Zireen music features a variety of different tunings, instruments, and musical styles. This page is a work in progress intended to eventually illustrate a representative sample of Zireen music. For the moment, it's mostly theory, but there are a few brief examples of unfinished music included to illustrate the tunings.

Zireen tunings go by a number of different names, but the most widely used ones are from the Yasaro language. The Yasaro word for a tuning scheme is *nai*; this is a set of rules for tuning that typically (but not always) corresponds to what tuning theorists call a *linear temperament*. In a linear temperament, one interval (an octave, in the simplest case) is taken as the period of repetition, and a smaller interval is taken as the *generator* of the temperament. Scales are produced by repeated transpositions of the generator interval. Our familiar tuning system (historically derived from meantone temperament) is produced in this way by repeating the interval of a perfect fourth or fifth: B - E - A - D - G - C - F. Zireen tunings, on the other hand, are often based on repeated intervals other than fourths or fifths, and periods of repetition that are not always exact octaves. Many Zireen scales repeat at a fraction of an octave, and Zireen tuning has a strong tendency to deviate from exact integer ratios, even in the octaves. Thus, there is always a certain amount of "beating" in Zireen music, but the degree of beating varies widely.

Zireen music uses a wide variety of scales (*yemet*), but the majority of them can be described in general terms as consisting of two different sizes of steps. The most basic of these scales is the pentatonic scale (Yemet Kerasi), with two large steps and three small steps to the octave; this configuration can be symbolized as 2L+3s. Zireen music also uses an "anti-pentatonic" scale, Yemet Veriko, with three large steps and two small steps (3L+2s); this scale has a very exotic and alien sound. Typically, the small and large steps in Zireen scales are distributed as evenly as possible in the basic configuration, but "chromatic" variations on the basic configuration are also common. For instance, a typical 2L+3s pentatonic scale C-D-E-G-A might be varied by substituting F# in place of G, or Bb in place of A. With the larger scales, melodies are often constructed from "sub-scales": a relatively even selection of 5 to 9 notes from the basic scale. A scale may also have different ascending and descending modes, for instance C-D-F-G-A-G-E-D-C.

tuning map: [[1, 2, 4, 7]], [0, -1, -4, -10]]

characteristic scales: 7L+5s, 12L+7s, 12L+19s, 19L+12s

Nai Seret is the Zireen tuning that should be most familiar, since it's based on a cycle of fourths or fifths tempered so that the major third is a close approximation to the fifth harmonic. But the Seret tuning used by the Zireen reaches higher into the harmonic series, by setting the size of the fifth so that an augmented sixth is a close approximation to the seventh harmonic. So this is a specific variety of meantone temperament, and we need a way to distinguish it from other temperaments. This is where the mathematical operation called a "wedge product" comes in handy; it produces a unique "wedgie" that uniquely identifies this temperament. See below for more technical information.

tuning map: [[2, 2, 5, 6]], [0, 3, -1, -1]]

characteristic scales: 6L+4s, 10L+6s

Lemba is a very popular Zireen tuning based on a half-octave period. This interval represents both 7/5 and 10/7, like the tritone in 12-note equal temperament. Music in Lemba tuning typically uses scales of 7 or 9 notes taken from a subset of the 16-note scale.

tuning map: [[2, 3, 5, 6]], [0, 1, -2, -2]]

characteristic scales: 2L+6s, 2L+8s, 10L+2s, 12L+10s

Telek is another popular half-octave tuning.

tuning map: [[1, 0, 1, 2]], [0, 6, 5, 3]]

characteristic scales: 4L+3s, 4L+7s, 4L+11s, 4L+15s

tuning map: [[1, 2, 1, 3]], [0, -2, 6, -1]]

characteristic scales: 5L+4s, 9L+5s

tuning map: [[1, 2, 3, 3]], [0, -2, -3, -1]]

characteristic scales: 1L+3s, 4L+1s, 4L+5s

tuning map: [[1, 2, 2, 4]], [0, -1, 1, -3]]

characteristic scale: 3L+2s (yemet veriko)

tuning map: [[5, 8, 12, 14], [0, 0, -1, 0]]

characteristic scales: 5L+5s, 10L+5s

tuning map: [[1, 2, 1], [0, -1, 3]]

characteristic scale: 3L+4s

The *suri* is a characteristic Zireen instrument, much like a hammered dulcimer. Suri come in a range of sizes from deep bass to beyond the upper range of a piano; the larger suri are set near the floor and sounded by forceful movements of the arms, and many of the smaller ones have piano-like keys that trigger the hammers with light finger motions.

The *kelora* is a small Zireen-size harp, typically strung with metal strings. The simplest kelora are tuned to scales of 7 to 9 notes per octave, but more complex models with sets of levers for varying the tuning (like an orchestral harp) are also popular.

The *vila* is a kind of reed organ with an array of keys like a button accordion. Vila have different arrangements of keys according to the varying natures of the *nai* (systems of tuning). Although they typically play the accompaniment in an ensemble, they may also double the melody, and there is also a tradition of intricately complex music for solo vila.

The *ketana* is a set of wooden bars similar to a marimba. Ketana are used with a variety of tuning systems that aren't very closely related to the harmonic series, such as Nai Hanaki or Nai Yulung. Ketana music often features complex interlocking rhythms.

The *thira* is a Zireen flute. Thira come in various sizes, but they all tend to be high pitched and shrill to human ears.

The *raiva* is double-row set of panpipes, with each row tuned to a pentatonic scale. This instrument is well-suited to Pilina tuning, although raiva tuned to Lemba or Telek are also common. Raiva are relatively lightweight and produce low-pitched tones, since the pipes are stopped at one end.

The *yama* is a kind of mouth organ, like the Chinese sheng. Yama come in a range of sizes covering the middle range of pitch in Zireen music.

The *vunya* is a compact bass reed instrument, similar to the medieval rackett.

The *tamuka* is a set of two large drums, which are played in alternation to establish the rhythmic pattern of the music.

One way to represent a tuning system is with a *tuning map*. This represents the number of iterations of the period and generator needed to approximate each of the prime numbers up to a certain limit (typically 5, 7 or 11). A typical 7-limit tuning map might look something like this: [[1, 2, 4, 7]], [0, -1, -4, -10]]. What this means is that the first prime number (2) is approximated by one period up (an octave); the next prime number (3) by two periods up and one generator down (two octaves minus a perfect fourth = an approximate 3/1 interval); the next prime number (5) by four periods up and four generators down, and the last prime number (7) by seven periods up and ten generators down. To represent any arbitrary interval in the tuning system, simply factor it and add up the approximations of each of the prime factors. Take 35/18 for instance: you can factor it as 2^{-1}3^{-2}5^{1}7^{1}. Consulting the tuning map, you end up with (1*-1)+(2*-2)+(4*1)+(7*1) = 6 for the number of periods, and (0*-1)+(-1*-2)+(-4*1)+(-10*1) = -12 for the number of generators. Tuning maps are convenient for determining which note to use as a representation of any desired rational interval, but they're not unique. You could just as easily use a fifth as a meantone generator instead of a fourth, and end up with a different tuning map: [[1, 1, 0, -3]], [0, 1, 4, 10]].

Another way to represent a tuning system is with a *wedgie*. The mathematics of wedgies is a highly technical subject. If you understand Grassman algebra, see here for a technical explanation of how it works. The important thing for the purposes of this page is that the wedgie uniquely identifies a temperament by representing the combination of all of the commas it tempers out. This is done by taking a "wedge product" of the commas. Take for instance the commas 81/80 and 126/125: you can wedge them to get <<1, 4, 10, 4, 13, 12]]. But what if you wedge 81/80 with 225/224? You end up with the same result. This should come as no surprise when you realize that 126/125 and 225/224 are the same in meantone temperament; they differ by a factor of 81/80, which is tempered out. But wait, there's more! You can also take the wedge product of a tuning map and get exactly the same result! The procedure for wedging a tuning map differs slightly from wedging commas, but the end result is the same: a unique representation for any temperament.

For higher-limit wedgies, it can be useful to line them up in the form of a triangle. Take a typical 11-limit wedgie, for instance: <<7, -3, 8, 2, -21, -7, -21, 27, 15, -22]]. This is what it looks like in triangular form:

7, -3, 8, 2, -21, -7, -21, 27, 15, -22

Notice that if you look at the triangle in the upper left, ignoring the rightmost column, you get a 7-limit wedgie: <<7, -3, 8, -21, -7, 27]]. The upper left corner of *that* triangle gives you a 5-limit wedgie, <<7, -3, -21]], which as a comma is notated [-21 3 7>. This represents a tempering out of the approximately 10-cent interval 2109375/2097152, or 2^{-21}3^{3}5^{7}.