Recap

In the last thrilling episode, we looked at the arithmetic involved in stretching scales, focusing on what to do if we want to make the octave so-much wider (stretched) or so-much narrower (compressed) than the perfectly pure 2:1.

The pure 2:1 dyad — favored interval of tyrants, math freaks and psychiatrists

These are nice effects since instead of having a dead, motionless, static sounding octave, your octave shimmers and sounds more alive. It makes your ears perk up and take notice instead of saying ‘ho hum’.

Stretched octave dyad — sweet freedom, carry me home!

Whether you stretch or compress and how much is up to you — sometimes for a given tuning stretching the octave will make some intervals not be the best sonic fit for what you want to do, yet compressing it will work just right. For other tunings, the opposite might be true for you. Thus it’s good to try both and see which you like better for a given scale. Personally, I prefer the sound of octaves stretched as wide as 1212 to 1217 cents, or compressed as much as 1188 cents.

Stretching into other Intervals

As we have seen, modifying the octave is one way to approach scale stretching. But it need not be the only way. If you like, you can stretch all the intervals of a scale uniformly in order to create a stretching effect for any interval in the scale. Let’s look at some examples of how to do this.

Example #1 - meantone stretched to give pure fifths

So you really like the baroque sound of quarter comma meantone with its smooth thirds and sixths. But the fifths are notoriously flat — nearly 6 cents so! What to do? Why not relax the constraint on the octave being in tune and stretch out the entire scale evenly until the fifths are in tune? Will it be a screeching monstrosity? Or will it delight the ears with its subtle nuances? There’s only one way to find out and that’s to tune it up and have a listen! So let’s round up this scale!

How far should you stretch it? What will the effect be on the thirds and sixths? Will they be rendered horribly dissonant and useless after this bizarre transformation?

Here’s how to accomplish this fiendish act of unholy desecration:

Step 1 — rope ye up one a’ them thar a meantone scales

Here’s one a’ them meantones, specified as cents from tonic:

  0,76.049,193.1569,310.2647,386.3137,503.4216,579.4706,696.5784,772.6274,889.7353,1006.8432,1082.8921,1200.0

Step 2 — identify the interval of interest

  0,76.049,193.1569,310.2647,386.3137,503.4216,579.4706,696.5784,772.6274,889.7353,1006.8432,1082.8921,1200.0

Step 3 — decide on the new desired value of the interval

  just fifth = 3:2  =>  ratio2cents(3:2) = log(3/2)/log(2^(1/1200)) = 701.9550 cents

Step 4 — calculate the stretching factor (see Part I for details)

       701.9550
  sf = -------- = 1.007719
       696.5784

Step 5 — multiple all values in cents by the stretching factor (or if working with ratios, raise to the power of the stretching factor)

   0.0000 * 1.007719             =    0.0000,
   76.0490 * 1.007719            =   76.6360,
   193.1569 * 1.007719           =  194.6478,
    310.2647 * 1.007719          =  312.6595,
     386.3137 * 1.007719         =  389.2956,
      503.4216 * 1.007719        =  507.3074,
       579.4706 * 1.007719       =  583.9434,
        696.5784 * 1.007719      =  701.9550,
         772.6274 * 1.007719     =  778.5911,
          889.7353 * 1.007719    =  896.6029,
          1006.8432 * 1.007719   = 1014.6147,
           1082.8921 * 1.007719  = 1091.2506,
            1200.0000 * 1.007719 = 1209.2623

Step 6 — Tune that girl up and see how she sound!

Ear A — Fantastic! Absolutely Pure Just Fifths and both the thirds are almost perfectly just too! In fact, the minor third is even more in tune now. It’s just like floating on a puffy cloud! And the resulting deliciously stretched 1209 cent octave is just what the doctor ordered for these tired old ears! And with my special 19 tone per octave keyboard, I can map this thing out great! Gettin’ funky now... Yee haw! Lookie them microtones go!

Ear B — Ug! Horrible! Turn it down! It offends my senses to the very core of my being! What are you doing?? You ain’t foolin’ nobody! That is very nasty!!

Well there’s no fooling the ear, is there? Now you try it! It’s fun and very, very naughty...

Example #2 - stretching an ET

Stretching intervals in equal division scales is a special case that is easy to handle.

Let’s suppose that nefarious BUPOs (Beings of Unknown Point of Origin) have recompiled your brain’s audio cortex to only process variants of 14-tone equal-temperament (85.7143 cent equal-temperament).

14th root of 2:1 (shown compared to 21-limit intervals) — Even the Devil won’t use this tuning

While despondently dilly-dallying with this tuning, you realize that 16 steps of this scale span an interval pretty close to 11/5 (1365.004228 cents) — your very favorite interval in the whole world!

Unfortunately, the actual tuning of this interval in 14tET is (2/1)^(16/14), or 1371.4826 cents, which is a nerve-wracking 6.4243 cents sharp of the proper tuning, making any music written in this scale well-nigh unbearable for you to listen to! If only it could be compressed to the sweet perfection of the lovely 11:5 interval!

Don’t panic now! No need to become frozen with fear and unable to cope with life. There’s good news: When working with equal scales, such an operation is simple. Don’t freeze, just do the squeeze. Instead of basing the tuning on the step size of 1/14th of 2:1, divide your beloved 11/5 (1365.004228 cents) 16 ways to get a step size of 85.3128 cents. The result is that the octave at the 14th step will be compressed a wee bit to become 14 * 85.3128 cents = 1194.3787 cents.

16th root of 11:5 (compared to 21-limit) — Welcome relief for those recompiled by BUPOs

So to summarize, instead of tuning up 85.714cET, tune up 85.313cET instead. This will result in a pure 11:5 instead of a pure 2:1.

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Some 13tET Modes to Try Out

Yikes! So many numbers made me head go bam bam! Let’s drop the math talk for now and wrap up with a quick change of subject.

Here are some heptatonic (7-note) modes of 13tET to play with, along with associations that occured to me while playing them on a piano timbre. Try compressing the octave a wee bit with these and see if you like it better that way.

impressionist,triadic,robust,pretty

intricate puzzle,exciting,thief’s escape

mysterious,arabic,rich deep colors,persian rug

Have fun with all this and see you next time!

— X. J. Scott

[<= Go to Part I]