User:FloraC/There is not a third side of the river

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Chapter I. Binary System

Donald Sauter, an advocate for octalism, made this famous verse: "humans are essentially binary beings."[1] Apparently, I have refuted it in Ingression, the earlier essay where I shared my takes on radicology and related issues. It remains true, though, that binary opposition is the principal form of categorical perception.

"There is not a third side of the river" is the analogy that hopefully helps to apprehend the context, which is extremely important whenever bringing up the topic of binary systems. One is either on this side of the river, or on the other side. Obviously, the reader may argue, one can be in the river. Actually, that case cannot be expected, since the probablity of being in the river as one is thrown into the world is zero. The reason is simple. The ideal river, that is, a boundary, contains no substance, and even if it does contain something, it is infinitesimal compared to either side of the river. That captures the essence of any binary system.

It is tempting to postulate that the continuum of real numbers comprises three distinct types: the positive, the negative, and the zero. However, we can see the zero is not to be regarded as on an equal footing as the positive or the negative because it is degenerate. In this case, zero is on the boundary. It is also possible to contruct binary systems whose boundary literally does not contain anything, or there is not even a concrete boundary.

We may conclude that, for any system, what we consider a part should contain nondegenerate substances; otherwise it is a boundary. It follows, surprisingly at first, that the rational vs irrational numbers is not a binary system. It ultimately makes sense if you think about it – the rational are literally degenerate substances compared to the irrational.

Chapter II. Parity

This concept is straightforward. If binary functions P and Q satisfies this analytical distinction:

  • P of P is P;
  • Q of P is Q;
  • P of Q is Q;
  • Q of Q is P;

we may say this binary pair has parity, and we may say P is even and Q is odd.

Parity is the analytical struture of positivity and negativity, and is significant because it partly answers how signs can morph into tyranny. For example, the pair of presence and absence clearly demonstrates parity, as

  • the presence of presence is presence;
  • the absence of presence is absence;
  • the presence of absence is absence;
  • the absence of absence is presence.

Some believe the parity of presence and absence can be swapped. It has an appeal, even though the semantic coherence is doubtful.

Still, we may notice that everyday binary pairs can be connoted with parity, like life vs death. Widely believed, life is the presence of being, and death is the absence of being. Meanwhile, we can also think of death as a stable state, and life as a changing state, and hence the parity of life vs death is quite variable.

Then there are other pairs that clearly do not show parity. Notably, the pair of femininity and masculinity does not. In fact, just like the imaginary units i and −i, there is no analytical distinction between femininity and masculinity.

Chapter III. Relation to Interval Categorization

The Pythagorean tuning shall be used as the frame of interval categorization. The justification is but clichéd, so I am skipping it.

The Pythagorean tuning naturally gives us MOS scales of sizes of 5, 7, 12, 17, 29, 41, 53, … For human-scale studies, we may well substitute 12 equal temperament for Pythagorean tuning. What is so special about 12? Of course, the 3-limit comma associated with 12 equal temperament, namely, the Pythagorean comma, is considerably smaller than others for its complexity. Not until 53 will be found the next equal temperament like that, which we will eventually get to. For now, it is safe to say 12 equal temperament is an excellent approximation of Pythagorean tuning.

The 12 equal temperament is the quintessential tuning of diatonic. It has, obviously, 12 explicit intervals up to octave equivalence. In the diatonic scheme, they are: minor second, major second, minor third, major third, perfect fourth, tritone, perfect fifth, minor sixth, major sixth, minor seventh, major seventh and octave. We may use a shorthand notation:

  • m2, M2, m3, M3, P4, tt, P5, m6, M6, m7, M7, P8.

It comes as no surprise that there are 12 implicit intervals as well, where the 12 explicit intervals are not. They are: semiaugmented unison, neutral second, hemifourth, neutral third, naiadic, semiaugmented fourth, semidiminished fifth, cocytic, neutral sixth, hemitwelfth, neutral seventh, and semidiminished octave. In shorthand notation:

  • sA1, n2, h4, n3, na, sA4, sd5, co, n6, h12, n7, sd8.

So the 12 equal temperament does indeed define 24 intervals, with respect to a given reference point. The key to make it a 12 equal temperament and not a 24 is as follows. Here is the taijitu, or the diagram of taiji. There is some yang in the domain of yin and some yin in the domain of yang, and they convert each other constantly. The 12 explicit intervals are regularly played, that is, present. When they are silent, they are absent, but they are regularly played. Traditionally, the 12 implicit intervals are not played at all, so always absent. However, they can be played, for the same reason why the 12 explicit intervals can be silent, and they should be played – but only sparingly – in order to reach an equilibrium.


If both implicit and explicit intervals are played just as regularly, the distinction is lost, and it is a 24 equal temperament. What if only the implicit intervals are regularly played? This is left to the readers as an exercise[2].

Therefore, the way I use what I consider the 12 equal temperament may be seen as 24 equal temperament by others. For the same reason, of some approaches to what they consider the 24 equal temperament I do not label them microtonal.

Chapter IV. Relation to Tonality

Harmonic dualism, nowadays reintroduced in the name of negative harmony, provides spectacular insight on how to approach tonality.

Clearly, the major key simulates the presence of overtone series. The just major triad – the fundamental building block of harmony – represents 1-3-5, which covers the first six harmonics when placed through octave transpostion, and can be extended to do more. Nothing is special here.

Then there is the minor key, an inferior second role that manifested itself as a supplement of major. The role unfolded through various facets. Minor was only spoken through the "grammar" of major. Minor entailed the harmonic/melodic minor scale, whereas major entailed the natural major scale. The lower dominant was called the subdominant, whereas the upper dominant was simply called the dominant. The D-T progression was considered "stronger" than the S-T progression. It was in the dominant key that fugues answered. It was to the dominant key that sonatas modulated.

So we see how the traditional music theory underrated the entirety of subdominant function and the entirety of the harmonic/melodic major scale. Should it be used to guide us to composition, all the possibilities were not empowered. Harmonic dualism fixed it excellently.

Even with harmonic dualism, however, there is a meaningful analytical distinction between major and minor. See, the overtone series has long existed in the real world, and itself signifies presence. The undertone series traditionally does not exist, and signifies absence. Harmonic dualism does not change the fact. Hence, major, simulating the presence of overtone series, is presence of presence. Minor, simulating the presence of undertone series, is presence of absence. Meanwhile, it, simulating the absence of overtone series, is absence of presence. Finally, major, simulating the absence of undertone series, is absence of absence. That makes major even and minor odd. It explains exactly why inverting all the notes in a piece of music makes for a nontrivially different piece.

Of course, with the help of technology, the undertone series can exist. The point is, again, using them only sparingly. If both overtone series and undertone series are to be expected, the expressiveness of each is neutralized. What if only undertone series is to be expected? With the practice in the section above, this should be easy[3].

Observably, tonality is the quality of presence–absence, and is determined by the following arguments. First, what the audience presupposes as present, i.e. the expectation; next, the actual timbres played; finally, the intent of simulation which follows from the chord construction scheme.

Below is a table of permutations of all three elements and the resulted tonality. In the quality column, P denotes presence and A denotes absence.

# Expectation Timbres
Quality Tonality
1 Harmonic series Harmonic Up PP Major
2 Harmonic series Harmonic Down PA Minor
3 Harmonic series Subharmonic Up AA Major
4 Harmonic series Subharmonic Down AP Minor
5 Subharmonic series Harmonic Up AP Minor
6 Subharmonic series Harmonic Down AA Major
7 Subharmonic series Subharmonic Up PA Minor
8 Subharmonic series Subharmonic Down PP Major

Traditional music theory only explored case #1 since it viewed the minor tonality more as a modal variation than constructing chords downwards. In other words, it did not have full minor tonality. Harmonic dualism is advantageous as it encompasses the first two cases.

Be it harmonic or subharmonic, the timbre alone does not change the tonality. By playing subharmonic timbres, we step into case #3 and #4.

After that, come the unfamiliar lower four scenarios. The reason why they are unfamiliar is that changing the expectation is no easy. Most of us have been trained with harmonic timbres all the time. Most sounds in the real world tend to be harmonic (as opposed to subharmonic). There is not a dynamic to propel us to such a change. Hence, those four are likely to remain obscure in the future.

Appendix. On Nature

Natural is a term that is used in so many ways that its effect has practically been nullified. Tons of disputes arose from claims of something being "natural" or "not natural". People enjoy dismissing something by asserting it is "not natural". People also enjoy countering it by saying everything is "natural". Of course, none of those contributes anything to understanding.

I use natural synonymously with simple. It agrees with academic usage, such as natural logarithm. It does not make sense if you think about it because logarithm, like all math, is man-made, but in this case it refers to the simplicity in the particular base where its differential has the unity coefficient, while for other bases there is a multiplier[4].


  • Figure
Gregory Maxwell, Public domain, via Wikimedia Commons.


  1. Base 8 – The Best Number System!. Donald Sauter.
  2. Answer is the implicit intervals should be relabeled explicit and vice versa.
  3. This is answered right next. The undertone series will take the role of presence and vice versa.
  4. In Mandarin, the diatonic scale is 自然音階 (zìrán yīnjiē), which literally means natural scale. This section exists as a defence.

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