
This book is Schoenberg's last completed theoretical work and represents his final thoughts on the subject of classical and romantic harmony. The earlier chapters recapitulate in condensed form the principles laid down in his Theory of Harmony, the later chapters break entirely new ground, for they analyze the system of key relationships within the structure of whole movements and affirm the principle of "monotonality," showing how all modulations within a movement are merely deviations from, and not negations of, its main tonality. Schoenberg's argument is supported by music examples, which range from entire development sections of classical symphonies to analyses of the experimental harmonic progressions of Strauss, Debussy, Reger, and Schoenberg's own early music. The final chapter, "Apollonian Evaluation of a Dionysian Epoch," discusses the music of our time, with particular reference to the possibility of new methods of harmonic analysis.
If you actually want to learn music theory for the purposes of playing I strongly recommend you just follow whatever tutorials for your instrument and style and if you get to the point where feel you want to learn more and take it seriously in and of itself, buy “The Jazz Theory Book” by Mark Levine[1]. In spite of what it says, there really isn’t any such thing as Jazz theory - it’s just music theory, and this book is a great presentation with lots of good examples. If you like jazz (or the western popular music tradition including rock, pop, funk, etc) you’ll come out understanding how that music works better than most people. If you’re doing a music degree, there will be books recommended by your music programme, but honestly, Levine will cover most of what you need if you’re not doing a straight classical course in which case you’ll want to augment with a more conventional source so you understand the “rules” they want you to follow when writing pastiche counterpoint etc.
If you want to get big into composition and/or want to understand late romantic and 20th century music, get Harmonielehre and “Structural Functions of Harmony” by Schoenberg. Together they are by far the best harmony books I have ever seen and go super-deep into why the western classical tradition works the way it does. If you compose there’s a decent chance they will change your life forever as his approach is to not take anything for granted but exhibit everything with examples from masterworks of the western tradition. Amazing books both of them. Harmony is not a spectator sport though - to get full benefit you have to get the books and work through stuff on the piano. It will reward you immensely.
If you want something else that will blow your mind, read “The technique of my musical language” by Messiaen. If you read Levine first this will have the additional benefit of showing you where the “diminished” and “augmented” scales and the whole “harmonic major” thing come from.
https://archive.org/details/structuralfuncti0000arno/page/n8...
It's "functional" because the chords in the harmony have the function of establishing key centres, moving towards and away from them, establishing transitions to remote key areas etc.
One of the best technical books about functional harmony is "Structural Functions of Harmony" by Arnold Schoenberg[1], which is much shorter than his "Harmonielehre" but really packed with interesting detail about key and chord relationships.
[1] Yes that Schoenberg but this isn't about serialism, it's about trying to explain all the harmony up to and including Mahler. Here is an online pdf of an early edition https://is.muni.cz/el/1421/podzim2007/VH_53/Schoenberg_Struc...
Saying there's nothing you can't do in serialism you can't do in free atonality seems unfair. Firstly free atonality came about after integral serialism. Secondly, there's nothing you can do in any type of musical composition that you can't do in free atonality, so you may as well say all the things you can do with functional harmony are a worthless constraint.
As for popularity - I personally don't think that's an important characteristic at all, but if you do, there are lots of serialist works that are very popular (eg Wozzeck[1]).
[1] And I deliberately chose that one rather than say Berg's violin concerto which is also very popular because that's one of the ones that you're accusing of mining serialism for tonal elements