Ultramajor and inframinor

This article is about "tendo" and "arto", the qualities more extreme than supermajor and subminor. For the neutral qualities, see Neutral (interval quality).

Ultramajor and inframinor, or tendo and arto, intervals are more extreme than supermajor or subminor intervals, with ultramajor being sharp of supermajor, and inframinor being flat of subminor. For example, ultramajor thirds are sharper than about 446 cents, and inframinor thirds are flatter than about 256 cents. This region is unique as it often coincides with the opposing region from another category, for example, ultramajor seconds are often also inframinor thirds. Common ultramajor and inframinor intervals are often in the 13-limit, for example:

• 15/13 (248 ¢), ultramajor second OR inframinor third
• 13/10 (454 ¢), ultramajor third OR infrafourth
• 20/13 (746 ¢), ultrafifth OR inframinor sixth
• 26/15 (952 ¢), ultramajor sixth OR inframinor seventh

Ultramajor and inframinor thirds and sixths are found in diatonic scales, specifically when fifths are tuned extremely sharp (to about 715 cents or so). Ultramajor seconds are not found in the diatonic scale. For a given neutral interval k in cents, the ultramajor quality is sharper than around k + 95 cents, and the inframinor quality is flatter than around k − 95 cents.

For the discussion of the topic as presented in Margo Schulter's theory, see Interseptimal. For the discussion of how to use ultramajor and inframinor intervals in chords, see Extraclassical tonality.

Optionally, the category of ultramajor or inframinor may be split into two smaller categories. Tuning ranges have been provided in terms of thirds:

• When used more precisely, ultramajor and inframinor may refer specifically to ranges that, for thirds, are between about 446–458 ¢ (for ultramajor), and between 244 and 256 cents (for inframinor). These are the "interseptimal" ranges, the ones that overlap with each other. Under this definition, ultramajor seconds range from about 242–254 ¢, demonstrating the overlapping nature. For a given neutral interval k in cents, the ultramajor version in this sense is found at around k + 100, and the inframinor version is found at around k − 100.
• Similarly, when used more precisely, tendo and arto may refer specifically to ranges that, for thirds, are sharper than roughly 458 cents (for ultramajor) and flatter than roughly 244 cents (for inframinor). Under this definition, arto seconds are sharper than about 254 cents, demonstrating that in most cases, arto and tendo intervals are better off relabelled as belonging to an adjacent category - the exception is functional cases, such as chord names and certain temperaments. For a given neutral interval k in cents, the tendo version in this sense is found at around k + 115, and the arto version is found at around k − 115.

In the theory of MidnightBlue, some of these ranges are considered "dead zones" as they introduce potentially unwanted ambiguities between interval categories. However, in other practices, this might be leveraged to musical effect.