Organ Study #1 in 26 Tone Equal Temperament

Here is another experiment in microtonality. This tuning features 26 equally spaced tones to the octave instead of the more traditional 12. This is not a very common tuning, even among experimental microtonalists. Its intervals tend to be a little more dissonant than we are used to. It is, however, one of my favorite tunings. (The term temperament is a little more accurate in this case, but less known.)

This study is fairly simple. I have avoided complex harmonies and modulations. It uses a simple seven note diatonic scale. You could easily "translate" this piece into a similar piece tuned to twelve tone equal temperament. (The common scale in use today.) I don't find the translation to be effective. This tuning has its own peculiarities and suggests different ways of organising the notes. My ears find this tuning to be somewhat darker and more mysterious. I have tried to take advantage of this by using the majestic sound of a pipe organ.

I welcome your comments. I neither expect nor hope that my microtonal music will produce any sort of consistent response in my listeners. The varied responses of others are as interesting to me as my own and help me to better understand the effects of harmony, dissonance and novel musical structures.

As the title suggests, I hope that there will be at least one more organ study in this tuning that explores some of its more exotic possibilities. This tuning has enormous untapped potential. Experimentation in it is rare and it doesn't have any established tradition or set of rules to follow. There are ways of handling this tuning that don't seem to have any strong parallels to more conventional tunings. Stay tuned and you may see what I mean.

I hope you will find Organ Study #1 to be enjoyable or at least interesting.


Michelle King said...

I liked the piece. It had a feel that grabbed my attention. By no means am I someone who is an expert on music, however, I can tell when a chord "rings", when overtones are produced or when there is dissonance (my body literally tenses when something is off...my friends think it's hilarious while I think it's a curse sometimes). I look forward to seeing/hearing future entries. If ever you want the opinion of someone who has an appreciation of music, I am around. I don't compose music, I just sing it sometimes :)

Anonymous said...

The harmonies are pretty cool. There are some neat resonance like effects happening.

Obviously 26 has some nice beating going on in there, and I think that's what's doing it. You can hear this better with a completely harmonic timbre like the organ you have here, and so it's more prominent.

dave said...

Interesting idea. I listened to the 15-toned piece and liked it. I'm curious how the factors of the numbers affect your ability to compose 'reasonable' music. Since 26 = 2x13, I'd think it not too easy. Obviously 12 = 2x2x3, but what about 2x3x5 = 30?

Daniel Thompson said...

You ask an excellent question. I wish I had a good answer. I may write a post on this someday. I need to do some more thinking and research first.

There are a few things that I can say on the subject. The divisibility of the number of an equal temperament is very important to its structure. The relationship between them, however, is far from obvious. 12 is divisible by the primes 2 and 3 and serves as a reasonable approximation for certain traditional tunings that use purely tuned intervals. However, 19, 31 and 53 tone equal temperaments are considered by many to be even better approximations, yet these are all prime numbers!

30 tone equal temperament has three prime factors, yet it is a fairly exotic tuning.

13 is a peculiar division of the octave, but I think this probably has more to do with the fact that it is one more than 12, which is a familiar reference, than it does with the fact that it is prime.

Temperaments with non prime numbers have symmetries that can be used creatively, but aren't always desired.

dave said...

I checked equitempered scales with 12-107 notes to see how close they can fit a 3:4:5 ratio. I'm not entirely sure I'm doing it right--I took powers of 2^(1/n), where n is the number of notes in the scale, and looked for 3.0 and 5.0.

Approx 3 = 2^{int[(n*ln 3)/ln 2 +.5]/n} for example. Replace 3 with 5 for the other one.

Then I subracted from 3, 5 resp. to get the error, which I took absolute value of and totaled.

The best one in my range is n = 87, followed by n=53.

For n=53, you get a 3:4:5 chord with any note plus the one up 22 semi-tones plus the one up 17 from that. Its total error is about 1/20 of the n=12 chord.

By the way, n=31 is the next best value smaller than 53. The best value smaller than 31 is 22 and then 19, and then 12.

If you're interested in the details, I can send the spreadsheet.

dave said...

Sorry--I just had another idea. The equitempered scales I've used all assume you want a perfect octave, so they use roots of 1/2. But you could use roots of 1/3 or 1/5 or whatever. The octaves would be approximate, but there might be some weird creative possibilities there. Is this something people have tried?

Daniel Thompson said...

You have been doing a lot of calculating! I don't completely understand your method, but your results are consistent with other techniques. You have isolated some important temperaments that a lot of microtonalists are interested in. I would be grateful if you could send me the spreadsheet. (My email address is listed on my profile page.)

I would like to mention a different method that is more traditionally used. An octave is commonly divided into 1200 cents (100 cents to a semitone). This is a logarithmic scale, so it provides an easy way to evaluate temperaments. You can just divide 1200 by the number of tones to find out the size of the smallest interval. When you calculate the other intervals, you can compare them to known values for purely tuned intervals.

For example, a purely tuned major third is represented by the ratio 5/4. This is about 386 cents. The twelve tone approximation is 400 cents. Some people consider this 14 cent difference to be trivial. Others consider it to be a sort of crime against nature. I view this 400 cent interval as a substantial deviation, but also as a separate interval with its own valuable qualities.

I am wondering if you play or write any music. Let me know if you need any help in turning your thoughts into sound!

Daniel Thompson said...

Concerning your other comment on octaves:

You are describing what is known as non octave music. It is an exciting field, although still somewhat obscure.

Many instruments such as a piano have octaves that are slightly stretched. This is called inharmonicity. Piano tuners have to allow for this.

Indonesian music uses octaves that are stretched even more. It can contribute to a lovely shimmering sound.

Of course, there are other more exotic possibilities. A pioneer in the field is X.J. Scott. He already left a comment on this post, so you can just click on his name at the top of his comment to be taken to his website.

I just asked him about a certain non octave scale that I have been working on. He said that it was virtually unexplored. This is an area of microtonal music where you can really do some unique things. Its pretty exciting.

High Power Rocketry said...

Just checking back Micro T

Nicore said...


Anonymous said...

Es exelente ke más personas en el mundo estén encontrando el verdadero élixir de la música. Ke personas concientes de ke la música va decayendo cada día más es precisamente el de no kerer salirse de akella escala de estética supérflua al oído (la típica E12-TE), desaprovechando al mismo tiempo akellas frecuencias ke también son válidas en el ámbito de la música.
Me gustó mucho tu composición en E26-TE (SPANISH: Escala de 26 Tonos Ecualizados; ENGLISH: 26-Tone Equidistant Scale or 26-TES).
Al igual ke tú, también estoy desarrollando mi propia teoría musical, creando piezas musicales en torno a E38-TE (38-TES), lo cual es muy fabuloso y como es más amplio el espectro musical, pues habrá muchísimas melodías ke componer, porke es un tema tan rico y de posibilidades innumerables, la música es VIDA.

Thank You

Jon said...

When I read 26 tones, I quickly calculated how sharp the fifth is, and thought, "yuck!" but hearing it, it's really beautiful... Great work.

Anonymous said...

An exceptional étude here!

26-tone is a very interesting temperament. While normal harmonic thirds are non-existant and the fifths are by most standards quite flat, the seventh and eleventh harmonics are perfect. This lends itself, as your piece shows, to some very intriguing effects, including some truly whacked out chords and melodies.

Very, very cool stuff you're doing here.

Anonymous said...

What do you think about of the 23-EDO?
23-EDO contains an excellent minor third and major Sixth, but all the rest is distorted.

Daniel Thompson said...

23 edo is a pretty interesting tuning. It can have a nice exotic flavour at times. I haven't done much with it. One of these days I'll probably finish a composition in it.

Anonymous said...

Very Interesting!
Thank You!

Anonymous said...

Great post, I am almost 100% in agreement with you