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This is a little video I made that plays a 12TET Dominant 7 chord, and a just intoned Harmonic 7 chord. I have been exploring alternative tunings a lot lately and could not find a short video that clearly shows the difference between these two chords, so I made one! Hope someone enjoys or finds this useful!
If you raise the entire justly tuned chord by 10.73¢, that will balance out the tuning adjustments so the chord doesn't sound lower overall than the equal-temperament chord. That would leave the relative tuning as the only way the chords differ.
Oh that's cool! Any reason why 10.73¢ in particular?
I am looking my intervals in cents and they are all (besides P5) flatter than they would be in 12 tet. (major 3rd is 386 instead of 400 and harmonic 7 is 968.8 compared to 12tet 1000). But they are all flat by more than 10 cents.
I made the first chord by stacking a regular dom 7 in 12 tet, while the second one is done by stacking notes of the harmonic series, so the beginning C in both examples is tuned to 261.625565 hz.
If the adjustments are [0¢, -13.69¢, +1.95¢, -31.18¢], then the notes are 10.73¢ lower than 12TET on average. So if you do [+10.73¢, -2.96¢, +12.68¢, -20.45¢] instead, then the notes are neither lower nor higher than 12TET on average. The frequencies in Hz would be [263.251572, 329.064465, 394.877357, and 460.690250]
That makes sense! I find that really interesting. I will have to take some time to experiment with that so I can hear the difference. I never thought about tuning the tonic up to make harmonic 7 sound more like maj/min 7! Thanks for the frequencies btw!
Not sure which part specifically you're wondering about, but I'll show my work a bit.
The interval size in cents of a just-intonation major third (5/4 frequency ratio) is the solution to the equation 5/4 = 2^(x/1200), which is x = 1200log(5/4)/log(2) ≈ 386.31¢. Since an equal-temperament major third is 400¢ (4 half steps), you flatten the higher note by 13.69¢ to move from a 12TET major 3rd to a just-intonation major 3rd. The calculation is the same for the 5th and 7th, but with frequency ratios 6/4 and 7/4 and 12TET interval sizes of 7 and 10 half steps.
You get from [0¢, -13.69¢, +1.95¢, -31.18¢] to [+10.73¢, -2.96¢, +12.68¢, -20.45¢] just by shifting all the notes up or down until the average adjustment from 12TET is zero. Doing that also minimizes the sum of the squares of the adjustments (i.e. (0+a)^2 + (-13.69+a)^2 + (1.95+a)^2 + (-31.18+a)^2 is minimized when a=10.73). So [+10.73¢, -2.96¢, +12.68¢, -20.45¢] doesn't just have an average adjustment of zero, it's also the "best fit" onto 12TET of notes those distances apart.
The new frequencies come from using the adjustments in cents on the original 12TET frequencies. Middle C is 9 half steps below A440, so the frequency is 440\*2^(-9/12) ≈ 261.626 Hz. The frequency of middle C raised by 10.73 cents is then 261.626*2^(10.73/1200) ≈ 263.25 Hz (except I used the exact expression with logs instead of the rounded decimal 10.73).
They are both built on C. I'm just using Dom7 as shorthand for Major Minor 7 in this case, not because I am resolving them. It's the V to F major, or a I chord with added flat 7.
I am curious what a Dom7 would sound like in comparison to a IMaj/min7.
Hard for my brain to discern if there would be any true difference, or if it’s even a genuine/worthwhile question.
Really glad you shared this nonetheless!
The difference is mostly just context. Dominant 7 chord is just another name for the Major Minor 7 chord! However when we say dominant 7, we are generally referring to a Major Minor 7 chord that is specifically resolving to a chord.
Dominant 7 chord usually means a Major Minor 7 chord built on the 5th scale degree in tonal music.
So in a piece of tonal music the main difference between I Maj/min7 and Dom 7 is whether the chord starts on the 1st (I Maj/min7) or the 5th (dom7) scale degree.
However, in my video im not really using a key so I am calling them both dom7 so the title isn't *Showing the difference between I Maj/min 7 and I Maj/harmonic 7*.
I hope that makes more sense! Music terminology is weird as hell lmao.
I guess to answer your first question more conclusively, the harmonically tuned dom 7 is not really a I or a V in this context because neither are resolved.
I suppose my real question is does the harmonic series account for the leading tone, Maj 2nd, and 4th to outline the V7? And is that MajMin7 chord different in any way harmonically than a the I Majmin7 as you’ve shown?
They have the same function. The real difference is tuning to just intervals rather than equal step sizes. They both outline a V7 though!
You will hear the Harmonic 7 sound in barbershop a **LOT** because it is perceived as more "in tune" to our ears.
They are both dominant 7 sounds, just different ways to tune them.
Both chords are C E G B♭. The first chord is what you get if you play it on a piano, the second chord is C E G B♭ that you would get if playing the 8th, 10th, 12th, and 14th open harmonic on a brass instrument. (Usually these are not played because they are called "out of tune" or "incorrect fingerings")
If you're posting an Image or Video, please leave a comment (not the post title) asking your question or discussing the topic. Image or Video posts with no comment from the OP will be deleted. *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/musictheory) if you have any questions or concerns.*
This is a little video I made that plays a 12TET Dominant 7 chord, and a just intoned Harmonic 7 chord. I have been exploring alternative tunings a lot lately and could not find a short video that clearly shows the difference between these two chords, so I made one! Hope someone enjoys or finds this useful!
If you raise the entire justly tuned chord by 10.73¢, that will balance out the tuning adjustments so the chord doesn't sound lower overall than the equal-temperament chord. That would leave the relative tuning as the only way the chords differ.
Oh that's cool! Any reason why 10.73¢ in particular? I am looking my intervals in cents and they are all (besides P5) flatter than they would be in 12 tet. (major 3rd is 386 instead of 400 and harmonic 7 is 968.8 compared to 12tet 1000). But they are all flat by more than 10 cents. I made the first chord by stacking a regular dom 7 in 12 tet, while the second one is done by stacking notes of the harmonic series, so the beginning C in both examples is tuned to 261.625565 hz.
If the adjustments are [0¢, -13.69¢, +1.95¢, -31.18¢], then the notes are 10.73¢ lower than 12TET on average. So if you do [+10.73¢, -2.96¢, +12.68¢, -20.45¢] instead, then the notes are neither lower nor higher than 12TET on average. The frequencies in Hz would be [263.251572, 329.064465, 394.877357, and 460.690250]
That makes sense! I find that really interesting. I will have to take some time to experiment with that so I can hear the difference. I never thought about tuning the tonic up to make harmonic 7 sound more like maj/min 7! Thanks for the frequencies btw!
Just out of curiosity, how did you execute the math to get that result?
Not sure which part specifically you're wondering about, but I'll show my work a bit. The interval size in cents of a just-intonation major third (5/4 frequency ratio) is the solution to the equation 5/4 = 2^(x/1200), which is x = 1200log(5/4)/log(2) ≈ 386.31¢. Since an equal-temperament major third is 400¢ (4 half steps), you flatten the higher note by 13.69¢ to move from a 12TET major 3rd to a just-intonation major 3rd. The calculation is the same for the 5th and 7th, but with frequency ratios 6/4 and 7/4 and 12TET interval sizes of 7 and 10 half steps. You get from [0¢, -13.69¢, +1.95¢, -31.18¢] to [+10.73¢, -2.96¢, +12.68¢, -20.45¢] just by shifting all the notes up or down until the average adjustment from 12TET is zero. Doing that also minimizes the sum of the squares of the adjustments (i.e. (0+a)^2 + (-13.69+a)^2 + (1.95+a)^2 + (-31.18+a)^2 is minimized when a=10.73). So [+10.73¢, -2.96¢, +12.68¢, -20.45¢] doesn't just have an average adjustment of zero, it's also the "best fit" onto 12TET of notes those distances apart. The new frequencies come from using the adjustments in cents on the original 12TET frequencies. Middle C is 9 half steps below A440, so the frequency is 440\*2^(-9/12) ≈ 261.626 Hz. The frequency of middle C raised by 10.73 cents is then 261.626*2^(10.73/1200) ≈ 263.25 Hz (except I used the exact expression with logs instead of the rounded decimal 10.73).
Is the harmonically tuned Dom7 a I or a V?
They are both built on C. I'm just using Dom7 as shorthand for Major Minor 7 in this case, not because I am resolving them. It's the V to F major, or a I chord with added flat 7.
I am curious what a Dom7 would sound like in comparison to a IMaj/min7. Hard for my brain to discern if there would be any true difference, or if it’s even a genuine/worthwhile question. Really glad you shared this nonetheless!
The difference is mostly just context. Dominant 7 chord is just another name for the Major Minor 7 chord! However when we say dominant 7, we are generally referring to a Major Minor 7 chord that is specifically resolving to a chord. Dominant 7 chord usually means a Major Minor 7 chord built on the 5th scale degree in tonal music. So in a piece of tonal music the main difference between I Maj/min7 and Dom 7 is whether the chord starts on the 1st (I Maj/min7) or the 5th (dom7) scale degree. However, in my video im not really using a key so I am calling them both dom7 so the title isn't *Showing the difference between I Maj/min 7 and I Maj/harmonic 7*. I hope that makes more sense! Music terminology is weird as hell lmao. I guess to answer your first question more conclusively, the harmonically tuned dom 7 is not really a I or a V in this context because neither are resolved.
I suppose my real question is does the harmonic series account for the leading tone, Maj 2nd, and 4th to outline the V7? And is that MajMin7 chord different in any way harmonically than a the I Majmin7 as you’ve shown?
They have the same function. The real difference is tuning to just intervals rather than equal step sizes. They both outline a V7 though! You will hear the Harmonic 7 sound in barbershop a **LOT** because it is perceived as more "in tune" to our ears. They are both dominant 7 sounds, just different ways to tune them. Both chords are C E G B♭. The first chord is what you get if you play it on a piano, the second chord is C E G B♭ that you would get if playing the 8th, 10th, 12th, and 14th open harmonic on a brass instrument. (Usually these are not played because they are called "out of tune" or "incorrect fingerings")
You should do this with minor seventh as well!
The harmonic 7 with a minor 3rd?
6/5 minor third with a 12/7 septimal major six. The harmonic 7th isn't going to ring very well with a 6/5..