Five-limit tuning
However, most tuning systems designed for acoustic instruments restrict the total number of pitches for practical reasons.The prominent notes of a given scale are tuned so that their frequencies form ratios of relatively small integers.It is evidently not possible to get all seven diatonic triads in the configuration (4:5:6) for major, (10:12:15) for minor, and (25:30:36) for diminished at the same time if we limit ourselves to seven pitches.To build a twelve-tone scale in 5-limit tuning, we start by constructing a table containing fifteen justly intonated pitches: The factors listed in the first row and first column are powers of 3 and 5 respectively (e.g., 1⁄9 = 3−2).For instance, in the first row of the table, there is an ascending fifth from D and A, and another one (followed by a descending octave) from A to E. This suggests an alternative but equivalent method for computing the same ratios.For instance, you can obtain A (5/3 ratio), starting from C, by moving one cell to the left and one upward in the table, which means descending by one fifth (2/3) and ascending by one major third (5/4): Since this is below C, you need to move up by an octave to end up within the desired range of ratios (from 1/1 to 2/1): A 12-tone scale is obtained by removing one note for each couple of enharmonic notes.Note that it is a diminished fifth, close to half an octave, above the tonic C, which is a disharmonic interval; also its ratio has the largest values in its numerator and denominator of all tones in the scale, which make it least harmonious: all reasons to avoid it.The first strategy, which we operationally denote here as symmetric scale 1, consists of selecting for removal the tones in the upper left and lower right corners of the table.It is possible to construct just intervals with even "juster" ratios, or alternately, with values closer to the equal-tempered equivalents.For instance, a 7-limit tuning is sometimes used to obtain a slightly juster and consequently more consonant interval for the minor seventh (7/4) and its inversion, the major second (8/7).Notice that the ratios 45/32 and 64/45 for the tritones (augmented fourth and diminished fifth) are not in all contexts regarded as strictly just, but they are the justest possible in the above-mentioned 5-limit tuning scales.An extended asymmetric 5-limit scale (see below) provides slightly juster ratios for both the tritones (25/18 and 36/25), the purity of which is also controversial.In other tuning systems, a comma may be defined as a minute interval, equal to the difference between two kinds of semitones (diatonic and chromatic, also known as minor second, m2, or augmented unison, A1).In late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant.
Tonnetztuningmusical instrumentinteger powersprime numbersjust intonationlatticepitch classesfrequency ratiodiatonic scaleG majorperfect fifthperfect fourthmajor triadsPtolemy's intense diatonic scalenatural numberssemiditonesyntonic commaminor triadsdiminished triadsemitoneminor tonemajor toneenharmonicoctavesfifthsmajor thirdsdiminished fifthminor thirdsmodulationwolf intervalsconsonance7-limitPerfect unisonMinor secondMajor secondMinor thirdMajor thirdAugmented fourthMinor sixthMajor sixthMinor seventhMajor seventhPerfect octavetritonesseptimal tritoneseptimal minor seventhsemitonesaugmented unisonequally temperedmeantone temperamentDiaschismameantonediesis1/4-comma meantonesyntonic temperamentdiminished seconddissonanttriadschordstextureMiddle Agesconsonantwolf fifthGioseffo ZarlinodiatonicchromaticMathematics of musical scalesMicrotonal musicMicrotunerPythagorean intervalList of intervals in 5-limit just intonationList of meantone intervalsList of musical intervalsList of pitch intervalsWhole-tone scaleRegular numberHexanyElectronic tunerConsonance and dissonanceJohnston, BenGilmore, BobJohn FonvillePerspectives of New MusicDon Michael RandelHarvard Dictionary of MusicSalinasWest, M. L.Music & LettersKyle GannTellus Audio Cassette MagazineUbuWebWayback MachineHelmholtzpitch notationMusical tuningsMillioctaveSavartIntervalInterval ratioPitch classMicrotoneEuler–Fokker genusHarmonic scaleHarry Partch's 43-tone scaleList of compositionsOtonalityPythagorean tuningScale of harmonicsTonality diamondTonality fluxTemperaments6-tone12-tone15-tone17-tone19-tone22-tone23-tone24-tonepieces31-tone34-tone41-tone53-tone58-tone