitone very much. 2 steps of 19edo sounded too large, while 1 step sounded too small... I liked the 17edo step much better - and, seemingly paradox, 1 step of 22edo sounded better again to me. I have no theoretic explication for his - but that's what my impressions were.
--
Hans Straub

> Hmm, when I played around with 19edo, I did not like the 19edo semitone very much. 2 steps of 19edo sounded too large, while 1 step sounded too small... I liked the 17edo step much better - and, seemingly paradox, 1 step of 22edo sounded better again to me. I have no theoretic explication for his - but that's what my impressions were.

I've come to similar conclusions. Check out this image:
http://upload.wikimedia.org/wikipedia/commons/b/bc/Syntonic_Tuning_Continuum.jpg
It orders tunings by their generator size in the section of the 2D continuum that generates 5L2S MOS scales, aka our diatonic scale.

As you approach the bottom, the major and minor second converge on the same "neutral" interval, where there is very little melodic expression because of a lack of contrast in the 7-edo scale. Sometimes melodies can be "smoother" towards that end of the continuum which is cool if it's what you want, but anyway I find it generally less expressive.

As you approach the top of the continuum, on the other hand, the major and minor second *diverge*, creating a starker and starker contrast between major and minor seconds and allowing more and more expressive melodies.This trend stops applying, however, when the minor second tends to 0 cents (at 5-edo). Even before that point, it becomes indistinguishable from the unison. I've found that, until a little above 22-edo, melodies just get more and more expressive. But after that point, getting closer to 5-edo makes the minor second TOO small and makes the diatonic scale a little unpleasant. For me this takes place around 713 cents, but I'm sure the effect would vary.

Anyway, it would make sense to me that the 19-edo step wouldn't sound to hot. The minor seconds of 22- and 17-edo fit into a diatonic framework where the single step of the tuning is the S in the 5L2S. In a diatonic context the single 19-edo step, however, wouldn't work the way you expected it to. (Theoretically it would work as an augmented unison VS minor second). And your finding that 22-edo has a better leading tone than 17-edo makes a lot of sense too. It's a starker contrast with the major second, without getting unpleasantly small.

John

19 rocks! for all these reasons.

15 steps of 19 equal is very septimalish sounding to me. Still cool, even though it doesn't nail it. I don't mind; it's a great sound.

AKJ

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> > Personally, 19TET does too many things too much like 12TET...grabs a little
> > bit of the 7-limit, but that's it.
>
> LOL, I used to think that too, and that's why it took me over 7 years to really give 19 a serious look. 19-EDO does a whole lot of things very differently than 12-tET. For starters, you've got about a bazillion more MOS scales in 19, most of which involve very different structures and harmonic resources. For instance, there's the scale you get by dividing the ~4/3 in half, which incidentally tempers out 49/48 (the difference between two 7/6's and a 4/3) and makes a nice 9-note MOS scale of 3 3 1 3 1 3 1 3 1 and has 7 consonant triads with two sizes of 3rd each, or the scale you get by dividing the major 3rd into 3 parts, giving 2 2 2 2 3 2 2 2 2, or the scale you get by dividing the 5/3 in half, giving 2 2 3 2 2 3 2 3, or the f'ed-up 7-note version of the whole-tone scale it does, 3 3 3 3 3 3 1. 19 does some cool 13-limit stuff pretty decently, like 13/7, 13/9, and 13/10, and it's the smallest EDO that approximates harmonics 1-31 within at least 30 cents. It's not great in the 7-limit but it at least gets in the ball-park. It fascinates me how, unlike in 12-EDO, the 5-limit and 7-limit are separated out in 19, but the 11- and 13-limit are wrapped up with both the 5- and 7-limit harmonies (particularly how 8/5 and 13/8 are approximated by the same interval of about 821 cents, and also that 7/5=11/8=18/13)...yet if you use the right harmonic configurations, you can totally distinguish the different prime identities. It's also not significantly worse than 22 in the 7-limit, despite what you might think, and to my ears actually sounds more "septimal" than 22 (even if less in-tune).
>
> Even the 5-limit diatonic sounds a whole lot different than 12-tET to my ears, much more of a departure than either the 31-EDO diatonic or the 17-EDO diatonic (probably because of the wide ~14/13 semitone).
>
> There's lot's to like about 19 that has little to do with meantone and 12-tET. You just gotta know how to look.
>
> -Igs
>

--- In tuning@yahoogroups.com, "akjmicro" <aaron@...> wrote:
>
> 19 rocks! for all these reasons.
>
> 15 steps of 19 equal is very septimalish sounding to me.

Why not? We can take it a either a 7/4 or a 12/7. Or maybe it's a 16/9, lowered by 36/35, if you think that's "septimalish".

> It is much safer to stick to the six consonances of the senario, in my opinion. Of course this is not to say that the thirteen dissonant intervals of the octave are of no importance.
>

I am replying to my own post, in order to clarify the above.

My reference to the senario and the thirteen dissonances is much too succinct and may create confusion; it may even appear to be an outright misstatement, including the octave with the dissonances.

Qua harmony, there are seven consonances within the octave inclusive, viz the senario and the octave itself. This leaves twelve dissonances.

Qua melody however, two of the seven consonances, the fourth and minor sixth, especially the former, have binary values (the binary code used by the deep brain to recognize intervals) which do not agree with their ratios. Hence the well-known difficulty of singers in learning the fourth, which despite its simple ratio 4:3 is actually little if at all easier to sing in melody than say, 25:18, which is a much more complex ratio, but that happens to almost exactly correspond with a binary-code equivalent.

Readers will have to excuse me for not elaborating upon this point, as my work regarding this is not yet published. I will however observe that the octave and its replicas form a binary sequence: 2 4 8 16 32 etc or 10 100 1000 10000 100000 etc and this is one of the simpler clues or elements in the development of a theory which can at last explain how the brain hears intervals. Obviously this is crucial to an understanding of tuning and temperament beyond say, the level of Mersenne, Huyghens or Sauveur.

It is of course highly necessary to insure that theory or speculation marches forward hand in hand with experimental results, or one will quickly find oneself in a morass of mere speculation, which however interesting, has no permanent value, except perhaps as a kind of quarry for future investigators.

Gregg Gibson

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
>
> I've come to similar conclusions. Check out this image:
> http://upload.wikimedia.org/wikipedia/commons/b/bc/Syntonic_Tuning_Continuum.jpg
> It orders tunings by their generator size in the section of the 2D
> continuum that generates 5L2S MOS scales, aka our diatonic scale.
>
> As you approach the bottom, the major and minor second converge on
> the same "neutral" interval, where there is very little melodic
> expression because of a lack of contrast in the 7-edo scale.
> Sometimes melodies can be "smoother" towards that end of the
> continuum which is cool if it's what you want, but anyway I find it
> generally less expressive.
>
> As you approach the top of the continuum, on the other hand, the
> major and minor second *diverge*, creating a starker and starker
> contrast between major and minor seconds and allowing more and more
> expressive melodies.This trend stops applying, however, when the
> minor second tends to 0 cents (at 5-edo). Even before that point,
> it becomes indistinguishable from the unison. I've found that,
> until a little above 22-edo, melodies just get more and more
> expressive. But after that point, getting closer to 5-edo makes the
> minor second TOO small and makes the diatonic scale a little
> unpleasant. For me this takes place around 713 cents, but I'm sure
> the effect would vary.
>
> Anyway, it would make sense to me that the 19-edo step wouldn't
> sound to hot. The minor seconds of 22- and 17-edo fit into a
> diatonic framework where the single step of the tuning is the S in
> the 5L2S. In a diatonic context the single 19-edo step, however,
> wouldn't work the way you expected it to. (Theoretically it would
> work as an augmented unison VS minor second).

Quite interesting!
But OTOH, in 19edo you can have a 5L2S scale with the interval of 2 steps as S - which I didn't like so much either...

> And your finding that
> 22-edo has a better leading tone than 17-edo makes a lot of sense
> too. It's a starker contrast with the major second, without getting
> unpleasantly small.
>

I wouldn't say 22edo is better than 17edo here. Not sure about leading tones, either - my observations were more generally concerning melodic work. I do not use 5L2S scales in 22edo, anyway (the superpyth major third sounds much too sharp to me). But who knows, maybe leading tone behaviour is the (or one) reason for my impressions.

Generally, I would qualify 22edo as melodically more difficult than 17edo, for which I blame mostly the interval of 3 22edo steps (minor whole tone - hard to avoid if you want to use the 7 steps major third and the beautiful 18 steps harmonic seventh).

But speaking of leading tones - 22edo is quite interesting here because it has indeed two intervals that can work well as leading tones, 1 step as well as 2 steps. Within different chord progressions, of course.
--
Hans Straub

Hey Gene,

Not sure you read correctly--I wrote that I *do* think it sounds like a decent 7/4; it least it strongly implies it.

AKJ

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "akjmicro" <aaron@> wrote:
> >
> > 19 rocks! for all these reasons.
> >
> > 15 steps of 19 equal is very septimalish sounding to me.
>
> Why not? We can take it a either a 7/4 or a 12/7. Or maybe it's a 16/9, lowered by 36/35, if you think that's "septimalish".
>