ABSTRACT

The author develops a syntax for meantone harmony based on the structural characteristics of the temperament. These features include a description of the Regions with their boundary functions, and also the three genera. An abstract topology of triad progression is introduced, as well as an introduction to the organization of functional tables for 31-ET.

The great one-quarter-comma meantone tuning system has had overriding importance in the history of European music culture, where it was put to use for longer than any other temperament. Moreover, it was the tuning system associated with the greatest flowering of that culture, being used from the high Renaissance through the 16th and 17th centuries and on well into the 18th century. The only evident reason for its demise was a problem of technology. No adequate keyboard was developed which enabled the performer to implement all 31 notes to the octave. Since the keyboard became more and more popular over this period of time, musicians moved to an inferior system which better suited the medieval keyboard. However, the long-term influence of this tuning can still be felt in the continued use of a musical notation which employs both flats and sharps, a legacy of the meantone era.

Modern electronic technology makes possible the fabrication of efficient music
keyboards for 31-ET. Remember that most other instruments, for instance wind
instruments and strings, have variable intonation and so can readily adapt to
whatever temperament is at hand. The keyboard was the *only* impediment
to the use of extended meantone music in ensemble settings. Enharmonic
keyboards would be happily accepted by violin players who have no qualms about
the use of the enharmonic, thus liberating the potential of the string
ensemble. Hence we have the real possibility of a new flowering of meantone
music. At the same time, many composers now feel that the 12-ET system has
been exhausted of its new harmonic possibilities. As a result there is a new
willingness to explore alternative systems of harmony. 31-ET offers immense
harmonic resources which are as yet largely untapped. With the aim of
promoting the exploration of meantone harmony I offer a systematic layout of
the structural features of the system. Such a layout will suggest a natural
syntax for meantone music, without dictating such matters as musical styles.

My aim in this endeavour is to develop a theory which is descriptive, and not
proscriptive (as was the case with 19th century music theory). Musical styles
change as the progressive working out of the implications of the structure
develop over time. I have already used this observation with regard to the
evolution of 12-ET harmony in the 19th century, an evolution which developed
often in spite of the theoretical structure which was overlaid on the
materials. We could say, for example, that the structural characteristics of
12-ET included an inherent tendency toward atonality. No matter how badly
the theoretical structures of the late century fit the musical patterns,
composers felt compelled, nevertheless, to explore the implications of the
*actual* musical structure with its functional ambiguity and potential
for rapid modulation. The theoretical language in use, meanwhile, was largely
inherited from the meantone era, with its distinction of sharps and flats.
Eventually this theoretical language became overly complex and stilted, since
it did not suit the actual structure of 12-ET. To a large extent, this
situation came about because of the lack of an adequate model with which to
compare the divergent structure of 12-ET and 31-ET.

In the same way as 12-ET, 31-ET has its own structural character which endows it with inherent tendencies. The history of the evolution of harmony during the 16th and 17th centuries attests to the working out of various implications of the extended meantone fabric. One prominant example which comes to mind was the increasing use of chord relations which depend on the mediant during the course of the 16th century, so that by the end of the century we see the 'avant-garde' progressions of Marenzio and Gesualdo. Another example was the Baroque distinction between the usual type of Dominant Seventh tetrad and another type which is typified by the function 6-1-3-4: called a "German Sixth Chord". This chord displays the Harmonic Series Seventh between 6 and 4, instead of the usual Seventh which would have the function 6-5, thus giving it a different structure and sound. (This distinction is meaningless in 12-ET). Hence we see a progressive working out of the implications of the true, underlying structure of the system; moreover, we see the conservative maintenance of this theoretical language long after musicians were using 12-ET in fact, and not 31-ET. The 31-ET distinction between sharps and flats remains 'in theory' alone, and not in the actual structure of 12-ET.

The appropriate theoretical language for a system of harmony should arise from its true underlying structure, and not from some other system. Such a practice results in confusion if there is little understanding of the divergencies between the systems. The music cultures of the 19th and 20th centuries have displayed a deplorable ignorance in their comprehension of meantone harmony. Although the modern situation has improved, there is still the need of an integrated overwiew of the system anatomy. The composer who may wish to work in the milieu of extended meantone tuning needs a theoretical structure which reflects the inherent tendencies of the tuning. With such tools he/she can better utilize the materials at hand. The proper aim of a theoretical structure is to clarify the nature of the materials so that the composer or improviser can command his medium to produce the most satisfying results.

I have already written at length concerning the extended
meantone fabric. In the article *The Meantone Series of Cyclical
Temperaments - An Introduction and Comparative Appraisal* the most
important structural features are presented within the context of
the other great tuning cycles, so that an understanding of the
structure can emerge from its contrast to Just Intonation. In the
article *The Extended Meantone Style of Gesualdo* I have focused on
historical issues so as to put the evolution of 16th century style
within the context of meantone harmony. Both of these articles
complement this one. In the interest of brevity, this article will
develop issues that are only just touched upon in those two
articles. In like manner, themes which are well worked in those
two articles will be presented in an abbreviated form here, so
that we may look upon these three articles as mutually complementary.
Of course, a certain amount of overlap cannot be avoided. We are observing a
jewel from many angles in order to appreciate its luminous structure.

31-ET is fundamentally a "microtonal" scale of 31 pitches to the octave. This "ultra-chromatic" scale uses the diesis or enharmonic (the difference between a sharp and a flat) as the basic step interval of the system. All other intervals are made up of aggregates of this basic step. Consequently, a good workable preliminary notation for the system is a numeration of the pitches from 0 to 30, pitch 31 being the octave equivalent to pitch 0. This numerated scale arbitrarily commences at 'C'=0. The Baroque notation for the system used naturals, sharps, flats, double-sharps, and double-flats. This notation is preserved in the modern Fokker notation, but it has been augmented in a clear and rational fashion so that double-sharps become semi-flats, and double-flats become semi-sharps. Additionally, flats may be renamed as sesqui-sharps, and sharps may be renamed as sesqui-flats, for purposes of clear staff notation. The complete rationality of this notation can be seen on the Circle of Fifths which this system exhibits. See Figure 1.

31-ET is a cyclical temperament, meaning that the fifths
are tempered (mis-tuned) in such a way that they form a closed
circle of fifths, thus allowing unlimited modulation. The regional
structure of the system emerges from the 'background' nature of
Just Intonation, which has its own structural features. The
anatomy of Just Intonation is carefully covered in the *Meantone
Series* article. Briefly, we may say that the framework of Just
Intonation stems from the restriction to the use of the pure 3rd
and 5th harmonics in setting up the tuning. Consequently, a field
or matrix of relations results which arises directly from the
lattice of just musical triads. This lattice is open-ended, it
does not form a closed cycle of fifths, due to the nature of
number relations. However, several musical intervals crop up which
suggest the possibility of alternative closed cycles. Such closed
cycles can be achieved by slightly mis-tuning the Just norms in
controlled ways resulting in an system of Equal Temperament. The
schisma interval lies at the very edge of our pitch discrimination
perception. Consequently, we could resolve the field into a scale
of schismas, by tempering to the closed cycle of 612-ET. Such a
scale is maximally accurate in its 'resolution' of the Just
lattice, but its complexity makes it musically impractical. The
presence of the Syntonic Comma interval (11 schismas in size), and
the Ditonic or Pythagorean Comma (12 schismas in size), leads to a
natural resolution of the field of Just Intonation into a scale of
Commas, if we temper into the closed cycle of 53-ET. This great
system of temperament can rightly be named Quasi-Just Intonation
since it preserves so many of the essential properties of pure
Just Intonation within a reasonably manageable complexity.

Figure 2: The field of Just Intonation showing the *comma based
notation*:

The regional structure of 53-ET discloses three primary
regions: a central region in which intervals are not
comma-altered, and two flanking regions in which intervals are
raised by a comma (on the dominant side), and lowered by a comma
(on the sub-dominant side). See Figure 2. The important boundary
between these regions of Just Intonation I have called the
*Meantone Corridor*. 31-ET tempers away the comma, but preserves
the Diesis, which is about two commas in size, as its basic step
interval. This is done by tempering the fifths by one-quarter of a
comma, so that the two pitches D and \D of Just Intonation form a
compromise pitch of about midway in size - the meantone. Such a
compromise strongly orients the lattice of triads along the Major
Thirds Axis of Just Intonation, since the major thirds are tuned
pure while the fifths and minor thirds are compromised. The notion
of a triad lattice is preserved, but it is given a decidedly
'vertical' cast. Figure 3 shows the alteration of the Just Field
in parts of a comma. Figure 4 indicates the resultant field of
pitches with the modern 31-ET notation.

We see that certain aspects of the Just Field are preserved, while other aspects are eliminated. The Schisma boundary (which is elsewhere also called the 'East-West Boundary' of the Field) is eliminated along with the comma-alterations, leaving the Meantone Corridor as the true 'east-west' boundary of the field. At the same time the Enharmonic Gateway (the North-South boundary) of 53-ET and 19-ET is eliminated, while preserving the Harmonic Antipodes with its neutral intervals. The orientation of the lattice of triads is quite 'vertical' along the Major Thirds Axis of Just Intonation, where the enharmonic displays itself.

This lattice of triads is the key to understanding the field of potential chord relations for the system, since it may also be viewed as the field of functional relations to a given Tonic. Figure 5 shows this Functional Field for 31-ET using the numerical notation (which may thus also be employed to describe functional relations) as well as the more standard functional notation in which "do-re-me" becomes "1-2-3". Thus the tonic major triad has the function 1-3-5, and the tonic minor triad has the function 1-3-5. It is important to remember that this chart is independent of any particular pitch, since any pitch in the system may act as the Tonic. It shows all of the ways in which various tones can relate to the Tonic. These tones combine to form triads, tetrads, etc. leading to the complex relations of functional harmony.

The 'vertical' nature of the functional field leads to a natural stratification of the lattice into eight regions with the Enharmonic Region occupying both the 'south' and the 'north' of the triad web. The Meantone Corridor forms the appropriate boundary between the regions. Triads may be classified according to where they 'sit' in the lattice. For instance, the 6 major triad (6-1-3) lies squarely in the middle of the Minor Region; whereas, the 6 minor triad (6-1-3) lies within the Major Region. Note that the Enharmonic Region lies at the furthest removed position from the Major Region. The top and the bottom of the chart meet in a 'strange loop' since it is a closed cycle. Consequently, the regions may also be indicated on the Circle of Fifths, as shown on Figure 6.

Certain functional triads seemingly lie in two regions at
once. For example, the 2 major triad (2-4-6) appears in the Major
Region as well as the Supra-Major Region, as does its Relative
Minor - the 7 minor triad (7-2-4). Such triad functions which
have two componants within the Meantone Corridor are called
*boundary functions*. These functions have peculiar significance to
meantone tuning. They do not have the same character in Just
Intonation or in 12-ET where this whole regional structure
collapses to essentially three: Major, Minor, and 'Enharmonic'.
Thus in 31-ET the 5 minor triad (5-7-2) lies on the boundary
between the Major and the Minor Regions, as does its relative
major triad, 7 major triad (7-2-4).
Suspended triads and
tetrads, for example the function usually called the 'Tonic
Suspended Fourth Triad' (1-4-5) straddle the Major and Minor
Regions. They are 'ambivalent' about their major or minor
character and do not 'disclose' their region. Certain other
harmonic structures, on the other hand, may cover two regions for
even more) at once. For example, the Diminished Triad called 6
Dim. Triad (6-1-3) covers the Major and Minor Regions at once.
The Tonic Wolf Major Triad (1-3-5) spans three
regions: the
Minor, Flat-Minor, and Semi-Major Regions. The two pitch functions
which stand at the furthest removed position from the Tonic, in
the very heart of the Enharmonic Region are the neutral third and
sixth, 3 and 6. Note their position on the
field lattice of Figure 5.

Figure 6:

The eight regions shown on the Circle of Fifths-Fourths. The functional
notation indicates major triads (outer part of the circle), and minor
triads (inner part of the circle). The majors and minors are paired by the
traditional relation known as "Relative Major-Relative Minor" (Type 3 pairing).

The eight regions are highlighted. Major-minor triad pairs associated with
the boundary between regions (that is, triads with two tones within the
meantone corridor) are marked by asterisks:

Major and minor triads naturally pair along these lines so that the circle of
fifths becomes the circle of major and minor triads in functional array
around the Tonic. Note the symmetrical nature of the regions. The
Enharmonic Region lies 'opposite' the Major Region in which the
Tonic resides. Modulations to related major and minor key centres
can be mapped on the circle of fifths according to movement in a
clockwise or a counter-clockwise direction. An awareness of the
regional structure of 31-ET leads to increased understanding of
the specifics of meantone progression.

The ancient Greek classification of musical 'types' according to the notion of Genus is perfectly suited to the meantone system. Intervals, chords, and scales can all be classified according to whether they are Diatonic, Chromatic, or Enharmonic. Such a classification is meaningless to 12-ET where the enharmonic is eliminated. This feature of 31-ET can already be seen in the functional field of Figure 5. The Ptolemaic structures which we associate with the diatonic major and minor scales of meantone and of Just Intonation form a 'double strand' of fifths on the graph. Within this graup of intervals, for example, is the diatonic semitone 3-4 (eg. E-F). If we now 'skip' a level we get the semitone associated with the Chromatic Genus - the chromatic semitone 5-5 (eg. G-G). Within the chromatic grouping are the important septimal intervals, for instance 2 and 6. Thus we can also associate the diatonic with the '5-Limit' intervals, and the chromatic with the '7-Limit' intervals. If we now jump one more 'vertícal' level we integrate the enharmonic intervals; for instance, we now have the enharmonic 'semitone' between 7 and 1 (eg. C-C). We have jumped these levels in a 'northerly' direction, but we can also do so symmetrically in a 'southerly' direction. The notion of the three genera is another result of the 'verticality' of the system.

The field of Just Intonation also embodies a similar structure, but it has in addition a 'horizontal' component in the comma-altered regions. In 31-ET this horizontal component is eliminated so that the 'vertical' component takes on much greater structural importance. Hence we have the classification of patterns in the system according to the three genera alone and not the additional comma-alterations. We will use the notion of Genus in investigating the intervals and also the more complex harmonic structures of the tuning system.

Musical intervals are the primary building blocks of a harmonic system. Hence an understanding of the interval types is essential in mastering the materials for composition. 31-ET offers a rich assortment of intervals, which may be grouped according to the notion of Genus.

It is helpful to group an interval with its inversion, so as
to form a *class* of interval type in aggreement with the principle
of octave equivalence. Such a grouping of the intervals of 12-ET
leads to seven interval classes, which may be ordered according to
degrees of consonance. This order stems from the progressive
appearance of the interval types within the Harmonic Series:

1. Unison | - Octave | } Perfect Consonances |

2. Perfect Fifth | - Perfect Fourth | |

3. Major Third | - Minor Sixth | } Medial (Imperfect) Consonances |

4. Minor Third | - Major Sixth | |

5. Wholetone | - Minor Seventh | } Dissonances |

6. Halftone | - Major Seventh | |

7. Tritones |

Figure 7: Overview of interval classification:

Class | Diatonic Genus | Chromatic (Septimal) Genus | Enharmonic Genus |

1 | 0 - unison | ||

31 - octave | |||

2 | 18 - fifth | 19 - "wolf" fifth | |

double-flat sixth | |||

semisharp fifth | |||

13 - fourth | 12 - "wolf" fourth | ||

sharp third | |||

semiflat fourth | |||

3 | 10 - major third | 11 - septimal major third | |

semisharp third | 9 - neutral third | ||

flat fourth | semiflat third | ||

23 - major sixth | 24 - septimal major sixth | doublesharp second | |

semisharp sixth | |||

doubleflat seventh | |||

4 | 8 - minor third | 7 - septimal minor third | |

sesquisharp second | sharp second | 22 - neutral sixth | |

sesquiflat third | semiflat sixth | ||

21 - minor sixth | 20 - septimal minor sixth | doublesharp fifth | |

sesquisharp fifth | sharp fifth | ||

sesquiflat sixth | |||

5 | 5 - wholetone | 6 - septimal wholetone | 4 - semiflat wholetone |

semisharp second | doublesharp prime | ||

doubleflat third | |||

26 - minor seventh | 25 - septimal minor seventh | 27 - neutral seventh | |

flat seventh | sharp sixth | semiflat seventh | |

sesquisharp sixth | sesquiflat seventh | doublesharp sixth | |

6 | 3 - diatonic semitone | 2 - chromatic semitone | 1 - diesis, doubleflat second |

flat second | sharp prime | enharmonic | |

sesquisharp prime | sesquiflat second | semisharp prime | |

28 - major seventh | 29 - flat octave | 30 - semiflat octave | |

semisharp seventh | sharp seventh | ||

7 | 16 - flat tritone | 15 - sharp tritone | 14 - semisharp fourth |

flat fifth | sharp fourth | doubleflat fifth | |

sesquisharp fourth | sesquiflat fifth | ||

17 - semiflat fifth | |||

doublesharp fourth | |||

The class 3 and class 4 intervals have been somewhat 'shuffled' to reflect the fact that the major third and major sixth embody more consonance than the minor third and the minor sixth. The chromatic and enharmonic forms of the thirds and sixths obviously fall within the status of dissonances rather than consonances, with the neutral (enharmonic) forms exhibiting strong dissonance.

In spite of the notion of octave equivalence, the particular voicing of the interval is a big factor in judging the strength of its consonance. The minor third is the most significant example. In closed position the minor third embodies more consonance than its chromatic cousin, the septimal (or sub-minor) third; however, in open or compound form (as a minor tenth) the sub-minor form gains in consonance over the minor form. This Sub-Minor Tenth is on the verge of being considered as a consonance, just as the diatonic minor sixth is on the verge of being considered as a dissonance. Some intervals gain in their compound expression (eg. fifth, major third, wholetone) and others lose (eg. fourth, major sixth, minor sixth, minor third).

Of course, the intervals can also be arranged according to their size, leading to Figure 8. This table is the appropriate place to include other technical information relating to the 31-ET scale. The fractional ratios are useful for tabulating positions in the layout of fretboards. The Frequency numbers are useful for synthesizer tuning programs which have the option of direct input of frequency data. The two columns on the right indicate the size of the interval in cents and also the size of closely related just intervals in cents, so that we can compare the deviation of 31-ET from Just.

Figure 8: Technical data on 31-ET intervals:

Numerical notation | Ratio in cents | Interval notation | Fractional ratio | Relation to just ratio | |

0 | 0.000 | C | 1.000000 | 1/1 | |

1 | 38.710 | C | D | 1.022611 | 64/63 |

2 | 77.419 | C | D | 1.045734 | 21/20 |

3 | 116.129 | C | D | 1.069379 | 16/15 |

4 | 154.839 | C | D | 1.093559 | 35/32 |

5 | 193.548 | D | 1.118286 | 9/8 | |

6 | 232.258 | D | E | 1.143572 | 8/7 |

7 | 270.968 | D | E | 1.169430 | 7/6 |

8 | 309.677 | D | E | 1.195873 | 6/5 |

9 | 348.387 | D | E | 1.222913 | 128/105 |

10 | 387.097 | E | F | 1.250565 | 5/4 |

11 | 425.806 | E | F | 1.278842 | 9/7 |

12 | 464.516 | E | F | 1.307759 | 21/16 |

13 | 503.226 | F | 1.337329 | 4/3 | |

14 | 541.935 | F | G | 1.367568 | 48/35 |

15 | 580.645 | F | G | 1.398491 | 7/5 |

16 | 619.355 | F | G | 1.430112 | 10/7 |

17 | 658.065 | F | G | 1.462449 | 35/24 |

18 | 696.774 | G | 1.495517 | 3/2 | |

19 | 735.484 | G | A | 1.529333 | 32/21 |

20 | 774.194 | G | A | 1.563914 | 14/9 |

21 | 812.903 | G | A | 1.599276 | 8/5 |

22 | 851.613 | G | A | 1.635438 | 105/64 |

23 | 890.323 | A | 1.672418 | 5/3 | |

24 | 929.032 | A | B | 1.710233 | 12/7 |

25 | 967.742 | A | B | 1.748904 | 7/4 |

26 | 1006.452 | A | B | 1.788449 | 16/9 |

27 | 1045.161 | A | B | 1.828889 | 64/35 |

28 | 1083.871 | B | C | 1.870243 | 15/8 |

29 | 1122.581 | B | C | 1.912532 | 40/21 |

30 | 1161.290 | B | C | 1.955777 | 63/32 |

31 | 1200.000 | C | 2.000000 | 2/1 |

It is useful to conceptualize the extended meantone fabric and 31-ET as equivalent because they are sonically almost identical. However, we should keep in mind that, strictly speaking, subtle distinctions exist between the two tunings. The 'true' extended meantone fabric is essentially a line of musical fifths each of which is flattened by exactly one quarter of a syntonic comma. Such a line of fifths will never meet to form a circle. In the 31-ET tuning the size of the musical fifth is slightly altered from 'pure' meantone so that a circle of fifths results. The 31-ET fifth is widened by about 0.19 cents (an amount which is sonically imperceptible) from the meantone fifth. All of the intervals of the two systems are hence slightly skewed to each other.

We can best compare the two tunings by projecting the line of fifths in the two directions from the Tonic, culminating in the Enharmonic Region. The interval sizes, expressed in cents, can then be contrasted and their differences tabulated. See Figure 9. Note that the greatest deviation of the 31-ET temperament from the 'true' me antone temperament amounts to only 3.13 cents, and occurs at the most remote functional relations of 3 and 6. We can consider these amounts to be negligable, especially for the more common functions. Consequently, the two systems may be considered as identical for practical purposes, sonic effect and in theoretical structure.

Figure 9: The difference in interval size between 31-ET and the
extended quarter-comma meantone tuning (all values are in cents):

Number notation | Interval notation | 31-ET | 1/4-comma meantone | Difference MT-31 | ||

22 | G | A | 851.613 | 854.746 | 3.133 | |

9 | D | E | 348.387 | 351.324 | 2.936 | |

27 | A | B | 1045.161 | 1047.902 | 2.741 | |

14 | F | G | 541.935 | 544.480 | 2.545 | |

1 | C | D | 38.710 | 41.059 | 2.349 | |

19 | G | A | 735.484 | 737.637 | 2.153 | |

6 | D | E | 232.258 | 234.216 | 1.958 | |

24 | A | B | 929.032 | 930.794 | 1.762 | |

11 | E | F | 425.806 | 427.373 | 1.566 | Subdominant |

29 | B | C | 1122.581 | 1123.951 | 1.370 | direction |

16 | F | G | 619.355 | 620.529 | 1.175 | |

3 | C | D | 116.129 | 117.108 | 0.979 | |

21 | G | A | 812.903 | 813.686 | 0.783 | |

8 | D | E | 309.677 | 310.265 | 0.587 | |

26 | A | B | 1006.452 | 1006.843 | 0.392 | |

13 | F | 503.226 | 503.422 | 0.196 | ||

0 | C | 0.000 | 0.000 | 0.000 | Tonic | |

18 | G | 696.774 | 696.578 | -0.196 | ||

5 | D | 193.548 | 193.157 | -0.392 | ||

23 | A | 890.323 | 889.735 | -0.587 | ||

10 | E | F | 387.097 | 386.314 | -0.783 | |

28 | B | C | 1083.871 | 1082.892 | -0.979 | |

15 | F | G | 580.645 | 579.471 | -1.175 | Dominant |

2 | C | D | 77.419 | 76.049 | -1.370 | direction |

20 | G | A | 774.194 | 772.627 | -1.566 | |

7 | D | E | 270.968 | 269.206 | -1.762 | |

25 | A | B | 967.742 | 965.784 | -1.958 | |

12 | E | F | 464.516 | 462.363 | -2.153 | |

30 | B | C | 1161.290 | 1158.941 | -2.349 | |

17 | F | G | 658.065 | 655.520 | -2.545 | |

4 | C | D | 154.839 | 152.098 | -2.741 | |

22 | G | A | 851.613 | 848.676 | -2.936 | |

9 | D | E | 348.387 | 345.254 | -3.133 | |

It is interesting to note that he 'true' meantone fabric is
made up of intervals which can all be expressed as powers and
roots of the numbers two and five. For example, the meantone
itself is the ratio 5 / 2, and the musical fifth is the ratio
^{4}5 (the
fourth root of five, which means the square root of the square
root of five). The number five and the fifth harmonic are, of
course, associated with the Major third axis. The entire extended
meantone field can be expressed in ratios which stem from the
numbers two and five. (See Figure 39 in the *Meantone Series*
article). Although 31-ET deviates from these ratios in slight
amounts, the essential quality is the same. Since these ratios
belong to the family o£ ratios associated with the Golden Section,
we can associate 31-ET with the Golden Section just as well as
the 'true' extended meantone tuning. Thus in practically every
aspect, we can equate 31-ET to the extended meantone tuning.

The cyclical nature of the system results in the ability to
'map' out all of the possible functions of the tuning in a manner
similar to that used for 12-ET in *The Dictionary of Harmonic
Functions for 12-ET* The organization is essentially that of
non-ordered sets or pitch class. The 'patterns' are first grouped
according to internal structure into a *class* of functions within a
modulating framework. For example, we have the functions
associated with the class of major triads, and in addition the
functions associated with the class of wolf major triads, and so
on. Each class of pattern occupies one page. Then the various and
sundry classes of patterns are organized according to structural
complexity, into duads, triads, tetrads, pentads, and so on
theoretically to the one and only 31-tone pattern (the
ultra-chromatic scale).

It is obvious that any given pattern must embody 31 possible Functions. For instance, any given tone can relate to the Tonic in exactly 31 different ways. These ways may be described by the usual functional notation of 3, 4, and so forth; or alternatively, the numerical notation is also possible. For the sake of convenience and brevity, I have used the latter approach in the tables. The tables are also accompanied (as in the 12-ET Dictionary) by a representative of the class on the Chromatic Wheel and the Wheel of Fifths circular graphs. These graphs make clear and visual the interval connections between complex patterns, as well as such structural characteristics as symmetry, complementarity, and sub-set groupings.

Note that all of these mappings of structure: the triad
lattice, the Wheels, and the functional table, assume the notion
of octave equivalence - a natural companion of the concept of
pitch class. Hence a vast amount of data is presentable in an
elegant form. For example, the totality of relationships possible
between pitches taken 'two notes at a time' (Duads) amounts to
only fifteen classes (shown at the bottom of Figure 8). These are
all of the possible intervals paired with their inversions. As a
first example of a representative page from the functional tables,
I present the duad associated with the neutral third and its
inversion, the neutral sixth. See Figure 10. Functions which
contain the Tonic are called *primary functions*, the others are
called *secondary functions*. Of course, here there are only two
Primary Functions: 0-9 and 0-22. The table can not only be read to
refer to functional relations, but also pitch relations. It shows
all of the ways that two pitches can relate to each other by the
internal structure of a neutral third and its inversion. From the
two Wheels we see that the pattern is symmetrical.

Figure 10: A sample from the book of functional tables for 31-ET.

As a practical application of this data, suppose we want to find the function that is a neutral third from the Dominant (function 18). Simply find 18 on the table and read 27 to its right. Function 27 is the neutral seventh. The table is simply a modulating framework for the internal relation which defines the class of pattern.

As in the 12-ET *Dictionary of Harmonic Functions*, the
various classes of patterns are organized according to whether
they are triads, tetrads, etc. As in that system, there is only
one pattern representing the empty set (silence). This pattern may
be linked to the one pattern which is the 'full set' - the
ultra-chromatic scale. Then there is only one class of patterns
called the Monad, associated with the interval of the Unison and
its inversion, the octave. The Monad and the fifteen Duads are
lumped together to constitute all of the possible interval
patterns.

The total number of possible patterns in the entire 31-ET system amounts to
2^{31} (two to the 31st power), a huge number! Hence a Dictionary of
all of the possible patterns in the system is impractical. It is, however,
possible to tabulate all of the patterns and organize them according to class,
consonance, symmetry, and so on. Such a comprehensive mapping is practical for
12-ET which has only 2^{12} (4096) patterns, which may be grouped into
only 352 classes. 31-ET, on the other hand, has 2^{30} classes.
Consequently, in my tabulation of the patterns, I have only mapped the most
prominant harmonies which are of greatest interest to me. However, the
methodology for a complete mapping exists.

In order to show some further examples from the functional tables, I present the data for two forms of the 'Dominant Seventh' tetrad (alluded to earlier in the article). They are presented as Figure 11 and Figure 12. Note that a typical resolution format is provided in brackets. These two patterns can be grouped according to genus as diatonic and chromatic respectively.

Figure 11: A sample tetrad from the book of functional tables for 31-ET.

In a similar manner, all possible functions for any given structure or 'pattern in the system may be conveniently mapped so that its full functional possibilities may be guaged with a minimum of time imput. Patterns of great complexity, such as Hexads, Heptads, and Ogdoads tend to be regarded as 'scales' rather than 'chords'. The table can be viewed in both a vertical and a horizontal manner. In complex patterns the primary functions become the 'modes' of the pattern. Unfortunately, the short format of this article does not allow me to discuss at length the specifics of all the types of patterns in the system. Such information can best be gleaned by examining the functional tables themselves, which speak for themselves. Two more examples of pattern classes appear as Figure 13 and Figure 14. The former figure shows a dissonant triad class of the enharmonic genus. The latter figure maps the Functions for an eleven note segment of the circle of fifths. Of course, very many other classes of patterns exist in the system. Every class of patterns has its own functional table, defining and delineating all of the possible ways in which that class of patterns can be used (independent of specific octave voicing). These tables must be considered as the practical heart of the meantone system.

In addition to the mapping of the triad lattice, a given harmonic pattern is also charted on the Chromatic Wheel and the Wheel of Fifths. These two maps each give useful and distinct information relating to the structure of the pattern. Hence the Wheel mappings are compiled together with the lattice graph and the functional table.

The Chromatic Wheel lays out all of the potential voicings of the pattern, and reveals the interval (subset) structure of the pattern. Remember that, even though octave equivalence is assumed in designing our model, it is still possible to look at the Chromatic Wheel as an 'octave-specific' model. Movement on the Wheel in a clock-wise direction indicates falling pitch, and counter-clock-wise movement rising pitch. A complete 'round' is an octave. Thus the interval vectors can be viewed in an octave specific manner. In addition, the vector sizes are colour-coded, so that the interval size is easy to 'read'. (Unfortunately, the colour code must be eliminated for this article). Hence the Chromatic Wheel gives a direct, visual 'analogue' of the interval relationship and potential layout of the pattern.

The Wheel of Fifths, on the other hand, is never associated with a 'Staff' voicing layout. The relations are truly 'abstract' and strictly octave equivalent. This model gives information on the harmonic spread' of the pattern, its consonance and its region of the circle. Strong harmonies tend to be tightly grouped clumps of vectors; whereas, dissonant harmonies tend to be 'spread out' forms of vectors. These wide 'splays', for example the tetrad in Figure 12, define a domain of the circle of fifths in which the dissonance has its most natural resolution. (Of course, some would argue, that this particular pattern is not a dissonance at all, and that it has no natural tendency to resolve like its diatonic cousin. Even if we grant this value to the Harmonic Series Seventh tetrad, it still most naturally 'progresses in the way outlined).

The Wheel of Fifths gives information which is purely 'harmonic or abstract, much like the triad lattice mapping of the pattern. The lattice and the fifths wheel are closely related. The study of the pattern on both models together indicates a clear mapping of the harmonic 'tendency' of the pattern. This inherent tendency to movement is then 'expressed' through a specific function and a specific voicing.

We have emphasized elsewhere that meantone tuning is more
conducive to an extended tonality (or modality) than to atonality.
Since it is a *prime system*, it embodies no 'reductionist'
functions which play such a pivotal role in the structure of
12-ET. Every type of chord displays 31 possible functions, and
there can be no ambiguity of functional intent. This situation
does not mean, however, that tonality must necessarily follow
traditional channels. The gravitational pull of tonality may be
greatly diffused by using dissonant patterns and distant
modulations. Nevertheless, some form of tonality has a tenuous
hold on meantone progression.

The tendency of 12-ET patterns toward atonality is also aided by the 'restless' nature of the major and minor triads, with their significantly tempered thirds. 31-ET triads, on the other hand, are much more euphonious and 'restful'. Hence the status of the triad patterns is characteristically enhanced in meantone harmony. This situation also contributes to the tendency for meantone music to explore 'extended tonality' rather than atonality.

The early history of the use of meantone harmony was closely associated with the choral traditions of Renaissance and Baroque music. It could even be argued that this choral tradition was made possible through the implementation of this tuning system over the medieval 'Pythagorean' tuning system. Anyone who has worked with meantone is aware of the eminantly 'singable' quality of the triads. The 31-ET triads are truly a treasure having a 'sweetness' unmatched by other systems of temperament.

The quality of these triads means that progressions connecting such chords are natural to the system. Indeed, the music of the 16th and 17th centuries was largely triadic for probably this very reason. In the article on Gesualdo I examined the types of chord progressions between major and minor triads which prevailed throughout the 16th century. These progressions may be conveniently and systematically organized according to their relations on the triad lattice. Such a structuring of progression types is a topology of relations between triad chords.

The topology presented in that article was not comprehensive, since my interest was specifically the 16th century. Only such experimental composers as Gesualdo at the end of the century ventured into the more 'distant' progressions. As usual, this abstract topology of triad progression can be organized according to Genus. Figure 15 is a comprehensive reworking of this topology to include all of the possible 'progression types' between major and minor triads. Note that the majority of the 63 progression types fall in the enharmonic genus. Most of these progressions were largely untried before meantone went out of fashion in the 18th century, although Gesualdo used some of the enharmonic progressions in the 16th century. Thus we can see the immense scope for original harmonic progressions in the meantone system.

As an example of an unusual triad progression from this topology, I refer you to the progression marked as number 56 in the series. Here we have the unique situation in which a major triad moves to a minor triad (or the reverse in which a minor triad moves to a major triad) in such a way that all of the intervals shift by an enharmonic alone. Thus, for example, we have the move from C minor triad (C-E-G) to C major triad (C-E-G). Such a progression is peculiar to meantone tuning.

Of course, I do not want to leave the impression that only triad progression is possible in the meantone system. The vast array of possible structures in this system is truly dazzling. A new and largely unexplored universe of harmony is opened up - a universe which has many inhabitants, both beautiful and bizarre. It is hoped that the use of the triad lattice and the Fifths Wheel introduces a sense of order to this complexity. The movements of meantone harmony can be guaged according to these criteria. Rather than dictate as to matters of style, the aim is simply to create an awareness of the structure of the movement, so that the composer can make intelligent choices.

This article has served as an introduction to the structure of the meantone system. Due to the vast assortment of materials in the tuning, only the conceptual framework has been given, along with a few examples to illustrate the concept. The real 'meat' of the system lies in the functional tables, with their delineation of 'mode' and pattern. These tables rightly accompany this article, and this article properly serves as an introduction to the intelligent use of these tables. Such tables are by their very nature voluminous, even if they are easily organized into books of triads, tetrads, heptads, etc. The composer will find in these patterns not only the material for new composition, but also the basis for an intelligent analysis of past composition, in whatever style the composer had used.

It is hoped that this conceptual framework complements the Fokker notation for the system, a notation which has done so much to improve the intelligibility of the system, and make it applicable to the musical Staff. The combination of a clear notation, a clear comprehension of structure, and a suitable keyboard technology for the composer's exploration, opens the pathway to a glorious future in the evolution of musical harmony.