Dave Keenan & Douglas Blumeyer's guide to RTT

In April 2021, Douglas Blumeyer first set out to study regular temperament theory (RTT), guided primarily by Dave Keenan, one of the co-founders of the modern RTT paradigm. Douglas (howdy partner!) came mostly with questions, and Dave (g'day mate!) mostly with answers, but nearly 2000 emails later (and not short ones either—we're estimating well over a million words exchanged here), our ratios of Q:A have gotten closer to unison by now (January 2023). The following series of articles distill the fruits of our protracted discussions down into what we hope you'll find to be a complex but refreshing brew of insights and solutions. For those already familiar with both microtonality and linear algebra, we have further distilled the first 6 articles of the guide down to 6 pages, in our conference paper entitled Regular Temperament Theory: Exploring the Landscape between JI and ETs with Linear Algebra.

[math] \LARGE\left[ \begin{matrix} \style{background-color: #C6DC67; padding: 5px;}{12} & \style{background-color: #8DC73E; padding: 5px;}{19} & \style{background-color: #3FBC9D; padding: 5px;}{28} \\ \style{background-color: #7977B8; padding: 5px;}{17} & \style{background-color: #AF72B0; padding: 5px;}{27} & \style{background-color: #D883B7; padding: 5px;}{40} \\ \end{matrix} \right] \left[ \begin{matrix} \style{background-color: #FDBC42; padding: 5px;}{1} \\ \style{background-color: #FFDF42; padding: 5px;}{{-2}} \\ \style{background-color: #FFF200; padding: 5px;}{1} \\ \end{matrix} \right] = \left[ \begin{matrix} (\style{background-color: #C6DC67; padding: 5px;}{12}×\style{background-color: #FDBC42; padding: 5px;}{1}) & + & (\style{background-color: #8DC73E; padding: 5px;}{19}×\style{background-color: #FFDF42; padding: 5px;}{{-2}}) & + & (\style{background-color: #3FBC9D; padding: 5px;}{28}×\style{background-color: #FFF200; padding: 5px;}{1}) \\ (\style{background-color: #7977B8; padding: 5px;}{17}×\style{background-color: #FDBC42; padding: 5px;}{1}) & + & (\style{background-color: #AF72B0; padding: 5px;}{27}×\style{background-color: #FFDF42; padding: 5px;}{{-2}}) & + & (\style{background-color: #D883B7; padding: 5px;}{40}×\style{background-color: #FFF200; padding: 5px;}{1}) \\ \end{matrix} \right] = \left[ \begin{matrix} 2 \\ 3 \\ \end{matrix} \right] [/math]

In order to best help all of the different types of members of the regular temperament community, we have broken our series into three levels, according to both the complexity ceiling as well as the target audience. The basic level is targeted for musicians, the intermediate level is targeted for engineers, scientists, and mathematicians, and the advanced level is targeted for theoreticians.

• Basic
  • 1. Introductions: where we each briefly introduce ourselves and RTT
  • 2. Mappings: learn how temperaments map intervals, and how to find mappings you like
  • 3. Tuning fundamentals: learn about minimizing damage to target intervals
  • 4. Exploring temperaments: learn the relationship between mappings and commas
• Intermediate
  • 5. Units analysis: to look at temperament and tuning in a new way, think about the units of the values in frequently used matrices
  • 6. Tuning computation: for methods and derivations; learn how to compute tunings, and why these methods work
  • 7. All-interval tuning schemes: the variety of tuning scheme that is most commonly named and written about on the Xenharmonic wiki
• Advanced
  • 8. Alternative complexities: for tuning optimizations with error weighted by something other than log-product complexity
  • 9. Tuning in nonstandard domains: for temperaments of domains other than prime limits, and in particular nonprime domains

• Reference tables (for all levels)
  • Appendix. Conventions for names, variables, units, and notations

Other works

We also produced this related article, though we personally concluded that exterior algebra is not important to study as part of learning RTT:

• Dave Keenan & Douglas Blumeyer's guide to EA for RTT

Our RTT collaboration also led to the creation of this code library:

• Dave Keenan & Douglas Blumeyer's RTT library in Wolfram Language

Our RTT collaboration has further led to creating or majorly contributing to the following RTT-related pages:

• Saturation, torsion, contorsion, and defactoring
  • Defactoring algorithms
  • The pathology of enfactoring
  • Defactoring terminology proposal
• Matrix echelon forms
• Projection
• Generator embedding optimization
• Generator form manipulation
• Temperament addition
• Temperament merging
  • Cross-domain temperament merging
• Domain basis
• Map
  • Uniform map
  • Simple map
• Extended bra-ket notation
• Unison temperament
• Eigenmonzo
• Secor
• Talk:Interior product
• Talk:Meet and Join

Several more RTT articles could be added to our series, such as ones to cover scales and chords and progressions, temperament complexity and badness and classification, lattices, timbre, and notation. But we will probably not get to this in our lifetimes.

Credits

We couldn’t have put together what we've got so far without a ton of help from people such as:

• Paul Erlich
• Mike Battaglia
• Graham Breed
• Steve Martin
• Herman Miller
• Keenan Pepper
• Flora Canou
• Inthar
• Sintel
• Kite Giedraitis
• Tom Price
• Scott Thompson
• Joshua Sanchez
• Vincenzo Sicurella
• Petr Pařízek
• Margo Schulter
• Stephen Weigel
• Juhani Nuorvala
• Frostburn

plus many more we can't all list. And while sadly both George Secor and Gene Ward Smith had passed away before we started this project, we owe a huge debt to both of them as well.

We take full responsibility for any errors or shortcomings of this work (in particular Douglas does, being the sloppier of us two, and — unfortunately for you, dear readers — also the one of us who had way more time on his hands and therefore the one who did the majority of the actual typing up of the final materials). Please do feel free to edit these articles yourself if you have something you'd like to correct, revise, or contribute. Otherwise, feedback to either of us directly is very welcome.

Happy tempering!