Pajara
Pajara, 22 & 32, is a regular temperament wherein the octave is split into two tritone periods, and the generator is a fifth (3/2). A fifth minus a tritone is 16/15 (Diaschismic tempering), and therefore the 5/4 major third is found two generators below the tritone. Pajara makes the further equivalence of 5/4 plus a period to 7/4 (Jubilismic tempering) and therefore twice 4/3 is 7/4 (Archy tempering). The result is a 10-form system generated by a fifth tuned somewhere around 710 cents. There are five patent tunings of Pajara: 12, 22, 54, 32, and 10 (which is also the 20edo val for the 7-limit); of these 22, 54, and 32 are considered "reasonable" and 22edo is the generally assumed tuning.
Extensions
There are two main extensions of Pajara to the 11-limit: Pajarous (10 & 22) and a temperament called "Undecimal Pajara" (12 & 22, which is supported by only those two patent vals) which is pending a rename. Undecimal Pajara is best flat of 22edo; Pajarous is best sharp of 22edo.
Tuning considerations
Most optimization methods place the optimal tuning of Pajara's perfect fifth at around 707 cents (hence the focus on 12 & 22 as an extension). However, Pajara is usually not the best interpretation of those structures. EDOs with tunings of fifths flat of that of 22edo do not support Pajara in the patent val (except for 12edo), and 12edo alongside rank-2 Pajara structures with that tuning are generally extremely inaccurate in the 7-limit due to the fact that Archy temperament forces 9/8 and 8/7 together.
Conversely, systems with fifths sharp of 32edo narrow the distinction between 5/4 and 6/5 considerably, detuning 6/5 to a neutral sound. Therefore, the tuning range of Pajara can be considered to lie between about 709 to 712.5 cents.
Compositional theory
Every note has a corresponding note a tritone apart. Pajara[10] solves the problem of representing intervals of both 5 and 7 in diatonic by introducing three new ordinal classes to provide space for 7-limit intervals to fit on their own degrees of the scale. That way, 7/4 isn't a subminor seventh, it's a major version of the Pajara 8-step. One can even define a notation system for Pajara, wherein the notes are numbered 0-9 and # and b (or ^ and v in 22edo or 32edo) represent alterations by a single step. Pajara retains several desirable properties of MOS diatonic:
- soft-of-basic tunings in the main harmonic temperament
- most notes have a fifth over them
- structurally similar to diatonic, with strings of steps of one size broken by two steps of a different size
- the fifth is divided into two thirds
Additionally, there is a MODMOS sssLsssssL, which is considered the "pentachordal" pajara MOS due to being constructed from an sssL pentachord.
To extend to the 11-limit, Pajara[12] (sLLLLLsLLLLL) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step. The MODMOS [LLLsLLLLsLLL] of Pajara[12] is the Tellurian scale; it may be derived by splitting a pentatonic scale ([LL][LsL][LL][LsL][LL]), or by splitting each whole tone of MOS diatonic ([LL][LL]s[LL][LL][LL][LL]s), which makes for a more xenharmonic way of generalizing 12edo music than simply retuning the chain of fifths. ("When the 12edo goes chromatic, equally divide the whole tone!")
Degrees of pajara[10]
Bolded names are preferred.
| Degree | Name (Leriendil) | Name (standard) | Function | Minor | Perfect | Major |
|---|---|---|---|---|---|---|
| 0 | Unison | Tonic | - | 1/1 | - | |
| 1 | Grade | Second | Supervicinant | 16/15 (perfect) | - | 10/9 |
| 2 | Unilatus | Chthonic | Subvaricant | 8/7 | - | 7/6 |
| 3 | Semitres | Third | Mediant | 6/5 | - | 5/4 |
| 4 | Bilatus | Fourth | Subdominant | 9/7 | - | 4/3 (perfect) |
| 5 | Median | Tritone | Antitonic | - | 7/5, 10/7 | - |
| 6 | Trilatus | Fifth | Dominant | 3/2 (perfect) | - | 14/9 |
| 7 | Semisept | Sixth | Submediant | 8/5 | - | 5/3 |
| 8 | Antilatus | Ouranic | Varicant | 12/7 | - | 7/4 |
| 9 | Degrade | Seventh | Subvicinant | 9/5 | - | 15/8 (perfect) |
| 10 | Duplance | Octave | Tonic | - | 2/1 | - |
Notation
Both degrees and notes in 10-form pajara should be notated with 0-indexed numerals in text: the tonic is always 0, and absolute pitch should be specified in relation to standard diatonic notation. 0-indexing is used so that systems such as figured bass that depend on numerals being a single symbol each still work (if 1-indexing was used, the number 10 would indicate a degree). Additionally, Roman numeral analysis in this case would use N for zero.
As for notating the 10-form on the staff, there are a few different approaches. The first adds an extra line to each staff so that an octave can span 11 staff positions, but comes at the cost of losing intuition for people used to reading intervals from standard notation. The second uses the mosdiatonic notation for pentic, but uses an extra symbol to mark an alteration by a 109c semitone, allowing the full range of notes in pajara to be provided at the cost of potential overloading on symbols as opposed to visual distance to denote pitch. Meanwhile, the third option is simply to notate it starting from mosdiatonic as a base, with ups and downs notation.
10-tone functional harmony
The antilatus and unilatus become the varicant and subvaricant, which sit between the mediant/submediant and supertonic/subtonic in terms of stability (and feature as elements of chthonic chords like 6:7:8, an alternative to standard diatonic chords available in the 10-form). Additionally, the tritone acquires the antitonic function. While the dominant serves as a stable "structural anchor" in diatonic, here the antitonic serves as an unstable structural anchor - the opposite of the tonic both in placement and stability. Note that in the 10-tone system, we return to having two distinct interval qualities down from four, so we go back to having two different keys. 10-tone harmony is also useful in modal music. Also, note that ~100-cent leading tones comprise the majority of the intervals in pajara[10], so their impact may be reduced and in fact one might depend more on the few larger nearmajor seconds that exist in the scale, or skip steps entirely and use subsets.
Another thing to note about the 10-tone system is that it is possible to constrain oneself entirely to chthonic harmony, in which case a lot of the familiar functional harmony language somewhat breaks. The role of the traditional dominant with respect to the tonic disappears completely (even if, for instance, the root position of a chord is assumed to be 4:6:7), with instead the chords on the tritone and the sixth including a leading tone up to the tonic (in fact, the dominant in this system becomes a stable chord rather than a tense one, assuming a 6:7:8 root position).
We may contextualize these differences by examining the 3-function analysis of functional harmony, which in the 7-form (as in 22edo) places the tonic function on the degrees (1-indexed) 1, 3, and 6, the subdominant function on 2 and 4, and the dominant function on 5 and 7. In the chthonic 10-form, however, it requires some amount of reorganization. A theory for Vector's abandoned Earth#Pajara project utilizes four functional categories for chords, rather than three (0-indexed): tonic (0, 2, 8), dominant (4, 6), antitonic (3, 5, 7) and antidominant (1, 9) - the "dominant" function here acts similarly to the heptatonic subdominant (stable, dynamic), with the antitonic and antidominant serving as tense "static" and "dynamic" functions respectively. This essentially splits the 10-form into two pentatonic subscales, one built on the tonic and one built on the antitonic (which is actually how pajara[10] is constructed, but in this tonal system you can pretty strongly feel that construction determining how harmony is structured).
If this theory is also altered to work with tertian harmony instead, the functions follow a chain of thirds rather than a chain of chthonics, so that tonic is (0, 3, 7), a more traditional subdominant is (4, 1), dominant is thus (6, 9), and the remaining degrees (2, 5, 8) constitute antitonic. In this case, dominant is the "tense dynamic" function and antitonic is the "tense static" function.
Modal harmony
The following chart shows the modes of pajara[10], in 22edo tuning:
| Chart | 2 | 3 | 4 | 6 | 8 | 9 | |
|---|---|---|---|---|---|---|---|
| Dynamic minor | ├─┴─┴─┴─┴──┴─┴─┴─┴─┴──┤ 2 2 2 2 3 2 2 2 2 3 | minor | minor | dim | perfect | minor | minor |
| Static minor | ├─┴─┴─┴──┴─┴─┴─┴─┴──┴─┤ 2 2 2 3 2 2 2 2 3 2 | minor | minor | perfect | perfect | minor | major |
| Static major | ├─┴─┴──┴─┴─┴─┴─┴──┴─┴─┤ 2 2 3 2 2 2 2 3 2 2 | minor | major | perfect | perfect | major | major |
| Dynamic major | ├─┴──┴─┴─┴─┴─┴──┴─┴─┴─┤ 2 3 2 2 2 2 3 2 2 2 | major | major | perfect | perfect | major | major |
| Augmented | ├──┴─┴─┴─┴─┴──┴─┴─┴─┴─┤ 3 2 2 2 2 3 2 2 2 2 | major | major | perfect | aug | major | major |
And of a MODMOS of pajara[10], ssLsssssLs, the "pentachordal" pajara scale:
| Chart | 2 | 3 | 4 | 6 | 8 | 9 | |
|---|---|---|---|---|---|---|---|
| (Minor) | ├─┴─┴─┴─┴─┴──┴─┴─┴─┴──┤ 2 2 2 2 2 3 2 2 2 3 | minor | minor | dim | perfect | minor | minor |
| Alternate minor | ├─┴─┴─┴─┴──┴─┴─┴─┴──┴─┤ 2 2 2 2 3 2 2 2 3 2 | minor | minor | dim | perfect | minor | major |
| (Minor) | ├─┴─┴─┴──┴─┴─┴─┴──┴─┴─┤ 2 2 2 3 2 2 2 3 2 2 | minor | minor | perfect | perfect | major | major |
| Standard major | ├─┴─┴──┴─┴─┴─┴──┴─┴─┴─┤ 2 2 3 2 2 2 3 2 2 2 | minor | major | perfect | perfect | major | major |
| (Major) | ├─┴──┴─┴─┴─┴──┴─┴─┴─┴─┤ 2 3 2 2 2 3 2 2 2 2 | major | major | perfect | aug | major | major |
| (Major) | ├──┴─┴─┴─┴──┴─┴─┴─┴─┴─┤ 3 2 2 2 3 2 2 2 2 2 | major | major | perfect | aug | major | major |
| Standard minor | ├─┴─┴─┴──┴─┴─┴─┴─┴─┴──┤ 2 2 2 3 2 2 2 2 2 3 | minor | minor | perfect | perfect | minor | minor |
| (Major) | ├─┴─┴──┴─┴─┴─┴─┴─┴──┴─┤ 2 2 3 2 2 2 2 2 3 2 | minor | major | perfect | perfect | minor | major |
| Alternate major | ├─┴──┴─┴─┴─┴─┴─┴──┴─┴─┤ 2 3 2 2 2 2 2 3 2 2 | major | major | perfect | perfect | major | major |
| (Major) | ├──┴─┴─┴─┴─┴─┴──┴─┴─┴─┤ 3 2 2 2 2 2 3 2 2 2 | major | major | perfect | perfect | major | major |
Some names are from Paul Erlich.
It is useful to consider the MODMOS as roughly on the same level as the MOS form of the scale (as the only additional variety it introduces is in the tritone), giving 15 distinct modes available to choose from, each with a degree of brightness or darkness to them, much like conventional MOSdiatonic modal harmony.
A pajara[10] pentachord may be considered to consist of five tones. To continue the theme of pajara mirroring conventional harmony with two qualities from 12edo, we may constrain this specific set of pentachords such that they must be comprised entirely of semitones and nearmajor seconds, which is an analogous constraint to the one stating that a 12edo tetrachord must be comprised entirely of tones and semitones, as it leads to four distinct pentachords.
Alternatively in the more general interpretation, there are four additional "harmonic" pentachords. Note that in either case, no note in a pentachord may occupy the first or last step of the perfect fourth.
Generator chain
The following chart assumes Pajarous.
| Period 1 | Period 2 | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Up | Down | Up | Down | ||||||
| # | Cents | JI | Cents | JI | # | Cents | JI | Cents | JI |
| 0 | 0 | 1/1 | 600 | 7/5 | 0 | 600 | 12/7 | 1200 | 2/1 |
| 1 | 110 | 18/17, 17/16, 16/15, 15/14 | 490 | 4/3 | 1 | 710 | 3/2 | 1090 | 28/15, 15/8, 32/17, 17/9 |
| 2 | 220 | 8/7, 9/8 | 380 | 5/4 | 2 | 820 | 8/5 | 980 | 7/4, 16/9 |
| 3 | 330 | 6/5, 11/9 | 270 | 7/6 | 3 | 930 | 12/7 | 870 | 5/3, 18/11 |
| 4 | 440 | 9/7 | 160 | 12/11, 10/9 | 4 | 1040 | 9/5, 11/6 | 760 | 14/9 |
| 5 | 550 | 11/8 | 50 | 25/24, 49/48 | 5 | 1150 | 96/49, 48/25 | 650 | 16/11 |
List of patent vals
Due to being a weak extension of Archy, the patent vals of Pajara are a subset of the Archy edos. Specifically, they must be even (and therefore, some edos are doubled compared to the Archy table).
| EDO | Extension to 11 | Generator tuning | 7/4 tuning | 25/24 tuning |
|---|---|---|---|---|
| 10 | Pajarous | 480c | 960c | 0c |
| 32 | Pajarous | 487.5c | 975c | 37.5c |
| 54 | Pajarous | 488.9c | 977.8c | 44.4c |
| 22 | Pajarous, Undecimal Pajara | 490.9c | 981.8c | 54.5c |
| 12 | Undecimal Pajara | 500c | 1000c | 100c |
While Pajarous may appear to be canonical from this chart, one important consideration is the historical prevalence of non-patent Pajara tunings and the fact that Pajarous necessarily tunes 10/9~12/11 flatter than 11/10, while undecimal Pajara instead equates 10/9 and 11/10. 22edo, the intersection of the two systems, equates all three and consequently supports Porcupine.
