Pental major and minor

This article is about two different but related interval qualities. For 5-limit major and minor intervals, see 5/4, 6/5, 8/5, and 5/3. For other uses of "pental", see Perfect fifth and 5-limit.

Pental major intervals are an interval quality denoting tunings close to ptolemaic major intervals. They are sharper than submajor intervals and flatter than novamajor intervals. Likewise, pental minor intervals are an interval quality denoting tunings close to ptolemaic minor intervals. They are sharper than novaminor intervals and flatter than supraminor intervals. Pental major thirds range from about 375–394 ¢, and pental minor thirds range from about 308–327 ¢.

Common pental major/minor intervals can be found as simple 5-limit intervals, and include:

• 10/9 (182 ¢), pental major second
• 6/5 (316 ¢), pental minor third
• 5/4 (386 ¢), pental major third
• 8/5 (814 ¢), pental minor sixth
• 5/3 (884 ¢), pental major sixth
• 9/5 (1018 ¢), pental minor seventh

Pental major/minor intervals are found in diatonic sales where the fifth is tuned slightly flat of just. If these intervals are interpreted as the 5-limit, the result is meantone temperament. For a given neutral interval k in cents, the pental major quality ranges from around k + 24 to k + 43, and the pental minor quality ranges from around k − 43 to k − 24.

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

• Magimajor and magiminor, for thirds, range between about 375–382 and 320–327 ¢, respectively. These are flat of the 5-limit thirds, and appear in 5-limit temperaments where the chromatic semitone 25/24 is tempered narrow, like in garibaldi, magic (hence the name), or the minor third in flattone. Magimajor seconds range from 171–178 ¢, and thus contain the upper part of the "equable heptatonic" region. For a given neutral interval k in cents, the magimajor version is found at around k + 28, and the magiminor version is found at around k − 28.
• Pentamajor and pentaminor, for thirds, range between about 382–394 and 308–320 ¢, respectively. These are the regions containing 5-limit intervals. Pentamajor seconds range from 178–190 ¢. For a given neutral interval k in cents, the pentamajor version is found at around k + 35, and the pentaminor version is found at around k − 35.