Array indexes starting at 0 is unintuitive for beginners, I would never recommend a programming student learn arrays starting at 1 just to make it easier.

Proper playing technique is often unintuitive for the beginning musician, but encouraging it for accessibility will be destructive to their progress.

Conceptualizing a new difficult concept in a way that makes sense to you is good, foregoing convention is bad.

We might say, that finding the Rosetta Stone hindered those who knew Greek trying to learn hieroglyphs.

I was deathly afraid to try woodworking until someone showed me how to construct familiar angles on the machines using basic trigonometry. After I had a tiny relationship formed, I was able to experiment on my own (and then adopt woodworking vocabulary, to become entrenched in that community).

I think using some arithmetic rules to entice someone who knows arithmetic, but otherwise is awkward around music theory, is an agreeable compromise to get them on the way.

The biggest win from that POV would actually be numbering the diatonic scale degrees starting at 0 for the tonic, and the diatonic intervals starting at 0 for the unison. Unfortunately, it heavily conflicts with all sorts of existing notations. But once you think of the third as a "2" interval, the fourth as a "3", the fifth as "4" etc. you can arbitrarily add intervals together and to scale-degrees, without having to constantly correct for the archaic 1-based counting.

  V7 -> I 

  (x+2, x+5, x+7, x+11) -> (x, x+4, x+7, x+12)

{2, 5, 7, 11} -> {0, 4, 7, 12}

There's actually established notation in music theory for this[0]. Because it IS a super convenient way to look at notes. Doing otherwise would be akin to using Roman numerals for calculus.

From personal experience, I worked with this notation a little bit. I spent a lot more time getting the hang of traditional notation. Not only is integer notation intuitive immediately, but number combinations like {0,4,7,12} become memorable very quickly and yes when I look at it (even if I saw it outside of musical context!), I would know it's a major chord.

That is, personally, I learned about the traditional notations (because you can't learn music theory and avoid those, or miss out on most of the material :) ). But that took me deliberate effort. On the other hand I wrote some python classes for converting between notes and chords and traditional notation and integer notation, seeing the patterns in those numbers required no effort at all, it's right there. The traditional notation has of course the same patterns, but they are obscured by layers of translation.

I think integer notation could be used a lot more, especially when teaching music theory. But traditional notation is also useful, partly because many people know it, partly because integer notation is less suitable for certain instruments.

[0] https://en.wikipedia.org/wiki/Pitch_class

The problem with "V7 -> I" is precisely that it hides all the internal structure and encourages rote learning. Rather than encouraging the musician/composer to think about what that resolution actually consists of in terms of shifting intervallic relationships within the chords, it encourages you to just learn the transition itself.

Which of course is also its strength!

You might think that after centuries of musical composition across many different cultures that we'd have fully explored all the 4096 scales that exist within 12TET or even all the 4096 scales that exist for any given 12 tone tuning system.

But this is far from the case - witness how revolutionary Messaien's modes were. The conventional language is great for conveying meaning/intent/practice if the goal is to remain solidly inside the parameters of western musical practice circa 1100 to the present day.

But it's fairly inadequate if your goal is explore the rest of the music possibilities presented by psycho-acoustics, or even just those derived from (say) musical cultures which use microtones.

I also note that there's not a single comment in this thread regarding rhymthic structure, again reflecting the western emphasis on a particular (simplistic) understanding rather than the highly developed music cultures built around rhythm across Africa and Asia.

This is so true. I've been wanting to learn about rhythmic theory for about as long as I've been studying music theory (harmony). But while you can find youtube videos and webpages full of explaining musical harmony all over the place, information about rhythmic theory is far and few between.

Most I've been able to find is either very abstract, like Euclidean Rhythms, which are interesting and weird but only explore equidistant grids (no swing). The other information is usually about really specific cultural rhythms. Both are good and useful, but I feel like they combine into less than 20% of the total theory about rhythm that I believe should be there. There's really a lot to it and I'm still looking for some fundamental theories about repetition and expectation and tension/release. The latter is very important in rhythm, but as far as I'm aware, the music theory about harmonic tension and release (chord progressions etc) seems so much more complete than what I've been able to find about rhythm.

If you got any good pointers, I'm all ears :)

Scale -> basis makes no sense to me. The scale climbs up and down, like how you scale a ladder. The basis… does not do that.

Tonic, dominant, subdominant are used mostly in discussions about functional harmony. The biggest reason why it would be a total disaster to rename tonic -> primary is because we already use the terms primary and secondary here! For example, V is the “primary dominant” but nobody actually says the word “primary”. Primary chords are taken from the scale of the tonic, and secondary chords are taken from a different scale. So V/V is a “secondary dominant”.

I had in mind a reform akin to how modern lisps have rename CAR and CDR to first and rest respectively. For example the Lydian Mode, could be referred to by a codeword that can be unwrapped to understand what exactly it is.

In general I am always in favour of trying to advance the expressiveness and meaningfulness of notations in all fields, and I think that is something that should be continuously evolved, trying to find better way to say things.

I can get behind car/cdr -> first/rest.

I could get behind renaming the modes, but I’m not sure how to rename them. I would be hesitant to name them after notes or using numbers, since that would be confusing. As it is, I can say “C dorian scale” and that’s a pretty concise and unambiguous name for a series of notes. Maybe I could say “C minor, natural sixth scale”. It’s a bit verbose and modes are fairly rare. It’s also a bit weird—would I then say “A dorian scale” as “A minor, sharp sixth scale”? Already I’m a bit confused by this terminology. Maybe “A minor, major sixth scale”? The “major minor” is a bit weird.

The trick with advancing expressiveness and meaningfulness is that 1) you are going to have to memorize new meanings even if the words are familiar and 2) if it’s not usable by experts, it won’t be adopted by experts and you’ll end up with at worst, multiple incompatible sets of terminology.

Face it—I’m going to have to do a fair bit of memorization to learn music theory. It is unavoidable with music theory as it is with other subjects. So I might as well pick nice, short words like “scale” which make some intuitive sense, rather than pick “ordered basis” which is a total mouthful. After all, I will only spend a short amount of time memorizing terms, and a much, MUCH longer time using them.

It's really that much more intuitive. Give somebody CEGC' and GBD'G' or give them {0,4,7,12} and {7,11,14,19}. And as a bonus, if you can read clock, you already know how to do modulo 12 arithmetic! 14:00 is 2pm, 19:00 is 7pm, therefore that last one can also be written as {7,11,2,7}.

I don't want to do away with traditional notation, it has its advantages. But I do think that integer notation can be used very effectively and should be used more, when teaching the fundamentals of music theory.

I'm a little put off by the way you wrote that comment.

That link provides an introduction to MST terminology, starting from somewhat math/nerd angles and moving gently.

You will find that MST provides clear terminology for the sort of thing you're trying to talk about/say.

The system is most useful for describing / exploring certain types of atonal music. I’m going to make an unfair generalization—most people are interested in tonal music. You can see that the references here are to Forte’s The Structure of Atonal Music and Rahn’s Basic Atonal Theory. So if that’s your jam, by all means, go and make atonal music. But I’m a fan of using modes of common scales, so I like having concise names like phrygian, mixolydian, and dorian.

It's still unclear how much "tonal music" is simply a shared set of cultural assumptions and practices, and how much arises from acoustic physics and acoustic perception. So ... rather than a priori priviledging a specific group of interval sets ("major", "minor" plus "a few modes"), why not start by exploring the full universe of all possible (12TET) interval sets (aka "scales"). By using an explicitly mathematical approach one can bring some analysis techniques to bear on the "full universe" that are not readily available when stuck in traditional western theory and notation. To me, this approach accomplishes two things. The first is that it offers a bridge to non-western musical traditions. The second is that it offers a common core from which to understand both "tonal" and "atonal" music, particularly their similarities and differences.

Finally, its a particular bug of mine when people connect "modes" and "scales" as you do in the your final sentence. What is important in (almost) all musical traditions are not the specific notes used to form the pitches that make a piece of music, but the intervals between the pitches, and their ordering. There's significant evidence that we are vastly more sensitive to relative pitch (intervals) than absolute pitch - play the same series of intervals starting from a different tonic/root and we experience it in almost the same way. Change the interval series (i.e. the scale), and we hear a much more noticeable difference. So, the major scale is just an interval series, and the "modes of the major scale" are also just interval series. There is no inherent relationship between them, other than one can construct them by rotation. The modes do not "belong" to the major or minor scales, nor vice versa (indeed, in fact the major and minor scale are just modes too ... just an interval series).

And now really finally: we actually don't have special names for most of the possible interval series that can constructed from the diatonic notes. We have a very limited number, and from my reading and understanding it is unclear if the ones we have names for are priviledged by physics and perception, or are mostly the result of historical precedent.

You actually want to learn these names, because they characterize the diatonic scale. Basically, pick seven contiguous notes on the circle of fifths, you get the pitch-class set of a diatonic scale. Then put the notes in the set in pitch order, and pick a tonal center. The default, naïve choice (pick the "flattest" note as your tonal center) is called Lydian mode. It can be interesting, but it has a drawback in that it forgoes the subdominant relationship. Picking the next-to-'flattest' note gives you Ionian, which solves this (the fourth scale degree forms a "perfect fourth" with the tonic, which is the flip side of a perfect fifth. Having more notes that can be related to the tonic makes for more musical possibilities). This is one simple explanation of the diatonic scale we ordinarily use.

One other quirk that also explains the "weird historic names": in traditional music theory, sharps and flats are definitely not treated equally, the way that would be implied by 12-equal temperament. The musically-relevant distinction is simple enough to explain: taking one example, F# "wants" to step up to G, whereas Gb "wants" to step down to F. This means that the "circle of fifths" turns into more of a helix of sorts that can be extended in both directions, in principle indefinitely. This difference cannot be "heard" directly; it's all about characterizing how the notes "work" in a piece of music.

After learning about the diatonic scale degrees, the next sensible step would be to start learning about counterpoint, which is based on simple definitions of consonance and dissonance between scale degrees. Then move on to thoroughbass and harmony.

For a total beginner, I might agree. But thinking about the scale degrees (Do, Re, Mi etc.) is also a totally viable approach (even as a starting point), and it's extremely helpful to learn about how the two relate ASAP so you aren't left holding a mess of seemingly-contradictory "theories" in your head!

whats the difference between them?

And apparently the human ear can be train to ignore this slight difference (which everybody does because we are used to 12-TET). And this, for me, kind of throws the whole "simple integer fraction ratio == pleasing harmony" a bit into question. It's probably not wrong, but there's definitely more to it. But it's hard to explore, because you need the exposure to get used to the new microtonals if you want to experiment with it. Definitely very hard to test scientifically because it depends so much on a particular person's musical background and education.

Uhhh....no? A flat or sharp can be the tonic, which does not lead to any other note. A note is named flat or sharp to maintain proper interval distance without repeating letters within the scale. A flat is typically not a leading tone because of interval distance, but can still want to resolve up or down. Any note can be consonant or dissonant, leading or resolved depending on the context.

You're right, and this is most often seen with flats; when Bb is the tonic, it doesn't "want" to resolve to A. --Of course, a modulation to F - the next-sharpest key in the cycle of fifths - is enough to change that. A "stable" sharp note is seen starting from the key of D, where F# is a stable third. But in the context of enharmonic notes, what I said generally holds. F# and Gb, to take the most common example, are so far in the 'extended' cycle of fifths that whenever both appear in the same piece, one can generally assume that the rule holds.

> A note is named flat or sharp to maintain proper interval distance without repeating letters within the scale.

Notes are not just named flats or sharps; at least in principle, there can be double, triple etc. sharps, and double, triple etc. flats. This is done in order to properly notate modulations in the cycle of fifths; one does not arbitrarily "switch" from sharps to flats, but just keeps adding to them.

I took a few years of music lessons (granted light on theory) but I never heard this idea.

Is this exclusive to notes that are not in the current key? For example the f# in Gmaj doesn't want to step up to G any more than B wants to step up to C in Cmaj.

(Actually, this was thought of most often in terms of hexachords, where there is no Ti and one would always use Mi-Fa for a half-step interval. I rewrote it in terms of scale degrees to avoid confusion.)

I.e. the F# in Gmaj does "want" to step up to G, in basic structural terms. We call the places where this happens most properly "cadences", and they are among the main structuring elements in a piece of music. (This Ti-Do - F#-to-G or B-to-C motion is then called a "cantizans", since it appears most prominently as a "canto" or "sopran" cadence. It's most typically seen as Do-Ti-Do, where the first appearance of Do is first "prepared" in a context that makes it a consonance, but then sounds as a dissonance as the other parts shift to a dominant chord (Sol and Re), so it's allowed to resolve to the leading tone.)

"tonic" is another way of referring to the "key" of the song.

If I'm writing a song and want to make La the tonic, what should I do?

In the context of an electronic synth or keyboard, C# and Db are perfectly the same. In almost all traditional music, though, instruments are almost never tuned to equal temperament, so that C# and Db then mean different things.

As a famous example, Bach was very fond of his well-tempered clavier, which was a tuning method that is to be found somewhere between pythagorean tunings (usable for only one scale) and equal temperament (everything sounds equally lifeless). Bach's tuning was therefore a very carefully chosen compromise between being able to play the most common scales cleanly and sacrificing some scales in exchange for more precise harmonic relationships.

All famous European composers were experts at squeezing nice harmonics out of an un-equal tuning system where some key combinations just so happened to always sound horrible. It was called "Wolfsquinte" if you hit the wrong combination. If you limit yourself to only equal tuning, your are missing out on the slight harmonic differences that are the historical basic for all contemporary harmonic progressions.

If you start with one fixed frequency and then derive everything else as pure harmonics, that is called "Just Intonation". Wikipedia has a table on how those perfect harmonics differ from the notes that are mapped to your 12 keys: https://en.wikipedia.org/wiki/Equal_temperament#Comparison_w...

And lastly, by only thinking about notes as the keys on your keyboard, you completely lose the concept of musical Commata, which are when the true frequencies of two notes happen to fall onto the same key on a keyboard.

Others in this thread already pointed out that C# wants to go up while Db wants to go down. The reason for that is that for violins, cello, some flutes, and Organs, they are not the same notes. They just happen to be rounded onto the same key on most modern electronic keyboards.

Since you asked to learn about harmonics, I would therefore advise against focusing on 12-keys, because that would hide the underlying complexities from your view. Harmonics are in my opinion best studied on analogue instruments like a violin, where you can actually play a musical comma, as opposed to pretending it doesn't exist.

For a simple introduction, I thought it would be better to skip the subject of tunings altogether and just focus on the structural implications that one would "read" in an actual piece of sheet music.

But in my opinion, how the notes that you play simultaneously sound together is the core of music theory. A good chord progression can feel like a journey and it will set the mood of the entire song. Therefore, I would argue that note frequencies and the reasons why certain tuning systems are preferred in certain situations are the key concept to understand if one wants to elicit a chosen mood in the listener.

But I brought up Just Intonation here because people tend to use that when they sing freely. So I would see this as a way to first understand why people (and famous composers) act the way they do before one tries to categorize things scientifically.

Start from a tuning system. 12TET (12 tones per octave, equal temperament) would be conventional if you live in a contemporary western culture, but there are others.

Next step: pick a set of intervals (any number, though 4, 6, 7 are common numbers). Congratulations, you have a mode.

Next step: pick a root/tonic. Congratulations, you have a scale.

Next step: play it, over and over and over again till you can do so without thinking about.

Next step: do the above steps again, with different choices. And again, and again, and again ...

Next step: understand the differences in the "feel" of the different intervallic relationships present within the scale.

Next step: understand the impact of presenting these intervallic relationships when ordered in time (i.e. melody, one note after another), or when presented all at once (i.e. harmony, chords)

Then, when you're ready, tackle Music Set Theory: https://ianring.com/musictheory/scales (sadly limited to 12TET tuning, but it's still a fabulous start)

Bon Voyage!

Naming stuff 1-12 and I, II, IV etc.

I found this much more relatable.

Only keyboards seem to favor C major with their layout, which makes applying these "relatives" a bit more cumbersome, if you don't want to use that scale.

At least on guitar everything looks the same.

Once you're familiar with the cycle of fifths, modulation becomes fairly trivial. You start to apply the appropriate corrections "sharpen this, flatten that" simply as a matter of habit.

In fact, this is precisely how modulation arose historically; it used to be the case that all music was notated diatonically or nearly so (only distinguishing between B and B-flat!), but performers would implicitly "add" sharps and flats to make it sound good depending on the context - a practice known as "musica ficta". You're right though that on a plucked string instrument everything looks the same - and historical intabulations (i.e. tablatures!) meant for plucked string instruments are actually an important source that gives us info about how musica ficta was played in many cases.

Sure, the steps are the same from the prime up, but with C major you end up on all white and with others you can end up on blacks here and there.

Yeah, they are largely a historical artifact from before the days of equal-temperament tuning. A few things they have going for them:

* They give you a visual reference point for an absolute pitch. It's easy to find any specific note by looking for it relative to the unique pattern of black keys. C is always to the left of the pair of blacks. Other instruments require different techniques to keep track of where you are since otherwise all note positions look the same. (For example, guitars have fretboard markers and players have to be careful to count frets correctly.)

* Having two rows of notes, even if not evenly distributed, lets a hand span a greater range of pitches. If all of the black keys were inline with the whites, it would be harder to span an octave without making the keys smaller which increases the risk of mistakes.

* The non-uniform layout means that each diatonic key has a different physical layout. This is a negative when it comes to transposing, but it can be a positive when it comes to composing. Picking a different key with its unique layout might get your hands to try different chord voicings or progressions you wouldn't think of otherwise. It increases the spatial variet of the instrument at the expense of less consistency.

And a zigzag would be nicer than here two and there three.

But yes, I probably have to learn the keys for a few days and things are good :)

True. Honestly, from an engineering and UX standpoint, guitars are just breathtakingly brilliant. We don't give enough credit for how amazingly expressive, flexible, simple, and robust they are.

It's an analog instrument that you can make for a few hundred bucks out of a couple of pieces of wood and some wire. It lets play chords of up to six notes with one hand while articulating each note individually with great expressive control using the strumming hand. It allows a player to cover a huge range of chords across the entire chromatic spectrum and even supports bends and slides. At the same time, a beginning can make something that sounds nice within a couple of hours. It is simple to adapt it to electronics. And even when acoustic, it sounds great.

Total mechanical marvels.

x + 12 = 880 Hz in your notation.

The reason and importance of intervals are due to the harmonic series.

Perfect 5th = 3/2 * x (or x + 7 in your notation)

Perfect 4th = 4/3 * x (or x + 5 in your notation)

The 12-note equal temperament is a little off from these ratios as a hack to allow multiple key signatures. You'll find that 3/2 ~= 12_sqrt(2)^7.

You can go into complicated chords too, and you'll still find dualities and the hormonic series behind it. A major chord is a stacked major 3rd with a minor 3rd and together they range over a perfect 5th.

Realizing that every note was basically the same and all semitone intervals are the same, I asked myself the "innocent" question, then why are they labeled black and white on a piano keyboard?

Trying to answer this question to my (full) satisfaction took me on a very deep google dive, several over a couple of years in fact. But it roughly led me through most areas of music theory. I'm still not entirely satisfied with the explanations I found, but some of the remaining questions are also kind of open in music theory.

It comes down to the question of what's so special about the major scale? And the answer is kind of in the circle of fifths and combinatorial music theory. If you have a modulo 12 system (because octave equivalence, which seems to be a physiological property of human hearing), there are two generating primes, 5 and 7. These correspond to a fifth down or up. Generating prime means that it generates all the 12 notes if you follow it modulo 12. Also it turns out that the complementary scales of the major (7) and the pentatonic (5) are "maximally even" (IIRC), .. and now I forgot why that was important. It's complex stuff.

There's also reasons why we got 12 notes instead of 10 or 16. Mainly to do with how close you can get to simple fractions of frequency ratios. You also have 19-TET, which has more notes and gets pretty close, but 12 is still superior in some ways afaik. This is the part that I found really interesting, but over time it's been nagging at me: Simple frequency ratios are special because their waves and harmonics coincide in periodic fashion. But if it's good enough to just be "close enough" to some ratio, that is actually equivalent to exactly hitting a much more complex fractional ratio. The accepted reasoning is, I guess, that human hearing is kind of fuzzy and not too fussy about these things. But that feels a little bit too hand-wavy to me. Especially cause the fuzzy can be trained, and most of us expect to hear the particular 12-TET tuning, and when they hear the exact ratios, they sound kind of "off". So I feel there's still some understanding missing from this theory, or at least more I'd like to learn (somewhere between physiological human hearing and cultural music theory of scales from all over the world and history).

I kind of feel like I learned about music "in reverse" this way, and I'm not sure I'd recommend it as the way to study music theory, but it sure as hell has been interesting.