Regular Tunings and Temperaments

A regular tuning system contains all notes created by the addition or subtraction of a (finite) set of intervals called generators. The minimum number of generators required to generate all the notes in the system is called the dimension of the tuning system. A 1-D system requires one generator, a 2-D system requires two generators, etc.

All equal temperaments are 1-D because all the intervals can be generated from just one interval. For example, all the intervals in 12-TET can be generated from multiples of the 100 cent semitone.

Non-equal temperaments require at least two generators. For example, all the intervals of 1/4-comma meantone can be generated by a fifth and an octave Ð there is no single interval that an generate all of its intervals. The dimensionality of a p-limit JI is equivalent to ¹(p), where p is the prime-limit and ¹() is the prime counting function. For example, 3-limit JI is 2-D, 5-limit JI, and is 3-D, 7-limit JI is 4-D, 11-limit JI is 5-D.

The Syntonic, Magic, and Hanson tuning continua are 2-D. Although 3-D or higher systems can provide more accurate tuning, 2-D systems represent a very good compromise because:

They can be isomorphically mapped to a 2-D button-lattice thus giving Transpositional Invariance.

They have a musically useful range of Tuning Invariance.

The generators are easy to control using a pair of tuning sliders or an X-Y pad.

They produce simple and aesthetically consistent scales.

They are easy to notate, learn and perform.

A regular temperament is a mapping of just intonation intervals to a regular tuning system. Usually the mapping is from a higher-dimensional Just Intonation (e.g. 3-D 5-limit JI, 4-D 7-limit JI, or 5-D 11-limit JI) to a lower dimensional 2-D or 1-D tuning system. The loss of dimensionality means that some intervals become mistuned (or tempered). In exchange, the structure of the system becomes simpler and easier to control, notate, play, and understand.

Every regular temperament is associated with an Ordinal Tuning Range over which its intervals can be said to play the same role and so remain essentially the "same" despite the differences in the precise tuning. In the TransFormSynth, three temperaments and their associated Ordinal Tuning Ranges are implemented: the Syntonic, Magic, and Hanson. Each temperament has:

A continuum of possibile tunings (i.e. ordinal and purer-tuning ranges with their associated n-TET and commatic landmarks).

A unique tempering of overtones (when the tone diamond is set to give tempered timbres)

A unique MOS scale, which gives a unique relationship between the triads embedded within that scale.