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Western modal and tonal music generally conforms to what is known as the 5-limit; this is often explained (e. g., by Hindemith) by stating that the entire tuning system is an expression of integer frequency ratios using prime factors no higher than 5. A more correct description is that all 5-limit intervals, or ratios involving numbers no higher than 5, aside from factors of 2 arising from inversion and extension, are treated as consonances in this music, and thus need to be tuned with some deg ree of accuracy. The diatonic scale of seven notes per octave has been in use since ancient Greek times, when harmonic simultaneities (other than unisons and octaves) were not used and the only important property of the scale was its melodic structure. As melodic considerations continued to play a dominant role in Western musical history, scales other than the diatonic had very limited application, and harmonic usage was intimately entangled with diatonic scale structure. It is fortuitous that the diatoni c scale allowed for many harmonies that displayed the psychoacoustic phenomenon of consonance, defined by small-integer frequency ratios, as well as many that did not.

Medieval music conformed to the 3-limit, the only recognized consonances being the octave (2:1) and the perfect fifth (3:2) (plus, of course, their octave inversions and extensions, obtained by dividing or multiplying the ratios by powers of 2; this "o ctave equivalence" will be implicit from here on.) The diatonic scales were therefore tuned as a chain of consecutive 3:2s (Pythagorean tuning). By the end of the 15^th century, the 5-limit had eclipsed the 3-limit as the standard of consonance and two new consonant intervals were recognized: the major third (5:4) and the minor third (6:5). It was now possible to create consonant triads, consisting of all three non-equivalent consonant intervals and no dissonant intervals. There are two such tri ads: major (6:5:4), and minor (1/4:1/5:1/6). The advent of the 5-limit brought with it a new temperament of the diatonic scale, the meantone. Typical varieties of meantone are well approximated in 19- and 31-tone equal temperament; 12-tone equal temperame nt is approximately 1/11-comma meantone, lying about halfway between typical meantone temperaments and Pythagorean tuning.

Soon anything less than a triad was deemed an incomplete harmony. Use was also made of linked 5-limit (or stacked-thirds) chords ("major and minor seventh chords") and linked 3-limit chords ("suspended chords"), which contained only one dissonant inter val but had to resolve. More dissonant triads ("diminished") and tetrads ("dominant, diminished, and half-diminished seventh chords"), formed from the same pattern of scale steps as the 5-limit sounds, were normally restricted to cadential or modulatory p rogressions. These chords resulted from the existence of a diminished fifth within the diatonic scale. Unlike most fifths in the scale, the diminished fifth was considered dissonant, since it did not approximate any 5-limit consonances. A complete resolut ion was not felt until both members of the diminished fifth had moved to a consonant chord. The major and minor modes gained supremacy because only in those modes was the diminished fifth disjoint from the tonic triad.

As early as the mid-eighteenth century, Tartini observed that the "harmonic 7^th is not dissonant but consonant. . . . it has no need either of preparation or resolution: it may equally well ascend or descend, provided that its intonation be true." So it seems natural to try to expand harmony by making 7-limit intervals the fundamental harmonic units. The new consonant intervals would be 7:4, 7:5, and 7:6. There would be a major tetrad (7:6:5:4) and a minor tetrad (1/4:1/5:1/6:1/7), each consisting of the six non-equivalent 7-limit intervals. However, these chords occur in only one position and with questionable accuracy in the diatonic system, relatively pure versions obtainable only by certain chromatically altered chords (such as the a ugmented sixth) when played in 31-tone equal temperament or similar meantone systems. Therefore an entirely new scale is required, in which the tetrads can act as the basic consonances.

Figure 1 shows the adequacy with which the simplest equal-tempered tunings approximate all justly tuned intervals of the 3- through 15-limits. The value assigned to the "accuracy" of each interval is proportional to an estimate of the likelihoo d of the nearest equal-tempered approximation being heard as the just interval it represents. A normal probability distribution is assumed, centered on the actual interval with an accuracy value of 1.0 assigned to the peak, and with a standard deviation o f 1% in log-frequency space. We do not go beyond 34 tones because this would allow two notes of the tuning to be within 1% of a given target pitch. The accuracy of the tuning as a whole is defined as the geometric mean of the accuracies of the intervals c onsidered so that (a) even if only one interval is badly out-of-tune, the entire tuning receives a poor rating; and (b) the rank-order of the results for a given limit is just that of the mean squared errors and is independent of the standard deviation as sumption

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In non-harmonic music, the purity of consonances is considerably less important, and intervals beyond the 3-limit are of negligible importance. Figure 1 suggests that among equal temperaments, scales of 5 and 7 notes per octave ought to be candidat es for tuning melodic music, and one indeed finds many approximations to these scales throughout the world. Thai music, which requires equal temperament so that it can modulate, is played in 7-tone equal temperament, and the 5-tone equal tempered scale is widespread in Africa. The huge gain in the accuracy of 3-limit intervals obtained by using a 12-tone scale (or approximation thereof) is what allowed Chinese and medieval Western music to develop dyadic harmony. It turned out that 12-tone equal-tempered scales, and the diatonic modes that were in use in the West, were conducive to 5-limit (triadic) harmony as well, with a decent degree of accuracy (about 78% of perfection, according to figure 1).

Although the move to 19- or 31-tone equal temperament has been proposed time and again for the greater consonance that these tunings would afford to modal and tonal triadic music, it has been felt that the potential aesthetic advantages are greatly out weighed by the practical disadvantages, which could include the need to invent and learn to play a new set of (more complex) instruments. These sorts of difficulties have been surmounted by Partch, Ben Johnston, and the Dutch 31-toners, and advances in el ectronic music have made such explorations more widespread. In any case, if we wish to move to the 7-limit without decreasing the overall level of accuracy from that of 12-equal in the 5-limit, fig. 1 tells us to consider 22, 26, 27, and 31-tone tunings. It will turn out that only 22-equal contains a system analogous to the diatonic system, but which involves harmonies of the 7-limit.

It may be objected that higher-limit intervals may need to be tuned more accurately than lower-limit ones, since they are more apt to be confused with other intervals. If the intervals are often sounded alone, this is indeed the case. Partch stated in his "Observation One" that the "field of attraction" of an interval is inversely proportional to its "Identity," or limit. A similar result can be derived from the work of van Eck, when errors in his reasoning are corrected. Figure 2 is a modification of fig. 1 in which the standard deviation used in evaluating each interval is inversely proportional to its limit. The standard deviation used for 5-limit intervals is still 1% (this again has no effect on the rank-order of the results within a given limit). The same four tunings appear best for the 7-limit, but only two are still more accurate than 5-limit 12-equal: 27 and 31. However, we are interested in a style where complete harmonies, not isolated intervals, are the norm, and in complete harmonies the context of simpler intervals seems to aid in the recognition of the more complex ones. For example, the tritone b-f in 12-equal is ambiguous when heard in isolation, but can be heard as 7:5, 10:7, or 17:12 in the chords g-b-d-f, c#-g#-b-f, and e-g#-d-b-f, respectively. Additionally, the argument can be made that although the recognizability of simpler ratios is reduced more slowly than that of more complex ratios when both are detuned by the same amount, the consonance of the simple ratios deteriorates at least as rapidly. Therefore, we will consider figure 1 to be more important for our purposes.

Many of the previous authors who have tried to generalize the diatonic scale have ignored the fact that 12-equal is not the historical basis for the diatonic scale; in fact the converse is true, and 12-equal was chosen over other meantone tunings o nly for convenience. Balzano focuses on certain group-theoretical properties of 12-equal and of the diatonic scales within it. For example, his view of harmonic structure hinges on the fact that in 12-equal, the size of the minor third (3) times the size of the major third (4) equals the size of the entire set. He proceeds by finding a nine-tone scale in 20-equal with many of the same properties. Had he taken the diatonic scale from 19- or 31-equal, systems that musicians actually considered before 12-equ al won out, he would not have found many of his properties. Likewise, Clough and Douthett’s maximal evenness property fails if the diatonic scale is taken from any tuning other than 12-, 19-, or 26-equal. Yasser also assumes a 12-tone tuning in defining t he diatonic system as "7+5=12" (seven diatonic plus five altered notes make the 12 chromatic notes), which allows him to posit the evolutionary series "2+3=5, 5+2=7, 7+5=12, 12+7=19, 19+12=31" where the chromatic notes in one system become the diatonic no tes in the next system, and the diatonic notes in one system become the altered notes in the next system.

Many other theorists, including Schoenberg, have explained the diatonic scale as arising from three consecutive triads in a chain of fifths. This is another reversal of historical facts, as the chords were constructed from the scale, and not the other way around. Von Hoerner constructed a scale from three consecutive 7-limit tetrads in a chain of fifths, using 31-tone equal temperament. The structure of this scale is too bizarre for it to function as a melodic entity.

A few other theorists have attempted to preserve some important features of the diatonic system while proposing musical tones with unusual harmonic spectra. Bohlen and Pierce independently discovered that the equal-tempered division of the 3:1 (perfect twelfth) into 13 parts contains a nine-tone scale with excellent 9:7:5:3 and 1/3:1/5:1/7:1/9 chords. Pierce suggests that using timbres without even-numbered harmonic partials would not only highlight the consonance of these chords but would also cause t he phenomenon of octave equivalence to give way to one of "tritave equivalence," or equivalence at the perfect twelfth. However octave equivalence seems pervasive, regardless of the overtone structure of the timbres used. Goldsmith proposed using tones wi th inharmonic overtone series to go along with a division of the octave into 16 equal parts. Proceeding by analogy from the major scale, a nine-tone scale is derived, with three disjoint triads, which could be rendered somewhat consonant with an appropria te overtone structure. However, these triads do not contain anything resembling a 3:2 interval, which is what confers stability (and a root) to the triads of diatonic music. We will restrict our attention to tones with harmonic overtone series because the y evoke the sensation of a single pitch and arise from the widest variety of acoustic and electronic methods of tone generation.

The diatonic scale has many properties that allow it to form the melodic basis for a 5-limit harmonic style. A few definitions will allow us to generalize these properties. Given any odd-number harmonic limit, we can define the set of consonant interva ls as all ratios of odd numbers less than or equal to the limit (as well as their octave equivalents, which will contribute factors of 2 to the numerator or denominator of these fractions), and we define the complete consonant chords as those which contai n all of the consonant intervals and no dissonant (non-consonant) intervals. We will use the following symbols for tempered intonations of just intervals and their octave equivalents: Q will represent any tempered intonation of 3:2 (or 4:3, 3:1, etc.); T— 5:4 (or 8:5, 5:2, etc.); S—7:4 (or 8:7, 7:2, etc.); and U—1:1 (or 2:1, 4:1, etc.). For example, any meantone system has 4Q=T. The Q, the strongest interval within the chord, determines the root of a consonant chord: the root is that member of the Q that r epresents the number 2 in the ratio 3:2. Let us define a "characteristic dissonance" as any dissonant interval in the scale that is the same size, as defined by number of scale steps, as a consonant interval. The only example in the unaltered diatonic sca le is the diminished fifth (and, of course, the augmented fourth). The properties that we would like to reproduce in our scale are as follows:

(0) Octave equivalence:

There is a basic scale (a subset of the tuning) which repeats itself exactly at the octave, extending infinitely both upwards and downwards in pitch.

Octave similarity is universally perceived, even by some animals.

(1) Scale structure:

(Version a - maximal evenness): The basic scale has two step sizes, and given these step sizes, the notes are arranged in as close as possible an approximation of an equal tuning with only as many notes per octave as the basic scale.

(Version b - tetrachordality): The basic scale has a structure emphasizing similarity at the Q. In particular, there is a "tetrachordal" structure, that is, within any octave span, the pattern of steps within one approximate 4:3 are replicated in anoth er approximate 4:3, with the remaining "leftover" interval spanned using patterns of step sizes (often just one step) found in the "tetrachord."

This scale defines the "key" or "mode"; the set of unaltered pitches used in any section of a composition.

These properties are enough to ensure an intelligible melodic framework. The following properties allow harmony to be based on this framework.

(2) Chord structure:

There exists a pattern of intervals (defined by number of scale steps, not specific as to exact size) which produces a complete, consonant chord on most scale degrees.

This condition provides for a formal rule governing the origin and use of the consonant chords, so that they can become recognized, after a reasonable period of exposure to the system, as the structural harmonies. Or, as Krumhansl puts it,

(3) Chord relationships:

The majority of the consonant chords have a root that lies a Q away from the root of another consonant chord.

This ensures the existence of simple chord relationships that could serve as the basis for comprehensible chord progressions and modulations.

(4) Key coherence:

A chord progression of no more than three consonant chords is required to cover the entire scale.

This restriction should suffice to ensure that the sense of "key" or position within the scale is never lost in the course of the music, and to allow a new key to be easily established.

While these properties may suffice for a "modal" style, the rest of the properties have to do with certain special, "tonal" modes of the basic scale.

(5) Tonicity:

The notes of the scale are ordered, increasing in pitch, so that the first note is the root of a complete consonant chord, defined hereafter as the "tonic chord."

This is more a numbering convention than a true property.

There are two ways of formulating the remaining properties. The stronger will define the primary tonal modes, while the weaker will define secondary modes whose tonic chords are points of intermediate stability.

(6) Homophonic stability:

All characteristic dissonances are disjoint from the tonic chord, with the following possible exception: A characteristic dissonance may share a note with the tonic chord if, when played together, they form a consonant chord of the next higher limit (3 Þ 5, 5Þ 7, 7Þ 9).

This ensures that the scale will sound at least as consonant against the tonic chord as against any other chord.

(7) Melodic guidance:

The rarest step sizes are only found adjacent to notes of the tonic chord, acting as "signposts" if not necessarily leading tones per se.

As Richmond Browne has pointed out, "Rare intervals aid position finding."

6. Homophonic stability:

At least one characteristic dissonance either is disjoint from the tonic chord, or shares a note with the tonic chord such that, when played together, they form a consonant chord of the next highest limit (3Þ 5, 5Þ 7, 7Þ 9).

This provides for a "tonicizing" interval, which bears a special structural relationship to the tonic chord.

When a characteristic dissonance is disjoint from the tonic, it resolves to it, producing a "dynamic" tonality; when a characteristic dissonance blends with the tonic chord to form a stable quasi-consonant sonority, it produces a "static" tonality.

Let's see how these rules apply to 5-limit and 3-limit harmony. The scales are illustrated by the division of the octave into large (L) and small (s) steps. We will use rules 5-7 as a "sieve" to remove those modes that do not satisfy all of them.

5-limit: The diatonic scale.

1. The two small steps are as far apart (in a cyclic sense) as possible.
2. The two tetrachords are each composed of one small plus two large steps, with an additional large step filling in the octave.

(2) The consonant intervals are Q=3L+s, T=2L, and Q-T=L+s. The pattern 1, 3, 5 produces a consonant triad 6 times out of 7.

(3) The six consonant chords form a chain of Qs.

(4) Any three consecutive chords in the chain will cover the entire scale.

(5) Removes the locrian (s L L s L L L) mode.

1. Removes the phrygian (s L L L s L L) and lydian (L L L s L L s) modes, since they contain a note a diminished fifth (s+L+L+s, the characteristic dissonance) above the 5^th and below the 1^st scale degrees, respectively. The mix olydian (L L s L L s L) and dorian (L s L L L s L) modes each contain a note a diminished fifth from the 3^rd scale degree, but it forms a rough 7-limit tetrad with the respective tonic triads in 12-equal.
2. Removes the mixolydian and dorian modes, leaving the usual major (L L s L L L s) and natural minor (L s L L s L L) modes.

1. Since there is only one characteristic dissonance, the result is the same as for (6) above: major, natural minor (both dynamic), and possibly in 12-equal mixolydian and dorian (both static).

3-limit: The pentatonic scale.

1. The two "trichords" are each composed of one small plus one large step, with an additional small step filling in the octave.

1. The consonant interval is Q=L+2s. The pattern 1, 4 produces a consonant dyad 4 times out of 5.
2. The four consonant chords form a chain of Qs.
3. Any three chords that are not all consecutive in the chain will cover the entire scale.
4. Removes the L s L s s mode.

Strong Version

5. The characteristic dissonance is s+s. The major (s s L s L) and minor (L s s L s) pentatonic modes contain a note that is this interval above the 1^st and below the 4^th degrees of the scale, but, especially in a meantone syste m, this note will form a 5-limit triad with the tonic chord.
6. Removes the s L s L s and s L s s L modes, leaving the major and minor pentatonic modes, at least in meantone tunings.

1. The s L s L s and s L s s L modes (both dynamic) both qualify in addition to the major and minor pentatonic modes (both static).

We can even consider a 3-tone scale, a bare "tetrachordal" framework, to be a sort of degenerate 1-limit case:

1. 2L+s=U. Any tuning where L is a recognizable Q will do, including 5-equal (L=2, s=1).

1. Is trivially satisfied.
2. May be questionable because the "leftover" interval has no counterparts within the empty "dichords".

1. A consonant "monad" exists on every scale degree.
2. The three notes form a chain of Qs.
3. The three monads are the notes of the scale.
4. Says nothing.

(6) The characteristic dissonance is s, removing the s L L and L L s modes.

(6) Only the L s L (dynamic) mode is weakly tonal.

With version (a) of rule (1), we can almost add three scales to our list: the augmented (hexatonic) scale (1, 4 is always consonant), the diminished (octatonic) scale (1, 4, 7 is always consonant), both of which fail criterion (3); and Blackwood’s ten- tone symmetrical scale in 15-equal, (1, 4, 7 is always consonant) which just misses property (4). The "just" major and minor diatonic scales satisfy properties (2) through (5), but fail (1a) since they have three step sizes, and fail (1b) because not ever y octave span contains two identical tetrachords.

Some identities for the equal tunings we will be considering are as follows:

In searching for a scale with tetradic harmonies, rule (4) limits us to scales of twelve or fewer notes. Rule (3) tells us that the scale will consist of strings of Qs. A single string of Qs, as in the diatonic scale, is not enough to produce sufficien t Ts and Ss in any of the tunings above. However, the last row of identities suggests that we might try deriving the notes of the scale from 2 or 3 strings of Qs. Such scales in 31-tone equal temperament cannot satisfy either version of property (1) witho ut exceeding fourteen notes. In 27-equal, a twelve-tone symmetrical scale satisfying (1b) exists, but it fails condition (2) miserably.

Only if we relax rule (4) does 26-equal show some promise. Two diatonic scales 2 or 13 degrees apart can be interlaced to satisfy (1a) or (1b). The pattern 1, 5, 9, 12 produces ten and twelve consonant tetrads, respectively, out of fourteen. 22-equal e xhibits a similar structure when interlaced pentatonic scales are used, the chord pattern being 1, 4, 7, 9. The resulting "decatonic" scales include modes that do satisfy all of our conditions for a tonal scale:

Let’s step these scales through our diatonicity rules.

(1) 8s+2L=U. Tuning: 22-equal (L=3, s=2).

1. is satisfied by the five modes of the symmetrical scale L s s s s L s s s s, since the large steps are a half-octave apart.
2. is satisfied by the ten modes of the scale L s s s L s s s s s, where the "pentachord" consists of one large and three small steps, and two small steps span the "leftover" interval.

1. The consonant intervals are Q=L+5s, T=L+2s, Q-T=3s, S=2L+6s, S-Q=L+s (or equivalently, Q-S=L+7s), S-T=L+4s (or equivalently, T-S=L+4s). The pattern 1, 4, 7, 9 produces a consonant tetrad 8 times out of 10 in the version (a) scale, and 6 times out of 1 0 in the version (b) scale.
2. In the symmetrical scale, there are two chains of Qs with four consonant tetrads each. In the pentachordal scale, there are two chains of Qs with two consonant tetrads each, and the other two consonant tetrads are not in a chain of Qs.
3. Verification of this property is left as an exercise for the reader.
4. Removes the symmetrical mode L s s s s L s s s s and the pentachordal modes L s s s L s s s s s, s L s s s L s s s s, s s L s s s s s L s, and s s s L s s s L s s.

Strong Version

5. The characteristic dissonance of the symmetrical scale is s+s+s+s. This removes all its remaining modes from consideration. The characteristic dissonances of the pentachordal scale are s+s+s+s+s and s+s+s+s. The 5s interval removes modes s s s s s L s s s L and L s s s s s L s s s, which contain a note that is this interval above the 1^st and below the 7^th degrees, respectively. The 4s interval removes modes s s s s L s s s L s and s L s s s s s L s s, which contain a note tha t is this interval above the 1^st and below the 7^th degrees, respectively. The 4s interval would also remove modes s s L s s s L s s s and s s s L s s s s s L, if it didn’t form a complete 9-limit pentad with the tonic tetrad in these modes.
6. Imposes no further restrictions, leaving the s s L s s s L s s s (standard pentachordal major) and s s s L s s s s s L (standard pentachordal minor) modes.

1. One of the two 4s intervals in the maximally even scale forms a complete pentad with the tonic tetrad in s s L s s s s L s s (static symmetrical major) and s s s L s s s s L s (static symmetrical minor). One of the 4s intervals is disjoint from th e tonic tetrad in s L s s s s L s s s (dynamic symmetrical major) and s s s s L s s s s L (dynamic symmetrical minor). The characteristic interval of the pentachordal scale is 5s, so the standard pentachordal major and minor modes defined above (both dyna mic and static), as well as alternate pentachordal major (s L s s s s s L s s) and minor (s s s s L s s s L s) modes (both dynamic), are tonal.

So we are left with the following possible decatonic modes (expressed as degrees of the 22-tone equal-tempered scale):

named "Major" or "Minor" after the quality of their tonic tetrads. The five types of (1, 4, 7, 9) tetrads found in these scales can be named as follows:

Major: 0 7 13 18

Minor: 0 6 13 17

Augmented: 0 7 14 18

Major-minor: 0 7 13 17

Minor-major: 0 6 13 18

These chords are encountered in the tonal modes at the following positions:

The symmetrical scale is useful as a basis for a keyboard mapping of 22-equal. To wit: leaving every "E" on the standard keyboard out of the mapping gives two keyboard octaves to one acoustical octave. The black keys then form a maximally even scale, w hich can be thought of as the "natural" scale. Four different pentachordal scales can be played with only one "accidental" -- that is, replacing one out of the ten black keys with a neighboring white key. We can use the numbers 1 through 9 and 0 for the " natural" notes, and the symbols ó , õ , and í to indicate chromatically raised, lowered, and natural notes. In the following table, we introduce names for intervals in, and notation for, the decatonic scale, in analogy with ordinary terminology and notation. A subscript of "10" is a reminder that the intervals are measured with respect to the decatonic scale, as opposed to the customary diatonic measurements. The "key signatures" for the tonal decatonic modes are given in the Appendix.

Table 1

In analogy with the linked 3-limit, or "suspended," triad of the diatonic scale, the decatonic scales contain linked 3-limit and linked 5-limit, as well as incomplete 9-limit, chords which have four notes and resolve to the 7-limit tetrads. These a re:

Linked 3-limit: 0 9 13 18 resolves to major

0 4 13 17 resolves to minor

Linked 5-limit: 0 7 13 20 resolves to major

0 6 13 19 resolves to minor

Incomplete 9-limit: 0 5 13 18 resolves to major

0 8 13 17 resolves to minor

In analogy with the linked 5-limit, or "major and minor seventh," chords, we may find pairs of tetrads in the decatonic scales that have two notes in common. Unfortunately, each chord has two, rather than one, unshared notes, resulting in four, rather than one, potentially dissonant intervals. The relevant six-tone chords possible in decatonic scales are:

Major+major: 0 2 7 11 13 18

Minor+minor: 0 2 6 11 13 17

Major+minor: 0 2 7 13 18 20

0 2 6 13 17 19

0 2 7 9 13 18

0 5 7 11 13 18

Perhaps a better analogue to the diatonic minor and major seventh chords is found through the "stacking" paradigm. Stacking a 3^rd₁₀ on a tetrad leads to a duplication of the root, and stacking another 3^rd₁₀ ( as it occurs in the scale) results in the following five-tone chords:

Major+maj. 3^rd₁₀: 0 7 13 18 (22) 27

Major+min. 3^rd₁₀: 0 7 13 18 (22) 26

Minor+min. 3^rd₁₀: 0 6 13 17 (22) 26

Aug.+maj. 3^rd₁₀: 0 7 14 18 (22) 27

Ma-mi+min. 3^rd₁₀: 0 7 13 17 (22) 26

Mi-ma+min. 3^rd₁₀: 0 6 13 18 (22) 26

These may be considered "safe" additions to the tetrads because they add neither ambiguity nor much dissonance, but merely provide "color," to the tetrads arising in a scale.

The author hears the standard pentachordal modes as most stable and most likely to define key centers and modulatory practice. The tonic chords of the alternate pentachordal modes may simply serve as points of intermediate harmonic stability within the standard pentachordal mode. The symmetrical modes have a weird, bitonal quality due to their symmetry at the half-octave. However, it is worth noting that one can consider all the modes above as arising from a major mode with mutable 3^rd a nd 8^th degrees and a minor mode with mutable 5^th and 10^th degrees. When only one of the two mutable degrees is allowed to vary, consonant tetrads can be formed at nine positions in the scale; when either is allowed to vary , consonant tetrads can be formed at all ten positions in the scale. This consideration may indicate two chromatic alterations that will be common when the key center is not changing.

Let us assume the standard pentachordal modes do clearly define the key centers. Moving between one key and another (of the same quality) a Q away often allows short chromatic passages in diatonic music, and the same holds true with the decatonic s cale. The resulting pattern of steps is 2 2 3 2 2 2 2 1 1 1 2 2 in 22-equal, allowing for a tonal harmonization of four consecutive notes of the tuning. A greater degree of chromaticism is often achieved by combining keys of both qualities on the same roo t (parallel keys). The decatonic result is 2 2 2 1 2 2 2 2 1 1 1 1 1 2, thus allowing a tonal harmonization of a micro-chromatic passage six notes long. In 12-equal, total chromaticism is achieved by combining major and minor diatonic modes on the root, t he Q above, and the Q below. The decatonic analogue leads to only 18 of the 22 chromatic notes. But adding the immediate micro-chromatic neighbors of the root and the 3:2 above fills in the gaps. Since the root and the 3:2 above are structurally the most important notes, their immediate neighbors can be given an important ornamental role. The result is a tonal framework for total 22-tone micro-chromaticism.

It is well worth noting that just as the 5-limit (diatonic) scale is the complement of the 3-limit (pentatonic) scale in 12-equal, so the complement of the 7-limit (decatonic) scale in 22-equal may serve as a basis for a 9-limit (dodecatonic) system. F igure 1 shows the accuracy of 22-equal in the 9-limit as slightly greater than that of 12-equal in the 5-limit. The interested reader is encouraged to discover how closely the "hexachordal" and symmetrical dodecatonic scales satisfy the requirements of a 9-limit tonal system. Note that the definition of "root" may need modification, since complete pentads will contain two Qs (a 3:1 and a 9:3). The union, starting from the same root, of the four major (or minor) decatonic modes described above is a symmetr ical dodecatonic scale.

The hexachordal dodecatonic scale contains within it seven consecutive notes in a cycle of Qs. This forms a diatonic scale where the three "minor" triads are tuned 9:7:6 and the three "major" triads are tuned 1/6:1/7:1/9 – a reversal of the usual harmo nic/subharmonic distinctions. For example, the "minor" mode of this scale, which could be termed "sub-minor" due to its small minor thirds, would be 0 4 5 9 13 14 18. This scale can be mapped to the white notes of a keyboard, with the remainder of the hex achordal dodecatonic scale mapped to the black keys. Then the white-note scale satisfies all the properties of the ordinary diatonic scale except that the triads are not complete sonorities (although they can all be partially completed since the 8 or 1/8 is always a scale tone, the required pattern of scale steps is not the same for the minor and major cases), and the dorian and mixolydian modes can no longer support a static tonality.

Table 1 shows that all ratios involving 9, 11, 15, and 17 – in addition to 1, 3, 5, and 7 – can be expressed consistently (and with a maximum error of 20.1 cents) in 22-equal, promising a great wealth of harmonic possibility when "micro-chromaticism" i s explored. The harmonies involving 11 should prove particularly novel, since they cannot be expressed in 12-equal (in fact, 22 is the simplest equal division of the octave capable of expressing all ratios through the 11-limit consistently – see footnote 8 for a definition of consistency). Finally, it is worth noting that 11-tone equal temperament, contained within 22-equal, is a ridiculously dissonant tuning, containing hardly any tonal (root-defining) sounds, and none whatsoever within the 5-limit. The serial composer who is willing to subtract one from the number of notes in the row could be freed from the constant effort to avoid those intervals in 12-equal which define a key center.

Tonal music utilizes not only diatonic scales in their pure form but also altered modes in which one note is displaced from its diatonic position. A general derivation of such scales is to replace one step size with another of a sort found in t he unaltered scale. Either the lower or upper note comprising the step may be moved in this operation. The altered modes thus produced must observe rules (2) through (7), with the following condition: If a certain number of scale steps produces an interva l that the same number of scale steps never produces in the unaltered scale, that interval will be largely avoided and so will not "count" for rules (6) and (7); i.e. the characteristic dissonances and allowed step sizes remain what they were in the unalt ered case. Let’s work out the implications for the scales we have already discussed.

The unaltered scale can be represented as L L s L L L s. Replacing a small step with a large one, there is only one new scale that can be produced:

1. L s L L L L s

The rest of the scales result from replacing a large step with a small one:

2. s (2L-s) s L L L s
3. (2L-s) s s L L L s
4. L L s s (2L-s) L s
5. L L s (2L-s) s L s
6. L L s L s (2L-s) s
7. L L s L (2L-s) s s

1. removes scale (i) modes s L L L L s L and L L L s L s L; the scale (v) modes s (2L-s) s L s L L, L s (2L-s) s L s L, and (2L-s) s L s L L s; and the scale (vi) modes s L s (2L-s) s L L, s (2L-s) s L L s L, and L s L s (2L-s) s L; these modes all c ontain an augmented fourth below the 5^th or above the 1^st scale degree. The scale (i) modes L s L L L L s and L L s L s L L both contain a note an augmented fourth from the 3^rd scale degree, but it forms a rough 7-limit te trad with the respective tonic triads.
2. imposes no further restrictions, leaving the usual harmonic minor (L s L L s (2L-s) s) and harmonic major (L L s L s (2L-s) s) modes, and possibly the melodic minor ascending (L s L L L L s) as well a less common mode known in Carnatic music as Mela C arukesi (L L s L s L L) and sometimes referred to simply as "the Hindu scale." The first two are dynamic, and the last two are both dynamic and static.

The unaltered scale can be represented as s s L s L. Replacing a large step with a small one, there is only one new scale that can be produced:

1. s s s L L

The rest of the scales result from replacing a small step with a large one:

2. L (2s-L) L s L
3. (2s-L) L L s L

1. removes all three scales, since the pattern 1, 4 produces only two consonant dyads in any of them.

(3) precludes any, since any alteration destroys both Qs, as the reader may verify.

The reader may verify that there are none.

Indian music is described in terms of a 22-sruti division of the octave. Whether this represents an actual set of pitches, or just a means of specifying that some scale steps were to be larger than others, is controversial. Some readings of early accounts suggest a tuning approaching 22-tone equal temperament. Modern accounts describe the three step sizes of the basic seven-tone Indian scales by taking 4-sruti intervals as 9:8, 3-sruti intervals as 10:9, and 2-sruti intervals as 16:15. This determines 3:2 intervals as 13 sruti, 4:3 as 9 sruti, and 5:4 as 7 sruti. Thus the harmonic structure of these scales is well represented in 22-equal. But the melodic structure can be somewhat different.

The two basic scales specified in ancient Indian theory are sa-grama and ma-grama. Their structure was described according to the following table (notes named with a + sign were chromatic tones recognized by the ancients; the derivati on of the ratios is discussed below):

Table 2

sruti position in sa-grama position in ma-grama

1