specific note ratios can always be debated within the context of this or that harmonic/melodic context. the problem we are assuming is that tuning must be fixed and cannot subtly modulate (multiple *possible* ratios for a major fourth, or what have you), in string and choral music, or with any non-fixed-pitch instrumentation, these complicated mathematical questions are intuited immediately and effortlessly, because music is a language primarily of emotions which can later be described and dissected in abstract intellectual terms, in fact, whether or not this or that ratio is exactly perfect or accurate takes a backseat to whether or not the player really *intended* the feeling, or simply played it well. personally, when writing/playing music digitally, where everything is basically shoehorned into the format of a piano (fixed pitch, putting aside the idea of pitch automation envelopes), I like to use 5-limit with some "secondary keys", notes not in the scale I'm using shifted to be alternate ratios of notes I am using, to fill in where certain ratios do not work. assuming C=256hz, there are many cases where either E=320 hz sounds best or E=324hz does, there is no reason why both cannot be used
another thing to take into account is the halving/double sequences of the frequencies of our notes in relation to accidentals/beating frequencies and combination tones, 320 is a few octaves above 5, 256 above 2, 384 above 3, 448 above 7, so on and so forth, our brainwaves/states of consciousness are influenced by these and exhibit certain characteristics or shifts when entrained to them via external stimulation over long periods, to fully and comprehensively make use of all the properties of sound and psycho-acoustic phenomena, from ultra low frequencies to tones to harmonious intervals to the distribution of frequencies across the broader hearing spectrum, I believe is key to unlocking the full objective truth and meaning of the music
@@pauwel9380 The problem with higher limit tunings above 3 is that 1, fundamentally they are illogical systems, and 2, they inevitably sound out of tune. Yes, in certain situations you as the composer can trick the ear and hide certain intervals, and coax the ear into hearing something as in tune, but it will always be at the expense of the holistic listening experience. Where the listeners mind is both relaxed completely and aware intently, accounting for the notes and harmonies in the moment as well as the notes and chords that came before and anticipating what may come next. Using higher limit tunings makes one only listen or primarily listen in the moment, and not over time. Music is something that happens both over time and in the moment. but because the melodic intervals are so out of tune in 5 and 7 limit and higher, the ear and brain start to only listen in the present moment, to the buzz of the chords, and all sense of melodic and interval awareness is lost. In terms of just adjusting by ear on the fly, this also is not right. Yes it can be done, and is done, but it is not ideal, it is the result of a compensation for errors, adjusting so the entire piece does not become more and more entropic. There were two groups in the past the harmonists and the canonists, the harmonists, had this approach you speak of, go by ear, it is emotional, nothing is exact, go by feel, they did not think notes had specific pitches to rigidly adhere to. The canonists said that pitches were specific frequencies, that and A was an A, an E was an E in a given piece, for example, always the same pitches per given name. Cb was a different note than B, which was a different note than Ax. For a harmonist Cb, Ax, and B are really essentially the same thing, the same note, that are written or spelled differently for the coherence of the score, but not 3 distinct exact pitches. The harmonist view won out in the world currently, and has lead to nilism and post modernism, and an anything goes mentality, expanding outside the realm of music. Not saying music is the sole cause of that, but definitely a major contributing factor. In 5 and 7 limit tunings and higher, one always has compromises to make, and if you go into it deeply enough & your listening has not been compromised into only listening in the present moment, you will eventually find 5 and 7 limit tuning to be unuseful in music. I say this as someone who worked exclusively in 5 and 7 limit tuning for nearly 10 years. In True Intonation, no compromise is ever needed. Notes are exact frequencies which you other hit exactly or not, and there is clarity in both harmony and melodic intervals. But in order to hear 3 limit as in tune, one must decondition their ears from 5 limit and 7 limit philosophy which if you are into tuning theory, you will have no doubt undergone this conditioning at some point. This conditioning of only listening vertically, as opposed to vertically and horizontally at once. For me I had to focus on more on horizontal listening at first, or my ear would go right back to vertical only. Even in Barbershop music if you probe the experts of that field deep enough you can eventually get them to reluctantly admit that 7 limit intonation is not a coherent system, it needs compromise and trickery of the ear, and hiding of melodic lines to be somewhat presentable. Because all they are looking for is presentability and a ringing chord, not a functional system of tuning. For a major chord on C, hearing the C, E and G, as a 1st degree, 3rd degree, and 5th degree of the scale sounding at once be present in your awareness and let the combined sound speak for itself, as opposed to looking for a 4:5:6 timbral sound. C is 3 to the 0th power, E is 3 to the 4th power, and G is 3 to the 1st power. All pitch classes as powers of 3, and powers of 2 for all octaves of those pitch classes. Growing up singing in choir, whenever I had to sing the 3rd of a major chord, I never knew EXACTLY where I was aiming for, the piano was off, and no tuning theory taught. With 5 limit and higher you need a LOT of theory to avoid pitch drifting unintentionally and weird intervals that sound out of tune and are melodically incoherent and harmonically fused and obfuscated. In 3 limit, True Intonation, you just sing what is written on the page as is.
What convinces me that Pythagorean tuning is a complete system is the fact that the Pentatonic scale is complete and is the worldwide system of music in which all the world's most popular melodies are composed e.g. Auld Lang Syne is said to be the most widely recognised tune in the world: and the Pentatonic scale cannot be constructed by linking together 4:5:6 triads without producing additional tones. However, when expanding the Pythagorean system beyond 5 notes using ascending 5ths, the tonic note shifts from the first tone (F) to the second tone (C) somewhere between 5 tones and 7 tones. I think it may be ambiguous with 6 tones. However, 7 tones seems to be the first scale which can be constructed out of triads or 5ths. The Baroque major/minor scale with 10 notes can also be constructed out of triads and so also can 12 notes. The transition of the tonic from 1st to 2nd needs explaining and I feel that it may have something to do with triads, with the CEG triad being central to the system as it expands beyond 7 notes. Another parameter of this inquiry is that I do not regard 9/8 as a harmony. 6/5, a just minor 3rd, is the smallest interval which I acknowledge to be a harmony. I do not feel a need to side with one system or the other as "truer" than the other, but Pythagorean does come first. Whether it is somehow succeeded isn't settled with me at all. I think what I really need is probably a wife or girlfriend to laugh at me when I try to explain it to her. Bless.
Good points about the pentatonic in 3 limit Pythagorean, but not in 5 limit, at least not without a wolf fifth. i think the hexatonic scale is indeed ambiguous whether F or C is the tonic. But the reason that the tonic more strongly moves to C with 7 notes is because of where the tritone lies in the scale in relation to the chord of C, if you are in C major. That F to B tension resolves by limmas in contrary motion to the root and major third of the C major chord. So there is a strong tension which steps by limma in contrary (balanced) motion to C and E, the 2 most important notes in a C major chord. I definitely would consider the 9/8 to be a harmony in many contexts, and a non chord tone in others. Much of the way we perceive consonance and dissonance is largely determined by rhythm and where changes in chords and melodies occur in a musical phrase. Even with notes from F to B in an unbroken chain of six fifths, F can still be heard as a stable tonic. All of the 7 diatonic modes can be heard as a stable tonic if you know how to put certain chord changes at certain places and with certain rhythms. Wherever the tritone is in relation to the tonic chord must be observed and handled accordingly. but even Locrian can be satisfactorily stabilized and heard as the tonic chord. I think Major is as prominent as it is because the tritone can be easily resolved to the tonic chord root and third by limmas in contrary motion. no other mode has this. I think minor was attributed to Aeolian because the i iv and v chords are all naturally minor as a mirror to the I IV and V major chords in Ionian. And to create that same tritone resolution we see in major resolving by contrarily moving limmas, the G is raised to a G#, moving that tension from F to B, to D and G#. Plus the major 2nd and b6 lend itself as generally easier to stabilize, One thing I did not mention in this video, is that in 5 limit, the brain has to track motion of pitch classes in 2 different dimensions and combinations of those dimensions. A voice can move in the 3 limit direction or in the 5 limit dimension and in different proportions of those. This makes the voices much more difficult to track and process. If all pitch classes are powers of 3, then suddenly it is much more straight forward. Then voices either move in the sharp direction or the flat, and there are no combinations of dimensions in different proportions, all pitch classes are in one dimension, with only octaves existing as another dimension. So would you say that a C69 chord voiced C E G A D sounds bad or out of tune to you? Considering you say that the 9/8 does not sound harmonic? I mean by it's definition is not the 9/8 a literal harmonic and thus a harmony? What do you hear as a better major second? And btw you can even hear my wife laughing at me in the background in this very video. I am teaching her, but having to go slow. I have been at this a long time and I have never really had the opportunity to teach it before, especially to someone who is starting music from the very beginning.
Sorry I couldn't follow your explanation of why C becomes tonic (starting at F) from 7 notes onward. Something about limmas in contrary motion (?). I don't know if anyone has ever tackled this subject convincingly but I haven't done much research either. My reason for suspecting the involvement of 4:5:6 triads is that 7 is the first number of notes where 3:2 steps and 4:5:6 triads produce a scale of the same form. The next such number is 10 which corresponds to the combination of both major and minor scales in the same key. After that the next is 12. 10 is of interest because David had a harp of 10 strings. Octaves can be played on the "harmonic" on harps, so it is quite possible to play 2 octaves of a key in both major and minor on a 10 string harp. Also, the tonnex diagram for this fits on a Star of David ✡️ with major triads pointing upwards and minor downward. I'm not sure I really think he had one carved on his harp to remind him how to tune it!
@@RememberGodHolyBible another observation that has just come to me is that there are no (approximate) triads (major or minor, if minor should be significant) until the 5th tone has been added by Pythagorean progression in 5ths e.g. starting with C, G, D, A there is no triad until E is added: both major and minor triads contain the E.
Ugh, you tube deleted my response. To put it shortly. Yes, I just discovered the same thing, it is pretty neat. I would not think the hexagram would be on David's harp considering it's ties in witchcraft and it's absence from the Scriptures apart maybe from the star of Rempham, the idol. I know what you mean with the tonnetz, but the tonnetz implies a 5 limit not 3 limit tuning. A 10 note major minor scale is neat, but if that was on a harp you could only play the high octave or the low octave, not both at once. I wonder if there is a compelling 9 note scale that repeats the octave on the tenth note. For a 10 note scale the Phrygian mixolydian scale may be interesting as well. On C: C, Db, D, Eb, E, F, G, Ab, A, Bb, (C) 10 is a strange and interesting number I have been thinking a lot about recently. Our system of counting is base 10 and God's system of numbers is centered on base ten. Base 12 and 6 are fascinating, but it is not what God ordained in scripture, neither in chapter in verse numbers, nor in the text itself. I do eagerly await the day to play the instrument of heaven and sing. To see and hear, and understand the tuning they employ.
specific note ratios can always be debated within the context of this or that harmonic/melodic context. the problem we are assuming is that tuning must be fixed and cannot subtly modulate (multiple *possible* ratios for a major fourth, or what have you), in string and choral music, or with any non-fixed-pitch instrumentation, these complicated mathematical questions are intuited immediately and effortlessly, because music is a language primarily of emotions which can later be described and dissected in abstract intellectual terms, in fact, whether or not this or that ratio is exactly perfect or accurate takes a backseat to whether or not the player really *intended* the feeling, or simply played it well. personally, when writing/playing music digitally, where everything is basically shoehorned into the format of a piano (fixed pitch, putting aside the idea of pitch automation envelopes), I like to use 5-limit with some "secondary keys", notes not in the scale I'm using shifted to be alternate ratios of notes I am using, to fill in where certain ratios do not work. assuming C=256hz, there are many cases where either E=320 hz sounds best or E=324hz does, there is no reason why both cannot be used
another thing to take into account is the halving/double sequences of the frequencies of our notes in relation to accidentals/beating frequencies and combination tones, 320 is a few octaves above 5, 256 above 2, 384 above 3, 448 above 7, so on and so forth, our brainwaves/states of consciousness are influenced by these and exhibit certain characteristics or shifts when entrained to them via external stimulation over long periods, to fully and comprehensively make use of all the properties of sound and psycho-acoustic phenomena, from ultra low frequencies to tones to harmonious intervals to the distribution of frequencies across the broader hearing spectrum, I believe is key to unlocking the full objective truth and meaning of the music
@@pauwel9380 The problem with higher limit tunings above 3 is that 1, fundamentally they are illogical systems, and 2, they inevitably sound out of tune.
Yes, in certain situations you as the composer can trick the ear and hide certain intervals, and coax the ear into hearing something as in tune, but it will always be at the expense of the holistic listening experience. Where the listeners mind is both relaxed completely and aware intently, accounting for the notes and harmonies in the moment as well as the notes and chords that came before and anticipating what may come next. Using higher limit tunings makes one only listen or primarily listen in the moment, and not over time. Music is something that happens both over time and in the moment. but because the melodic intervals are so out of tune in 5 and 7 limit and higher, the ear and brain start to only listen in the present moment, to the buzz of the chords, and all sense of melodic and interval awareness is lost.
In terms of just adjusting by ear on the fly, this also is not right. Yes it can be done, and is done, but it is not ideal, it is the result of a compensation for errors, adjusting so the entire piece does not become more and more entropic.
There were two groups in the past the harmonists and the canonists, the harmonists, had this approach you speak of, go by ear, it is emotional, nothing is exact, go by feel, they did not think notes had specific pitches to rigidly adhere to. The canonists said that pitches were specific frequencies, that and A was an A, an E was an E in a given piece, for example, always the same pitches per given name. Cb was a different note than B, which was a different note than Ax. For a harmonist Cb, Ax, and B are really essentially the same thing, the same note, that are written or spelled differently for the coherence of the score, but not 3 distinct exact pitches.
The harmonist view won out in the world currently, and has lead to nilism and post modernism, and an anything goes mentality, expanding outside the realm of music. Not saying music is the sole cause of that, but definitely a major contributing factor.
In 5 and 7 limit tunings and higher, one always has compromises to make, and if you go into it deeply enough & your listening has not been compromised into only listening in the present moment, you will eventually find 5 and 7 limit tuning to be unuseful in music. I say this as someone who worked exclusively in 5 and 7 limit tuning for nearly 10 years.
In True Intonation, no compromise is ever needed. Notes are exact frequencies which you other hit exactly or not, and there is clarity in both harmony and melodic intervals. But in order to hear 3 limit as in tune, one must decondition their ears from 5 limit and 7 limit philosophy which if you are into tuning theory, you will have no doubt undergone this conditioning at some point. This conditioning of only listening vertically, as opposed to vertically and horizontally at once. For me I had to focus on more on horizontal listening at first, or my ear would go right back to vertical only.
Even in Barbershop music if you probe the experts of that field deep enough you can eventually get them to reluctantly admit that 7 limit intonation is not a coherent system, it needs compromise and trickery of the ear, and hiding of melodic lines to be somewhat presentable. Because all they are looking for is presentability and a ringing chord, not a functional system of tuning.
For a major chord on C, hearing the C, E and G, as a 1st degree, 3rd degree, and 5th degree of the scale sounding at once be present in your awareness and let the combined sound speak for itself, as opposed to looking for a 4:5:6 timbral sound. C is 3 to the 0th power, E is 3 to the 4th power, and G is 3 to the 1st power. All pitch classes as powers of 3, and powers of 2 for all octaves of those pitch classes.
Growing up singing in choir, whenever I had to sing the 3rd of a major chord, I never knew EXACTLY where I was aiming for, the piano was off, and no tuning theory taught. With 5 limit and higher you need a LOT of theory to avoid pitch drifting unintentionally and weird intervals that sound out of tune and are melodically incoherent and harmonically fused and obfuscated. In 3 limit, True Intonation, you just sing what is written on the page as is.
What convinces me that Pythagorean tuning is a complete system is the fact that the Pentatonic scale is complete and is the worldwide system of music in which all the world's most popular melodies are composed e.g. Auld Lang Syne is said to be the most widely recognised tune in the world: and the Pentatonic scale cannot be constructed by linking together 4:5:6 triads without producing additional tones. However, when expanding the Pythagorean system beyond 5 notes using ascending 5ths, the tonic note shifts from the first tone (F) to the second tone (C) somewhere between 5 tones and 7 tones. I think it may be ambiguous with 6 tones. However, 7 tones seems to be the first scale which can be constructed out of triads or 5ths. The Baroque major/minor scale with 10 notes can also be constructed out of triads and so also can 12 notes. The transition of the tonic from 1st to 2nd needs explaining and I feel that it may have something to do with triads, with the CEG triad being central to the system as it expands beyond 7 notes. Another parameter of this inquiry is that I do not regard 9/8 as a harmony. 6/5, a just minor 3rd, is the smallest interval which I acknowledge to be a harmony. I do not feel a need to side with one system or the other as "truer" than the other, but Pythagorean does come first. Whether it is somehow succeeded isn't settled with me at all. I think what I really need is probably a wife or girlfriend to laugh at me when I try to explain it to her. Bless.
Good points about the pentatonic in 3 limit Pythagorean, but not in 5 limit, at least not without a wolf fifth. i think the hexatonic scale is indeed ambiguous whether F or C is the tonic. But the reason that the tonic more strongly moves to C with 7 notes is because of where the tritone lies in the scale in relation to the chord of C, if you are in C major. That F to B tension resolves by limmas in contrary motion to the root and major third of the C major chord. So there is a strong tension which steps by limma in contrary (balanced) motion to C and E, the 2 most important notes in a C major chord.
I definitely would consider the 9/8 to be a harmony in many contexts, and a non chord tone in others. Much of the way we perceive consonance and dissonance is largely determined by rhythm and where changes in chords and melodies occur in a musical phrase.
Even with notes from F to B in an unbroken chain of six fifths, F can still be heard as a stable tonic. All of the 7 diatonic modes can be heard as a stable tonic if you know how to put certain chord changes at certain places and with certain rhythms. Wherever the tritone is in relation to the tonic chord must be observed and handled accordingly. but even Locrian can be satisfactorily stabilized and heard as the tonic chord. I think Major is as prominent as it is because the tritone can be easily resolved to the tonic chord root and third by limmas in contrary motion. no other mode has this. I think minor was attributed to Aeolian because the i iv and v chords are all naturally minor as a mirror to the I IV and V major chords in Ionian. And to create that same tritone resolution we see in major resolving by contrarily moving limmas, the G is raised to a G#, moving that tension from F to B, to D and G#. Plus the major 2nd and b6 lend itself as generally easier to stabilize,
One thing I did not mention in this video, is that in 5 limit, the brain has to track motion of pitch classes in 2 different dimensions and combinations of those dimensions. A voice can move in the 3 limit direction or in the 5 limit dimension and in different proportions of those. This makes the voices much more difficult to track and process. If all pitch classes are powers of 3, then suddenly it is much more straight forward. Then voices either move in the sharp direction or the flat, and there are no combinations of dimensions in different proportions, all pitch classes are in one dimension, with only octaves existing as another dimension.
So would you say that a C69 chord voiced C E G A D sounds bad or out of tune to you? Considering you say that the 9/8 does not sound harmonic? I mean by it's definition is not the 9/8 a literal harmonic and thus a harmony? What do you hear as a better major second?
And btw you can even hear my wife laughing at me in the background in this very video. I am teaching her, but having to go slow. I have been at this a long time and I have never really had the opportunity to teach it before, especially to someone who is starting music from the very beginning.
Sorry I couldn't follow your explanation of why C becomes tonic (starting at F) from 7 notes onward. Something about limmas in contrary motion (?). I don't know if anyone has ever tackled this subject convincingly but I haven't done much research either. My reason for suspecting the involvement of 4:5:6 triads is that 7 is the first number of notes where 3:2 steps and 4:5:6 triads produce a scale of the same form. The next such number is 10 which corresponds to the combination of both major and minor scales in the same key. After that the next is 12.
10 is of interest because David had a harp of 10 strings. Octaves can be played on the "harmonic" on harps, so it is quite possible to play 2 octaves of a key in both major and minor on a 10 string harp. Also, the tonnex diagram for this fits on a Star of David ✡️ with major triads pointing upwards and minor downward. I'm not sure I really think he had one carved on his harp to remind him how to tune it!
@@RememberGodHolyBible another observation that has just come to me is that there are no (approximate) triads (major or minor, if minor should be significant) until the 5th tone has been added by Pythagorean progression in 5ths e.g. starting with C, G, D, A there is no triad until E is added: both major and minor triads contain the E.
Ugh, you tube deleted my response.
To put it shortly. Yes, I just discovered the same thing, it is pretty neat.
I would not think the hexagram would be on David's harp considering it's ties in witchcraft and it's absence from the Scriptures apart maybe from the star of Rempham, the idol. I know what you mean with the tonnetz, but the tonnetz implies a 5 limit not 3 limit tuning.
A 10 note major minor scale is neat, but if that was on a harp you could only play the high octave or the low octave, not both at once. I wonder if there is a compelling 9 note scale that repeats the octave on the tenth note. For a 10 note scale the Phrygian mixolydian scale may be interesting as well. On C:
C, Db, D, Eb, E, F, G, Ab, A, Bb, (C)
10 is a strange and interesting number I have been thinking a lot about recently. Our system of counting is base 10 and God's system of numbers is centered on base ten. Base 12 and 6 are fascinating, but it is not what God ordained in scripture, neither in chapter in verse numbers, nor in the text itself.
I do eagerly await the day to play the instrument of heaven and sing. To see and hear, and understand the tuning they employ.
ref
why does the woman randomly laugh?