Tuning recommendations for beginners

This article contains lists of various users' optimal tunings for a new xenharmonist to start with. It is recommended to pick one early and focus on it so that tuning theory does not become more important than composition.

To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. "Subgroups" are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.

• 2.3.5: Known as "pental", widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.
• 2.3.7: Known as "septal", more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.
• 2.3.5.7 (2...7): Known as "septimal", the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.
• 2.3.5.7.11.13 (2...13): Known as "tridecimal", an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).

The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:

• Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.
• Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.
• Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.

User lists

Ground

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This section is written from the perspective of User:Ground and may feature proposed concepts.

This is my list of the most versatile edo tunings, split into two tiers. All of them are diatonic to have a baseline level of harmonic stability:

Absolute best: 12edo (M/P), 19edo (M), 22edo (A), 24edo (M/P), 31edo (M), 41edo (P).

Second best: 17edo (A/P), 26edo (M), 27edo (A), 29edo (P), 37edo (A).

I intend to add sound examples to make the selection process easier.

The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.

Tuning descriptions (incomplete):

• 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.
• 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.
• 26edo: Debatable 2...13.
• 27edo: Good 2...7.13, all tuned sharp.
• 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.
• 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.
• 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of straddling the fifth to be used on the same level as the others.
• 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.

Main types of scale-based composition theory required for the list, according to me:

• Diatonic: Already known from 12edo, affects everything
• Dicoid: 17, 24, 27, 31, 37, 41
• Slendric: 26, 31, 41
• Semiquartal: 19, 24, 29
• Porcupine: 22, 29, 37
• Aberrismic: 22+

I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.

Dicoid is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset mosh is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.

Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.