Motivation for Scale Construction

Recently I finished a recording for the first piano piece I was commission to perform.  It was called Persian Glances by Jesse Maw.  Hear it here (https://www.patreon.com/posts/persian-glances-17963263).  And check out some of Jesse’s other music too (jessemaw.com).

The piece uses a scale generated by 3/2s and 7/4 (octave reduced 3rd and 7th harmonics).  On the Patreon page I give some examples of how to derive the scale, but here I will give a quick example of how this could create a scale before I go onto the main topic.

(( for reference:

“Persian Glances” JI tuning-

1:1 F @ 344 hz

567/512
9/8
5103/4096
81/64
21/16
729/512
189/128
3/2
1701/1024
27/16
7/4
15309/8192
243/128
63/32
2/1

))

Let’s start with the unison 1/1, this can be any note, as long as it is some note that we will use as the reference point for the scale.

Now let’s decide to make the scale repeat at the octave, so now we have pitch set 1/1, 2/1

Since, it is an octave repeating scale, it is convenient to express all notes between 1/1 and 2/1 (between one and two).

Let’s decide (as is the case for the Persian Glances scale) that we want to focus on combinations of the 3rd and 7th harmonics.

So lets triple our initial frequency and we get 3/1 (or 3).  Now, to put it within 1/1 and 2/1, we need to put some power of 2 on the bottom.  In this case it is just 2, so we have 3/2, which is 1.5 and is between one and two.  So now we have 1/1, 3/2, 2/1 (the root, P5 and octave).

Lets do the same for the 7th harmonic: multiply the initial frequency of vibration by 7, and divide by 4 (2 to the power of 2) to put it within the octave and we get 7/4.  Now we have: 1/1 3/2 7/4 2/1

Now lets combine these 3/2 * 7/4 = 3*7/2*4 = 21/8 … octave reduce by dividing another 2 we get 21/16.

We could combine two 7/4s to get 49/16 .. octave reduced to 49/32, but we see this scale doesn’t have it.  Instead we combine two 3/2s to get 9/4 = 9/8 which is in the scale, and then add a 7/4 to get 9*7/8*4 = 63/32 .. so on we can repeat this process.

Anywho, the central idea I wanted to discuss was that of motivation for scale design.  Persian Glances was both a good starting point for an example here, and also part of a conversation I had with Faras Almeer that inspired me to make this post.  We had been speaking recently about scale design and he had given me his specs for possible tunings for the Arabic Maqam of Rast and Hijaz.  He asked about the motivation for the Persian Glances tuning, and Jesse told me it was to explore the comma of 64/63 which is the interval between 63/32 (9/8 and 7/4 combined) and the octave, and this applied various times throughout.

Let’s show how this works:  If we apply another 3/2 to 9/8, following the Pythagorean method of scale construction, we get 3*9/2*8 = 27/16.  Let’s add the 7/4 again — 27/16 * 7/4 = 27*7/16*4 = 189/64 = 189/128.  This interval on the scale list is one step below 3/2.  This makes sense as 27/16 is a kind of major sixth and 7/4 is a kind of minor seventh, a flat one: the harmonic 7th.  So it is intuitive, as based on 12tone music that the 7th of the major 6th will be the 5th, and so a flat one would be flat of that.  And in just intonation, with the 7th harmonic, this does end up being true.  And lets test if this comma is there.  If it is then 189/128 * 64/63 should = 3/2.   We can see that 63*3 = 189 and 64*2= 128 and so this comma of 64/63 appears between 189/128 and 3/2 or between three perfect fifths + a harmonic 7th and a perfect 5th.  This pattern is repeated on each of the pythagorean fifths that are generated, essentially creating a 7/4 from each of these 5ths and then the respective 64/63 commas between each new part of the pattern.

Another perspective of how this works is to focus on the 3/2 and the 64/63 comma.  This was the method of interpretation of Faras Almeer.  You can take the octave and subtract the 64/63 comma and generate the tone of 63/32.  Then you can take the 3/2 and subtract the 64/63 to generate the 183/124 and so on.  This is an example of commas and generators being a different way to fundamentally express the same thing.  This is of music advantage to the composer and performer who can use these functions to their musical and creative advantage.

I have spoken with Faras Almeer a number of times recently about representations of maqam through just intonation.  What follows is not meant to be an accurate portrayal of maqam, not as historically or culturally accurate, but a framework in which some of the methods of scale construction may be analysed.

In Rast maqam, the third is often described as a neutral third (in just intonation, it is more simple to write this as 11/9 .. 347¢), or as sharp from a minor third, or flat from a major third.  These last two definitions both can describe a neutral third but also create an amount of flexibility or ambiguity.  In 12tone music, the minor third is 300¢ and the major third is 400¢, in 5-limit just intonation (ratios using up to the 5th harmonic), the minor third is 6/5 which is about 314¢ and the major third is 5/4 which is about 384¢ — sharp of the minor third or flat of the major third from the reference point of 12tone.

In addition, we can consider a pythagorean method or deriving the major or minor third: 32/27 (three 4/3s) for the minor third at 294¢ or the 81/64 (four 3/2s) for the major third at 404¢.  These don’t quite work for this definition.

Another approach is to approach this by melodic combinations of tones.  Two 3/2s will give us a 9/8 as a whole tone, and two of these will give us 81/64, which is expected as we just derived this interval as four of the 3/2s.  The minor whole tone of 10/9 combined with 9/8 will give us 5/4 which we already considered.  What about two 10/9s?  This gives us 100/81 at 365¢.  This could work.

What about a smaller representation of a tone?  The neutral third or neutral tone is often spoke of in Rast and the neutral sound often has correspondence with the 11th harmonic.  We have thus considered the 11/9 neutral third of 347¢,  what about the 11/10 neutral second (at 165¢)?  Two of these give us 121/100 which is 330¢, which could work.  With 11/10 and 10/9 we get 110/90 = 11/9, hey the neutral third (good to note that this is a way this interval can be generated).  What about 11/10 and 9/8, we get 99/80 which is 376¢.

We can also use another method of finding a note between a note.  Say between 6/5 and 5/4 we can calculate this mean as 6+5 / 5+ 4 = 11/9 which we already know to be between these.  We can continue the process with 6/5 and 11/9 as 6+11 / 5+9 = 17/14 which is 336¢, and another option.  We can also do this with 11/9 and 5/4 as 11+5 / 9+4 = 16/13 as 356¢, another option.

If you can imagine this many senses in which you can generate types of 3rds, you can imagine the spectrum of possibilities that exist in total.

For quick ratio to cent conversion, this place is great:

http://robertinventor.com/software/tunesmithy/help/cents_and_ratios.htm

Hope to write more on this soon.

Stay tuned ❤