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I'd like to self-study for the Advanced Placement Music Theory Examination in the United States of America (offered by the College Board, a nonprofit corporation in the USA). This is an examination taken by high school students in preparation for university study in music, and which can enable one to exempt certain first-year university music courses if the score is high.

What are the topics of AP Music Theory and what is the best way to study for them?

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For the OP - Taking / Passing the AP Music Theory test does not guarantee that the college / university you eventually go to will accept the score. While many institutions obviously accept the scores, many others prefer to teach their brand of music theory. Make sure you check with your prospective school. – jjmusicnotes Feb 18 at 4:57
    
This level of music theory you are unlikely to able to do without the aid of a seasoned professional. – Neil Meyer May 9 at 17:18

The AP Music theory test consists of a lot of different concepts including :

  • being able to read all clefs (including alto, tenor, etc.)
  • knowledge of musical terminology (there is a lot covered on the test)
  • knowledge of keys, scales and modes
  • knowledge of basics about the instruments like what they sound like
  • basic chord construction
  • basic voice leading concepts
  • Roman numeral analysis, form and analysis, figured bass
  • harmonization
  • transcription of a melody
  • sight singing

Pretty much everything you would go over in an intro college level music theory/ ear training class. Taking lessons from someone who teaches music theory would be very helpful along with getting a good theory book and drilling all the concepts.

The test itself is broken into two sections which are a multiple choice section and a free response section. Each section is further split into an aural section where you need to listen to something and respond, or in the case of the sight singing create sounds based on what is given, and a concept section where your basic theory knowledge and concepts are tested.

Here is the PDF of the College Board Music Theory Course Description, with a full explanation of what the test covers, and sample questions.

In terms of preparing, there is a practice test on the College Board web site, and it says you can "talk to your counselor" about taking the AP Music Theory course online from an "approved provider" if your school does not offer it. You should also do a web search for something like "AP music theory practice questions". There are plenty of sample and prep questions peppered around the Internet.

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The topics are listed on the College Board's website, here. Follow the link for several PDF documents you can download that provide further details.

Topics:

I. Musical Terminology

  • Terms for intervals, triads, seventh chords, scales, and modes

  • Terms pertaining to rhythm and meter, melodic construction and variation, harmonic function, cadences and phrase structure, texture, contrapuntal devices, small forms, and musical performance

II. Notational Skills - Rhythms and meters

  • Clefs and pitches

  • Key signatures, scales, and modes

  • Intervals and chords

  • Melodic transposition

III. Basic Compositional Skills

  • Four-voice realization of figured-bass symbols and Roman numerals

  • Composition of a bass line (with chord symbols) for a given melody

IV. Musical Analysis (from a printed score or with aural stimulus)

  • Small-scale and large-scale harmonic procedures

  • Melodic organization and developmental procedures

  • Rhythmic/metric organization

  • Texture

  • Formal devices and/or procedures

V. Aural Skills

  • Sight-singing (major and minor modes, treble and bass clefs, diatonic and chromatic melodies)

  • Melodic dictation (major and minor modes, treble and bass clefs, diatonic and chromatic melodies)

  • Harmonic dictation (notation of soprano and bass lines and harmonic analysis in a four-voice texture)

  • Identification of isolated pitch and rhythmic patterns

  • Detection of errors in pitch and rhythm in one-and two-voice examples

  • Identification of processes and materials in the context of music literature representing a broad spectrum of genres, media, and styles

Update

In the USA, many public and private high schools and private schools that teach music offer a course to prepare you to take the Advanced Placement Music Theory Examination. You said that you want to do self-study for the exam, but if you are in school, see if your school offers such a course, and ask the teacher if you can have a look at their textbook and study materials on your own. Ask around and see if you can find people who have taken the preparatory course and the exam, and ask for their suggestions.

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You might take a look at this online Music Theory course. It is free but you can pay $50 or so for a certificate of completion. This certificate may not count for much, and it certainly would not exempt you from taking the foundational music theory course at whichever college you went to. But Berklee is a very distinguished music school so you need have no doubts about the content. This is a MOOC. MOOCs work much better for you if you actively participate. And you should still find someone to help you with the aural training. This edition of the course started 25th January so you could still join in.

https://www.edx.org/course/introduction-music-theory-berkleex-oharm100x

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