Musical Temperaments

Preface What you are now reading is the written version of an electronic document that explains the mathematical principles for different musical temperaments. The electronic version contains many music examples that you can listen to while you are working with this document at a computer. The written version obviously cannot offer this possibility. It serves therefore merely as a parallel study aid and guide and cannot replace actually working with the electronic text. Musical Temperaments Contents • V II Contents Introduction and Fundamental Properties 1 Pitch and Frequency 1 Preliminary Remarks 1 Frequencies and Intervals 2 Tuning Systems and Frequencies 5 Musical Scales in Different Tunings 5 Pure Tuning 5 Intervals and Triads in Pure Tuning 12 Pythagorean Tuning 23 Intervals and Triads in Pythagorean Tuning 31 Meantone Tuning 34 Intervals and Triads in Meantone Tuning 39 Equal Temperament (Tuning) 42 Intervals and Triads in Equal Temperament (Tuning) 47 Summary 50 Appendices 53 Pictorial explanations 53 Tables of Frequencies and Intervals 54 Operating Instructions 62 Glossary 67 Musical Temperaments Introduction and Fundamental Properties • 1 Introduction and Fundamental Properties Pitch and Frequency Preliminary Remarks It is well known that tones consist of periodically recurring phenomena, that is, beats repeating in a regular pattern. The number of repetitions of beats per second is measured in Hertz: 440 Hertz mean 440 beats per second. This number is also called the frequency of a beat. This multimedia product (book) lets the reader investigate the mathematical foundations of musical tuning systems as theory and as audible experience. The embedded acoustical examples help to establish immediate connection between the formulas and the acoustical characteristics of different musical scales. The musical temperaments covered in the document are Pure, Pythagorean, Meantone, and Equal, the historically most important tuning systems. For each one of these temperaments the construction principles are explained in detail and with interactive visual aids, and whenever a mathematical formula for an interval is derived, this interval is made audible and can easily be compared with similar intervals in the different tuning systems. The sound examples in the multimedia version cover intervals, scales, and chords in the different tuning systems, and some examples even allow to listen to different scales simultaneously on the two channels of a stereo system. Besides being an introductory text with sound the document also contains a reference part which gives easy access to all the sound examples in a systematic way. System requirements: IBM PC or compatible, 8 MB RAM, CD-ROM drive, Soundcard, VGA Graphic Adapter, 16 colours, 40 MB free diskspace; Windows 3.x, Windows 95, Windows NT 4.0 European Academic Software Award 1996! Mathematical Foundations Of Intervals And Chords Combined With Sound Examples Embedded In The Cd-rom-based Multimedia Version Connect Aural Experience With The Construction Principles Of Four Different Historically Significant Temperaments. Introduction And Fundamental Properties. Pitch And Frequency. Frequencies And Intervals -- Tuning Systems And Frequencies. Musical Scales In Different Tunings. Pure Tuning. Intervals And Triads In Pure Tuning. Pythagorean Tuning. Intervals And Triads In Pythagorean Tuning. Meantone Tuning. Intervals And Triads In Meantone Tuning. Equal Temperament (tuning). Intervals And Triads In Equal Temperament (tuning). Erich Neuwirth. Includes Bibliographical References. Lets the reader investigate the mathematical foundations of musical tuning systems as theory and as audible experience. This title covers such musical temperaments as: Pure, Pythagorean, Meantone, and Equal, the historically most important tuning systems. It explains the construction principles for these temperaments with interactive visual aids. Front Matter....Pages I-VII Introduction and Fundamental Properties....Pages 1-4 Tuning Systems and Frequencies....Pages 5-51 Back Matter....Pages 53-71