Jimi Hendrix

Graham,

I'm 40 now and I started playing when I was 16. Once I knew a few chords and could fool around with the blues scale I had no desire to delve further into guitar playing. The maths behind music though caught my imagination and since 1995 I've spent an average of at least six hours a day either at my computer programming or thinking about the maths behind music.

I know next to nothing about the prevalent theories of Just Intonation. I bought Doty's Primer a few years ago but to be honest I couldn't quite grasp a lot of the ideas there so I just ploughed on on my own.

As I said before, my system does not use or depend on prime numbers.

Since I first made contact with other members of the Just Intonation community (January of this year) I have learned a great deal more and my book will need some revising. At this point though, I still think my formulae have some merit and I also think that even if the formulae are wrong, my 12 key tuning system (NPT) is the best possible 12 key tuning system that has two different tonics of equal strength (1/1 and 3/2). One tonic works better for ascending music and the other for descending music, see chapter 7 of my book.

If you haven't done so already I would urge you to read chapters 4, 6 and 7 of my book and let me know what you think of my reasoning.

My formula for melody is 2/x + 2/y and (If I'm right) any melodic interval with a value of 1.0 or higher is a Major melodic interval. So any interval 1/n will have a minimum value of 2.0, so yes, all 1/n intervals, in melody, should be strong.

John

On 2 April 2010 00:24, jfos777 <jfos777@...> wrote:
> Graham,

> Since I first made contact with other members of the Just Intonation community (January of this year) I have learned a great deal more and my book will need some revising. At this point though, I still think my formulae have some merit and I also think that even if the formulae are wrong, my 12 key tuning system (NPT) is the best possible 12 key tuning system that has two different tonics of equal strength (1/1 and 3/2). One tonic works better for ascending music and the other for descending music, see chapter 7 of my book.
>
> If you haven't done so already I would urge you to read chapters 4, 6 and 7 of my book and let me know what you think of my reasoning.

Chapter 4 is where you look at different formulas. It depends on your
own ranking of consonances, which you haven't cross checked with
anybody else. And once you have a ranking the formula's useless
anyway. You also don't consider chords of more than 2 notes.

Chapter 6 is more of the same.

Chapter 7, then. Firstly you're assuming 12 notes which isn't the
only way to do it.

I see "prime number" on p.25. Maybe that's why you sent a
clarification. But if you're going to exclude 4/7, why keep any 7s at
all?

You turn your fractions upside-down after a while. So the scale now
contains 7/5 instead of 5/7. Anyway, that 7/5 would go with the 7/4.
By taking the 7/4 out you leave 7/5 orphaned. I don't know what the
argument is for 15/14 over 16/15, but they're only 8 cents apart.

Oh, wait, we do have 4/7 now? (p.32) I can't actually find the
listing of Natural Pan Temperament, which suggests it isn't presented
clearly. Is it the same as Natural North?

1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1

Most of them are obvious. It would be nice to have 7/4, but it's
nicer to have 9/5. Then the only question is why 15/14 and 7/5 are
still in there because they don't work with anything else. They don't
even work with each other because the 7 comes in the numerator of one
and the denominator of the other. You should be able to replace them
both and keep some symmetry.

Anyway, this is a lot of talk about finding a JI scale with simple
ratios. The result is reasonable but not special.

You haven't considered temperament. If you've quantified mistuning,
you might find something like schismatic temperament gives better
results because it can be perfectly inverted.

> My formula for melody is 2/x + 2/y and (If I'm right) any melodic interval with a value of 1.0 or higher is a Major melodic interval. So any interval 1/n will have a minimum value of 2.0, so yes, all 1/n intervals, in melody, should be strong.

This suggests something wrong, because these intervals aren't used
much in melody, and I note you don't have a radical melodic style that
changes this.

Graham

Graham,

the premise I began with was that it seems that, in general, the smaller the numbers in the ratio, the stronger the interval sounds. From this I deduced that there must be a mathematical formula for consonance that uses the numbers in the ratio. I *have* crossed checked the formula with John Chalmers whom I wrote to in January. He said: "your consonance functions ... seem to reproduce qualitatively, at least, my subjective feelings about sensory consonance". He even went as far as to program my functions in BASIC to test them and seemed enthusiastic about them.

You said that once you have a ranking then the formula is useless anyway. It isn't, the formula is incorporated into my formula for identifying the strength values (and key notes) of chords.

You also said that I don't consider chords of more than 2 notes. I do, see chapter 10 "Harmony & Chords". I even have formulas for identifying the key chord of a group of chords and the "progression" value of two chords. I also have 5 classifications of chords which are based on my consonance formulas.

I also have a chapter on Melody and Scales which has some new ideas.

Chapter 14 deals with stretch tuning and is one of the more intriguing chapters if you want to take a look.

As regards the 12 keys per octave (and not more or less than 12) I said before that, on a guitar, if extra frets are included then a lot of barre chords would be harder to play. For me, 12 is the golden mean between simplicity and complexity. Also, for someone making the transition from 12 key Equal Temperament to my 12 key Natural Pan Tuning, the transition should be relatively simple.

You said:"if you're going to exclude 4/7, why keep any 7s at all?" Good question. I excluded 7/4 because it "goes with" very few other notes and 9/5 is more versatile and is almost as strong. I kept the 7/5 however because there is no other note between 4/3 and 3/2 that is more versatile (except perhaps 10/7 but 7/5 goes better with 1/1 than 10/7).

My convention of writing intervals x/y (where x<=y) where the numerator is less than the denominator is clearly confusing. When I began I used this way of writing intervals because on a piano or guitar the lower notes are to the left and the higher notes are to the right. I didn't know that this was not the norm. For me 5/7 is the same as 7/5 when describing an interval (which may or may not contain the tonic, 1/1). When writing the pitches of a scale however then obviously the numerator should be higher than the denominator.

Leaving the 7/5 orphaned...Within an octave range going up and down, 7/5 "goes with": 3/4, 4/5, 9/10, 1/1, 2/1, 12/5, 5/2 and 8/3. All these intervals have a "harmony value" of 0.75 or higher. That's nine intervals excluding the octaves and unison, not too bad.

You said "Oh, wait, we do have 4/7 now? p.32" Yes we do, but not among intervals that contain the tonic. 4/7 does occur a few times among some intervals that don't contain the tonic.

Yes Natural North is the same as Natural Pan.

Going up, 15/14 goes with: 3/2, 5/3, 15/8 and 2/1 and goes with a few other notes going down.

I use 15/14 instead of 16/15 because (i) some of the more unusual and exotic (not found in ET) intervals occur, (ii) 15/14 goes better (in melody) with 1/1 than 16/15 but (iii) most importantly, with 15/14, the notes between 1/1 and 1/2 (Natural North) are the mirror image of the notes between 3/2 and 3/1 (Natural South). 1/1 and 3/2 are both tonics. So you have two strong temperaments of equal strength (both contained in one system) and one is the inverse of the other (herein lies the symmetry). As I said, one temperament is better for ascending music and the other for descending music.

Melody? again, see my chapter on melody and scales.

Thanks for the elaborate and considered reply,

John.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 2 April 2010 00:24, jfos777 <jfos777@...> wrote:
> > Graham,
>
> > Since I first made contact with other members of the Just Intonation community (January of this year) I have learned a great deal more and my book will need some revising. At this point though, I still think my formulae have some merit and I also think that even if the formulae are wrong, my 12 key tuning system (NPT) is the best possible 12 key tuning system that has two different tonics of equal strength (1/1 and 3/2). One tonic works better for ascending music and the other for descending music, see chapter 7 of my book.
> >
> > If you haven't done so already I would urge you to read chapters 4, 6 and 7 of my book and let me know what you think of my reasoning.
>
> Chapter 4 is where you look at different formulas. It depends on your
> own ranking of consonances, which you haven't cross checked with
> anybody else. And once you have a ranking the formula's useless
> anyway. You also don't consider chords of more than 2 notes.
>
> Chapter 6 is more of the same.
>
> Chapter 7, then. Firstly you're assuming 12 notes which isn't the
> only way to do it.
>
> I see "prime number" on p.25. Maybe that's why you sent a
> clarification. But if you're going to exclude 4/7, why keep any 7s at
> all?
>
> You turn your fractions upside-down after a while. So the scale now
> contains 7/5 instead of 5/7. Anyway, that 7/5 would go with the 7/4.
> By taking the 7/4 out you leave 7/5 orphaned. I don't know what the
> argument is for 15/14 over 16/15, but they're only 8 cents apart.
>
> Oh, wait, we do have 4/7 now? (p.32) I can't actually find the
> listing of Natural Pan Temperament, which suggests it isn't presented
> clearly. Is it the same as Natural North?
>
> 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1
>
> Most of them are obvious. It would be nice to have 7/4, but it's
> nicer to have 9/5. Then the only question is why 15/14 and 7/5 are
> still in there because they don't work with anything else. They don't
> even work with each other because the 7 comes in the numerator of one
> and the denominator of the other. You should be able to replace them
> both and keep some symmetry.
>
> Anyway, this is a lot of talk about finding a JI scale with simple
> ratios. The result is reasonable but not special.
>
> You haven't considered temperament. If you've quantified mistuning,
> you might find something like schismatic temperament gives better
> results because it can be perfectly inverted.
>
> > My formula for melody is 2/x + 2/y and (If I'm right) any melodic interval with a value of 1.0 or higher is a Major melodic interval. So any interval 1/n will have a minimum value of 2.0, so yes, all 1/n intervals, in melody, should be strong.
>
> This suggests something wrong, because these intervals aren't used
> much in melody, and I note you don't have a radical melodic style that
> changes this.
>
>
> Graham
>

On 4 April 2010 21:06, jfos777 <jfos777@...> wrote:

> the premise I began with was that it seems that, in general,
> the smaller the numbers in the ratio, the stronger the interval
> sounds. From this I deduced that there must be a mathematical
> formula for consonance that uses the numbers in the ratio. I *have*
> crossed checked the formula with John Chalmers whom I wrote to
> in January. He said: "your consonance functions ... seem to
> reproduce qualitatively, at least, my subjective feelings about
> sensory consonance". He even went as far as to program my
> functions in BASIC to test them and seemed enthusiastic about them.

Sure, that's where everybody starts, and it basically works. If you
could present a table showing the different results for different
formulas we could talk about which are better. But it's really not
that important. It isn't the main issue in the mathematics of tuning.
To talk about it authoritatively you need to do a proper double blind
survey and nobody has the funding for that. The answers in the
literature depend a lot on which questions you ask.

> You said that once you have a ranking then the formula is useless
> anyway. It isn't, the formula is incorporated into my formula for
> identifying the strength values (and key notes) of chords.

Moving on to chords is something.

> You also said that I don't consider chords of more than 2 notes.
> I do, see chapter 10 "Harmony & Chords". I even have formulas for
> identifying the key chord of a group of chords and the "progression"
> value of two chords. I also have 5 classifications of chords which
> are based on my consonance formulas.

Sorry, yes, you do that. But you don't use this to inform your choice
of scale. So it only optimizes for intervals relative to the tonic,
not chords.

Your formula's a function of overtone numbers, which makes it
"otonal". That means it should give a lot of microtonal chords
precedence over a regular 5-limit minor triad. But your scale's a
combination of "up" and "down" harmonies. It isn't optimized for
otonal chords.

> I also have a chapter on Melody and Scales which has some new ideas.

It looks like that's dyadic (two notes at a time) again.

> Chapter 14 deals with stretch tuning and is one of the more
> intriguing chapters if you want to take a look.

I don't think it's intriguing at all. It talks about tuning
stretches, which are well known.

> As regards the 12 keys per octave (and not more or less than 12)
> I said before that, on a guitar, if extra frets are included then a lot of
> barre chords would be harder to play. For me, 12 is the golden
> mean between simplicity and complexity. Also, for someone making
> the transition from 12 key Equal Temperament to my 12 key
> Natural Pan Tuning, the transition should be relatively simple.

You have everything easy to play, but some chords out of tune. It's
easy to optimize the frets for the easiest chords. The point of barre
chords is that you can play them anywhere on the neck, which is always
a problem with an unequal fretting. To get all chords working, you're
looking at circular temperaments, which there's a lot of history of.

> You said:"if you're going to exclude 4/7, why keep any 7s at all?"
> Good question. I excluded 7/4 because it "goes with" very few other
> notes and 9/5 is more versatile and is almost as strong. I kept the
> 7/5 however because there is no other note between 4/3 and 3/2
> that is more versatile (except perhaps 10/7 but 7/5 goes better with
> 1/1 than 10/7).

Yes, that show's you're optimizing intervals relative to the tonic,
rather than chords. But, fair enough, I think that 7/5 gives you a
full 4:5:6:7:8:9 chord, but with the tonic as 5 instead of 4. So it
earns its keep as long as you don't care about modulation.

> Going up, 15/14 goes with: 3/2, 5/3, 15/8 and 2/1 and goes with a
> few other notes going down.

These would give "utonal" chords. The whole chord written as
frequency numbers doesn't look simple, so I think it'd score badly by
your formulas.

> I use 15/14 instead of 16/15 because (i) some of the more
> unusual and exotic (not found in ET) intervals occur,
> (ii) 15/14 goes better (in melody) with 1/1 than 16/15 but
> (iii) most importantly, with 15/14, the notes between 1/1 and 1/2
> (Natural North) are the mirror image of the notes between
> 3/2 and 3/1 (Natural South). 1/1 and 3/2 are both tonics.
<snip>

Sure, but why care about this "mirror image" business when chords
don't follow it, and you don't get any modulation? And does 15/14
really sound better than 16/15 in melody to you?

Graham

> the premise I began with was that it seems that, in general,
> the smaller the numbers in the ratio, the stronger the interval
> sounds.
One general question I have is what does "strong" mean in this context?
I've seen/heard 2/1 as the interval that sounds "most like" 1/1, pretty much replicating the mood and thus is very predictable/easy-to-listen-to...but it provides little tonal color.
Meanwhile intervals like 11/6 or even 12/11 seem to provide much more tonal color, yet "warps" the sense of mood a bit...but still comes across to me as stronger emotionally.

> "I also have 5 classifications of chords which are based on my consonance formulas."
Where can I find a copy of your book and/or at least a summary about these formulas?

>"I excluded 7/4 because it "goes with" very few other notes"
Really? Anything with a denominator of either 2,3,12, 4 (LCD = 12) <or> 2,4,8 (LCD = 8) works for me...and thus any chord that can be phrased at x/12 (IE 4:5:6, 6:7:8, 12:14:17) or x/8 (IE 4:5:6, 8:9:10, 2:3:4) becomes fair game.
The only thing I see that jumps at me with 7/5 (1.4) is it nears 17/12 (1.41666, within not much more than 10 cents of 7/5)...and I think you should be able to get more chords by using 17/12 as it aligns with the x/12 LCD.

>"I kept the 7/5 however"
Oddly enough I see only 2,5,10 (LCD = 10) here...numerically (at least on the surface) it seems less versatile.

>"It (the scale) isn't optimized for otonal chords."
Just thinking that keeping the LCD of the scale low (or at least all intervals within 8 cents or so of intervals that would form a low LCD) would help accommodate for this.