Historic Temperaments

Here are some examples of a part of a Bach chorale played in a number of different tuning systems common during the 16th, 17th and 18th centuries. They were made using Mathematica 3.0, and to hear them you need to download MathReader from http://www.wolfram.com/mathreader/ and configure your browser to use MathReader. Alternatively, download the notebooks and just view them in MathReader itself.

The Mathematica package which was used to generate these sounds contains modules for building a chromatic scale in a number of different tunings. It provides functions for you to experiment with your own tuning, and also to create your own sound (you need Mathematica 3.0 for this). It allows transposition so you can hear how temperaments behave in different keys. The transposition comes in two flavours - the normal sort, and transposition of the tuning with respect to the notes so you can hear the result of transposition without changing pitch (this is the sort used in the examples). There are functions for building up chords with as many notes as you like, and an example which is a whole Bach chorale, part of which was used for the sounds below.

The fake organ sound will be ok on some systems (generated on a PowerMac) but may sound bad on others. I have more examples of these tunings, as well as Pythagorean, 1/2 comma meantone (Fogliano), 1/5 comma meantone (Rossi), Kirnberger, and the whole chorale in Aron.


Complete notebooks for MathReader

Introduction

A description, with examples to listen to, of the problems encountered in tuning the musical scale, some of the ways these problems have been dealt with, and the effect these solutions have on the way music is written. This notebook also contains a brief set of instructions for using the Mathematica 3.0 package, tuning.m (for those who have Mathematica).

Equal Temperament

Example in equal temperament

Just Intonation

Example in just intonation, C major, with description of the tuning.

Example in just intonation, C sharp major

Example in just intonation, B flat major

1/4 comma meantone tuning - Aron

Example in 1/4 comma meantone tuning, C major, with description of the tuning.

Example in 1/4 comma meantone tuning, C sharp major

Example in 1/4 comma meantone tuning, A major

Well temperament - Werckmeister III

Example in Werckmeister III well temperament, C major, with description of the tuning.

Example in Werckmeister III well temperament, C sharp major

Example in Werckmeister III well temperament. A major


The package and other example files

If you want to make up more sounds and have Mathematica 3.0 or later, you can download the tuning.m package and the other notebooks (without any sounds evaluated so the files are quite small)

Meantone tunings

Fogliano 1/2 comma

Aron 1/4 comma

Rossi 1/5 comma

Well temperaments

Werckmeister

Kirnberger

Barnes

Other tunings

Equal temperament

Just intonation

Pythagorean

The package and auxiliary files

Tuning.m package

scales.nb notebook

Source file for Bach chorale

Bach chorale in Fogliano

Bach chorale in Aron

Introduction (no sounds evaluated)