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JI Primer Ed.3 THE JUST INTONATION PPRIRIMMEERR An Introduction to the Theory and Practice of Just Intonation by David B. Doty Third Edtion, December 2002 ISBN 0-9726810-0-0 ©1993, 1994, 2002 The Just Intonation Network A Project of Other Music, Inc. 535 Stevenson Street San Francisco, CA 94103 Phone: (415) 864-8123 Fax: (415) 864-8726 www.justintonation.net [email protected] iii Contents The Harmonic Series and the Subharmonic Series Preface to the Third Edition (2002).................... iv ........................................................................... 28 Preface to the Second Edition (1994)................. iv Converting Ratios to a Harmonic or Introduction .....................................1 Subharmonic Series Segment ..................... 28 Identities...................................................... 30 What is Just Intonation? ...................................... 1 Prime Numbers, Primary Intervals, A Little History.................................................... 2 and Prime Limits ............................................... 30 Antiquity ....................................................... 2 What is a Chord? ............................................... 30 The Middle Ages and the Renaissance ......... 2 Interval Names .................................................. 32 The Common-Practice Period Notation............................................................. 32 and the Rise of Temperament ....................... 3 Anomalies.......................................................... 33 The End of Common Practice....................... 5 Tetrachords and Tetrachordal Scales ................ 33 The Twentieth-Century Just Intonation Revival The Ladder of Primes, Part One:Two, ............................................................................. 6 Three, and Five................................ 35 The Purpose of this Publication........................... 7 One, the Foundation .......................................... 36 Acoustic and Psychoacoustic Background........................................9 Two, the Empty Matrix ..................................... 36 The Three Limit (Pythagorean Tuning)............. 36 Introduction ......................................................... 9 The Pythagorean (Ditone) Diatonic Scale .. 36 Periodic Vibrations .............................................. 9 Pythagorean Chromatic Scales What is an Interval?........................................... 10 and the Pythagorean Comma ...................... 37 Superposition of Pure Tones...............................11 Chords......................................................... 38 Phase Relationships ................................... 12 The Five Limit................................................... 38 Interference Beats ....................................... 12 Constructing a Five-Limit Major Scale ...... 39 The Harmonic Series......................................... 14 The Supertonic Problem ............................. 40 Difference Tones................................................ 16 The Five-Limit Lattice................................ 42 Periodicity Pitch ................................................ 17 The Great Diesis ......................................... 44 Complex Tones with Harmonic Partials............ 19 The Harmonic Duodene.............................. 45 Coincident or Beating Harmonics............... 20 Five Limit Chords....................................... 46 Arthur H. Benade’s “Special Relationships” Condissonant Chords in the Five Limit ...... 47 .................................................................... 22 Defi ciencies of the Five Limit .................... 48 Special Relationships Beyond the Octave .. 23 The Ladder of Primes, Part Two: On the Consonance of Relationships Seven and Beyond............................. 51 Involving Higher Harmonics ...................... 24 Seven-Limit Intervals........................................ 51 Basic Definitions, Conventions, and Procedures ............................... 25 New Consonances....................................... 51 Whole Tones ............................................... 54 Introduction ....................................................... 25 Semitones.................................................... 54 Rules and Conventions...................................... 25 Microtones .................................................. 54 Calculations with Ratios.................................... 25 False Consonances...................................... 54 Addition ..................................................... 25 Two Dimensional Planes in the Subtraction ................................................. 25 Seven-Limit Fabric ........................................... 55 Complementation ....................................... 26 Seven Limit Chords........................................... 56 Converting Ratios to Cents ........................ 26 Subsets of the Dominant-Seventh Dividing Just Intervals ............................... 26 and Ninth Chords........................................ 56 Calculating Absolute Frequencies (Hz) ..... 28 Diminished Triad ........................................ 56 Contents iv Half-Diminished Seventh Chord ................ 57 Electronic Organs ....................................... 70 “4-6-7” Chord ............................................. 57 Stringed Instruments with Seven-Limit Subharmonic Chords.............. 57 One String per Note .................................... 70 Condissonant Chords in the Seven Limit ... 58 Fretted Instruments ..................................... 71 Fixed Scales in the Seven Limit........................ 59 Wind Instruments .............................................. 73 The Higher Primes: Eleven, Thirteen, Continuous-Pitch Instruments ........................... 74 and Beyond........................................................ 59 “Natural Tendencies’’.................................. 74 Eleven and Thirteen .................................... 59 Bowed Strings............................................. 75 Primary Intervals......................................... 61 Voices.......................................................... 76 Chords and Harmony.................................. 62 Trombones .................................................. 76 Seventeen and Nineteen.............................. 64 Midi Synthesizers.............................................. 76 Beyond Nineteen......................................... 65 Pitch Bend................................................... 76 Practical Just Intonation with Real Instruments with User-Programmable Instruments ..................................... 66 Tuning Tables.............................................. 77 Tuning Tables plus Pitch Bend ................... 78 Introduction ....................................................... 66 The Midi Tuning-Dump Specifi cation........ 78 Fixed-Pitch Instruments .................................... 67 Some Thoughts on Obtaining Acoustic Keyboards.................................... 67 Satisfactory Performances................................. 79 Piano Problems ........................................... 67 Reed Organs................................................ 70 Notes................................................ 81 Pipe Organs................................................. 70 Index ................................................ 85 Electroacoustic Keyboards ......................... 70 Preface to the Third Edition (2002) Intonation Primer; any future editions will be in the The third edition of The Just Intonation Primer contains form of multimedia products. There seems to be little no signifi cant changes in content from the previous edi- purpose in merely writing about unfamiliar intervals, tions, though a few minor errors have been corrected. scales, and chords when it is now possible to present The type, however, has been completely reset and all of students with the actual sounds. the art has been recreated from scratch. Why? The fi rst Thanks to Lucy A. Hudson for creating software and second editions of The Primer were produced with that simplified the conversion of the text files from the software that has long since become obsolete, necessitat- previous edition. ing the resetting of the type. As for the art, both my skills and the available tools have greatly improved since the Preface to the Second Edition (1994) production of the previous editions, and my standards, The writing and publication of The Just Intonation as well, have risen considerably. Primer were made possible by a gift from a friend of This is not to say that I regard The Primer, as it now the Just Intonation Network. stands, as needing no improvement — quite the contrary. The material in this publication was excerpted The problem is that once I began making improvements, from a larger work in progress, tentatively entitled A there is no telling where the process would end. This Composer’s Guide to Just Intonation. would be contrary to my purpose for creating this third Thanks to Carola B. Anderson and Dudley Duncan edition: to keep the publication available until I am able for proofreading and commenting on The Just Intona- to complete the work that will supersede it. This is tion Primer. Thanks to Jim Horton for the use of his intended to be the last “print-only” edition of The Just collection of obscure publications. The Just Intonation Primer introduction What is Just Intonation? They are what the
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