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Two-octave scales catalog Ia.

The complete collection of notes, intervals and note groups of the 24-tone system

Introduction

The logic behind the two-octave scales

Issues with demonstrating the scales

Related sections

The Western music system defines intervals, out of which the most harmonious is, not taking into account the perfect unison, the octave. Each culture has considered this interval significant; some of them have even made it a reference in their music system, e.g. see the Pythagorean tuning.

The word scale has Latin roots; it bears different meanings. One of its meanings is a sequence of musical notes arranged pitchwise. Other than this, scales have a pretty obvious characteristic; these pitchwise-ordered notes have a range of one octave, always one octave. In other words: a scale structure within an octave is always constant, for example what we call a major scale is always a sequence of notes starting from the tonic lasting to its octave.

I would raise a question that no one has raised until now; What if the structure of a scale ranges to two octaves instead of just one? Then the sequence of the intervals defined by the scale's notes would start all over again after two octaves.

Let's see how it works!

Engineers, computer programmers and others thinking globally might be aware that there are many things to consider when planning a project, which in our case is the following:

• Why are you the first to come up with this, and not one of those so-called experimental jazz musicians? The answer is in the question: classical music theory inherently limits the musical mind from discovering new perspectives. No wonder musicians rigorously sticking to classical music rules cannot use their creativity to find new ways to music.

• How many notes and intervals should be within that two octaves?

• Do you want to make certain examples, or want to know the exact number of variations to these scales? The latter, with certain restrictions.

• All note-combinations that range to one octave, including all scales that define one octave, must be ruled out. In other words, all the 4096 scale variations of the binary scale catalog (all scales that ranges under one octave, for example 101000000000, including all scales with the first note and the last note being the first/last digit, i.e. 1xxxxxxxxxx1. It is not difficult to find out that since the two-octave note-combination, i.e. 24 notes, automatically covers the twelve-note scale catalog's notes, thus all 4096 scale variations are ruled out.

• The total number of variations of the two-octave scale catalog is, according to the formula 224, 16777216, out of which subtracted the 4096 twelve-note scale variations (212) will give a final result of 16773120. It is still a huge number of scale variations.

• Out of this great number of variations, we are only interested in the ones that range the whole two octaves, meaning the first and the last note of this two-octave range must be in the note-combination. So the two notes at the start and at the end of the two-octave range would literally form a frame around the two-octaves, looking like this:

100000000000000000000001

The scale catalog starts with a scale consisting of 2 notes, in standard position. The next one would be the scale catalog consisting of 3 notes, the scales of which would look like this:

110000000000000000000001

101000000000000000000001

100100000000000000000001

100010000000000000000001

100001000000000000000001

100000100000000000000001

100000010000000000000001

100000001000000000000001

100000000100000000000001

100000000010000000000001

100000000001000000000001

100000000000100000000001

100000000000010000000001

100000000000001000000001

100000000000000100000001

100000000000000010000001

100000000000000001000001

100000000000000000100001

100000000000000000010001

100000000000000000001001

100000000000000000000101

100000000000000000000011

Even though these structures could barely be called real scales, the concept behind it is now clear. Thus, the two notes forming the "frame" of the scale constantly remain there...

100000000000000000000001

...it gives us a total number of 222 = 4194304 variations. (Explanation: there are 22 digits between the two notes on the sides).

Plotting all the scale catalogs was a hard work. To process this huge amount of data, Tamás Tóth,... ...the creator of OSIRE, helped me with some algorithmic calculations. Below you can see the result, one being an .exe application listing up all 4194304 variations, and the other a zipped .txt file incorporating all the scale catalogs.

72 KB - .exe

20,2 MB - .zip

NB: the uncompressed size of the .txt file exceeds 140 Mbytes, which may result in slow decompressing and loading.

Another issue that may arise: since HTML-based web pages, like this one, may face difficulties with loading and displaying exceptionally high number of characters, which may result in extremely slow performance, any scale catalog that contains more than 10,000 characters is published in a .txt file, in the following way:

• I indicate the first and the last numbered scales of the respective scale catalog, while the rest of the scales can be downloaded in .zip format and decompressed into a .txt file. The below sections contain all the 4194304 variations, classified into scale catalogs by number of notes. The twelve-note scale catalog section contains a calculation, which can also apply here. Here comes the formula... ...where:

• n = 22

• k = variable

So let's see the result for the scale catalog consisting of 3 notes:

• n = 22

• 22! = 1124000727777607680000

• k = 1

• 1! = 1

22! / (1! x (22-1))! = 1124000727777607680000 / (1 x 51090942171709440000) = 22

• 0 pc â€“ theoretical reference point

• 22! / (1! x (22-1))! = 1124000727777607680000 / (1 x 51090942171709440000) = 22 pcs

• 22! / (2! x (22-2))! = 1124000727777607680000 / (2 x 2432902008176640000) = 231 pcs

• 22! / (3! x (22-3))! = 1124000727777607680000 / (6 x 121645100408832000) = 1540 pcs

• 22! / (4! x (22-4))! = 1124000727777607680000 / (24 x 6402373705728000) = 7315 pcs

• 22! / (5! x (22-5))! = 1124000727777607680000 / (120 x 355687428096000) = 26334 pcs

• 22! / (6! x (22-6))! = 1124000727777607680000 / (720 x 20922789888000) = 74613 pcs

• 22! / (7! x (22-7))! = 1124000727777607680000 / (5040 x 1307674368000) = 170544 pcs

• 22! / (8! x (22-8))! = 1124000727777607680000 / (40320 x 87178291200) = 319770 pcs

• 22! / (9! x (22-9))! = 1124000727777607680000 / (362880 x 6227020800) = 497420 pcs

• 22! / (10! x (22-10))! = 1124000727777607680000 / (3628800 x 479001600) = 646646 pcs

• 22! / (11! x (22-11))! = 1124000727777607680000 / (39916800 x 39916800) = 705432 pcs

• 22! / (12! x (22-12))! = 1124000727777607680000 / (479001600 x 3628800) = 646646 pcs

• 22! / (13! x (22-13))! = 1124000727777607680000 / (6227020800 x 3628800) = 497420 pcs

• 22! / (14! x (22-14))! = 1124000727777607680000 / (87178291200 x 40320) = 319770 pcs

• 22! / (15! x (22-15))! = 1124000727777607680000 / (1307674368000 x 5040) = 170544 pcs

• 22! / (16! x (22-16))! = 1124000727777607680000 / (20922789888000 x 720) = 74613 pcs

• 22! / (17! x (22-17))! = 1124000727777607680000 / (355687428096000 x 120) = 26334 pcs

• 22! / (18! x (22-18))! = 1124000727777607680000 / (6402373705728000 x 24) = 7315 pcs

• 22! / (19! x (22-19))! = 1124000727777607680000 / (121645100408832000 x 6) = 1540 pcs

• 22! / (20! x (22-20))! = 1124000727777607680000 / (2432902008176640000 x 2) = 231 pcs

• 22! / (21! x (22-21))! = 1124000727777607680000 / (51090942171709440000 x 1) = 22 pcs

• 1 pc - theoretical ending scale

www.music-instrument-guitar.com - Two-octave scales catalog Ia. - Introduction