# Carl Lumma

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**Carl Lumma** (b. 1977) is a man with varied interests, including nuclear engineering, macroeconomics, and music theory. Making his home in silicon valley, he has worked as an editor at Keyboard magazine and as a program manager at Apple. Carl has been a moderator of the Tuning and Tuning-math mailing lists. Some of his recent posts are collected here and some ancient posts are here.

**Some of his music theory articles:**

- 28-string Cosmolyra after Ivor Darreg
- Isomorphic keyboard mapping for 5-limit just intonation
- Spectral analysis reveals accurate 7-limit intonation in barbershop music
- Archimedean keyboards
- Rothenberg stability in pitch space
- Stellated Combination Product Sets: How many tones?
- Xenharmonic Moving Windows: Extended JI with conventional keyboards
- MIDI-based adaptive tuning by common-tone matching
- Generalizing diatonicity
- Successive improvements in 17-limit TOP damage for the first 100 ETs
- Logflat-best rank 1 temperaments up to 100 notes/octave
- Classification of microtonal notation systems
- No limits: Optimal subgroups for equal temperaments
- Measuring the strength of chord progressions
- Maximum dyadic error of triads in equal temperaments
- The MOS theorem: Well-formed scales are tempered Fokker blocks

**Stuff of a more tutorial nature:**

- Xenharmonic Glossary
- The Alternative Tuning FAQ
- Cross-sets Primer
- The Too-Condensed Tuning-Math Outline

**Elsewhere on the web:**