What Is a Major Chord?
jefftk.com
jefftk.com
But I've noticed that the people who tend to have the biggest problem with music theory are engineers. I think it comes from an assumption that music should be learned like engineering, where you start with fundamentals, and then build upon them. And then they're horrified to discover that even the fundamentals are arbitrary and approximate.
Most musicians don't start out with fundamentals. We started by choosing an instrument and somehow learning to play it. For the most part, musicians don't care about temperaments. Equal temperament only applies to a handful of instruments that can be kept in tune long enough for it to matter, and that can play the same note, the same way, twice. An orchestra, jazz band, or solo fiddler, have no temperament in any meaningful sense.
Yes, musical notation is a cluster. But it serves its purpose well enough to be useful to support a symbiosis between the composer and the performer. If any composer chooses to try a bespoke notation, they'll discover that nobody will ever perform their composition. It's like a programming language that nobody likes, but that is supported by a job market and labor pool.
My advice for any engineer wanting to get into music is, get an instrument and learn to play it. Preferably a non-electronic one, so you won't be tempted to turn music into a numbers game.
I had the impression that fundamentals are much easier if you played around with some real world tech.
But to go back to music theory. I think, it helps when you already started a song and you're stuck. If you got a catchy melody, some theory gives you the tools to add fitting harmonies to it, etc.
Playing an instrument is nice and fun too, of course, but I think it's more helpful for intuition rather than actual understanding -- and those are usually best to combine.
I'm confused. If I'm not mistaken, equal temperament is the norm for all instruments since Bach.
Any keyboard instrument is tuned that way. Or any instrument with frets. Or any instrument that must be played alonside with one of the previous ones.
If you meant something else, could you ellaborate?
Woodwind instruments have "bad" notes, and I once saw an advertisement for a flute with a "new improved scale." Brass instruments have their main valves, but also little tuning slides with hooks on them so you can work them in and out with your extra fingers while playing. It's fascinating to see someone play tuba, it takes both hands to work the thing properly.
Intonation is a lifelong journey for most musicians. Near the end of his life, they interviewed Ray Brown and asked him if he still needed to practice. He said that he still practiced every day, to improve his intonation. If players sound in tune, it's because of an almost obsessive effort to make that happen.
What I'm saying is you can't begin to talk about temperaments until you can play the same note at the same pitch, in any octave, at least twice. I'm a double bassist, and I'm still working on that, after playing for nearly 40 years.
In that sense, your violin's strings might be tuned to match an equal tempered piano, but your own intonation will not (probably) follow any "equal temperament" for the notes you play on the fingerboard, unless you follow an external reference, not your ear.
When you try to play a fifth on a violin, your ear will help you to play it exactly (ratio 1.5), not the equal temperament approximation. Of course, for the fifth the difference is very small, but the same idea applies to the other intervals.
"Natural" tuning by integer fractions would sound very unusual to everybody.
Take the example barbershop's use of justly intonated chords. It doesn't sound unusual, it sounds very clean and pure.
Take the example of Jacob Collier's vocal arrangements, he likes to sing the chords justly intonated too. He has various explanations of this on youtube. Also gifted with an insane ear so that he can clearly sing/explain the difference.
Maybe this is an ok explanation of Collier. https://www.youtube.com/watch?v=NHC2XNGerW4
(I don't know much about playing the violin!) Intonation is a living thing - you use your ear, it's a feedback system - listen to your own instrument and those you are playing with.
> Intonation is relative. When playing in an ensemble, care must be taken to blend with the others in the group.
Temperament methods evolved towards ever-better approximations of equal temperament. They all had names. Bach's "Well Tempered Klavier" was written for an instrument tuned in the Well temperament, and explored the unique tone quality of each key.
String and wind instruments can't be played in tune consistently enough to worry about temperament. And musicians will push notes slightly sharp or flat for emphasis. On the cello, I was taught to slightly stretch whole-step intervals, and squeeze half-step intervals together when playing scales. It made a major scale sound "more major" for instance. Every wind instrument has one or two notes that are known to be bad and must be corrected on the fly, though it's less noticeable on faster passages. The musicians play in tune by listening to one another.
Fretted instruments in Bach's time were an odd situation. Gut strings vary in density along their length, and go out of tune as they age. The viola da gamba had frets, but they were movable and required periodic adjustment.
Even the temperament on a modern steel-stringed guitar is approximate. The compensated bridge is an attempt to correct the scale, but it's not perfect. And it's not systematically tempered, as in, adjacent octaves are not in tune with one another.
About guitars, I can believe the bridge, or even finger pressure provide adjustement for individual notes. But if you're playing a chord, you just trust the frets, that are uniformly spaced, following the logarithmic scale.
I wrestled with temperaments for a bit before realizing that the "engineering rules" of music (at least at the compositional layer) are not dictated by physics but by culture. Equal temperament is in some sense "arbitrary", but because it's what people are used to hearing and performing music in, you have to just accept it as an axiom and move on (unless you're setting out with the express intent of challenging that norm, in which case more power to you).
Also when you delve into world music (particularly from India or the Middle East) you quickly learn that twelve-tone octaves are also arbitrary and that musicians in those regions regularly perform tones that don't fit in that model and have other common musical idioms that western music theory is poorly equipped for because our common music theory originated from a particular western European style from a particular era.
My hope for the future of music notation is that we recognize this and welcome more forms of music theory/notation for different styles of music just like we recognize different strengths of different programming languages.
What aspects did you find inadequate?
Now, you have to choose an instrument. There's a good reason why the guitar remains popular. The learning curve for tone production is extremely gentle, and there's a huge variety of styles that you can enjoy, plus online learning materials. It's portable, affordable, and alcohol powered. ;-) It looks like Fender is moving towards a new business model that revolves around learning to play.
Learn by mimicking, then learn structure after you've already had fun just making some noise.
My problem with music theory is that in the presentation there is no clear separation of what can be derived from first principles (so, similar to physics theory), and what is based on observations of what people seem to like (so, similar to literary and film theories).
Granted, both physics theory and literary theory are called theories. But they are very different approaches. And then music theory is a mix of both. Some parts can be explained physics-like from psychoacoustic fundamentals, and other parts are like anthropological observations: This is what people seem to like.
And the way it's presented, you don't know which is which.
I'm a classically trained violinist, and when learning there was a huge sense of "just follow the rules and the sheet music exactly until you get to composing".
In that sense, I think there's a lot of similarities to a lot of engineering practices. There's a straight line to follow on a lot of tasks, adhere to the rules and leave the magic/cleverness to the system architects.
When I switched to guitar, and subsequently discovered the world of prog/math metal, it felt incredibly freeing. However describing that kind of freedom to my friends who adhered to sheet music exclusively was a struggle. Polyrhythms, improvisation, playing by feel were blasphemous to that world.
In that sense I liken it to ratatouille. You must cook the recipe the master chef created exactly, and it's not our place to iterate and experiment.
As a software engineer, I see a lot of the same mindset often. There's a world of engineers who are rigid about sticking to the tried and tested way of doing things. There's nothing wrong with it, in fact it's very valued.
However you need the counterpoint. You need the innovative composers, you need the recipe creators, you need the system designers/architects. Having only one or the other is an imbalance.
I think that's what a lot of engineers get hung up on. Music is taught as rules.
but those rules quickly fall apart when you start having to account for feel, for emotion, for cultural differences etc... Things that aren't able to be captured in rules cleanly.
That's not what I see (as the author of a cross-platform DAW for the last 22 years).
Engineers get hung up on the fact that despite the reality that almost all western (and, realistically) almost all of the world's musical tradition(s) are in fact rooted in clear rules and numerical relationships, music culture wants to use terminology and practice that pretend that it's not.
The simplest example: the major scale. Like all scales, it's just a pattern of intervals. What intervals? Why is the scale not named after the interval series, which would allow you to see how similar it is to the minor scale, and precisely how it differs (ditto for comparisons between any other scale).
Another example: ordinal naming of scale degrees, "the third" etc. These have no consistent meaning other than when you write out the notes of the scale, you count through them. But the third of a minor scale is a different interval from the root than the third of a major scale. If they were named by interval, you'd have consistent naming no matter what the scale is.
There are dozens more examples like this, where consistency in naming would ease many a musician-engineer's mind.
I agree that the naming and accessibility could be significantly easier. That's just the nature of language though, and music theory is a language. One that's a cumulation of many cultures.
However the rules (in western theory) are very consistent and clear if you've learned your foundations. This is no different than programming too.
If you know the circle of fifths, and the WWHWWWH series of modes, there's a very strong consistency.
Now if we're arguing about why fifths, and why we name our modes after Greek groups rather than something numerical, that's valid. But it's just like the names of the days of the week. Once you know them, it's just an arbitrary term that you internalize.
The rules I'm bemoaning in my post are things like sticking to a fixed time signature or scale, playing perfectly aligned notes. Versus things like polyrhythms and drunk drumming, exploring disharmony or using alternates to octaves.
The problem with the day of the week comparison is that, yes, they are arbitrary and you just learn them and be done with it, but with music the names are just the building blocks for a vocabulary/dialog about a creative process, and so you just don't throw down "play the 5th" the way you might say "it's Wednesday". Instead, that language is interacting with a conceptual process which the language doesn't mirror. If the four components of music are rhythm, timbre, melody and harmony, the last two are all about intervals, and almost none of the music terminology in the west reflects that.
I think it's a common mistake, especially in classical music, to think about musical traditions as "rooted in clear rules and numerical relationships." It's analogous to thinking that language is rooted in clear rules and structures.
The truth is just the reverse, the rules, numbers, and structures are rooted in music. They exist to describe an organic, emergent cultural mechanism that is continuously changing. Ask yourself: which came first, music or theory? Language or grammar? There are plenty of musical styles with no formalized structure or numeric relationships, just as there are plenty of languages with no formalized structure or even spelling. Grammar and music theory/notation are fingers pointing at the moon, and we are looking at the finger.
And when we try to engineer systems to help people point at the moon, we should be focused on the finger. Musical structure and numeric relationships are the way people communicate about music, and the tools should speak their language. Unfortunately they have to grapple with the painful inconsistencies in that language and those structures. Computer music comprehension has problems analogous to computer language comprehension - emergent complexity so high it took neural nets to finally achieve real utility.
PS - it's true that Western music has, near the bottom, some relationships that were mathematically derived by none other than Pythagoras, among others. Relationships of fourths, fifths, octaves, and equal temperment all had numeric justification... But that justification was still only there to describe the practice which had already become common, an explanation of "why this sounds harmonious" as well as a proscription for "how to sound harmonious". That some rules are internalized by some generations and broken by others illustrates exactly the problem with the system.
However, I disagree quite a lot with what you've written above. Not really as a matter of historiography - yes, of course, I agree that the history of musical cultures is not the same as the history of mathematics. I agree that musical practice is not rooted in an understanding of numerical relationships. I also agree that the higher level elements of musical composition/improvisation/performance are not sensibly described with math.
However, one half of what defines music as a sensory experience (melody and harmony, the other half being timbre and rhythm/time) is totally, fundamentally, absolutely embedded in the nature of the intervals between pitches. The intervals we use (regardless of any actual absolute pitch selection) and how we use them seem to me to be almost indisputably rooted in the physics/mathematics of the harmonic series even if the development of the musical practice surrounding intervals had been done in complete ignorance of this.
Western musical culture uses a set of terminology and even concepts that obscure the fundamental importance of intervals. The use of scale degrees rather than intervalic terminology means that there is a lack of consistency in naming that isn't necessary (it's really a historical accident), but also a lack of focus on the most important concept for understanding the nature of melody and harmony.
I was really impressed to discover last year that Byzantine musical culture doesn't notate pitch, only intervals. Song melodies are remembered and notated as a series of intervals between the pitches, so that you can start from any note and play something that almost every human being alive will recognize as the same melody (they also do not have octave equivalence, but that's a nother story), even though the actual pitches are totally different. Of course, western musicians can do this too, but consider how much more cognitive load there is in understanding the process when you consider a melody to be made up of specific set of pitches, or a specific set of scale degrees, rather than just a set of intervals.
> But that justification was still only there to describe the practice which had already become common, an explanation of "why this sounds harmonious"
Yes, I agree that the historical ordering is as you describe. We do, however, change our terminology and practice in many other fields as we gain new insight or come to understand better ways of approaching things. I would argue that we could learn from both other musical cultures, and from the numerical/physical understanding of intervals to get to much more rational, consistent terminology and practice when it comes to melody and harmony.
Music, of any genre, is never about reproducing the notes and dynamics accurately. MIDI does that, and the day we can create historically informed MIDI we will have it perfected.
Rather, music is about the spontaneous communication and individuality that occurs WHILE reproducing those notes and dynamics. Genres differ in which dimensions of freedom the musician can use, but not in that fundamental objective. In classical music it's what makes it worthwhile to hear different interpreters of the same piece. It's why it's worth the 5 year waiting list to see Wagner in Bayreuth, when arguably the greatest rendition of The Ring in 100 years (Solti, Chicago, fight me) is available relatively cheaply everywhere.
If that's the case...
1) Encourage your classical friends to break out of the mindset of "cooking the recipe the master chef made exactly." Audiences complain that classical music is sterile and doesn't speak to them, precisely because this mindset is so common among all but the most elite performers. Also, notice that NO elite performers approach their musicianship that way.
2) Notation and theory must have a slightly different purpose than the one you articulated. I believe they're simply there as tools of communication between musicians, to describe recurring patterns we hear. Otherwise we'd be forever saying things like "Beethoven 3, is the one that starts with that thing where it sounds like it's going to end but really doesn't." Calling it a "deceptive cadence" or even a "surprise VI" is just way easier and lets us communicate (and operate) on a higher level of abstraction. It's the same way "dependency injector class" tells you about a given chunk of code, and lets us reason about its place in the larger structure. A great engineer isn't defined by their ability to reproduce a textbook dependency injection class, but rather by their ability to adapt the concept of dependency injection well, in the right places and times.
I've seen some rather niche notation types that focus on emotion and relative energy while describing notes instead of the usual Italian volume notations alongside the bar notations.
But more interestingly (to me) are things like Indian classical music that doesn't have explicit notation, and instead has a taught relative theory, or other countries like Indonesia that have non octave based scales who's music can only be expressed in western notation by microtones.
Anyway I very much agree that music is a communication language at its core, with music notation existing to aid the spread/consistency of its playback, but it shouldn't stiffle the intention.
The most useful music theory I've ever learned was the stuff that I needed to play jazz guitar: to improvise solos and comp chords. That course of study has continued to pay dividends many years later.
The second most useful was an esoteric but extremely approachable exercise called "arch mapping", where you draw arches on graph paper to represent the form of a piece of music in terms of its subdivisions. For music with a steady tempo and constant time signature, basically anybody can try it: https://williamwieland.midcoip.net/theory/form/archmaps.htm
I also like arch mapping because it explodes the possibilities of what "music theory" is. It can be applied to literally any genre. You don't need to read music. You don't need to know western harmonic theory. You don't even need to know what a major chord is!
Not debating your advice but just saying that some brains work differently.
I play both guitar and piano and also know the "math numbers" of music theory and after decades of learning, I've concluded that the mechanical aspect of hands playing an instrument often limited my understanding of music while the numbers aspect always enhanced it.
I was about 11 years old when I first learned about mathematical relationships of note frequencies from an computer programming book. So to make the computer's sound chip play a C Major arpeggio would be coding something like this in the BASIC language:
10 CALL SOUND (1000, 262, 0)
20 CALL SOUND (1000, 330, 0)
30 CALL SOUND (1000, 392, 0)
40 CALL SOUND (1000, 523, 0)
I've always liked knowing that octaves are 1/2x or 2x multipliers frequency. And the guitar frets spacing is a visual representation of the 2^(1/12) formula. And because a B1 is 61.7Hz, that's why a 60Hz hum of electricity sounds like a slightly flat low B. And math group theory perspective of chord constructions helped me see how Chopin piano voicings transformed in a coherent way.
For people like me, I'd slightly reword your advice as "My advice for any engineer wanting to get into music [enjoyment feedback loop] is, get an instrument and learn to play it."
If you're engineering-minded like me, the "systems" approach of the numbers behind the music explains it in a way that the fingers will not.
Would love to hear more about this if you know of any good sources
A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice
Perhaps we are observing the same thing still: engineers gravitate towards music but then get stuck because the rules are so arbitrary.
Oh, and speaking of making music a numbers game: check out Tool's Lateralus!
How so?
Strong agreement there. And make sure at least some of your learning is by ear, getting the feel for the music.
For me, music theory was only helpful (or even interesting) after I had already been playing for a while.
Theoretically, though, there will always be some error: any non-integer power of two is necessarily irrational. Thus, there is no perfect way to divide an octave into equally hz-spaced increments so that you hit the fifth along the way to doing the whole octave.
When using a tuner to tune a guitar, you get the equal temperament normally desired for it's ability to play equally in any key. When tuning by ear, if you are just minimizing the beating you hear (instead of targeting the beating that exists in equal temperament, like good piano technicians are trained to do) you'll end up with integer ratios. Accidentally ending up at the integer ratio of a major third is "farther off" than doing so for a perfect fourth. Typically this manifests as a difficult time tuning the B string though, not the G string, if you go low to high. Solution: tune low to high for the E,A,D,G and tune high to low for the high E,B (doing a perfect octave for the E to E where the equal temperament ratio [4] perfectly matches the integer ratio [5]).
[0] 1.334840:1 or 2^(5/12):1
[1] 1.333333:1 or 4:3
[2] 1.259921:1 or 2^(4/12):1
[3] 1.25:1 or 5:4
[4] 2:1 or 2^(12/12):1
[5] 2:1
I assume you are talking about fretless instruments. With a guitar you just play a unison using the frets and the thicker string.
Rather, I'm talking from a point of view of playing two adjacent open strings. Or harmonics thereof (as some people like to tune with 7th fret harmonics and so forth). If you aren't fretting and you minimize beating down to nothing, then you end up with integer ratios.
I think you're right about the "beating"... in general this post has been extremely instructive for me. My teachers were piano players. Either they didn't know better (because they didn't need it) or they didn't bother to explain the fine details to me. Even the books I consulted later don't tell the whole story. That's the magic of the Internet :-)
In contrast to matching the same note on the next string?
But recently I discovered the most helpful trick, use a fifth from G down to the A string, and also use a fifth from the D to the low E string. Those last two intervals can be the most surprisingly out of tune intervals when using either ear tuning or an electronic tuner.
When you play B and E open and tune them to eliminate beating, you're in okay shape because you're making a "just" (integer ratio) perfect 4th which is not terribly far from an equal temperament perfect 4th.
See my earlier comment [0] with all the numbers if you want to quantity "pretty bad" vs "okay". In that comment, my initial references to footnotes 3 and 4 should actually be 2 and 3, but it's too late to correct it.
Octaves are actually 100% right regardless of equal temperament versus perfect/just/etc.
The only real exception is when you get into octave stretching to compensate for string inharmonicity, but that's a whole separate matter.
Is it arbitrary? Perhaps not. There has been a lot of research on that as well, for example: https://www.newscientist.com/article/dn20930-why-harmony-ple...
So, once you figure out what a major and minor are, then the next question is why does that sound better than other combinations of notes?
How a chord sounds also depends greatly on what came before it. For example, in typical common practice period harmony, dominant seventh chords do not sound resolved. On the other hand, many songs in popular music have moments that resolve to a dominant seventh chord.
A power chord is root + fifth, which is a 1:1.5 ratio of frequencies (exactly in just intonation, and very nearly in equal temperament). Intermodulation distortion produces tones of frequency ratio 1.5 + 1 and 1.5 - 1, giving 2 and 0.5. Combine them with the existing tones and you get 0.5:1:1.5:2. This is just the harmonic series 1:2:3:4, but pitched down an octave (all frequencies divided by two). A distorted power chord contains the harmonics of a single note pitched an octave lower, which is why it sounds "heavy".
There are many other harmonics in the notes, so the timbre is much more complicated that that, but the fundamentals are most prominent, so the distorted power chord sounds consonant.
And if you have a guitar with a hexaphonic pickup and distort each string individually you don't get intermodulation distortion between the notes, so you can play much more complicated chords without it sounding dissonant.
http://sethares.engr.wisc.edu/paperspdf/consonance.pdf
Informal overview:
http://sethares.engr.wisc.edu/consemi.html
This is almost certainly an oversimplification (e.g. the listener's familiarity with the sounds also matters), but the results match my own perception of consonance and dissonance reasonably accurately. It doesn't address the question of "why", but intuitively it seems that dissonance is associated with sounds that take more mental effort to understand. If the partials line up closely then they are perceived as part of the same sound, and if they are spaced far apart then they are obviously different, but there's an awkward intermediate spacing that's uncertain (the "roughness" on the Plomp and Levelt curve). Dissonant sounds contain a lot of this uncertainty.
Unfortunately the first satisfactory answer that I got needed much more knowledge. The foundation is about proximity and dynamics (thinking about scales as a progression, not simultaneous notes).
But it's much better to just accept that major is "happier" than minor when you're starting, and later understand why intuitively.
Pitch, melody and harmony are integral aspects to music but they are by no means sufficient to provide an understanding of music. Just as a purely mathematical description of the optics of color is not sufficient to understand painting. I would argue that the overwhelming majority of music you have enjoyed in your life was written with literally zero regard to these details. In my own opinion, music programs that raise these mathematical details above all others lead to the dullest and least inspired music.
The best place to get the answer is Schönberg’s Theory of Harmony, but we can do something quick here.
The main harmonic series of music is what happens when one applies integer multiplication to a given frequency, which we’ll call a root. As the author described, an octave, or the same perceived sound but higher, is a 2x, and a 3x, or 1.5x the previous value, is called a “fifth” due to it being the fifth major scale degree in the twelve tone system described by the author. 4x is 2^2, another octave, and then 5x is the (major) third.
Because all of these integer harmonics are present in the root, though at greatly reduced force, these sounds serve to bolster the natural harmonics of the root.
He prefaced his derivation with the statement: "Therefore, whenever I theorize, it is less important whether these theories be right than whether they be useful as comparisons to clarify the object and to give the study perspective."
This is something I find compelling about Schoenberg. He doesn't claim to know more than he does, he's just presenting a useful mental model.
An overarching theme of the book was his philosophy that no eternal aesthetic laws were known to exist, so he was careful not to make definitive claims unless he could really be sure about them.
The idea that mhz is even worth mentioning…
Fwiw, huge shout out to https://improviseforreal.com which for me was the happy medium between “major chord is happy” (huh?, whyyy?) and “C0 is 16 mhz”.
A rudimentary question is a wonderful an opportunity to distill everything down to a simple understandable concept.
Here's how I would do it:
---
Western music is based on simple frequency ratios. 2:1 is so basic that we call it the same note, but an octave up.
Three pitches with ratios 4:5:6 make a major chord. These are the simplest possible ratios within an octave, so a major chord is the most basic chord, and all other chords are derived from that
Western music is based on simple frequency ratios, played by instruments with harmonic timbres (meaning that when you decompose the sound of a single note into sine waves using a Fourier transform, the frequencies of the sine waves are all integer multiples of the lowest frequency sine wave).
This is true for wind and string instruments, but not true with tuned percussion unless you go to special effort to shape the instrument to make it so (as is done by bell makers). Musical styles that use inharmonic timbres, e.g. Indonesian gamelan music, need different tuning to sound best.
And this also happens in Western music. Piano is technically tuned percussion, and the sound of a piano is slightly inharmonic. Pianos are tuned with stretched octave to compensate:
But the harmonic or inharmonic content of the instrument does not address the original question.
And it leads to confusion; putting your first and last sentences together implies that a piano is not a western musical instrument. We know that's not true.
I'm just struck by the zenlike simplicity of the original question, "What is a Major Chord?", compared to the random bits of technical jibber-jabber that would confuse the hell out of a poor soul who would dare to ask such a question, both in the linked blog post, and the discussion here.
I picked one of them for the title of my post, but I'm trying to answer them all.
A bugle has no valves, it can only play the natural harmonics of the horn, as determined by lip pressure.
x3, x4, x5, and x6
Thus the ratios of the available pitches are 3:4:5:6.
That's a major chord (with the "fifth" down below).
All your iconic bugle calls ("Taps", "Reveille", etc.) are the notes of a major chord.
They're approximated in the 12TET scale so that they harmonize across keys and octaves.
I always say I hate musical notation (letters and classical notation) being it so horrifically obscures the mathematical relationships of the 12TET scale. It's literally just approximations of:
Major Scale
Root frequency (C 262hz, whatever)
2: 9/8 root (D)
3: 5/4 root (E)
4: 4/3 root (F)
5: 3/2 root (G)
6: 5/3 root (A)
7: 15/8 root (B)
2x root (C, 524 hz)
2: 9/8 2x root
and so on.
Minor scale is simply the starting the whole thing at the Major 6th instead of at the root.
12TET in comparison to just temperament is slightly harder to explain because of the way it allows harmonies to scale without distorting when changing keys, but there really isn't much more to western music than that.
I feel like it has much more to do with convention than music theorists like to confess.
It’s comprehensive. It’s also frequently on sale for 40% or more off. He sends you updates when he releases new versions.
But then again, pi.
Join the discord https://discord.gg/a5ttYuG
The fact that 7 of the 12 notes in an octave are assigned letters has always bothered me. It's quite arbitrary, we could just as easily have 6 letters + 6 in-between notes, or just 12 discrete letters.
H J K A C E F
And calculating the intervals is annoying! You could go 2122212, which doesn’t correspond to major or minor — it’s dorian.
Meanwhile, in current notation, you’d just look at the sharps (C# and F# = D major) and the starting note (E).
I think in an alternate universe where a 12 (or 6+6) notation had historically been used instead of 7+5, the 7+5 option would seem quite unnatural to most people. In any case, starting from a blank slate without historic context, the 7+5 naming scheme just isn't intuitive.
Anyway, you can count the intervals now. For example, I know that there are five half steps between F and A#. It is slightly annoying to keep the B and E exceptions in mind, but it’s definitely possible — most people just find it easier to use the sharps and flats as a shortcut.
A 6+6 system makes more sense in terms of fundamentals - there's the exact same distance between each "full" letter. That doesn't limit what kind of scales you can annotate with the letters at all.
C D E E# F# A# B#
If you start with C major (no sharps), and ascend in 5ths (G, D, A, etc.) then you add one sharp to the previous major scale to get the major scale starting on the new note. The added sharp each time is the 7th of the scale. So G major just has F#, D major adds also C#, A major adds also G# etc.
If you start with C major (no flats), and descend in 5ths (equivalently: ascend in 4ths) (F, Bb, Eb, etc.) then you add one flat to the previous major scale to get the major scale starting on the new note. The added flat each time is the 4th of the scale. So F major just has Bb, Bb major adds also Eb, Eb major adds also Ab etc.
Key signatures put the sharps or flats in this order.
The two meet in the middle:
F# major: F#, G#, A#, B, C#, D#, E# (key with 6 sharps)
Gb major: Gb, Ab, Bb, Cb, Db, Eb, F (key with 6 flats)
(For simplicity’s sake I assume the Cmaj/Amin scale here. In non-Romance countries we typically teach basics of music using relative solfege aka “movable do”, where notes always have the same names (Do, Re, Mi etc) no matter what the absolute pitch of the tonic is)