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Mike's Lectures on Regular Temperament Theory

These are a series of posts that explain, in the perspective I've found most useful, some of the ins and outs of tuning theory. I will be adding to these posts over time.

In these posts, I assume that you have a basic understanding of
  1. High school algebra (e.g. what a vector is), but not matrix algebra
  2. What ratios are
  3. What cents are
  4. What monzos and vals are, but nothing else

If you don't understand monzos and vals, check out the General Theory pages on those - they're very simple, and this is supposed to build on that. I'll keep adding to this as I write more stuff.

The Lectures

  1. Vector Spaces and Dual Spaces
  2. The First Fundamental Law of Tempering




Some notes on terminology:

  • There are many times when more than one term has been used for the same thing, with there being no real preference one way or another. When this happens, I've picked the term that I find to be the most intuitive description of what the concept is.
  • Additionally, there are a number of instances where theorists disagree on various minutia involving the precise definition terms. Sometimes people have very subtle differences in their usage of "temperament," "MOS," etc. I've chosen the most middle-of-the-road naming convention possible, and where it's still unclear, I've left footnotes specifying how other people view these things.
  • I've seen firsthand that most people reading this are not only capable of understanding the theory as it's currently being talked about on places like tuning-math and XA, but also have interesting ideas to contribute. I believe that the lack of a resource simply stating what some of the terminology means in very clear terms is the simple greatest obstacle that this theory faces towards greater adoption. I hope that this guide can serve as such a resource, and as such I've placed an emphasis on defining "standard" terms as clearly as possible, so as to facilitate the understanding of what, exactly, everyone's saying.
  • There are a number of people who advocate completely scrapping the terminology commonly used, e.g. things like "val," "monzo," etc, and starting over from scratch with more intuitive terms. There is a place for that sort of argument, and consequently a separate guide I'd like to write someday for musicians who don't care about math at all and hate it. However, this is not that guide - this is meant to be a primer for the folks who aren't afraid to get a LITTLE bit mathematical, and still want to understand tuning theory. As such I've chosen to use the "standard" terminology when possible, as it's currently being used on tuning and tuning-math, on this wiki, on XA, etc. However, I have always taken the effort to define my terms in clear, intuitive ways when necessary, using lots of examples.

Some final notes about my goals with this series of articles:

Sometimes tuning theorists differ subtly in the ways they like to think about tuning theory. This is a result of this field of study being a hugely interdisciplinary collaboration between mathematicians, musicians, engineers, physicists, psychoacousticians, etc. Many of these disagreements derive from the fact that different fields emphasize different skills and ways of thinking.

For instance,
  • Mathematics often concerns itself with understanding how different problems and questions interrelate and derive from the same fundamental questions. Much emphasis is placed on finding mathematical commonalities between seemingly disparate ideas and questions, and handling as many of those questions at once. Ideal descriptions of theoretical concepts thus tend to focus on this interrelation and hence "compress" an enormous amount of information and potential into a much shorter description.
  • Musicians and artists, if they're into theory at all, are typically concerned with what "the point" of the model is, or what the model "means." They tend to be more interested in the different motivating ideas behind the development of any such models, and tend to care much less about any underlying mathematical ideas internal to the model, unless those ideas themselves have musical or artistic relevance. For an artist, a good description of a theoretical concept would be one that emphasizes the different motivating ideas behind the concept, gives lots of examples, simplifies the math where necessary and is redundant when clarity dictates.
  • Engineers often find themselves striking a middle ground: they typically need to find different ways to USE the models, thus having to figure out all sides of the picture. Emphasis is placed on understanding not only how to use the model to solve problems, but on how to identify the fundamental commonalities that the different problems in any field of study tend to share. For an engineer, a good description of a theoretical concept would be one that gives the motivating ideas behind the concept, gives explicit examples about how those ideas reduce to a much simpler set of questions, and finally goes into the mathematics of how to solve that set of questions with numerous examples.

These differing modi operandi often lead to one group advancing the most "compact" or "efficient" mathematical representation that unifies a wide range of ideas and phenomena, whereas another group asks "what does this -mean-," and another group says, "how can I use this?"

Thus there's been, historically, a lot of nitpicking over almost all aspects of this theory: people tend to disagree on the best way to state what it is, the best way to define terms, the best way to teach it to people, and so on. Sometimes these disagreements are subtle, sometimes they lead to large arguments, and sometimes they lead to a total breakdown of communication for bystanders. If you've asked a question on XA only to have it degenerate into a 300 post thread involving an argument about "contorsion" or something similar, leaving you more confused than you started, you've unfortunately fallen prey to this phenomenon.

I've tried my best to strike a middle ground between these different camps: I've placed a great emphasis on what these models "mean" and "how to use them," but have also tried to make sure my mathematical usage is correct (save for a few areas where I've simplified things intentionally and stated so explicitly). Additionally, many of these disagreements seem to be fundamental enough that it's not clear that people will ever agree on the best way to explain things leaving. When such a problem arises and no pedagogical solution seems to exist, I've attempted to innovate one, sometimes introducing new notations and terms when I think it's really necessary to fill a void in the way this is currently being explained. I've avoided introducing any new concepts unless they're absolutely necessary, and anything I do introduce will be something that most theorists ought to be able to easily understand; thus, you can be sure that the paradigm I'm presenting here is reasonably "canonical" enough to allow for communication between you and other parties.

Keep in mind that my understanding is always evolving, so I may change these pages as I come up with more clever and useful ways to explain things in the future!