Neutral (interval quality)

"Tendoneutral" redirects here. For the ultramajor quality (called "tendo"), see Ultramajor and inframinor.

Neutral intervals are between major and minor intervals. For example, neutral thirds fall between roughly 341 and 361 cents. Neutral intervals are the "center" of an interval category, and are the point around which the qualities of that category are symmetric. Common neutral intervals may be found as the square roots of common major or minor intervals:

• sqrt(32/27) (147c), neutral second
• sqrt(3/2) (351c), neutral third
• sqrt(8/3) (849c), neutral sixth
• sqrt(27/8) (1053c), neutral seventh

They may alternatively be found as 11-limit intervals:

• 12/11 (151c), neutral second
• 11/9 (347c), neutral third
• 18/11 (853c), neutral sixth
• 11/6 (1049c), neutral seventh

Neutral intervals are found in diatonic scales when the fifth is significantly flatter than just (flatter than around 689 cents). For a given "central" neutral interval k in cents, the range of the neutral interval quality is from around k-10 to k+10 cents.

Optionally and rather usefully, the neutral interval quality is split into two, based on the more "major" or "minor" side of neutral, since often (especially in JI or regular temperaments) you aren't working with perfect neutral intervals. (Examples are provided with thirds):

• Artoneutral intervals are flat of the "central" interval. Artoneutral thirds range from about 341 to 351 cents. 11/9 is the best-known example of an interval in this range. They also function as the "pure" neutrals when the interval that doubles them is tuned flatly, such as the neutral third in meantone or 7edo. For a given pure neutral interval k in cents, the corresponding artoneutral interval is found at around k-5 cents.
• Tendoneutral intervals are sharp of the "central" interval. Tendoneutral thirds range from about 351 to 361 cents, and 16/13 is the best known example of an interval in this range. They also function as "pure" neutrals when the interval that doubles them is tuned sharply, such as the neutral second in meantone. For a given pure neutral interval k in cents, the corresponding tendoneutral interval is found at around k+5 cents.