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Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression

W. A. Mathieu · 4 HN comments
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Amazon Summary
An exploration of musical harmony from its ancient fundamentals to its most complex modern progressions, addressing how and why it resonates emotionally and spiritually in the individual. W. A. Mathieu, an accomplished author and recording artist, presents a way of learning music that reconnects modern-day musicians with the source from which music was originally generated. As the author states, "The rules of music--including counterpoint and harmony--were not formed in our brains but in the resonance chambers of our bodies." His theory of music reconciles the ancient harmonic system of just intonation with the modern system of twelve-tone temperament. Saying that the way we think music is far from the way we do music, Mathieu explains why certain combinations of sounds are experienced by the listener as harmonious. His prose often resembles the rhythms and cadences of music itself, and his many musical examples allow readers to discover their own musical responses.
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A book that I came across the other day (via a recommendation from YT-ber Adam Neely): "Harmonic Experience" by W.A. Mathieu. It's a little expensive as a first dive into this stuff, but it does look like a very natural way to lead you into and through the actual experience of western tonal harmony, often by singing along with a drone (easy to generate on today's computers).

I remember getting Semiology of Graphics from the Palo Alto library around 2006. At the time it was sort of legendary and out of print, but it looks like it's since been reprinted. I think you can get most of the ideas from newer books, but it's well done and clearly ahead of its time.

Interestingly another relatively unknown book I like (and bought/read 20 years ago) is also about harmony:

I would say there's two kinds of harmony: harmony in equal temperament, and "alternative" harmonies based on physics, and this is about the latter. I can't tell from the link what the other harmony book is about. What's good about it?

As far as computer books, I've read a lot of recommendations here over the years like "thinking forth", "Computer Lib" by Ted Nelson, etc. They are well known to some audiences but not others.


I also enjoy reading what people though the computing future would be like. I have "Superdistribution" by Brad Cox:

And "Mirror Worlds" by Gelertner:

I'm pretty sure Gelertner claims that the Facebook feed is identical to his "life streams". I guess taken literally it's hard not to see the current Internet as a "mirror world" that's becoming the real world.

Thanks for the recommendations. Many look interesting but are not books I would organically bump into, which is an alternative description of what I was looking for.

As for a Sadai's book: it is an extremely thorough book about western harmony from first principles. It treats what is perceived - what we hear - as the anchor, and not what we see when we analyse the notes on paper. A good example of that is how we decide to give names to chords. We tend to name chords based on the notes in them, but this can sometime lead to misunderstandings because the context and how those notes are spread through the chord are also very important. Basics like which note is in the bass is taken into consideration, but otherwise these factors are often ignored. Sadai shows many examples for that throughout the book - as well as such "Mistakes" in other famous books. A quote from the book about the approach taken: "The conventional analytic approach as taught in academies is based primarily upon the depiction of the WRITTEN content of a composition by means of symbols and concepts inherent to the accepted analytic code. This analysis however, which describes mainly what is SEEN, does not always succeed in describing what is HEARD - the perceptual musical essence".

> I also enjoy reading what people though the computing future would be like

This prompts me to propose (although it's not obscure) "Artificial Intelligence: The Very Idea" by John Haugeland. It's an AI textbook that is extremely readable and inviting - the best I've seen as a purely readable text, though probably far too basic for most readers here - but that is entirely drawn from the realm of "good old-fashioned AI", i.e. things like logic systems that have very little in common with what is understood as practical AI nowadays. Combine the readability of the book with the apparent hopelessness of its premise, and you have a perfectly nostalgic experience.

I would recommend "Graphics and Graphic Information Processing" by Bertin over La Semiologie simply because the latter reads more like a reference book where Bertin is extremely thorough. But GGIP gets straight to the point and can frame your thinking while going through Semiologie such that you won't lose your way.

Unfortunately GGIP is expensive so I would try to find it at your local library. (French copies are online).

Which book are you reading, out of curiosity? I recently bought Harmonic Experience, the physical book, and thought it was funny that it is even offered in the Kindle format.

(By the way, it's not at all an introductory or comprehensive music theory book. It's about the history of harmony, temperament systems, and psychoacoustics.)

I'm reading

which was chosen because I hoped that the "angle" would be useful for me as a DJ who needs to start making some remixes to go to the "next level". I'm about 4 chapters in so far and I'm pretty happy. I've also been supplementing it with Tenuto, Theory and Fiddlewax Blue (All iOS apps). I plan on getting FiddleWax Pro soon as it seems really fun.

That's interesting. I don't listen to any electronic music so I've never contemplated music theory from that angle. I'm also not aware of any music theory apps other than EarWizard [0], which does relative pitch training. I'll have to check out the ones you mentioned.


I wanted to put in a plug for Open Music Theory [1], for anybody who's interested. It's an open-source [2], Creative-Commons licensed music theory text designed for use in college music theory classrooms. It's written primarily by some friends of mine, all with teaching positions and Theory PhDs, so the material is good. I'm using the 20th-century materials in my class this semester instead of assigning a $100+ textbook. It's not quite as comprehensive as a dead-tree book (yet?), but is a great resource for beginners, or people looking to supplement their knowledge.

[1] [2]

Out of curiosity, have you studied just intonation? I'm currently reading the mostly excellent Haromonic Experience [0]. I'm reasonably well educated on music theory, but I had never looked into just intonation and the way frequency relates to our perception of pitch and harmony. I'm finding this stuff enlightening.

An ideal string (or any oscillator) when disturbed (e.g. plucked) will tend to vibrate at a fundamental frequency of x Hz as well as modes of 2x, 3x, 4x, etc. These higher frequencies at integer multiples of the fundamental frequency are called harmonic partials. This is a physical phenomena unrelated to music and human hearing.

But the basic elements of harmony come from the fact that our auditory system seems to be "tuned" to identify harmonic partials.

If we start with a fundamental frequency x Hz (let's call it C), the first partial is 2x Hz, and it is another C an octave above (multiplying or dividing a pitch by a power of 2 will give you the same pitch in another octave).

The next partial is 3x Hz, and it sounds like a G. This is the interval called a perfect fifth, and is the strongest, most stable sounding interval (other than the octave). The next partial is 4x Hz, which is just another C, two octaves above the C we started with.

The next partial is 5x Hz, which sounds like an E. This is a major third, which is another strong and stable interval which is ubiquitous in most music.


Yeah, a little bit! What's interesting is that we're so used to equal temperament (i.e. a bit of error added to most intervals in exchange for greater flexibility) that pieces played in just intonation sound "off" to us, even though they're more mathematically correct.

Also, I believe instruments with arbitrary pitch (violin, voice) tend to naturally drift closer to just intonation.

I've been trying to train myself to hear the JI resonances (Harmonic Experience has tons of exercises for this). I can hear the sharpness of the ET major third and the flatness of the ET minor third fairly easily.

Barbershop quartet music (and other a capella music) is a great place to hear just intonation. The consonant barbershop seventh chord was a relevation to me when I first learned of it. It's really close to the dissonant dominant 7th interval, but serves a very different purpose harmonically. The minor seventh is also a different interval. Thus there are actually 3 seventh intervals, all of which are approximated in ET by a single note. This blog post has a good summary:

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