GMT3B: A brief reintroduction to writing music in 12 ED2 <12 19 28 34 ...]

# GMT3B: A brief reintroduction to writing music in 12 ED2 <12 19 28 34 ...] > Other installments: > [Section. I Parts 1-4](https://hackmd.io/@euwbah/GMT3A) > \> <u>**Section. I Parts 5-**</u> ###### tags: `GMT` [TOC] ## Interludium I - [x] The universal human ability to cognize and rationalize sounds. - [x] What is 'tonal' - [x] Why parallel fifths are 'bad' - [ ] **What 'chords', 'progressions', 'keys' really are** - [ ] **The pursuit of tonic clarity** - [ ] The unique properties of having 12 multiplicatively equal intervals that sum up to an octave. - [ ] The naming convention of notes - [ ] How to use consonance and dissonance - [ ] What 'scales' really are. - [ ] Zooming in on functional melody (on the harmonic implications of melody instead of themating development/phrasing) - [ ] How to 'voice' chords Recall the issue of rationalizing about 'color changes' in [GMT 3A I4: "The origin on functional/tonal harmony"](https://hackmd.io/@euwbah/GMT3A#I4d-Answering-the-main-question-The-origin-of-functionaltonal-harmony): the matter of being able to convey a 'color' using only one note at a time. Recall that we have already defined our own concept of 'tonality', but have yet to find an efficient way of administering and expressing 'tonality' towards the listener in our musical writing/improvisations. We have only dabbled with tonality in the monophonic context. Recall that we have the end goal of a constructive, personalised understanding of how to work in this tuning system, 12ED2, in a way that respects the nature of hearing. In this installation, we face the challenge of making sense of polyphony, and the challenge of being able to invoke **tonic clarity**: a succint conveying of a 'tonality' such that it leaves exactly as much room for interpretation for the listener as desired, assuming the listener conforms to the conventions of a particular musical culture. We continue our journey from _ars antiqua_ to see how these challenges were addressed. ## I5. What is a chord > **_Master:_ What is music? _Disciple:_ The science of regulating properly the movement of sound? _M:_ But what does it mean to regulate properly the movement of sound? _D:_ To control melody so that it sounds sweet. But this must be done in full conformance with the rules (_ad artem_). It is clear to me that one who misuses the sweetness of this art for worthless purposes, just as one who does not know how to apply the discipline (_ars_) where it is necessary, does not regulate sound properly. Rather, only someone with a heart full of devotion sings sweetly to the Lord. _M:_ You are right in thinking that sweet melodies are well-made only when they serve.** > ~ [Scolica Enchiriadis c. 900 (R. Erickson Translation, 1995)](https://archive.org/details/erickson-1995-musica-enchiriadis-and-scolica-enchiriadis/page/32/mode/2up?view=theater) [Etymologically](https://www.etymonline.com/word/chord), it refers to the cord of wire/string that is struck/plucked to produce a sound, it also refers to 'accord', which is etymologically similar to 'concord', which means 'in agreement' and 'harmonious'. Once again, the monochord comes back to mind. The word chord could also refer to the [geometric interpretation](https://en.wikipedia.org/wiki/Chord_(geometry)) (a straight line segment across any two points on a circular arc), which comes from _chorda_ which is bowstring in Latin. However, using the [Bernstein's "The Unanswered Question" excerpt](https://www.youtube.com/watch?v=Gt2zubHcER4&ab_channel=paxwallacejazz) as our map to this madness, we haven't even arrived in [_ars antiqua_](https://en.wikipedia.org/wiki/Ars_antiqua) yet. Out of the 6 minute lecture excerpt, we have only explored until the 1m 21s mark. Though, I hope this article's strong foundation in the first principles of music and tonality will make progress exponentially faster as we go along. Still being stuck in ancient times, this also means we first have to dig deeper than the etymological definitions given above. ### I5a. το 'τέλειο σύστημα' και 'αρμονία' (the _systema teleion_ and _armonia_) The Greek concept of [_systema teleion_](https://en.wikipedia.org/wiki/Musical_system_of_ancient_Greece) (i.e. "the perfect system") defines αρμονία (armonia/harmony) completly different than we would make of this word: > $\Large ``$ the concept of αρμονία did not have quite the same meaning that we lend to "harmony". It is not the pleasing effect of simultaneously-sounded concordant strings --- "harmony" in that sense was absent from classical Greek music --- but something more austere: αρμονία is simply the attunement of the strings to the intervals in the scale, and the pattern of the scale itself. It means that balance and order, not sweet pleasure, are the law of the world. $\Large "$ > ~ The Sleepwalkers (Arthur Koestler, 1959) To make _systema teleion_ a bit more concrete, here's a watered-down explanation of how it works: A 'scale' (_merely a collection of notes, no octaves required_) is constructed using two to four tetrads -- four notes spanning an interval of a fourth that include various specifically chosen ratios. There are 3 intervals between the 4 notes, which an appear in any order, but these 3 intervals are always fixed. Depending on the _genus_ (type/gene/class/"difficulty level") of the music, there are three different sets of intervals these 3 intervals could pick from. The first written down interval ratios for all three genera were given by [Archytas](https://plato.stanford.edu/entries/archytas/). Once the genus and mode was picked, the tetrachords would usually be stacked in fourths and fifths (but not necessarily in a way that always allows for octaves). Here are Archytas' _genera_: <iframe width="560" height="315" src="https://xenpaper.com/#embed:(2%2F3)(env%3A0267)%0A%23Diatonic_genus%0A1%2F1_8%2F7_9%2F7_4%2F3%0A%7Br4%2F3%7D_1%2F1_8%2F7_9%2F7_4%2F3%0A%0A%23Chromatic_genus%0A%7Br3%2F4%7D_1%2F1_32%2F27_9%2F7_4%2F3%0A%7Br4%2F3%7D_1%2F1_32%2F27_9%2F7_4%2F3%0A%0A%23Enharmonic_genus%0A%7Br3%2F4%7D_1%2F1_5%2F4_9%2F7_4%2F3%0A%7Br4%2F3%7D_1%2F1_5%2F4_9%2F7_4%2F3" title="Xenpaper" frameborder="0"></iframe> Of course, this is only a depiction of the musical practice of his peers in that time, but you get the idea. Eventually, [Aristoxenus](https://en.wikipedia.org/wiki/Aristoxenus#:~:text=Aristoxenus), a disciple of Aristotle, went on to make what ancient Greek music scholars deem the most '_influential_' edition of the _systema teleion_, which introduced the concept of an octave-repeating scale and standardized the modes (but not as we know them today at all). One of the closest decendents that etymologically model after this way of thinking about music, and is still a musical culture that is very much alive, is the **macam/maqam** system: <iframe width="560" height="315" src="https://www.youtube.com/embed/p8k72AgG-yI" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> :::info The Arabic word جنس (jins, plural أجناس ajnaas) is hypothesized to originate from γένος (genus), which shouldn't be surprising. The jins discussed in the video above are analogue to the tetrachords of a particular genus, and a 'scale' would be the result of 'pasting' various ajnaas in fourths or fifths apart, just like in _systema teleion_. ::: Finally, let us conclude that for the purpose of discussion in this article, a _'chord'_ refers to any collection of one or more notes which are _'αρμονία'_-ous, which simply means that they follow some Pythagorean idealist structure governed by [_the harmony of the spheres_](https://en.wikipedia.org/wiki/Musica_universalis). More concretely, any non-empty set of notes that can be thought of as rational ratios, or approximating rational ratios (more on this later), between frequencies. Just know that αρμονία doesn't necessarily mean harmonious, nor does it mean there has to be polyphony. > $\Large ``$ The word "harmony" ...signified to the Greeks the arrangement or ordering of sounds considere with respect to the melodic relationship between their pitches. It was not at all [concerned with] the mixture of several sounds striking the ear at the same time... We do not mean to say by this that music [involving] simultaneous sounds was excluded from Greek treatises on music, or that it is only a question of melody; one finds there in fact more that one passage where [the word αρμονία] is used in the sense we call "harmony". We only wish to demonstrate that the word "harmony" did not have for the Greeks the restrictive meaning that it has today, and that one would be in error if one one took it in this sense. $\Large "$ > ~ Histoire de l'Harmonie au Moyen Age (Edmond de Coussemaker, 1852) ## I5b. How to use 'chords' Now that we have established semantics and pragmatics for what constitutes a 'chord', let us peer into different culture's perspectives on polyphony. Bernstein is right in saying that the first instance of polyphony was that of parallel fifths and fourths. However, note that this is merely the first extant instance of written/composed excerpts that intentionally utilizes of this kind of polyphony, which are detailed in [Musica enchiriadis and Scolica enchiriadis](https://archive.org/details/erickson-1995-musica-enchiriadis-and-scolica-enchiriadis) -- two anonymous treatises c. 900. Although that is the case, I'm pretty certain that the concept of 'playing two different notes at once' predates that by at least 1500 years: as earlier discussed, in 600BC, Pythagoras and friends have already mentioned of the notion of sonority between two concurrently sounding pitches. Those who are familiar with the Classical theory syllabus should know the infamous 'no parallel fifths/fourths' rule that Bach infamously broke [at least 54 times](https://www.bach-chorales.com/ConsecutivesInChorales.htm). _At least_, because the only copies of BWV 253–438 we know of are not the original manuscript, and editorial corrections removed parallel fifths/fourths. Was Bach wrong? The editors? The no parallel fifths rule? It is time to take a trip in the sub-sub-sub-section time machine. The goal: figure out the role of 'chords' in modern music. :::warning Disclaimer: the following musical cultures/references only serve to convey a narrative of logical problem solving regarding the construction of polyphonic harmony. However I definitely wouldn't say that these were the actual thought processes behind each musical culture's decision behind a specific aesthetic. We merely examine, only in hindsight, a way in which western-centric harmony could have been constructed as a sum of these ideas. ::: ### I5bi. Polyphony v0.1 alpha: Parallel harmony Earlier in this article ([I3a](#I3a-The-harmonic-series)), we discussed the notion of naturally occuring overtone partials that conform to the harmonic series. The octave and fifth are the first two naturally occuring intervals in the harmonic series, and we know that if we divide the octave ratio by the fifth ratio, we can subtract a fifth from the octave to get a perfect fourth. Since it was already discussed that the lower harmonics are more prominent in nature, it is hence no surprise that these intervals are the first to show up in written history. Parallel harmony is _trivially_ the act of taking the first few harmonics in the harmonic series, slapping them on to every note, just because they are _already there_, so it is akin to the act of _amplifying_ these already naturally existing harmonics. Recall the virtual fundamental phenomenon when we were discussing [harmonic entropy](#I3e-Harmonic-Entropy): that the presence of harmonics of a non-present pseudo-fundamental pitch can induce a psychoacoustic fundamental pitch that is not actually in external stimulus. <iframe width="560" height="400" src="https://xenpaper.com/#embed:(1)(env%3A0267)(osc%3Asawtooth)%0A%7Br1%2F2%7D%0A1%2F1%0A1%3A2%0A1%3A2%3A3%0A1%3A2%3A3%3A4%0A1%3A2%3A3%3A4%3A5%3A6%0A%7Br2%2F1%7D_2%3A3%3A4%3A5%3A6%3A7%3A8_%23_virtual_fundamental%0A%7Br3%2F2%7D_3%3A4%3A5%3A6%3A7%3A8-_%23_virtual%0A1%2F3-______________%23_real%0A%7Br4%2F3%7D_4%3A5%3A6%3A7%3A8-_%23_virtual%0A1%2F4-______________%23_real%0A%7Br5%2F4%7D_5%3A6%3A7%3A8-___%23_virtual" title="Xenpaper" frameborder="0"></iframe>   Now, there should be a bell ringing, because it is painfully obvious how timbre and parallel harmony are related: - Timbre is the summation of sine waves of various amplitude at integer multiple frequencies of a fundamental pitch to evoke a type of 'sound' texture/vowel formant/instrument characteristic - Parallel harmony is the summation of pitches (themselves being summmations of sine waves) at integer multiples of a virtual or real fundamental pitch to evoke 'fullness' of sound. This means you could theorize that it is a summation of summation of sine waves. This brings us to our third _music theorym_: ### _music theorym 1.3_: :::success Parallel harmony is second-order timbre. Just as one can append integer mutliple frequencies of sine waves to affect 'brightness' and 'character' of a timbre and to create an arbitrary waveform/timbre of choice; one can also append integer multiple frequency of that arbitrary waveform to invoke a 'fullness' surrounding the fundamental. This is the result of prior discussed virtual fundamental/combination tones phenomenon. This is much more effective when the integer multiples of choice are simple, concordant ratios that are prominent in nature and easily evoke the virtual fundamental by means of [nonlinearity of the ear](https://agilescientific.com/blog/2014/6/9/the-nonlinear-ear.html) or otherwise. Note that whilst [Musica Enchiriadis/Scolica Enchiriadis](https://archive.org/details/erickson-1995-musica-enchiriadis-and-scolica-enchiriadis/page/24/mode/2up?view=theater) did mention specific rules as to how to 'harmonize' using diatessara (through fourths), diapentes (fifths), and diapasons (octaves), its instructions were not rigorously defined. Singing _organum_, i.e. embellishing the melody in fourths/fifths, was an improvisatory process. Further reading reveals that parallel harmony relegated the effect our 'sesquisharp cancellation conjecture' had to subjective 'aesthetics'. Rather, its rigour was focused on the vertical pleasantness of sonority at each instantaneous point in time. ::: And to nail that point in, here is mary had a little lamb in MASSIVELY parallel harmony (volume down recommended): <iframe width="560" height="315" src="https://xenpaper.com/#embed:(1)(env%3A4357)%0A%7Br12%7D4%3A5%3A6%3A7%3A8%3A9%3A10%3A11%3A12%0A%7Br%6010%7D4%3A5%3A6%3A7%3A8%3A9%3A10%3A11%3A12%0A%7Br%6010%7D4%3A5%3A6%3A7%3A8%3A9%3A10%3A11%3A12%0A%7Br2%7D4%3A5%3A6%3A7%3A8%3A9%3A10%3A11%3A12%0A%7Br2%7D4%3A5%3A6%3A7%3A8%3A9%3A10%3A11%3A12%0A%7Br0%7D4%3A5%3A6%3A7%3A8%3A9%3A10%3A11%3A12%0A%7Br0%7D4%3A5%3A6%3A7%3A8%3A9%3A10%3A11%3A12." title="Xenpaper" frameborder="0"></iframe>   There are already instruments that use 'parallel harmony' for the effect of a 'fuller sound', namely: - Harpsichord with a 'choirs' of strings, where one can activate a stop/lever/coupler to make one key sound 2 notes an octave apart. - 12-stringed guitar/stringed instruments with courses such that the two 'unison' strings are tuned an octave apart. - Pipe/Tonewheel/Synth Organs that has 'stops' which sound a pitch (with varying individual timbres) at integer multiples apart, up to the 8th or even 16th harmonic. - Additive synthesis and layered synths work on essentially the same principle as the xenpaper demo above. So, back to the issue of answering why 'parallel fifths' are bad in the common practice era? Here's a hypothesis: - Chorale music and small-ensemble music in the common practice era had the huge task of evoking the most harmony out of the fewest notes/instruments possible. - Thus, every single part has to be efficient in 'making color changes'/resolving tensions/creating dissonance/harmonic movement. - If we use parallel voices, the above phenomenon occurs when we don't want it to: it reduces the multiple voices into one conglomerate 'super voice', which brings out the virtual/real fundamental and makes it fuller, but it also reduces many distinct parts into one 'super part', reducing the polyphonic ability of the ensemble. - Thus, it is generally not an efficient use of voices to use parallel harmony when voices are limited and polyphony/counterpoint is ideal. - But if you have some contrapuntal voices lying around with nothing to do for a specific part of the song, there is nothing wrong with put them in parallel to richen any particular voice that would benefit from being brought out. Then, there's this: <iframe width="560" height="315" src="https://www.youtube.com/embed/W71rF1hKJOM" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>   Now that _that_ is answered, its evident that parallel harmony -- without concern for other factors like mode/aesthetics/scale -- is still quite useless when it comes to attempting to convey a clear 'tonic' to the listener. Hearing a bunch of perfect fifths and fourths wouldn't provide the ear with much new pitch information, since these basic intervals are all already contained in the overtone series of the melody. Harmonizing a melody in parallel does not help with the main issue of **the ear choosing whatever it wants to hear as the tonic**, any more so than melody alone would be able to achieve tonic clarity. It is time to graduate and proceed to drone music. ### I5bii. Polyphony v1.0 Release Candidate 1: Drone music In the 1500s, the Baltic, Byzantine, South Asian/Indian regions developed music that all shared a similiar trait -- a mainly monophonic/stratific melody (poly-monophonic, without regard for vertical inter-melody harmonic interactions), set over a 'drone', ['_tambura/tanpura_'](https://musicwithaditi.com/taanpura/), ['_ison_'](https://en.wikipedia.org/wiki/Ison_(music)) to evoke something that is known as 'iso-polyphony', i.e., polyphony due to the result of a slow moving/constant low drone part. Recall the discussions in section [I4d](https://hackmd.io/@euwbah/GMT3A#I4d-Answering-the-main-question-The-origin-of-functionaltonal-harmony): that the notion of changing colors/moving harmony is inherently enabled by relative pitch memory. Recall how finnicky the concept of 'tonic' is. In a monophonic context, the 'tonic set' of notes can simply be replaced by any distracting, conflicting, contrary stimuli that can cause the listener to forget/reset their auditory context. Hence, it makes sense to think of the drone as a natural problem solving step: serving to ground the tonic with a constant note that is always present, such that melodies can freely explore without the perceiver losing the intended context of the melody. A common question that would arise here: Why do the drones generally seem to be a lower note? Couldn't a note above the melody be used as a drone? The same could be asked about _pedal points_ vs _inverted pedal points_ in classical convention. Well, when a lower note is droned, it contains harmonics, of which a higher melody note can choose to lock in to. Earing out intonation becomes trivial when the performer can 'lock' into the drone -- if a higher drone/inverted pedal were to be used, one cannot rely on the already existing overtone partials in the drone and must rely purely on audiation and small corrections. Recall the [Hayden Chisholm intonation masterclass](https://www.youtube.com/watch?v=CCC_WvEfeZE&t=1213s&ab_channel=StrongWaterPressure) video and attempt to do the exercises with a higher drone instead, and you will feel the difference. Before we continue, check out some present-day music that rely on this type of 'polyphony': - [Ustad Bismillah Khan - Live in London](https://www.youtube.com/watch?v=kHaEkkOhMB4&ab_channel=navrasrecords) - [Visoko multipart singing from Dolen and Satovcha, South-western Bulgaria](https://youtu.be/975ROm5PbK4) - [the MAQAM project #1a - Bereshit (Maqam Rast)](https://youtu.be/XJVxUBHWNQk) - [Sandeep Mishra Sarangi Vivek Mishra Tabla (Thumari Raag Mishra Tilang)](https://www.youtube.com/watch?v=ucxOXqOLToI&ab_channel=SandeepMishra) - [Throat singing from different continents](https://www.youtube.com/watch?v=TPSq-_jOLJ4&ab_channel=KUULAR) So how are changes in 'color' evoked in this system? We know that we are able to distinguish concordance/discordance/sonority of two or more simultaneously sounded pitches, so in this system, we can attribute color using the concordance-discordance metric. <iframe width="560" height="180" src="https://xenpaper.com/#embed:(1%2F2)1%3A1_1%3A2_2%3A3_3%3A4_4%3A5_3%3A5_5%3A6_%0A5%3A8_8%3A9_8%3A15_9%3A16_32%3A45_15%3A16" title="Xenpaper" frameborder="0"></iframe> Above, you can listen to the 12 familiar intervals within an octave, sorted approximately by concordance (there isn't enough context to rigorously define concordance yet). It is clear that having a drone at the bottom helps to give more context to the 'melody' and 'color'. After all, without the drone, we will hear nothing more than a 12-tone row. Being optimistic, let us bring back the fair-share sequence and our previous experiment we did with one note: can we decide on a tonic? <iframe width="560" height="315" src="https://xenpaper.com/#embed:(3%2F2)_%0A2%3A3_32%3A45_32%3A45_2%3A3%0A32%3A45_2%3A3_2%3A3_32%3A45%0A32%3A45_2%3A3_2%3A3_32%3A45%0A2%3A3_32%3A45_32%3A45_2%3A3%0A32%3A45_2%3A3_2%3A3_32%3A45%0A2%3A3_32%3A45_32%3A45_2%3A3%0A2%3A3_32%3A45_32%3A45_2%3A3%0A32%3A45_2%3A3_2%3A3_32%3A45" title="Xenpaper" frameborder="0"></iframe> So which two-note set felt more 'tonic' to you? $2:3$ (P5) or $32:45$? (Aug4) Repeat the experiment starting on the 5th line (click `>` on the left column) with loop mode on. Is the answer still the same? How about these? <iframe width="560" height="315" src="https://xenpaper.com/#embed:(3%2F2)%7Br3%2F2%7D%0A3%3A4_4%3A5_4%3A5_3%3A4%0A4%3A5_3%3A4_3%3A4_4%3A5%0A4%3A5_3%3A4_3%3A4_4%3A5%0A3%3A4_4%3A5_4%3A5_3%3A4%0A4%3A5_3%3A4_3%3A4_4%3A5%0A3%3A4_4%3A5_4%3A5_3%3A4%0A3%3A4_4%3A5_4%3A5_3%3A4%0A4%3A5_3%3A4_3%3A4_4%3A5" title="Xenpaper" frameborder="0"></iframe> <iframe width="560" height="315" src="https://xenpaper.com/#embed:(3%2F2)_%0A9%3A16_3%3A5_3%3A5_9%3A16%0A3%3A5_9%3A16_9%3A16_3%3A5%0A3%3A5_9%3A16_9%3A16_3%3A5%0A9%3A16_3%3A5_3%3A5_9%3A16%0A3%3A5_9%3A16_9%3A16_3%3A5%0A9%3A16_3%3A5_3%3A5_9%3A16%0A9%3A16_3%3A5_3%3A5_9%3A16%0A3%3A5_9%3A16_9%3A16_3%3A5" title="Xenpaper" frameborder="0"></iframe> <iframe width="560" height="315" src="https://xenpaper.com/#embed:(3%2F2)%0A3%3A4_4%3A5_4%3A5_3%3A4%0A4%3A5_3%3A4_3%3A4_4%3A5%0A4%3A5_3%3A4_3%3A4_4%3A5%0A3%3A4_4%3A5_4%3A5_3%3A4%0A4%3A5_3%3A4_3%3A4_4%3A5%0A3%3A4_4%3A5_4%3A5_3%3A4%0A3%3A4_4%3A5_4%3A5_3%3A4%0A4%3A5_3%3A4_3%3A4_4%3A5" title="Xenpaper" frameborder="0"></iframe> <iframe width="560" height="315" src="https://xenpaper.com/#embed:(3%2F2)%7Br5%2F4%7D%0A2%3A3_5%3A8_5%3A8_2%3A3%0A5%3A8_2%3A3_2%3A3_5%3A8%0A5%3A8_2%3A3_2%3A3_5%3A8%0A2%3A3_5%3A8_5%3A8_2%3A3%0A5%3A8_2%3A3_2%3A3_5%3A8%0A2%3A3_5%3A8_5%3A8_2%3A3%0A2%3A3_5%3A8_5%3A8_2%3A3%0A5%3A8_2%3A3_2%3A3_5%3A8" title="Xenpaper" frameborder="0"></iframe> Now play through them again, but in reverse order. Did you choose the same tonics both times? Take a break, touch grass and come back to it in the reverse order first, then forward order second. Did you choose the same tonics every time? Could you rationalize those decisions in terms of concordance score? Most probably, there should still be discrepancies. For example, if you picked $4:5$ over $3:4$ as the tonic, you've chosen the more _discordant_ interval of the two as your tonic. What gives? And changing your mind after listening to the excerpts in a different order? If your ears are trained in the western tradition, then the global context outside the scope of each individual excerpt would still affect your choices: <iframe width="560" height="315" src="https://xenpaper.com/#embed:(3%2F2)%0A4%3A5%3A6--.%0A3%3A4_4%3A5_4%3A5_3%3A4_4%3A5_3%3A4_3%3A4_4%3A5%0A4%3A5_3%3A4_3%3A4_4%3A5_3%3A4_4%3A5_4%3A5_3%3A4%0A%7Br3%2F4%7D%0A3%3A4_4%3A5_4%3A5_3%3A4_4%3A5_3%3A4_3%3A4_4%3A5%0A4%3A5_3%3A4_3%3A4_4%3A5_3%3A4_4%3A5_4%3A5_3%3A4%0A%7Br4%2F3%7D%0A3%3A4_4%3A5_4%3A5_3%3A4_4%3A5_3%3A4_3%3A4_4%3A5%0A4%3A5_3%3A4_3%3A4_4%3A5_3%3A4_4%3A5_4%3A5_3%3A4" title="Xenpaper" frameborder="0"></iframe>   You ear utilizes the harmonic context of more than just two intervals or three notes. If you hear the dyad $4:5$ (maj3 dyad) of $1/1$ in the first section as tonic, you will latch on to those notes. Then, in the second section that is transposed by $3/4$ (down a P4), the $4/3$ (P4) of the new key should sound familiar to your ears, since it is the same $1/1$ from before. Even if you were to start listening from the second section (click `>` on the left column), you would still find yourself hearing the $3:4$ (P4 dyad) transposed down $3/4$ as the 'tonic'. In other words, the [sesquisharp cancellation conjecture](https://hackmd.io/@euwbah/GMT3A#I4e-Music-Theorym-12-SesquisharpSemitone-Cancellation-Conjecture) persists between contexts: We do not completely 'forget' what we chosen as the tonic, since every 'cancellation' from the tonic set will be felt as a cancellation, and every 'return' to the tonic set can be recalled and felt like a return. Depending on your accustomed musical culture, there's a high chance that it would take more than a single cancellation to make you truly forget and reset the notes in the initial 'tonic set' (and if you have perfect pitch and strong willpower, nothing can possibly change the 'tonic' if you didn't want it to). To add, you could probably assume that your 'tonic set' usually contains 5, or even 6 notes, after all, we only 'cancel' a pitch from the 'tonic set' if it is a around a semitone/sesquisharp away. Take this for example: <iframe width="560" height="315" src="https://xenpaper.com/#embed:(3%2F2)(env%3A0097)%0A8%3A9_4%3A5_2%3A3_16%3A27%0A%7Br9%2F8%7D_9%3A10_3%3A4_2%3A3_9%3A16%0A%7Br10%2F9%7D_5%3A6_3%3A4_5%3A8%0A%7Br6%2F5%7D_9%3A10_3%3A4%0A%7Br10%2F9%7D_5%3A6%0A%7Br3%2F5%7D_1%3A2" title="Xenpaper" frameborder="0"></iframe>   Despite all these 'key changes' (fundamental pitch shifts denoted by `{rx/y}`) and intervals being thrown around, you have most probably heard everything in 'one key', instead of each new 'drone' being a new key. In fact, you would probably recognize the entirety of it as the sound of a _pentatonic scale_. Do musicians in the middle ages do that? Is this a phenomenon that is only restricted descendants of the european classical music culture? Would other music cultures perceive music 'in the moment' without relying on past context? These are all hard to answer questions, but worth figuring out. Whatever the case, it is clear that a drone is insufficient information if the goal is **tonic clarity**. Because in the second last example, we have shown that both $4:5$ (maj3 dyad) and $3:4$ (P4 dyad) can equally be given the role of 'tonic' depending on context. (For example, think about the resolution of the fourth to the third of any _'plagal IV-I cadence'_, and compare that with the first three notes of Stevie Wonder's _"Overjoyed"_) The issue of two different intervals with varying concordance-discordance scores being equally 'tonicizable' is also made evident by the _genus/systema teleion_ and _maqam_ which allow for their 'tonics' to be either fourths, fifths, even sixths/thirds apart, showing that these intervals are on equal footing in terms of 'tonicizability'. And no, these systems do not assume octave equivalence, so fourths are not inverted fifths. Now the next logical step would be to ask: What is one small change we can take to make it absolutely clear that we only want one specific color to be heard as the 'tonic' over all the others? Perhaps 'modes', 'scales', or 'key centers', since these always have been synonymous to the defintion of 'tonic' haven't they? Thus, what we need lies in cultural entrainment. In other words, a bunch of _seemingly_ random rules that form a common language (_vocabulary_) for everyone to agree on. Is there a method to this randomness? Could music have turned out differently if we agreed on a different set of rules? Does the phrase 'music is a universal language' hold? ### I5biii: Polyphony V1.0 Release Candidate 2: Renaissance Cadences, Gamut, and 'the modes'. We now teleport to Europe in the 16th century. First, let us look at how this musical culture approaches the idea of 'resolution' and counterpoint and see what we can take from it. <iframe width="560" height="315" src="https://www.youtube.com/embed/jaCRUdxTRSM" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>   The important takeaway is that harmony is merely a biproduct of individual parts/voices working in tandem. At this point in time, the concept of 'chords', let alone 'chord scale theory' is completely foreign and not developed. It would take another 400 years for such ideas. :::warning This is not to say that 'chords' or 'chord scale theory' is the one true correct™ idea. In fact, there is much to learn from the 16th century conception of harmony when it comes to efficiently evoking **tonal clarity**, which modern theories completely relegate to 'aesthetic decision' and 'taste'. In other words, there are much simpler, fundamental ways to think about, and construct harmony, which can be extrapolated to create an even fuller understanding of modern harmony. ::: Going back to the Renaissance musical culture: 'modal theory' was well and alive in monodic chants/hymns (i.e. the general idea of having more than distinct part/note at once, aka, polyphony). So, theorists of that time had to devise a way to combine recent advancements in polyphony/counterpoint and modal music culture (which we can find similarities in their construction to the construction of the genus/maqam): <iframe width="560" height="315" src="https://www.youtube.com/embed/lyq48eybjZw" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>   Thus, a completely different way of conceiving what notes are 'accepted': <iframe width="560" height="315" src="https://www.youtube.com/embed/IRDDT1uSrd0" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>   All 'arbitrary rules' were rather stringent, which meant that we have to conjure up a new strain of music which broke those rules, but was still acceptable to listen to, and make rules about breaking those rules: <iframe width="560" height="315" src="https://www.youtube.com/embed/6VF6YkCNRyE" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>   So many arbitrary rules, so many arbitrary rules about breaking the rules. All this in a musical culture whose written music would leave _accidentals_ up to the performers educated guesses informed by their musical culture. But one thing is certain: these rules would enforce a culturally-entrained ear to perceive the 'tonic' in a certain way. If you dug deeper still into the theories and treatises in the 16th-17th centuries, you would be able to spot seeds and saplings of the present day conception of music. Though, the details are beyond the scope of this writing, so we may go into that another time. So how do we make sense of these arbitrary rules in the modern context? We can naively summarize (leaving out a lot of details) the general idea of the 16th century European concept as such: