I don't know very much about music! Well, maybe I know a few things... but not much. I don't play any instruments at all, and I am not a mathematician or scientist of sound frequencies. But... at the same time, in spite of my lack of musical education, the seven note "Do-Re-Mi" octave of musical sounds, so popular to Western music, has been adequately drilled into my head just as much as any other person living in a world that uses the 7 note diatonic / 12 note chromatic musical system popular throughout much of the Western world. And... certainly... the idea of an octave - or tones coming into and out of phase with each other at a predicable interval of frequency - is, of course, inescapable to anyone - even those who don't know the first thing about music.

A so-called "octave" is an interesting phenomenon of nature. The idea of a progression that ends on a note that is similar, but still different, reminds me of the Geometrical Progression presented elsewhere in this study, where a nine step progression of geometrical shapes leads us from a point to a sphere... where, by way of our acknowledgment of an infinitely relativistic scale of sizes, we are capable of viewing the sphere as just another beginning point to an endless progression of ascension... (or decline if moving in reverse). It was this idea of point and sphere being in essence the same, while also carrying obvious differences, combined with the idea of a musical octave, that lead me to contemplate alternative ways to span that magical phenomenon of an octave in a way that might be compatible with the nine step progression of geometry and the nine numbers of numerology.

The seven notes of the "Do-Re-Mi" scale are indeed beautiful, but... as a numerologist, I could not help but wonder what would happen if someone were to span that space between a scale's ends with eight notes instead of seven; i.e. eight notes with a ninth that is in essence the same as the first, but still different, by way of its ascension along a scale of frequency? I was curious. I had to hear for myself, just exactly how reasonable or how awful a scale of musical notes would sound if they completed themselves in eight steps, with the ninth note in phase with the first, instead of seven steps, with the eighth note in phase with the first. Would it be possible to insert just one more note into an existing octave? Or would inserting just one extra note so completely screw up the rhythm of the sacred geometry of music as to render the resulting scale a useless atonal mess of unharmonious noise? I wanted to know.

In the process of finding out, I did some searching for nine step musical scales, but I did not find much. I found some instances of people playing different styles of music within the limitations of nine selected notes... but they were always just nine notes selected from the standard twelve notes of chromatic musical notation, and not nine completely unique frequencies that span the phenomenon of an "octave" in their own way. I encountered something called the "Secret" Solfeggio Scale but that was a less than satisfactory solution. So I continued to search. I did find some other methods of tuning that included more than seven or twelve tones, but none that were based on nine. Admittedly, I did not search all that hard, and... after failing to find any existing nine note scales that look like the one I've created here, I decided to craft this nine tone "octave" of my own. If I have recreated something that is already known by the world of music, that is a complete coincidence. Since this scale simply divides an "octave" logarithmically by nine, it is entirely possible that such an octave already exists and I just never found it. In either case, whether I am the first to do this or just another in a long line of curious people, the resulting series of tones are what this study of tarot, numerology, consciousness, cosmology and music considers compatible with the nine numbers used to label the theoretical model of existence being presented everywhere in this study. If these tones are not usable for making music of any kind, that is alright. They can still stand as tonal or audiophonic representations of the nine numbers of numerology.


A Tonal Message

I am not a historian, especially of music. But, from what I've learned so far, it appears as though the twelve note chromatic system of Western music is a more or less fixed system of established sound frequencies that a lot of instruments use to tune themselves equally. It's called an Equal Temperament scale because of how it distributes notes evenly, in a logarithmical way, across the range of an octave. Obviously, to insert even one extra note into such a fixed system would require an adjustment of all the established frequencies in order to make room for that one extra note. To an ear that has been trained to recognize established tonal frequencies that have been chosen for their beauty, it would not be surprising to hear complaints about any alteration of frequency as being "wrong" or "out of tune." At the same time, though, there are those other methods of tuning just mentioned, which are out there being used... or not being used, but at least invented. In fact, there are people who think that the Equal Temperament scale is the one that is "out of tune" and prefer to promote an alternative that makes slight adjustments to these established frequencies in order to achieve a more perfectly pure tone for each note. So the idea of playing music with a series of tones that don't exactly match those found on the average Western piano keyboard is not new, and therefore not necessarily a "wrong" thing to do. That is certainly a comfort, because a nine tone scale would otherwise be considered out of tune, were it not for the flexibility of music to accept different kinds of tuning and different kinds of notes. In the same way, it will be this kind of flexibility of mind, and ear, that will be required of anyone reading this essay or listening to the resulting tones of a nine step scale. Anyone who thinks that the seven notes of the twelve note system of Western music is the single most perfect and sacred utterance of sounds ever conceived and that they should not be altered in any way should probably stop reading right now... and let the curious carry on.

As I was experimenting with tones, and trying to "fit" another tone into the sequence of seven, I encountered some interesting anomalies of audio perception. First, because my ear, like so many others, has been tuned to the seven & twelve note system of Western music, all my initial attempts to insert an extra note failed, simply because that new note had nowhere to go in my prejudiced mind of "beautiful tones." All attempts to adjust existing notes, to make room, only caused those once beautiful notes to sound flat and out of tune. No matter how close I got to adjusting those seven notes so that they still sounded close enough, there was always one note along the way that sounded like a clunker. It seemed that spanning an "octave" with eight notes instead of seven was impossible, without at least one of those notes sounding flat and out of place. But what was really strange was how in some instances I could get a series of eight notes to sound pretty good as an ascending scale, but... when I played them back in descending order, there would be that one clunker that didn't fit. And... if I adjusted the frequencies so that the same scale sounded good in descending order, when I played it in ascending order, that same note that was supposedly fixed and seemed to fit, would now sound like a clunker in the other direction. I went back and forth like that for a while, making slight adjustments to the existing seven in order to fit an eighth note in. It never worked. Some might say that it didn't work because the seven notes I was altering are the most perfect and sacred utterance of sounds ever conceived and to try to "adjust" them is an abomination. But I encountered another anomaly of perception that made me question that wisdom as well.

The subjective nature of perception is interesting, especially when it comes to musical notes that have been declared to be beautiful. As my experiments continued, I encountered another interesting result that made me wonder about the beauty of special notes. This is an experiment that anyone can do with any standard keyboard instrument, like a piano. Try playing a tune that only uses the black keys. Or, just fiddle around with the black keys, playing any random sequence at all. Play with those notes for thirty second or a minute or more. Then... try suddenly including one of the white keys. Just randomly hit one of the white keys after thirty second or so of only hearing black key notes. When I did this, the note I selected sounded horribly flat and out of place... and yet, I had included a note that was supposedly one of the "beautiful" notes of a sacred sequence of perfectly beautiful notes. How is it possible that such a beautiful note could sound so flat and awful? The answer, I think, is... context... and how an established context can create a prejudice of comfort that does not want to be violated. What's amazing is how it only takes seconds for a mind to adjust itself to a new context of sounds, for which a member of a previous context of comfort can then become an intruder. In the same manner, it is this idea of contextual prejudice that must be overcome in order to hear any other scale of tones as their own context of tones and not as something that is just plain "wrong" or "out of tune" relative to some other prejudice of tonal preference.


A Traditional Octave

After trying to squeeze an extra note into the existing scale of seven beautiful notes... and failing, I decided to approach the task of making a new scale - or context of tones - in a way that did not necessarily concern itself with prejudice of comfort for tradition. I decided to just divide the space between the ends of an "octave" mathematically, and just see what tones appear. As mentioned above, I am not a mathematician or scientist of sound, I can't explain how an octave is divided in the twelve note system of Western music. All I could do is observe how each of those twelve notes appeared to be about 105.94% larger than the previous... leading to a scale that was not divided equally, but logarithmically. In music, this is supposed to create a scale that is equal in temperament, with smooth predictable ratios between notes, and thus establish a reasonable compromise between various tuning methods. That sounded like a good approach, so I decided to do the same with my scale. I also decided that my eight notes should all have half notes, so I divided the space between two corresponding "octave" tones into sixteen logarithmically spaced segments of 104.4273782426% between each ascending note. To help people with a prejudice of comfort originating from the context of the twelve note scale, I decided to start and end my "octave" with the established frequencies of the nearest piano I could find. Knowing, from the Do-Re-Mi song, that "Middle C" is an important starting place, I decided to span the C4 octave. I don't know who decided that "Middle C" should always be exactly 261.63Hz in frequency or why (apparently it comes from starting at "A" with 440.00Hz), but I accepted that as a starting point, and developed my scale from there. The results are seen in the diagram below.

Octave Comparison C4
I started with an existing octave, and divided it sixteen times instead of twelve. The result was five tones that were exactly the same as the standard octave, and several others there were very close to existing tones. The ones that are off are all off by about 1.45%. That's enough to notice a difference!

The results of dividing an existing octave by sixteen instead of twelve was interesting. Four notes (or five, if we count the ending note) were exactly the same frequency as existing notes of a standard C4 octave (an octave that begins at "Middle C" on a piano). Unfortunately... two of those four notes were not among the seven so-called perfectly beautiful notes of the "Do-Re-Mi" scale, but were in line with the flat/sharp notes played by the black keys of a piano. So overall, there was one third agreement. But in the critical area of beautiful notes, there was not as much agreement. But, again, agreement between these scales was not the primary goal. Comparing them to see how much agreement there was still mattered, but the new scale was only going to be what is was and nothing more. To appreciate the new scale we would eventually want to abandon the context of tradition and get to know this new scale for what it could or could not do on its own... musically, and as a device for describing the nine numbers of numerology.

Are these eight notes as smooth and beautiful as the seven notes of a traditional scale? That is a subjective matter for individuals to answer for themselves, relative to their own aesthetics and their own ability to adjust their context of comfort away from tradition and over to something new. Those with a finely tuned ear, who have spent a lifetime playing music with the standard seven & twelve notes of tradition might never be able to hear anything more than an abomination of atonal mess. While those without such prejudice might hear something of value, and be able to utilize this scale in making beautiful music. After all, as "off" or "out of tune" as some of these tones may seem from that of tradition, in many cases they are still very close. One could actually play a piece of music written in the twelve note scale and find approximations of those same notes in this new scale. To someone who already knows that piece of music, it might sound out of tune when played with this new scale. However, to someone who has never heard that piece of music it might sound perfectly fine with these new tones. Or... based on the sacred geometry of the traditional scale, these new notes might be intrinsically, empirically and objectively "ugly" rather than "beautiful" and not be any good for anything. I don't think we will know until someone builds an instrument capable of playing them! Or... we find someone who already has.


B The Magenta

The notes indicated in the chart above could work. However, after further consideration, I decided to start my scale on a B note instead of that C note, so that an even better alignment might be found to traditional notes. Thus, as a cycle of notes, my note that would be equivalent to a B would reside at the top or "12 o'clock" position if this cycle of notes, were to be presented in a circular form... which is exactly what I wanted to do, and did.

The ways in which an octave resembles the Geometric Progression, found elsewhere in this study, is interesting. Another place where we find a similar phenomenon of ends that meet, enabling an endless cycle of ascending or descending repetition, is the world of color. When the visible light spectrum of electromagnetic energy is bent around into a circle, such that the Red and Violet ends meet, they produce an intermediary color known as Magenta (see The Mystery of Magenta in the essay Enjoy the Pretty Colors, elsewhere in this study). Magenta is a color that is not found in the linear splitting of light into a rainbow. Magenta only appears when this otherwise linear sequence of colors is either juxtaposed next to another linear spectrum, or, bent around so that its own ends meet. By producing this intermediary color, an otherwise linear progression is made into a circular progression that is endless. In this way, Magenta becomes equivalent to the Point that is also a Sphere, or a C note that graduates to another C. So, by including Magenta in our sequence of colors, we establish three graduating progressions that are endless: color, music and geometry. Can they be fused as one?

Lots of people associate colors of the rainbow to music. However, most of the time they connect the seven notes of the traditional scale to the seven colors of light as defined by Sir Isaac Newton... skipping right over the vital color of Magenta as the link between Red and Violet that enables this otherwise linear spectrum to be cyclical. Where is the Magenta between Red and Violet on these scales? It has been conveniently ignored, in order to match up seven colors with seven notes. Seven has always been a popular number for people to obsess over; seven notes, seven colors, seven chakras, seven planets etc. But this study of numerology, color and music does not obsess over the number seven... we obsess over the number nine! By creating a musical scale with eight notes and a ninth that is in phase, or the same as the first only elevated or diminished, we are able to equate musical notes to the nine numbers of numerology and turn an octave of sound into a true "point-to-sphere-like" cycle... by showing the vital linking step that enables one cycle to repeat over the same terrain in its elevated or diminished state. And when we apply this idea of a vital linking step to the world of color we see that, as a cycle, there are in fact eight colors - not seven! In fact... as a single linear progression of light being split up into colors, there are actually a multitude of possible colors that we could label. We could say there are seven, or more than seven. But unless we acknowledge the existence of Magenta as a linking color, this linear progression of colors - however many we choose to label - does not become circular or cyclical... it remains linear and finite. Magenta is what makes it cycle.

If... in my quest to find an instrument capable of playing these tones, I am able to manufacture any kind of piano-like instrument I will not make the keys black and white... I will color them with their appropriate color and label them with their appropriate number - in accordance to how they would line up together in a circular chart of colors and notes. To serious musicians, an instrument crafted with such a color and number scheme might appear juvenile or excessively flamboyant, but I would welcome it as a celebration of the marriage between these three cyclical systems, and how each one reinforces the expression of the other.


B is for Binary

After conceiving of such a circular presentation of colors, geometry and music, I thought it might also be nice to have my scale align with the symmetrical pattern of a 12 Tone Chromatic scale, when it too was arranged in a circular fashion. In such an arrangement, I saw the black key note of G#/Ab as the apex, or point, of a pentacular arrangement of black keys, and subsequently adjusted my notes, once again, to begin and end on this critical point instead of a C or B. I did this for the sake of symmetry and so that comparisons between systems might be easier to see, especially when considering the spacing of notes that are considered harmonious in triads and chords. However... when I started to examine the subject of triads, chords and harmonics, something emerged that changed my direction once again!

Because my scale divides an octave into 16, or 8 notes, it is intrinsically binary, as 16 and 8 are part of the binary sequence of numbers, where each number in the binary sequence is half or double its neighbor (1, 2, 4, 8, 16, 32, 64, 128, 256 etc.) This is the kind of numerical thinking used by the digital world of computers to speak the language of the universe. It is a very basic, ordinary and easy to understand pattern. We like basic, ordinary, easy to understand universal patterns in this study of cosmology and patterns of nature. With this in mind I began to look for other ways in which the binary sequence might play a role in the formation of a musical scale.

In earlier iterations of my 16 notes, I was concerned with how close or far my notes were to traditional 12TET notes. I wanted my notes to be accepted as usable by everyone, and wished that they could even be used to play existing music. But ultimately there was no way to make the notes close to tradition any closer that 1.45%. I could make some closer, but it would only make others further away. I tried abandoning the logarithmic pattern and adjusting all my close notes to be closer. But the unequal temperament made the progression of notes sound bad. Eventually I abandoned all interest in having my notes align with tradition at all, and began looking for a scale of notes that just sounded good, and if possible had some connection to the binary sequence.

Along the way, I had also acquired a desire to move away from the irrational numbers that appear when a straight and strict logarithmic progression is applied to any starting frequency. So I searched for a sequence of patterns that used nothing but whole numbers for the frequencies and nothing but whole numbers for the interval between frequencies. I managed this feat several times, but, unfortunately, each time I did, the sequence only worked for a single octave. Meaning, that when the sequence crossed that octave "threshold" and the next frequency needed to be double its counterpart in the previous octave, the jump from the last frequency of the previous octave to the first frequency of the next octave was not smooth. Sometimes there was a big jump, sometimes there was a negative jump. I encountered this problem with several attempts. In my search, I acquired an oscilloscope application and began searching for standing wave frequencies. I thought it might be cool to base a musical scale on standing waves. I made a standing wave octave that began on the frequencies of 344hz to 688hz (rounded down from 344.4 and 688.8 so as to be whole numbers). This almost worked. I tried starting at a nice round number like 400 to 800. That almost worked as well, but not quite. The chart below shows how attempts to increase from the standing wave idea to lessen the "Jump" between octaves, and attempts to decrease from the 400 idea to lessen the "Jump" between octaves lead me to a single solitary solution... apparently, the ONLY possible solution for a scale wanting whole number frequencies and whole number intervals! Because of this miraculous revelation, and, because the whole number intervals ended on numbers that came from the binary sequence, I decided this arrangement of frequencies must be cosmically inspired, and decided to adopt it as an option to pursue.

Music Evolution
I tried a standing wave scale with rounded whole numbers. I tried starting with the pentacular peak frequency of a rounded G#/Ab at 416. I tried a nice round number like 400. The "Jump" between octaves was not good. Until I moved one up and the other down, to find this one perfect sequence of whole number frequencies and whole number intervals between frequencies. The fact that the intervals ended in 16 steps at an interval of 32 caused me to view this as a cosmically inspired, "Binary" scale!

So, after many dead ends, I found it (eureka!). And... happily... it is a very “Binary” kind of pattern. However, the binary nature of the scale can’t be seen in the display of the notes alone, because it is in the this underlying structure of how each note came to be. But it’s kind of cool the way it works out. Here is another chart showing how "Binary" this scale really is:

Underlying Structure
This whole number frequency/whole number interval scale is very "Binary" in its underlying structure, producing a scale that is not exactly logarithmic, but very close... quasi-logarithmic. Do whole number frequencies sound better? I don't know. But this is way cool! It will be noticed that that frequencies and intervals that go below the 392hz mark are not exactly whole numbers. But... the fractional amounts they use are all from the very binary nature of cutting a whole into 1/2, 1/4, 1/8, 1/16 etc. which itself fits with the binary pattern of 2, 4, 8, 16 etc. So even as fractions we remain binary in pattern, as we also do in the amount of increase from one interval to the next; increasing by 1 in one octave, then 2 in the next octave, then 4 in the next, then 8 etc. Cool.

The end result is not a perfectly logarithmic increase of notes, but is instead a series of whole number increases that result in a series of whole number frequencies... until it gets to the lower notes, where the fractions are all 1/2, 1/4, 1/8, 1/16... which is itself binary. And as each series of whole number intervals reaches the end of its octave, it ends on a binary number. And as the first interval of that octave is doubled in the following octave, the jump from the last interval to the first interval is smooth (30, 31, 32 to 34), where the following octave increases by a doubled factor in each interval (34, 36, 38 etc). So... not *perfectly* logarithmic, but close enough for things to work. And the notes themselves sound pretty good. Still hard to tell, because of how trained our ears are for 7 instead of 8, but they definitely sound better than any previous progression. And I think the harmonic triads formed by them are adequate. They might not be good enough for the experts, but I think the “Binary” nature of this solution makes this the only reasonable resting point for this experiment.

So... see what you can do with THESE notes.... Here is a link to another page where you can play these notes!

Some of these notes still line up with tradition. That is unavoidable when superimposing a 16 notes system over a 12 note system. But please keep in mind that there is no attempt here to duplicate traditional notes. So it will probably not be good to try and play existing music with these notes. Make up something completely unheard, and see if we like it.

Here is another graphic, showing 96 of these cosmically inspired notes in a spiral:

Binary Scale Spiral
Here are 96 cosmically inspired, binary sequenced notes to use in playing music! They don't line up with traditional 12TET notes so perfectly as previous arrangements did. But they do line up with a traditional G note, so some comparisons can be made from that point of commonality. Because it lines up with a G and not G#/Ab, it also misses in lining up with the symmetry of the traditional 12TET scale - a necessary sacrifice for such a cool arrangement of notes!

These notes don't line up with traditional 12TET notes so perfectly as previous arrangements did. But they do line up with a traditional G note, so some comparisons can be made from that point of commonality. Because it lines up with a G and not G#/Ab, it also misses in lining up with the pentacular symmetry of the traditional 12TET scale - a necessary sacrifice for such a cool arrangement of notes!


Harmonics Are Binary!

After having established a cool series of notes, I decided to explore the subject of Harmonics. Another subject I know little about! But explorations lead me to more interesting encounters with binary patterns.

I think my Binary Scale is pretty cool because it uses the binary sequence (1, 2, 4, 8, 16, 32, 64 etc.) to define its form and create notes. I don’t now if I understand harmonics correctly or not, but I came up with this chart where I extended the multiplications that are used to create relationships between notes. Many interesting things emerged:

Harmonics Calculator
When you double a note, you get harmony because of the octave phenomenon. When you multiply 4, 8 or 16 times you get the same harmony. When you multiply 3 times you get a different kind of harmony. In this case, the note is harmonious because it is exactly 150% more than the root. When you multiply by 5, the same thing happens except the note is now harmonious because it is exactly 125% more than the root note. When you multiply by 6, you are duplicating the same harmony as multiplying by 3. When you multiply by 7 the note is now harmonious because it is exactly 175% more than the root. These percentages all correspond to common fractions of a whole; 125% being 1/4, 150% being 1/2 and 175% being 3/4... which is essentially the same as 1/4, depending on which way you look at things. Subsequent multiplications reveal that all unique frequencies have percentages that equal binary fractional amounts! 112.5% is 1/16. The pattern of harmonics is binary. And when frequencies that are duplicate are removed or grayed out, we see a binary pattern in the number of unique frequencies existing between each duplicate octave note; between 2 and 4 times there is one unique frequency. Between 4 times and 8 times, there are 2. Between 8 times and 16 times, there are 4. Between 16 times and 32 times, there are 8 unique frequencies. This pattern continues. Harmonics are binary!

First of all... even though a 12 step scale is pretty good, I think that basing a scale on the idea of doubling has its own charm. An octave is important to sound and music. You get an octave by doubling. So a scale that uses the idea of doubling (1, 2, 4, 8, 16, 32, 64 etc.) to create its notes seems appropriate. So it was interesting to see how the "unique" notes of this harmonic sequence (colored in blue) came in groups of 1, 2, 4, 8 and 16! (wherever you see a grey row, it means that that sequence of numbers has already been established in a previous row). It was also interesting to see how the percentage of increase from the root frequency to each unique frequency came as a binary fractional amount and not some irrational number.

With this pattern in hand, I compared the unique numbers from my chart on calculated harmonics to my current Binary Scale of whole numbers... and to a 12 TET scale as seen in the chart below:

Harmonics Percentages

In this chart, it can be seen that some of my notes are a little further off from “perfect” than I’d like, but many of my notes are pretty close to a blue harmonic note of one kind or another. I had to go to x57 to get a number close to a 12 tone F and down to x43 to get a decent C. So, likewise for my scale. But overall, my whole number notes and whole number interval scale is not that far from “perfect” in representing perfect harmonic notes. That is... if I am looking at this correctly! In my scale, most of my notes are not more that about 5hz off. The 12 Tone scale might be better, with only a couple that are far off, like the B and f/g. But for my scale to be as close as it is, is encouraging.


C What I Mean

As stated, no attempt is being made to duplicate 12 Tone notes with this Binary Scale. But comparisons are inevitable, so I proceeded further to compare how my scale achieves harmonic blending of notes as triads and chords, compared to the 12 Tone system of tradition.

When Harmonic Frequencies were calculated (and reduced to fit within a single octave), the first harmonic was the Octave Frequency two times the root or “Root x 2” or “200% of the Root.” Subsequent harmonics continued from there. After that, we were meant to multiply by 3. That gave us a frequency that was 150% of the Root. The next harmonic (x4) repeated the same frequency as x2. The next unique frequency came when we multiplied by 5. There we got a frequency that was 125% of the Root. Next, multiplying by 6 repeated the same frequency as x3. The next unique frequency came when we multiplied by 7. There we got a frequency that was 175% of the Root. Frequencies after that all had percentages of increases that had fractions that were binary!

When we look at a Major Triad from the 12TET scale, starting with G, it asks us to combine G with B and D, to produce a pleasing combination. The note B is just a little more that 125% of G, and D is almost exactly 150% of G. Does this combination sound good because these notes are close to this ideal harmonic association of 125% and 150%? If so... harmonics are binary! One harmonic doubles. The next is 1/2 of that. The next is 1/2 of that 1/2, or 1/4. Even a Minor Triad ask us to move that second note to a A#Bb, which is almost exactly 3/16ths (118.75% or 1.1875) away from the Root of G.

One might think that it would be ideal to make a scale that hits these ideal, binary, harmonics exactly. It would. I tried that... with 16 notes... one for each 1/16th step. It sounded just fine. The harmonics were there. One problem though... each interval was the same, so... as stated above, when you reach that Octave Frequency that is twice the Root, and start another octave, you need to double each successive frequency... and each interval... which means that there is a noticeable jump between the smaller intervals of the first octave and the larger intervals of the next one. Lesson being... that you kind of have to be logarithmic in the spacing of frequencies. Or, as in my scale “quasi-logarithmic” so that the last interval of one octave progresses to the first interval of the next octave with little or no “jump.”

So... if these binary fractional amounts are important to harmony, the question in evaluating a scale could be centered around which logarithmic scale comes closest to hitting important binary marks of harmonics. My “whole number note/whole number interval” “quasi-logarithmic” scale has a tiny jump to consider. But overall, it hits some important frequencies with about as much accuracy as the 12TET scale. We’re both about the same amount off of a perfect 125% note. 12TET is very close to 150%, while I am just a bit short. But then... I am way closer to the 175% note, if anyone needs that! And we both hit that 3/16ths mark almost exactly.

Here is a diagram, to better see how notes seem meant to be 1/2, 1/4, 1/8 and 1/16th of a Root/Octave, or Binary!

Harmonics Comparison Circular
This diagram shows how two logarithmic scales line up with the not-so-logarithmic pattern of binary fractions.

With this diagram, we can see that the 12TET scale does a better job of hitting ideal frequencies at the beginning (upper left quadrant). But... between my note 3 and 7, I have seven notes; one for each "ideal" note, where 12TET only has five. Mine are all slightly short of ideal, but at least I am following the ideal with a note for each one. Actually from my note 2.5 to 7.5 I have one for each. It is only at the very beginning and very end that I don’t line up with the ideal. Which is not that critical, because those notes are too close to the Root Frequency to produce harmony anyway. You really need to go that 3/16ths point before things start sounding close to harmonic. And going beyond that 175%/3/4 mark also becomes questionable. So, overall, my scale follows the binary harmonic ideal pretty closely! So far so good.


C is for Chords

After observing how my quasi-logarithmic 16 note scale covers the not-logarithmic 16 segments of a whole when divided into 1/2, 1/4, 1/8 1/16 better than the 12 Tone system of tradition, I decided to explore how close this scale comes to producing actual triads and chords. This lead to yet another interesting chart:

Harmonics Comparison Chord Chart

Because my scale is not perfectly logarithmic, but has that tiny jump between octaves, the percent of increase from a root number to a harmonic note is not consistently the same. It might be off by another percentage point or so, depending on whether the triad crosses that octave threshold or not. A strictly logarithmic progression would not have any “threshold” to cross, and percentages of increase would be the same from any root. But the variance of my scale is minimal (or appears to be, based on experimentation with sound files so far). So these are basically my percentages of increase, compared to the percentage of increase required to achieve triads and chords in the 12TET system.

In this chart, the idea is that harmonics are binary. So I divided an octave into 16ths, and calculated the percent of increase required to hit each 16th perfectly. Then... plotted the position of each type of chord that I found on a big chart of piano chords. The chart showed 216 images of chords (12 images for each of 18 types of chord). But really, the pattern is the same across each of 12 notes. That pattern can be seen many ways. I chose to show it as a percentage of increase from the root note, to see how close those percentage come to the supposed ideal of binary increments.

As stated earlier, my scale comes up a bit short of the critical 1/2 or 150% mark. But I think it is still close enough to work (even when it varies to be a little further off). And... my note 7.5 is much closer to another critical point - the 3/4 or 175% mark. In general, it can be seen in this chart that the patterns of chords cluster around those three main marks of 1/4 or 125%, 1/2 or 150% and 3/4 or 175%. Which makes sense, that if you want to make something harmonious, you combine it with a pattern that is the same... only double, 1/2, 1/4 or 3/4. And if you want to make some interesting exceptions, you still don’t stray too far from those critical points.

So I have a scale that is binary because it has 8 or 16 notes. And because its underlying structure uses binary increments to establish those 16 frequencies and intervals. And, because it has more notes lining up with the binary fractional pattern of harmonics! But... how does it sound? Go HERE and experiment with some sound files. Or, make an instrument that can play these notes! And then... make one for me!!

Binary Scale w G
Here is another mandala that shows this new Binary Scale lines up with the frequencies of a 12 Tone system. Unfortunately, it doesn't line up with the top point of the pentacle of symmetry that exists in the 12 Tone system. This mandala shows the notes equally spaced and the critical binary fractional points of harmony logarithmically, instead of the other way around (as seen in a previous diagram). It is important to acknowledge that the halfway point of the circle is not the same as the halfway point of the frequencies. Use this to find various forms of harmony.

All words and images Copyright © 1980 - 2016 by Guy Palm

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