TromboneBoi9
Joined 2 May 2023
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Since the third harmonic is an even number of steps in 19 (30 steps), splitting evenly into two harmonic (subminor) sevenths. If you take a major pentatonic scale, put a harmonic seventh in between each fifth, and reduce the whole scale into an octave, you get a nine-tone scale somewhat similar to the [[wikipedia:Double_harmonic_scale|double harmonic scale]] in 12. | Since the third harmonic is an even number of steps in 19 (30 steps), splitting evenly into two harmonic (subminor) sevenths. If you take a major pentatonic scale, put a harmonic seventh in between each fifth, and reduce the whole scale into an octave, you get a nine-tone scale somewhat similar to the [[wikipedia:Double_harmonic_scale|double harmonic scale]] in 12. | ||
My thought was, if you use pure 3-limit Just Intonation, you can split the third harmonic into two "ratios" of √3. What would that sound like? | My thought was, if you use pure 3-limit Just Intonation, you can split the third harmonic into two "ratios" of √3, measuring around 950.978 cents. What would that sound like? | ||
{| class="wikitable" | {| class="wikitable" | ||
!Degree | !Degree | ||
!Ratio | !Ratio | ||
!Decimal | |||
!Cents | !Cents | ||
|- | |- | ||
|1 | |1 | ||
|1/1 | |1/1 | ||
|1.0000 | |||
|0.000 | |0.000 | ||
|- | |- | ||
|2 | |2 | ||
|9/8 | |9/8 | ||
|1.1250 | |||
|203.910 | |203.910 | ||
|- | |- | ||
|3 | |3 | ||
|81/64 | |81/64 | ||
|1.2656 | |||
|407.820 | |407.820 | ||
|- | |- | ||
|4 | |4 | ||
|3√3/4 | |3√3/4 | ||
|1.2990 | |||
|452.933 | |452.933 | ||
|- | |- | ||
|5 | |5 | ||
|27√3/32 | |27√3/32 | ||
|1.4614 | |||
|656.843 | |656.843 | ||
|- | |- | ||
|6 | |6 | ||
|3/2 | |3/2 | ||
|1.5000 | |||
|701.955 | |701.955 | ||
|- | |- | ||
|7 | |7 | ||
|27/16 | |27/16 | ||
|1.6875 | |||
|905.865 | |905.865 | ||
|- | |- | ||
|8 | |8 | ||
|√3/1 | |√3/1 | ||
|1.7321 | |||
|950.978 | |950.978 | ||
|- | |- | ||
|9 | |9 | ||
|9√3/8 | |9√3/8 | ||
|1.9486 | |||
|1154.888 | |1154.888 | ||
|- | |- | ||
|10 | |10 | ||
|2/1 | |2/1 | ||
|2.0000 | |||
|1200.000 | |||
|} | |||
But of course Saga wasn't looking for a √3 interval, he meant to use the [[7/4|harmonic (subminor) seventh]]. The two intervals are rather close though: the seventh is about 17.848 cents sharper. | |||
Using harmonic sevenths of 968.826 cents: | |||
{| class="wikitable" | |||
!Degree | |||
!Ratio | |||
!Decimal | |||
!Cents | |||
|- | |||
|1 | |||
|1/1 | |||
|1.0000 | |||
|0.000 | |||
|- | |||
|2 | |||
|9/8 | |||
|1.1250 | |||
|203.910 | |||
|- | |||
|3 | |||
|81/64 | |||
|1.2656 | |||
|407.820 | |||
|- | |||
|4 | |||
|21/16 | |||
|1.3125 | |||
|470.781 | |||
|- | |||
|5 | |||
|189/128 | |||
|1.4766 | |||
|674.691 | |||
|- | |||
|6 | |||
|3/2 | |||
|1.5000 | |||
|701.955 | |||
|- | |||
|7 | |||
|27/16 | |||
|1.6875 | |||
|905.865 | |||
|- | |||
|8 | |||
|7/4 | |||
|1.7500 | |||
|968.826 | |||
|- | |||
|9 | |||
|63/32 | |||
|1.9486 | |||
|1172.736 | |||
|- | |||
|10 | |||
|2/1 | |||
|2.0000 | |||
|1200.000 | |||
|} | |||
Using harmonic sevenths inverted around the third harmonic, or just [[12/7|supermajor sixths (12/7)]], of 933.129 cents: | |||
{| class="wikitable" | |||
!Degree | |||
!Ratio | |||
!Decimal | |||
!Cents | |||
|- | |||
|1 | |||
|1/1 | |||
|1.0000 | |||
|0.000 | |||
|- | |||
|2 | |||
|9/8 | |||
|1.1250 | |||
|203.910 | |||
|- | |||
|3 | |||
|81/64 | |||
|1.2656 | |||
|407.820 | |||
|- | |||
|4 | |||
|9/7 | |||
|1.2857 | |||
|435.084 | |||
|- | |||
|5 | |||
|81/56 | |||
|1.4464 | |||
|638.9941 | |||
|- | |||
|6 | |||
|3/2 | |||
|1.5000 | |||
|701.955 | |||
|- | |||
|7 | |||
|27/16 | |||
|1.6875 | |||
|905.865 | |||
|- | |||
|8 | |||
|12/7 | |||
|1.7143 | |||
|933.129 | |||
|- | |||
|9 | |||
|27/14 | |||
|1.9286 | |||
|1137.039 | |||
|- | |||
|10 | |||
|2/1 | |||
|2.0000 | |||
|1200.000 | |1200.000 | ||
|} | |} | ||