Mike Sheiman's Alternative Interval Categorizations
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People often say xenharmonic intervals like 16/11 are "sour" and mathematically similar intervals (e.g. octave inverses like 1/(16/11) or 11/8) are "sweet". Doesn't that seem a bit counter intuitive? We've been raised in music theory to accept everything, even xenharmonic/microtonal intervals, be pigeon-holed into some sort of diatonic category. In 12EDO C is the tonic/"first". C# (apx. 17/16) is a minor second D (apx. 9/8) is a major second D# (apx. 6/5) is a minor third E (apx. 5/4) is a major third **F (apx 4/3) is a perfect fourth** (Why not a major or minor? Inconsistency...) **F# (apx. 7/5) is on the borderline between a fourth and fifth** **G (apx. 3/2) is a perfect fifth** (Again, no major or minor. Inconsistency...) G# (apx. 8/5)is a minor sixth A (apx. 5/3) is a major sixth A# (apx. 9/5) is a minor seventh B (apx 15/8) is a major seventh **Notice how...even in 12EDO, interval categories seem a bit shaky and inconsistent.** So how, then, to you categorize something like an **11/8 or 16/11 between a fourth and a fifth?** Or an interval like 14/9, between a fifth and a sixth? **Furthermore, how do explain when, for example, a 16/11 feels "sour" while an 11/8 slightly below it feels upbeat/sweet?** Usually we simply add additional names as necessary and further complicate the system. 16/11? That's sour because it's a **diminished** fifth. Around 14/9? That's upbeat because it's an **augmented** fifth. Why not just stick with major (**more upbeat**) and minor (**more downbeat**) and neutral (**in-between upbeat and downbeat and a bit sour**)...equally distributed among 4ths, 5ths, 6ths...? Here's a proposal C is the tonic/"first". (15/14 and less) is a minor second (13/12 to 11/10) is a neutral second (10/9 to 9/8) is a major second (7/6) is a minor **second-half** **(15/13) is a** neutral **second-half** (8/7) is a major **second-half** (apx. 9/8) is a major second D# (apx. 6/5) is a minor third E (apx. 5/4) is a major third **F (apx 4/3) is a perfect fourth** (Why not a major or minor? Inconsistency...) **F# (apx. 7/5) is on the borderline between a fourth and fifth** **G (apx. 3/2) is a perfect fifth** (Again, no major or minor. Inconsistency...) G# (apx. 8/5)is a minor sixth A (apx. 5/3) is a major sixth A# (apx. 9/5) is a minor seventh B (apx 15/8) is a major seventh
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<html><head><title>Mike Sheiman's Alternative Interval Categorizations</title></head><body>People often say xenharmonic intervals like 16/11 are "sour" and mathematically similar intervals (e.g. octave inverses like <br /> 1/(16/11) or 11/8) are "sweet". Doesn't that seem a bit counter intuitive?<br /> We've been raised in music theory to accept everything, even xenharmonic/microtonal intervals, be pigeon-holed into some sort of diatonic category.<br /> <br /> In 12EDO C is the tonic/"first". <br /> C# (apx. 17/16) is a minor second <br /> D (apx. 9/8) is a major second <br /> D# (apx. 6/5) is a minor third<br /> E (apx. 5/4) is a major third<br /> <strong>F (apx 4/3) is a perfect fourth</strong> (Why not a major or minor? Inconsistency...)<br /> <strong>F# (apx. 7/5) is on the borderline between a fourth and fifth</strong><br /> <strong>G (apx. 3/2) is a perfect fifth</strong> (Again, no major or minor. Inconsistency...)<br /> G# (apx. 8/5)is a minor sixth<br /> A (apx. 5/3) is a major sixth<br /> A# (apx. 9/5) is a minor seventh<br /> B (apx 15/8) is a major seventh<br /> <strong>Notice how...even in 12EDO, interval categories seem a bit shaky and inconsistent.</strong><br /> So how, then, to you categorize something like an <strong>11/8 or 16/11 between a fourth and a fifth?</strong> Or an interval like 14/9, between a fifth and a sixth? <strong>Furthermore, how do explain when, for example, a 16/11 feels "sour" while an 11/8 slightly below it feels upbeat/sweet?</strong><br /> <br /> <br /> Usually we simply add additional names as necessary and further complicate the system. 16/11? That's sour because it's a <strong>diminished</strong> fifth. Around 14/9? That's upbeat because it's an <strong>augmented</strong> fifth. Why not just stick with major (<strong>more upbeat</strong>) and minor (<strong>more downbeat</strong>) and neutral (<strong>in-between upbeat and downbeat and a bit sour</strong>)...equally distributed among 4ths, 5ths, 6ths...?<br /> <br /> Here's a proposal<br /> C is the tonic/"first". <br /> (15/14 and less) is a minor second<br /> (13/12 to 11/10) is a neutral second<br /> (10/9 to 9/8) is a major second<br /> (7/6) is a minor <strong>second-half</strong><br /> <strong>(15/13) is a</strong> neutral <strong>second-half</strong><br /> (8/7) is a major <strong>second-half</strong><br /> <br /> (apx. 9/8) is a major second<br /> D# (apx. 6/5) is a minor third<br /> E (apx. 5/4) is a major third<br /> <strong>F (apx 4/3) is a perfect fourth</strong> (Why not a major or minor? Inconsistency...)<br /> <strong>F# (apx. 7/5) is on the borderline between a fourth and fifth</strong><br /> <br /> <strong>G (apx. 3/2) is a perfect fifth</strong> (Again, no major or minor. Inconsistency...)<br /> G# (apx. 8/5)is a minor sixth<br /> <br /> A (apx. 5/3) is a major sixth<br /> <br /> A# (apx. 9/5) is a minor seventh<br /> <br /> B (apx 15/8) is a major seventh</body></html>