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One can find many books on "tonality"during the common practice period. e.g., Piston.

Tonality seems to be definable as the way in which composers express the chromatic scale in a organized way around one central pitch. E.g., There are 12 notes, if we express any one single pitch as "central" in some way(which way?) with the other tones falling in place in some way then we have tonal organization(e.g., just like chess, we have the king, obviously the central piece with the other pieces having specific relations to it)

Now, harmony books tend to give all kinds of rules about how to express tonality(voice leading, establishing the tonic, note resolution, etc).

The question I have, is there some specific fundamental concept that all this is built off of or is it just some arbitrary set of rules to create a structure on that produces a specific result?

Much of modality and modern music contrast and even contradict many of the tonal principles but in some ways overlap with tonality. Even many tonal pieces "break the rules".

Is all this hoopla simply to create a logical structure that has inherent stability and instability? e.g., without tonality we generally don't have a sense of a "home key", which means we don't have a place to go and a place to come back to. This seems to mimic human behavioral aspects. Hence, tonality is simply more "human like" than non-tonality.

If that is the case, are the rules of tonality born out of centuries of experience on how to achieve tonality, codified in our harmony books... or is there a mathematical foundation that the rules are derived from or are people just making it up as they go along?

When I read Piston, I tend to find a lot of circular reasoning about why things are done the way they are done...

e.g., Bach wrote a lot of music and obviously would have a certain functional pattern of composing. We find some example that backs up/demonstrates rule X. We ignore/neglect all examples that go against rule X and/or create new rules to explain why rule X was violated.

Just to be clear on what I'm trying to get at: In physics, all things essentially are derived from the basic laws governing how mass and energy interact. These laws are generally codified gravity and the nuclear forces. Most other things in physics macroscopic simplifications to make thinking about that sort of stuff easier. But fundamentally everything is derived from a few fundamental concepts. The same holds true in math and I also believe it probably holds true in all life(since, after all, painting, music, psychology etc are fundamentally based in physics). Does tonality also have some fundamental laws?

Any ideas on the subject?

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    Physics also tells us that if one verified example of a rule being broken is found, the rule is invalid. The "Rules" of tonality are far inferior by comparison - they are dictated by opinion and by the amount of harmony and discord that a certain set of people regard as "musical". Purely opinion, in other words. – Andy Mar 30 '16 at 14:36
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    There is the idea of tonality and its system and how best to utilize it, but there are no "rules". Sure classes set them up as rules but the main purpose of that so you understand the concept and the effect it has on what you compose. – Dom Mar 30 '16 at 14:46
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    There are no laws or rules per se. There is lots of theory, but thankfully it stays as theory because there are few absolutes in music. We learn theory and some then try to adhere to it religiously, only to find others taking no notice and having fun. – Tim Mar 30 '16 at 15:08
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    What is apparent is that physics and maths are sciences, whilst music and art are not. They're arts. Thank goodness! – Tim Mar 30 '16 at 15:40
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tl;dr: Yes, there are probably relatively simple underlying principles, but we're still working on what they are.

You make a really common and really valid criticism of music theory. Nothing seems to be founded on clear and simple principles, or core truths about human perception of sounds. It's all "this is just how it's done," or at best we're told that this or that musical device sounds sad or happy or stable or unstable, but not why. Most of the work going on in music theory just piles more and more abstractions on the existing framework without really dealing with the fact that in a sense it's all castles built on air.

But here's the good news: for hundreds of years there has been ragtag group of people around the edges trying to develop theories that re-construct common practice theory and harmony from simple rules based on the core facts of human cognition. It ranges from music theorists with an interest in perceptual foundations, to neuroscientists studying music, to experienced musicians just trying to better understand what they do.

Here are the best examples I know of:

  • Helmholtz was a 19th century doctor and physicist who built up a theory of many of the basic ideas of harmony from simple principles of physics and psychoacoustics. His book On the Sensations of Tone is celebrated and studied by mainstream music scholars. I haven't read it, but I have the sense that he gives a convincing explanation for triads and the major scale, but not the nitty gritty of Piston-type harmony and voice leading.
  • To deal with the Piston-type stuff, a lot of people in music theory have tried to explain harmonic progressions in terms of movement through geometric spaces. A music theorist named Dmtri Tymoczko has gotten a lot of attention lately, even in the mainstream press. He's the first music theorist ever to be published in Science, and he claims his theory encompasses all of the other geometric approaches so far and can explain almost all of Western harmony. He has a book about it called The Geometry of Music. This book is probably your best place to start because it's relatively simple, written for a general audience, and it's probably the best comprehensive theory we've got so far. You can also read his papers in Science if you want to cut straight to the hairy math stuff.
  • W.A. Mathieu's Harmonic Experience. Mathieu is a respected expert in both jazz and Hindustani classical music and he has a grand unified theory of harmony based on the overtone series and the sensations we feel hearing different intervals. He doesn't try and reduce things to a small set of core principles, but he is good at making sure everything is grounded in how different sounds makes us feel, and even asks the reader to sing intervals and pay close attention to how they affect your feelings and state of mind.

There are a lot of common threads through all these explanations and I wouldn't be surprised if things start to congeal into a consensus theory in the next 10 or 20 years. But for now, there's just a lot of good ideas swimming around. Tymoczko is probably the closest thing we've got right now.

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    +1 for Harmonic Experience, and for Tymoczko, whose theories seem very interesting! – Johannes Mar 31 '16 at 18:54
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The question I have, is there some specific fundamental concept that all this is built off of or is it just some arbitrary set of rules to create a structure on that produces a specific result?

I suggest that the answer is "yes, there is a specific fundamental concept", and the fundamental concept is consonance versus dissonance.

Western functional harmony and the common practice period as a whole are typically taught based on a foundation of consonance versus dissonance, and I think that is because that foundation is central to how tonality is created and expressed, and even central to the choice of the twelve notes.

In addition, the preception of consonance versus dissonance is only partly subjective. I'm not aware of any culture that has pitched music where two tones an octave apart are not seen as being the same or at least more closely related than any other pairs of tones. Fifths are also widely used, and there are some largely objective aspects of consonance farther along the harmonic series, certainly including fourths and, to a lesser extent, thirds and sixths.

A survey of different scales and tunings from cultures around the world, along with understanding of the acoustic properties of the instruments used in those cultures and how music is made in those cultures supports the idea that dealing with consonance and dissonance is central to all pitched music. In music, humans seem to seek a balance between the pleasing aspects of consonance tempered with the more interesting or exciting aspects of dissonance, and this is certainly true with the common practice period.

One thing that distinguishes Western common practice theory when it comes to dealing with consonance and dissonance is that both consonance and dissonance are created mainly by the selection of tones to be played or sung, regardless of the instruments, and that the instruments almost always play well-defined pitches. Instruments with less clear pitch definition, such as tubular bells, are not central to Western harmony or common practice, and are even relegated to a separate category, like "pitched percussion". Contrast that with Balinese gamelan where some amount of dissonance is created by deliberate tuning variations between instruments played in pairs to add "shimmer" to the overall sound that is essentially a just tuning composed of relatively consonant intervals. The sound of a gamelan has less motion between consonance and dissonance, rather the two exist simultaneously to create a sound that is beautiful and interesting even though it is fairly static.

Western harmony deals with consonance and dissonance in several ways, but the main way is by arranging in time mostly consonant chords with more disonnant chords to create both pleasing sounds and interesting sounds, as well as flows of tension and release. A more subtle aspect of this is the use of cadences and harmonic resolution, where a chord that would sound consonant on its on actual creates harmonic tension in a particular context because of the dissonance between the tones of that chord and tones that are well established in the piece that are expected and or remembered by the listener.

To give a specific example, suppose a piece firmly establishes the key of C major, thereby fixing in the mind of the listener both the central note of C and major tonality. In that context, a G major chord will have a kind of implied or remembered dissonance between the B natural of the G major chord and the key note of the piece, namely C. When the G major chord is followed by a C major chord, the implied dissonance is dispelled in the most consonant way possible, and there is a feeling of resolution.

Note that this answer is about tonality and expression of tonality and leaves aside any discussion of rhythm and meter.

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Basically, as the answers of the others have already suggested, the answer is "yes and no". Yes, there is a great deal of order imposed upon our ideas of tonality based upon the mathematics of music, which we can perceive to some extent in our feelings of tonal relationships (octaves, fifths, etc) and consonance vs. dissonance of intervals. No, there is also a lot of historical/accidental culture behind particular choices of what is considered "good" vs. "bad" tonality. It's not a simple question to answer.

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Well, I cannot say that I truly have the answer to your question exactly, but I can at least give you my take on a possible answer. There seem to be many questions that you have asked, and I'll try to investigate each one of them.

FIRST, In tonal music, the very fact of having a key and having a tonic sets a hierarchical structure of chords in their relationship to the great almighty tonic. The tonic is, like you said, the king in chess, and I would argue that therefor the dominant is the queen. The dominant contains the leading-tone, which is an incredibly important note due to its relation to the tonic. All other chords (i.e. pre-dominants, secondary-dominants, even momentary or expanded modulations and tonicizations) are all (in the big picture) eventually functioning for the purpose of decorating those two fundamental tonic and dominant chords.

SO, your question, then, seems to ask where and why exactly humans have created these rules and ways of thinking about music. Well, I'm not sure of the historical reasoning for this. But, When speaking of the WHY aspect to your question, it is here where I think that music might not best be compared to physics or math, but psychology. If you read up on any Schenkerian analysis, Schenker talks about notes being bodies of masses which have their own gravitational pull. This isn't necessarily something we recognize by way of algebraic induction or anything, but more as a human psychological response to pitches and their relations to others. For example, when a tonic is set in stone in our ears, the tonic has a strong and dense mass that will require other notes to be drawn towards it. It is only when these notes move toward this tonic mass, or RESOLVE, that we feel RESOLUTION. In this description, we see why it is that the leading-tone is so important in common practice period music. Our ears have this pull to want to resolve up to that densely important tonic note, and that half-step resolution becomes one of the key concepts in voice-leading, as voice leading helps to make transitioning sound clean from chord to chord, or in other words, contain those half-step resolutions that we want to here so intuitively. So when we break the rules of tonal music, we are manipulating these resolutions and stretching the concept of what defines as being dissonant and consonant. One of the all important biggest differences between Jazz and Classical Music is that Classical Music's most important chord in any key is "I", whereas in Jazz, the "I7" chord becomes a staple of total consonance. That seventh scale degree becomes less of a passing-tone, deriving form Fuxian species counterpoint, and more as an actual diatonic note.

SECOND - It seems to be that when you look at old Medieval and Renaissance (or even acient Greek), and Baroque, Classical, Romantic, and post-tonal music, all of these eras hold different rules and boundaries. However, they all seem to never change when it comes to understanding INTERVALS. Intervals are extremely important to observe in the Greek modes and are vital to understanding post-tonal music. You might look towards the history and development of how we have set rules about intervals because these don't seem to have changed as much through history. Combinations of intervals make up triads, they make up chords. They are like the protons and neutrons to the atom (the atom being the triad itself). They might even the bases for which we can draw similarities between different genres and eras of music. I know that Schoenberg stressed incredible importance on avoiding repeated imperfect consonances (3rds and 6ths). By doing so, it too closely implied some sort of tonality in his works. There really might be something going on here with the importance of our responses to intervals. I don't know.
Not sure if this helped, but I hope so!

  • There is some sever formatting issues on this answer. Can we get you to introduce paragraphs? It is very hard to read as is. – Neil Meyer Mar 31 '16 at 8:28
  • I know, seriously! It didn't look like that when I was typing it. I'll see if I can fix it. – Kalander Mar 31 '16 at 15:49
  • You have to leave a completely blank line between paragraphs for the formatting to work here, so press Enter twice rather than once :) – Matthew Read Apr 2 '16 at 4:30
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The pentatonic scale exists in nature. (1 2 3 5 6 1) The greatest common factor of the distances between these notes is 1/2 step. 12 of these distances span an octave. (Chromatic scale) Educated humans used math to expand the scale by adding 2 notes (4 and 7) creating the major scale. These tones have a strong tendancy to resolve. You might imagine their tension is from being forced in to the scale. the 4 resolves downward by 1/2 to the 3. The 7 resolves upward by 1/2 to the 1.

Tonality is all about tensions and resolutions.

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