Harmonics Calculator

[democracy]

Studying Pythagorean arithmetic and contemplating harmonics on a number table.
https://g.co/gemini/share/5025e97e9ada

[democracy]

I am looking at the harmonic relations on this familiar table.


EDIT: Doesn't look like images show in the post.

[democracy]

I am looking at the harmonic relations on this familiar table. But I don't see it in the post.

[democracy]

I took some time to look carefully at this number table and there is so much going on, much more than I ever realized. The table depicts linear, plane and harmonic relations. For example, 4 is on the basic number line and is only a linear number on that line (we can call it R1, for Row 1.) In that position it is not a square number. This is easy to prove. Just take those 4 linear units and assemble a plane from them and you have 1 square unit. Plane numbers occur only when two linear numbers intersect. So let's take the number 16 on R1 and it is linear. When 16 occurs on the diagonal (D1) it is a plane number and square. If 16 occurs by the intersection on 2x8 it is a plane number and heteromecic (rectangular.) Therefore, the only square numbers on this table are on D1. Although 1, 3, 6, 10... are on the table they are never triangular numbers. Triangular numbers (or what Diophantus sometimes referred to as "triagonal") can only be depicted on an equilateral and triangular table. I will show later how D2 is a diagonal of heteromecics and each is double of the an apparent triangular number, but since these numbers are oblong and contain right angles, they are in fact not triangular numbers. to investigate the properties of triangular numbers we will produce an equilateral triangular multiplication table and reveal a flaw in classical Greek polygonal number theory that hangs on even today.

I say that 4 on R1 is only linear because it is produced through the operation of addition. Multiplication requires an angle. We will take a look at some classic texts on this, such as Rules for the Direction of the Mind by Descartes. Remind me of this.

[democracy]

Now any row is an arithmetical progression. Beginning from 1 and going on to 2, 3, 4, these are consecutive numbers and comprise a class that the Greeks called "superparticular." Today we express this as n, n+1, n+2... So this a linear sequence. Look at R2 and we see the doubles of the superparticulars. These too are an arithmetical sequence because they progress by the same interval, 2, 4, 6, 8, 10... where the superparticular progess by one unit, the double progress by 2 units. The same holds for the triples R3 and so on.

[democracy]

So I am disputing the idea that 36 is always a square number. If 36 occurs on D1 it is homomecic (having equal sides) and square, but if 36 occurs on R1, R2, R3, R4, R9, R12, R18, R36 alone and without intersecting with a column (C) then it is a linear number. If the intersection is of unequal lines, then it is heteromecic (rectangular).

[democracy]

We have 3 kinds of numbers which correspond to lines, planes and cubes, if we extend the table into 3D. These also correspond to arithmetic, geometric and harmonic ratios and proportions. All relations can be found in the superparticular because 1:2 is arithmetical and it is geometrical being the double, and furthermore it is the octave relation in harmonics. If you know something about music you will recognize this. The piano is tuned with pure octaves only on the note A because we have 55, 110, 220, 440, 880 and so on... all doubles. The relation 2:3 is arithmetic but also harmonic because this is the 5th or perfect 5th, the tonic being 2 and the 5th being 3. 3:4 is the 4th or perfect 4th and 4:5 is the major 3rd. This is all on one line.

[democracy]

This is where it gets weird and you wonder whether this subject belongs in Math, Science or Philosophy? Because every new superparticular on R1 introduces a never before heard interval. That means that we have heard 1:2 and 2:4 is just another octave in a higher range, so we have heard this before, but 2:3 introduced a P5 and 3:4 introduced the P4 and this continues to infinity. The hearing apparatus can only distinguish up to about 15:16. The question is does 16:17 make a musical interval? This is reminiscent of the old question - If a tree falls in the woods and no one is there to hear it, does it make a sound? I am reminded of the Platonic idea of a harmony of the universe. What do you think?

[democracy]

Take a close look at the diagonals, like D1 the central diagonal. These are not lines but planes. We can not really look at this as a line because the table is bisected by the series of square numbers 1, 4, 9...100 leaving two isosceles right triangles in which the diagonal as a line would be an irrational, but it is not because the series ends with a whole number. Therefore D1 is commensurable in square only or in plane only. Viewed this way, the diagonals are strongly suggestive of a next dimension.

The table also distorts the true dimensions of these numbers. It depends upon how we read it. The squares are all occupying equal square units. To read it properly we have to refer to rows and columns that intersect and this reveals the true shape of D1. The apex is 1 but the true base is 100 times that. The table cannot depict this accurately. You could place the table on a floor and then draw the diagonal with an actual base of 100. You would have a square unit as the apex, followed by 4 square units, followed by 9, 16 etc until the last square of 100 units. Our table is only 100 square units, yet the true dimensions of the central diagonal would be 385 square units. 385 is a pyramidal number. You cannot layout the numbers to extend beyond the end of the diagonal D1 which terminates at 100. They must enter into a solid dimension and that dimension is rooted in the triangular form. Thus, we are directed to the construction of a triangular multiplication table.

But we still have oodles and oodles of stuff to explore in this simple 2D table. It is absolutely mind boggling.

[democracy]

http://aleph0.clarku.edu/~djoyce/ele...I/propII4.html

Euclid 2.4 If a straight line is cut at random, then the square on the whole equals the sum of the squares on the segments plus twice the rectangle contained by the segments.

Take a look at D1 and see if you can find this proposition depicted along the diagonal.

[democracy]

To be precise, 385 is a square pyramidal number, having a square base and triangular sides. Looks familiar to your imagination? Obviously, the base is the table we began with having 100 square units. How can we logically distribute the remaining 285 units? What shape are the units? They must be square based pyramids throughout. In order to fit them together, we will have to invert every other one of them? Consider that one will serve as the apex then we have to distribute the other 284 between the base and the apex. What do you figure?

[democracy]

While we are trying to process this, it might be useful to construct the 2D triangular multiplication table because this is the table that will serve for all 4 sides of our pyramidal extension of D1 on the square table.
GeoGebra

According to Descartes, an angle is necessary for multiplication but the angle is arbitrary. We can multiply using 60° angles as well as 90°.

[democracy]

GeoGebra
A quick glance at the grid will show you that there is a such a thing as triangular numbers but they are the same numbers as the square numbers. They are not 3, 6, 10, 15... because if they were we would not get the same products from a triangular table as we do from the square table. If we did get those numbers we would contradict Descartes claim that the angle of multiplication is arbitrary.

With that in mind, we are completely off track thinking that 385 is a pyramidal number. The reports concerning triangular numbers are incorrect and have always been incorrect and any extensions of them into solid geometry will be equally false. We will need to go back and do a complete review of classical Greek polygonal number theory to discover their error.

[democracy]

The simple claim is that the addition of the series of consecutive numbers (superparticulars) results in a triangular number. The problem is that they laid out the figures with points and never enclosed the area by connecting the points with lines. Points do not enclose areas. They also began their number line with 1 which is another flaw. They counted the points. Our number line is just that - it is a line made of line segments and it begins with 0.

The key to understanding how to proceed is this: 1x1=1 square unit when the lines intersect. Likewise, the foundation of triangular theory should be 1x1=1 tri unit. In both cases we have constructed plane units.

Now we know that the triangular numbers are the same as the squares and that helps us connect back to D1 of the square table. Then what are the true square pyramidal numbers? We need those in order to proceed.

[democracy]

Feel free to correct any particular errors in my thesis. I am a generalist and just dash off my ideas. I leave the particulars for correction. Example: I said that we cannot hear an interval smaller than 15:16. This is very dubious but the general point should not be missed. At intervals smaller than 15:16 we are getting close to the limit.

So I have no argument with anyone who corrects a particular statement of mine.

[democracy]

Originally Posted by democracy

we are completely off track thinking that 385 is a pyramidal number. The reports concerning triangular numbers are incorrect and have always been incorrect and any extensions of them into solid geometry will be equally false. We will need to go back and do a complete review of classical Greek polygonal number theory to discover their error.

This statement is incorrect. I was assuming that the pyramidal numbers are always derived from the triangular numbers. I constructed the number 385 by adding the square numbers from 1 to 100. 385 is a square pyramidal number.

Feel free to correct any other particulars. I am aware that the Pythagoreans used points for their polygonal theory, but it was fallacious to call three points a triangular number. Triangles are trilateral figures. Naming arrangements of points after plane figures is where they went wrong. Their theory lacked the mediating lines between points and areas.

I am also versed in the modern formula for triangular numbers n(n+1)/2. If you inscribed this on the Washington Monument and Martians were to land in the year 5000 when we are extinct, there is no way that they could construct an equilateral triangle from this formula. The angle of operation is not included in the formula therefore the default angle of multiplication is a right angle.

I am sure that I am on a collision course with mathematics because it is a very messy subject by nature. Feel free to jump into the debate.