Beginner inquiries about changing the tonal center in Scala

Hi guys

I just poke my head out of the 12-ET, found a list of ratios and now I'm trying to figure out how to make sense of JI.

Let's say I want to play something really common, like the I V vi IV (C major, G major, A minor, F major). I take the 5-limit, 12-tone rational tuning ratios and create a custom tuning in Scala. We leave the reference pitch at A=440Hz for now. This means I ain't touching the Freq button for the Base Frequency dialog.

Just press Input, paste the ratios from the note after 1/1 until 2/1 inclusive, press OK and Export synth tuning (eventually Save).

Fairly intuitive I think. Now I can use the custom patch with my DAW. Everything is working perfect and I can hear the slight difference between this and 12-ET, for the first chord (C major). I am also testing it with a Lissajous Oscilloscope and it looks fantastic!

Now I need a new tuning/patch for the other 3 chords. And I do not know how to change the tonal center in Scala. What do you think?

B

"Bogdan" <baros_ilogic@...> wrote:

> Now I need a new tuning/patch for the other 3 chords. And
> I do not know how to change the tonal center in Scala.
> What do you think?

You don't *need* different tunings. Why do you want one?
The chords you need are all in this tuning:

C 1/1
D 9/8
E 5/4
F 4/3
G 3/2
A 5/3
B 15/8
C 2/1

You have an E minor as well.

Graham

Thanks for replying, Graham. I really appreciate it.

Not different tunings, yes. It is the same tuning, I only need to define G, A and then F as 1/1.

I may be of course, very wrong.

If after the C major I'll play a G major with this same configuration, I'll end up creating something else. Doesn't G have to become the new root?

My question is how to modulate, how to change the tonal centre from one key note to another?

Bogdan

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> You don't *need* different tunings. Why do you want one?
> The chords you need are all in this tuning:
>
> C 1/1
> D 9/8
> E 5/4
> F 4/3
> G 3/2
> A 5/3
> B 15/8
> C 2/1
>
> You have an E minor as well.
>
>
> Graham

--- In tuning@yahoogroups.com, "Bogdan" <baros_ilogic@...> wrote:
>
> Thanks for replying, Graham. I really appreciate it.
>
> Not different tunings, yes. It is the same tuning, I only need to define G, A and then F as 1/1.

Perhaps you're looking for the "key" command in scala? F, 4/3, is the fourth note of the below scale, so if you type "key 4" it will rearrange the scale to give all the notes as intervals from F rather than from C.

This is not necessary to play chords at all. It is only necessary to look at a scale differently inside Scala. After you put this scale on some instrument you don't need to "change the tonal center" at all; you just play the different notes.

I hope that explains it, and if not let's keep going until everything is clear.

> I may be of course, very wrong.
>
> If after the C major I'll play a G major with this same configuration, I'll end up creating something else. Doesn't G have to become the new root?
>
> My question is how to modulate, how to change the tonal centre from one key note to another?
>
> Bogdan
>
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > You don't *need* different tunings. Why do you want one?
> > The chords you need are all in this tuning:
> >
> > C 1/1
> > D 9/8
> > E 5/4
> > F 4/3
> > G 3/2
> > A 5/3
> > B 15/8
> > C 2/1
> >
> > You have an E minor as well.
> >
> >
> > Graham
>

<< There is an art to selecting the tuning root. If one tunes a standard synthesizer to a just scale in the key of A, and then plays a Bm chord, the intervals of this chord are too narrow, and either the B must be flattened or the D and F# must be sharpened. The choice is an aesthetic one. If the melodic priority is to keep a pure whole tone between an A note in an A chord followed by a B note in the B chord, then the tuning root for the second chord is B, and the other notes are sharpened. If, however, the melodic priority is to keep a pure fourth between a C# in the first chord and an F# in the second, then the tuning root of the second chord would be F#, and the B would be flattened. There are other harmonic considerations as well, and a correct system must allow for a manual selection of tuning root... >>

I hope the passage above explains what I'm trying to find out. If the choice is an aesthetic one, I would like to hear all the options. Kind of like creating with Rationale (http://www.badmuthahubbard.com/cgi-bin/rationaleinfo.py) but on another platform, using Scala generated content.

So one thing I don't understand is how "changing the root" works. When using the Key command in Scala, ratios change somehow and I do not know how. Where did the new ratios came from, what is the mathematics behind it (adding/multiplying or substraction/inversion/division between some ratios and some other ratios)?

And how are the frequencies calculated? For example, in the scale below, we start with 256Hz for C=1/1 and so we get the other tones. This is because 256 is simply 1, taken 8 octaves higher. In this case, A is 432Hz. If we set A to 440, C and all other notes will change frequency values accordingly.

But how are frequencies calculated when we "change the tonal center", even though I don't have to? It may be useless but I have to hear the difference between a F major with C=1/1 for example, and an F major with F=1/1.

Will there be any difference? And how is F's frequency calculated when it becomes 1/1, in rapport with what?

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, "Bogdan" <baros_ilogic@> wrote:
> >
> > Thanks for replying, Graham. I really appreciate it.
> >
> > Not different tunings, yes. It is the same tuning, I only need to define G, A and then F as 1/1.
>
> Perhaps you're looking for the "key" command in scala? F, 4/3, is the fourth note of the below scale, so if you type "key 4" it will rearrange the scale to give all the notes as intervals from F rather than from C.
>
> This is not necessary to play chords at all. It is only necessary to look at a scale differently inside Scala. After you put this scale on some instrument you don't need to "change the tonal center" at all; you just play the different notes.
>
> I hope that explains it, and if not let's keep going until everything is clear.
>
> > I may be of course, very wrong.
> >
> > If after the C major I'll play a G major with this same configuration, I'll end up creating something else. Doesn't G have to become the new root?
> >
> > My question is how to modulate, how to change the tonal centre from one key note to another?
> >
> > Bogdan
> >
> >
> > --- In tuning@...m, Graham Breed <gbreed@> wrote:
> > >
> > > You don't *need* different tunings. Why do you want one?
> > > The chords you need are all in this tuning:
> > >
> > > C 1/1
> > > D 9/8
> > > E 5/4
> > > F 4/3
> > > G 3/2
> > > A 5/3
> > > B 15/8
> > > C 2/1
> > >
> > > You have an E minor as well.
> > >
> > >
> > > Graham
> >
>

The Key command divides every scale member by the value of the scale degree that you tell it. So, if you pick the scale degree that is 4/3, Key divides everything by 4/3, does the octave equivalence calcuations, and resorts the scale.

Frequencies are calculated quite easily. If A=440 Hz, then a 3/2 E is 440Hz*3/2=660Hz. It's just a straight multiplication of the root to derive all the notes of the scale based on the integers provided.

Changing the tonal center does not change how the frequencies are calculated, except that the root changes. Given the A=440Hz example, if I wanted to instead form an E major scale with the same intervallic relationships as I had before in A, I just take the exact same set of ratios and multiple them by 660Hz.

The only difference between your F major triad at C=1/1 and your F major triad at F=1/1 is the pitch range; the triad itself will have the same relationships. However, if you mean the scales instead of the triads, the F major scale (standard 5-limit) will be slightly intervalically different than the C major scale unless you recalculate all of the scale degrees at the new pitch center. Specifically, when going from the F major scale to the C major scale, two pitches will be different: the Bb changing to B natural (and also adding a syntonic comma) and the D raises by a syntonic comma as well.

Does that answer your questions?

Andrew

--- In tuning@yahoogroups.com, "Bogdan" <baros_ilogic@...> wrote:
> So one thing I don't understand is how "changing the root" works. When using the Key command in Scala, ratios change somehow and I do not know how. Where did the new ratios came from, what is the mathematics behind it (adding/multiplying or substraction/inversion/division between some ratios and some other ratios)?
>
> And how are the frequencies calculated? For example, in the scale below, we start with 256Hz for C=1/1 and so we get the other tones. This is because 256 is simply 1, taken 8 octaves higher. In this case, A is 432Hz. If we set A to 440, C and all other notes will change frequency values accordingly.
>
> But how are frequencies calculated when we "change the tonal center", even though I don't have to? It may be useless but I have to hear the difference between a F major with C=1/1 for example, and an F major with F=1/1.
>
> Will there be any difference? And how is F's frequency calculated when it becomes 1/1, in rapport with what?
>

> The Key command divides every scale member by the value of the scale degree that you tell it. So, if you pick the scale degree that is 4/3, Key divides everything by 4/3, does the octave equivalence calcuations, and resorts the scale.
>

OK, got it. I take the scale (5-limit, 12-tone rational) and change the Key to the 4/3 scale degree. Now F=1/1 and our scale, first column, becomes the second:

C=1/1 F=1/1

1/1 1/1
16/15 135/128
9/8 9/8
6/5 6/5
5/4 5/4
4/3 27/20
45/32 45/32
3/2 3/2
8/5 8/5
5/3 27/16
9/5 9/5
15/8 15/8
2/1 2/1

This looks kind of strange, as it doesn't quite match the explanation but anyway, let's test it. Playing F together with C. This should be a perfect fifth but it is not. I am confused. I was expecting something more like the 3rd column below:

C=1/1 F=1/1 -> F=1/1, resorted to C

1/1 3/2 -> 1/1
16/15 8/5 -> ?/?
9/8 5/3 -> ?/?
6/5 9/5 -> ?/?
5/4 15/8 -> ?/?
4/3 1/1 -> ?/?
45/32 16/15 -> ?/?
3/2 9/8 -> ?/?
8/5 6/5 -> ?/?
5/3 5/4 -> ?/?
9/5 4/3 -> ?/?
15/8 45/32 -> ?/?

>
> Changing the tonal center does not change how the frequencies are calculated, except that the root changes. Given the A=440Hz example, if I wanted to instead form an E major scale with the same intervallic relationships as I had before in A, I just take the exact same set of ratios and multiple them by 660Hz.
>

Makes perfect sense. Here is how I understood the changing of tonal center (changing root), from C to F:

1/1 264,00 -> 3/2 264,00 C
16/15 281,60 -> 8/5 281,60 C#/Db
9/8 297,00 -> 5/3 293,33 D
6/5 316,80 -> 9/5 316,80 D#/Eb
5/4 330,00 -> 15/8 330,00 E
4/3 352,00 -> 1/1 352,00 F
45/32 371,25 -> 16/15 375,47 F#
3/2 396,00 -> 9/8 396,00 G
8/5 422,40 -> 6/5 422,40 G#/Ab
5/3 440,00 -> 5/4 440,00 A
9/5 475,20 -> 4/3 469,33 A#/Bb
15/8 495,00 -> 45/32 495,00 B

Now what happens if the next change will be to D, right in the middle of the performance? Which value will D have as the new 1/1: 297 or 293,33 Hz?

>when going from the F major scale to the C major scale, two pitches will be different: the Bb changing to B natural (and also adding a syntonic comma) and the D raises by a syntonic comma as well.
>

This can be seen in the Hz values in the columns above! Alright, I start to get it now, even though this is the first time I hear about "syntonic comma".

From what I read I understood that equal temperament grew out of the need of mediaeval musicians to be able to play in any key. This implies that when using rational tunings, one cannot freely modulate. So the need for changing the pitch center in real time led to 12-ET. If this is not true, I don't know what to believe anymore.

The main issue remains using Scala to change the tonal center.

--- In tuning@yahoogroups.com, "Bogdan" <baros_ilogic@...> wrote:
> OK, got it. I take the scale (5-limit, 12-tone rational) and change the Key to the 4/3 scale degree. Now F=1/1 and our scale, first column, becomes the second:
>
> C=1/1 F=1/1
>
> 1/1 1/1
> 16/15 135/128
> 9/8 9/8
> 6/5 6/5
> 5/4 5/4
> 4/3 27/20
> 45/32 45/32
> 3/2 3/2
> 8/5 8/5
> 5/3 27/16
> 9/5 9/5
> 15/8 15/8
> 2/1 2/1
>
> This looks kind of strange, as it doesn't quite match the explanation but anyway, let's test it. Playing F together with C. This should be a perfect fifth but it is not. I am confused. I was expecting something more like the 3rd column below:

In the second column you say that 1/1 is F. (You have "F=1/1" at the top.) So the eighth note of that column should be C, and it is indeed 3/2 relative to F. I don't see what the problem is.

> C=1/1 F=1/1 -> F=1/1, resorted to C
>
> 1/1 3/2 -> 1/1
> 16/15 8/5 -> ?/?
> 9/8 5/3 -> ?/?
> 6/5 9/5 -> ?/?
> 5/4 15/8 -> ?/?
> 4/3 1/1 -> ?/?
> 45/32 16/15 -> ?/?
> 3/2 9/8 -> ?/?
> 8/5 6/5 -> ?/?
> 5/3 5/4 -> ?/?
> 9/5 4/3 -> ?/?
> 15/8 45/32 -> ?/?

Try doing "key 7" instead of "key 5" and you might get what you're looking for. If it works, I can explain why, but it might take a while since you're thinking of everything backwards from Scala.

> Makes perfect sense. Here is how I understood the changing of tonal center (changing root), from C to F:
>
> 1/1 264,00 -> 3/2 264,00 C
> 16/15 281,60 -> 8/5 281,60 C#/Db
> 9/8 297,00 -> 5/3 293,33 D
> 6/5 316,80 -> 9/5 316,80 D#/Eb
> 5/4 330,00 -> 15/8 330,00 E
> 4/3 352,00 -> 1/1 352,00 F
> 45/32 371,25 -> 16/15 375,47 F#
> 3/2 396,00 -> 9/8 396,00 G
> 8/5 422,40 -> 6/5 422,40 G#/Ab
> 5/3 440,00 -> 5/4 440,00 A
> 9/5 475,20 -> 4/3 469,33 A#/Bb
> 15/8 495,00 -> 45/32 495,00 B
>
> Now what happens if the next change will be to D, right in the middle of the performance? Which value will D have as the new 1/1: 297 or 293,33 Hz?

The difference is 81/80, the syntonic comma. If you want to write music in just intonation that uses "normal" intervals and chords, you're going to get VERY familiar with it. It's no exaggeration to say it's the single most important comma in all music.

From your comments as a whole it seems to me like you're looking for something that automatically adjusts the intonation of notes in real time based on the music you're playing. Is that right? Scala does not do this (as far as I know...); it only lets you create static scales. For example you can make a scale with both D (9/8 above C) and D- (10/9 above C), but you have to map them to different keys on a MIDI controller and actually choose which one to use. It doesn't do it automatically for you.

> This can be seen in the Hz values in the columns above! Alright, I start to get it now, even though this is the first time I hear about "syntonic comma".
>
> From what I read I understood that equal temperament grew out of the need of mediaeval musicians to be able to play in any key. This implies that when using rational tunings, one cannot freely modulate. So the need for changing the pitch center in real time led to 12-ET. If this is not true, I don't know what to believe anymore.

It seems like you have the basic idea right, but you actually *can* freely modulate in JI. You just need an infinite number of different pitch classes. And sometimes the pitch continues to drift in a certain direction unexpectedly. Ask us what a "comma pump" is.

> The main issue remains using Scala to change the tonal center.

As I said, I think you're just conceiving of it backwards from Scala. So do "key 7" instead of "key 5" and see how that turns out. I can explain but it will be confusing.

Keenan

It looks like the column formatting got messed up...

> In the second column you say that 1/1 is F. (You have "F=1/1" at the top.) So the eighth note of that column should be C, and it is indeed 3/2 relative to F. I don't see what the problem is.

So you are saying that if I play it with a regular keyboard, when pressing C I will get an F? The problem was me expecting something else, you figured it out below.

> Try doing "key 7" instead of "key 5" and you might get what you're looking for. If it works, I can explain why, but it might take a while since you're thinking of everything backwards from Scala.

YES. This is it, this is what I wanted from the beginning. A "magic" algorithm for changing root while leaving keyboard mapping intact.

> > Now what happens if the next change will be to D, right in the middle of the performance? Which value will D have as the new 1/1: 297 or 293,33 Hz?
>
> The difference is 81/80, the syntonic comma. If you want to write music in just intonation that uses "normal" intervals and chords, you're going to get VERY familiar with it. It's no exaggeration to say it's the single most important comma in all music.

It is not the first time I see the ratio 81/80. I can search about this myself, I'm only curious what happens when root changes occur mid-stream: is the new value taken as the 1/1? If this is true, after changing keys a few times one might notice that frequency for a given note has changed, and by the end of the song the reference pitch will be another. Totally cool.

> From your comments as a whole it seems to me like you're looking for something that automatically adjusts the intonation of notes in real time based on the music you're playing. Is that right?

The software you're talking about exists. What I am doing is using the Scala files as custom synth tuning patches in my DAW, which can be automated.

> Ask us what a "comma pump" is.

I just found "comma pump" on xenharmonic.wikispaces.com
Thank you (all of you) for having the patience to explaining these things to me. I have searched extensively and what I'm asking you are things I couldn't find anywhere. I really appreciate it.

> As I said, I think you're just conceiving of it backwards from Scala. So do "key 7" instead of "key 5" and see how that turns out. I can explain but it will be confusing.

I just need Scala for making 12 files with the same tuning, but with different roots. For example, for changing root to F while keeping keyboard layout, "key 7" instead of "key 5" is what I was looking for.

I would love to understand this in order to do the same with the other 10 notes.

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> > From what I read I understood that equal temperament grew out of the need of mediaeval musicians to be able to play in any key. This implies that when using rational tunings, one cannot freely modulate. So the need for changing the pitch center in real time led to 12-ET. If this is not true, I don't know what to believe anymore.
>
> It seems like you have the basic idea right, but you actually *can* freely modulate in JI. You just need an infinite number of different pitch classes. And sometimes the pitch continues to drift in a certain direction unexpectedly. Ask us what a "comma pump" is.
>

Yes, but I prefer to think of how 12-ET came about as a direct result of - or at the same time as - keyboard development. Composers wanted to be able to play every key implied on what the keyboard eventually evolved to, with 12 pitches per octave. And more specifically, it is perfectly permissable to modulate even with a non-tempered keyboard as long as you're willing to have different keys have different intervallic relationships. Baroque, classical, etc. composers wanted the 12 keyboard keys to sound the same intervals. That useful if, for example, you are playing an accompaniment for a singer (much, much more common in pre-record times than now) who cannot handle the pitch range of one given arrangement, and you need to transpose the arrangement to a different key, yet keep the arrangement with the same general flavor.

I'll try to refrase my initial question, having in mind what I learned since I first asked.

If A=440 Hz, and the distance from C (1/1) to A is 5/3, then the value of C in Hz will be equal with the difference between the two intervals. To subtract 5/3 from 1/1, we divide (cross multiply) the numbers: 1/1 ÷ 5/3 = 3/5. The process of “subtraction” is in fact one of division because of the logarithmic nature of the scale of pitch. So 440 * 3/5 = 264 Hz - the value of C. Now we know 1/1 and we can easy calculate the rest of our scale. G will be 264 * 3/2 = 396 Hz.

Now let’s make G=1/1. I take the first column (C=1/1) in Scala and use the Key command. Instead of degree 7 I will use degree 5. I do this so I can represent the G=1/1 scale in a C=1/1 scale with intervals that reflect this change. I am still searching for my words here, so let me try to put it in another way.

We now make G=1/1, while keeping the keyboard layout. First step is to change the root to G. This is column 2 below. Because the first note cannot be something different than 1/1, we resort the scale. This is column 3, and it’s done by substracting the new head of the list 4/3 from all other members. Finally, we do the octave equivalence calculations.

To get to this result directly from column 1 to column 3, we change the Key degree in Scala not by our Current Degree (CD) which is 7, but with 12-CD. 12-7=5. I’m not sure why I’m using this algorithm, Keenan pointed me in doing so.

(Maybe because The Key command divides every scale member by the value of the scale degree that you tell it. So, if you pick the scale degree that is 3/2, Key divides everything by 3/2, does the octave equivalence calcuations, and resorts the scale. But we see in the second column that the first ratio is 4/3, and in order to make it 1/1 we have to substract 4/3 from it and from the others. It turns out that 4/3 is scale degree #5, so maybe that’s why we use 5 instead of 7.)

C=1/1 G=1/1 G=1/1 expressed as C=1/1

1/1 4/3 1/1
16/15 45/32 135/128
9/8 3/2 9/8
6/5 8/5 6/5
5/4 5/3 5/4
4/3 9/5 27/20
45/32 15/8 45/32
3/2 1/1 3/2
8/5 16/15 8/5
5/3 9/8 27/16
9/5 6/5 9/5
15/8 5/4 15/8

What happens with the base frequency when we make G=1/1? Which of the 2 options we choose:

a) We take the G value from the list with C=1/1, 396 Hz and calculate the rest of the notes (meaning that C=264 Hz is our reference), or

b) We take A=440 Hz as reference and calculate G, knowing that if G=1/1 then A=9/8 and using the same principle as described in the second paragraph, G will be 440 * 8/9 = 391,11 Hz and we can calculate the remaining notes using column 2 above. Conversely, we can use directly column 3 so if A=27/16 from C, then C will be 440 * 16/27 = 260,74 and from here we can find the other values.

I understood that if we change key while playing, we might end up by the end of the song with another reference pitch (due to 81/80, the syntonic comma).

So my question, and the reason for writing all this, is:

If I want to play music in the key of G, in JI, ussing the 3rd column of ratios, which of the options above I’m using, a) or b) ? If my song doesn’t change key and I start directly from G, which is the refference pitch? Is it A=440 â€" the b) case, or is it A=445,5 â€" the a) case ?