Topic: Equally dividing the pure fifth into 7
1 scales
| File | Description | Notes | Period (¢) |
|---|---|---|---|
| div28 | Dividing 5 into 28 equal parts | 28 | 2786.3 |
Thread (30 messages)
From: Ozan Yarman (2007-03-23) Subject: Equally dividing the pure fifth into 7 And extending to the octave yields the following 12-tone system: 0: 1/1 C unison, perfect prime 1: 100.279 cents 2: 200.559 cents D 3: 300.838 cents E 4: 401.117 cents 5: 501.396 cents F 6: 601.676 cents 7: 3/2 G perfect fifth 8: 802.234 cents A 9: 902.514 cents A 10: 1002.793 cents B 11: 1103.072 cents 12: 1203.351 cents C The fifths remain pure at every degree. Octaves sound fine too.
From: Cameron Bobro (2007-03-23) Subject: Re: Equally dividing the pure fifth into 7 http://aredem.online.fr/aredem/page_cordier.html --- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote: > > And extending to the octave yields the following 12-tone system: > > > 0: 1/1 C unison, perfect prime > 1: 100.279 cents > 2: 200.559 cents D > 3: 300.838 cents E > 4: 401.117 cents > 5: 501.396 cents F > 6: 601.676 cents > 7: 3/2 G perfect fifth > 8: 802.234 cents A > 9: 902.514 cents A > 10: 1002.793 cents B > 11: 1103.072 cents > 12: 1203.351 cents C > > > The fifths remain pure at every degree. Octaves sound fine too. >
From: Petr Parízek (2007-03-23) Subject: Re: [tuning] Equally dividing the pure fifth into 7 Hi Ozan. May I ask how long ago it was that this idea came to your mind? I was thinking about just the same thing about four months ago but so far I haven't found a way to realize this. The interesting thing about this kind of temperament is that 7 octaves are exactly (3/2)^12. Petr
From: Petr Parízek (2007-03-23) Subject: Re: [tuning] Re: Equally dividing the pure fifth into 7 Something like an English translation would be fairly appropriate, I think. Speaking for myself, I'm afraid a Czech translation doesn't exist. Petr ----- Original Message ----- From: Cameron Bobro To: [email protected] Sent: Friday, March 23, 2007 12:32 PM Subject: [tuning] Re: Equally dividing the pure fifth into 7 http://aredem.online.fr/aredem/page_cordier.html --- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote: > > And extending to the octave yields the following 12-tone system: > > > 0: 1/1 C unison, perfect prime > 1: 100.279 cents > 2: 200.559 cents D > 3: 300.838 cents E > 4: 401.117 cents > 5: 501.396 cents F > 6: 601.676 cents > 7: 3/2 G perfect fifth > 8: 802.234 cents A > 9: 902.514 cents A > 10: 1002.793 cents B > 11: 1103.072 cents > 12: 1203.351 cents C > > > The fifths remain pure at every degree. Octaves sound fine too. >
From: Cameron Bobro (2007-03-23) Subject: Re: Equally dividing the pure fifth into 7 --- In [email protected], Petr Paríº¥k wrote: > > Something like an English translation would be fairly appropriate, I think. Speaking for myself, I'm afraid a Czech translation doesn't exist. > > Petr It's Cordier's temperament from the '70s. Just checked- it's in the Scala archives. Doesn't mean it's not a groovy discovery by Ozan, but a re-discovery.
From: Danny Wier (2007-03-23) Subject: Re: [tuning] Equally dividing the pure fifth into 7 ----- Original Message ----- From: "Ozan Yarman" <[email protected]> To: "Tuning List" <[email protected]> Sent: Friday, March 23, 2007 4:21 AM Subject: [tuning] Equally dividing the pure fifth into 7 > And extending to the octave yields the following 12-tone system: > > > 0: 1/1 C unison, perfect prime > 1: 100.279 cents > 2: 200.559 cents D > 3: 300.838 cents E > 4: 401.117 cents > 5: 501.396 cents F > 6: 601.676 cents > 7: 3/2 G perfect fifth > 8: 802.234 cents A > 9: 902.514 cents A > 10: 1002.793 cents B > 11: 1103.072 cents > 12: 1203.351 cents C > > > The fifths remain pure at every degree. Octaves sound fine too. And octave-stretching is recommended for pianos, so you also have that advantage. A while ago, I recommended 19 equal divisions of a perfect twelfth, which has a step size of 100.10289 cents. But what I really like is 49 divisions of the twelfth as a replacement for 31-tet. Fifths are a little better (now 698.67735 cents), to the detriment of the fourth and of course the octave, the latter being stretched to 1203.27765 cents. ~D.
From: threesixesinarow (2007-03-23) Subject: Re: Equally dividing the pure fifth into 7 --- In [email protected], "Danny Wier" <dawiertx@...> wrote: > > ----- Original Message ----- > From: "Ozan Yarman" <ozanyarman@...> > To: "Tuning List" <[email protected]> > Sent: Friday, March 23, 2007 4:21 AM > Subject: [tuning] Equally dividing the pure fifth into 7 > > > > And extending to the octave yields the following 12-tone system: > > > > > > 0: 1/1 C unison, perfect prime > > 1: 100.279 cents > > 2: 200.559 cents D > > 3: 300.838 cents E > > 4: 401.117 cents > > 5: 501.396 cents F > > 6: 601.676 cents > > 7: 3/2 G perfect fifth > > 8: 802.234 cents A > > 9: 902.514 cents A > > 10: 1002.793 cents B > > 11: 1103.072 cents > > 12: 1203.351 cents C > > > > > > The fifths remain pure at every degree. Octaves sound fine too. > > And octave-stretching is recommended for pianos, so you also have that > advantage. > > A while ago, I recommended 19 equal divisions of a perfect twelfth, which > has a step size of 100.10289 cents. But what I really like is 49 divisions > of the twelfth as a replacement for 31-tet. Fifths are a little better (now > 698.67735 cents), to the detriment of the fourth and of course the octave, > the latter being stretched to 1203.27765 cents. > > ~D. > http://www.piano-stopper.de/html/onlypure_tuning.html Clark
From: Ozan Yarman (2007-03-23) Subject: Re: [tuning] Equally dividing the pure fifth into 7 Seems I just revamped Cordier's concert piano tuning without realizing it. The idea came to me just today, Petr. Oz. ----- Original Message ----- From: "Petr Par\ufffdzek" To: Sent: 23 Mart 2007 Cuma 13:28 Subject: Re: [tuning] Equally dividing the pure fifth into 7 > Hi Ozan. > > May I ask how long ago it was that this idea came to your mind? I was > thinking about just the same thing about four months ago but so far I > haven't found a way to realize this. The interesting thing about this kind > of temperament is that 7 octaves are exactly (3/2)^12. > > Petr > >
From: Carl Lumma (2007-03-23) Subject: Re: Equally dividing the pure fifth into 7 --- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote: > > And extending to the octave yields the following 12-tone system: > > > 0: 1/1 C unison, perfect prime > 1: 100.279 cents > 2: 200.559 cents D > 3: 300.838 cents E > 4: 401.117 cents > 5: 501.396 cents F > 6: 601.676 cents > 7: 3/2 G perfect fifth > 8: 802.234 cents A > 9: 902.514 cents A > 10: 1002.793 cents B > 11: 1103.072 cents > 12: 1203.351 cents C > > > The fifths remain pure at every degree. Octaves sound fine too. This is a tuning I've tried on pianos, which appreciate octave stretch anyway. I call it the 7th-root-of-3/2 temperament. Concise, isn't it? The problem with this tuning is that it makes major 3rds and 10ths worse. Some top-notch piano tuners on the usenet claim to tune a similar kind of thing, namely 19th-root-of-3 temperament. This one is easier on the 3rds. -Carl
From: Carl Lumma (2007-03-23) Subject: Re: Equally dividing the pure fifth into 7 > May I ask how long ago it was that this idea came to your mind? I was > thinking about just the same thing about four months ago but so far I > haven't found a way to realize this. The interesting thing about this > kind of temperament is that 7 octaves are exactly (3/2)^12. I first learned of it in 1998 from Julius Pierce. I posted on it here several times in the late '90s. I set it by using a strobe tuner to tune C-B, then tuning pure fifths by ear from there. -Carl
From: Gene Ward Smith (2007-03-23) Subject: Re: Equally dividing the pure fifth into 7 --- In [email protected], "Carl Lumma" <clumma@...> wrote: > The problem with this tuning is that it makes major 3rds > and 10ths worse. Not for pianos, but for situations where you might consider 12-et TOP: tuning so that 5 is pure; that is, with steps of 5^(1/28). Especially nice for chords which avoid octaves and don't cover too great a range. The major triad in close root position is 0-398.044816-696.578428. Another triad is 0-696.578428-1592.179265. You can get interesting seventh chords too, for example 0-696.578428-1592.179265-2189.246489. ! div28.scl Dividing 5 into 28 equal parts 28 ! 99.511204 199.022408 298.533612 398.044816 497.556020 597.067224 696.578428 796.089633 895.600837 995.112041 1094.623245 1194.134449 1293.645653 1393.156857 1492.668061 1592.179265 1691.690469 1791.201673 1890.712877 1990.224081 2089.735285 2189.246489 2288.757694 2388.268898 2487.780102 2587.291306 2686.802510 5
From: Danny Wier (2007-03-23) Subject: Re: [tuning] Re: Equally dividing the pure fifth into 7 From: "threesixesinarow" <[email protected]> To: <[email protected]> Sent: Friday, March 23, 2007 9:28 AM Subject: [tuning] Re: Equally dividing the pure fifth into 7 >> A while ago, I recommended 19 equal divisions of a perfect twelfth, > which >> has a step size of 100.10289 cents. But what I really like is 49 > divisions >> of the twelfth as a replacement for 31-tet. Fifths are a little > better (now >> 698.67735 cents), to the detriment of the fourth and of course the > octave, >> the latter being stretched to 1203.27765 cents. >> >> ~D. >> > http://www.piano-stopper.de/html/onlypure_tuning.html > > Clark I was thinking someone would've come up with the idea before me. But where does he get the name "OnlyPure"? There are fewer pure intervals in 19-edt than in 12-edo, aren't there? ~D.
From: Carl Lumma (2007-03-23) Subject: Re: Equally dividing the pure fifth into 7 > I was thinking someone would've come up with the idea before me. > But where does he get the name "OnlyPure"? There are fewer pure > intervals in 19-edt than in 12-edo, aren't there? > > ~D. I hate to state the obvious, but anyone who brands a tuning as obvious as this is an idiot. I'm almost as bad with the Alaska tunings, but geez. -Carl
From: Ozan Yarman (2007-03-24) Subject: Re: [tuning] Re: Equally dividing the pure fifth into 7 My, you are a bit touchy nowadays it seems. ----- Original Message ----- From: "Carl Lumma" <[email protected]> To: <[email protected]> Sent: 24 Mart 2007 Cumartesi 1:11 Subject: [tuning] Re: Equally dividing the pure fifth into 7 > > I was thinking someone would've come up with the idea before me. > > But where does he get the name "OnlyPure"? There are fewer pure > > intervals in 19-edt than in 12-edo, aren't there? > > > > ~D. > > I hate to state the obvious, but anyone who brands a tuning > as obvious as this is an idiot. I'm almost as bad with > the Alaska tunings, but geez. > > -Carl > >
From: Danny Wier (2007-03-24) Subject: Re: [tuning] Re: Equally dividing the pure fifth into 7 From: "Carl Lumma" <[email protected]> To: <[email protected]> Sent: Friday, March 23, 2007 6:11 PM Subject: [tuning] Re: Equally dividing the pure fifth into 7 >> I was thinking someone would've come up with the idea before me. >> But where does he get the name "OnlyPure"? There are fewer pure >> intervals in 19-edt than in 12-edo, aren't there? >> >> ~D. > > I hate to state the obvious, but anyone who brands a tuning > as obvious as this is an idiot. I'm almost as bad with > the Alaska tunings, but geez. > > -Carl He also said he wanted to patent it. Can you patent a mathematical formula? ~D.
From: Petr Parízek (2007-03-24) Subject: Re: [tuning] Re: Equally dividing the pure fifth into 7 Carl wrote: > I hate to state the obvious, but anyone who brands a tuning as obvious as this is an idiot. I'm almost as bad with the Alaska tunings, but geez. though I'm sure you have mentioned Alaska tunings some time ago, I'm not sure what they are. Petr
From: Carl Lumma (2007-03-24) Subject: Re: Equally dividing the pure fifth into 7 > though I'm sure you have mentioned Alaska tunings some time ago, > I'm not sure what they are. Circulating temperaments with flat octaves in the 1997-cent range. -Carl
From: Tom Dent (2007-03-24) Subject: Re: Equally dividing the pure fifth into 7 --- In [email protected], "Carl Lumma" <clumma@...> wrote: > > > though I'm sure you have mentioned Alaska tunings some time ago, > > I'm not sure what they are. > > Circulating temperaments with flat octaves in the 1997-cent > range. > > -Carl > Love the typo. Anyway, the piano tech mailing list agrees with Carl about the patenting of a piano tuning method using pure twelfths - namely, that it's probably both bogus (in the sense of not patentable) and useless (ie the patent is unenforceable). Search Google for 'Onlypure'. The promulgator of it, Mr. Stopper, is now promoting the existence of a 'Supersymmetry' between beats and frequencies. But you can't get an explanation of what this could mean until you buy an Onlypure licence. As a theoretical physicist who actually deals with genuine supersymmetry (graded Lie algebra, for those in the know) I find this half amusing and half pathetic. ~~~T~~~
From: Gene Ward Smith (2007-03-24) Subject: Re: Equally dividing the pure fifth into 7 --- In [email protected], "Tom Dent" <stringph@...> wrote: > As a theoretical physicist who actually deals with genuine > supersymmetry (graded Lie algebra, for those in the know) I find this > half amusing and half pathetic. Have you ever taken a look at tuning-math? Not much in the way of Lie algebras, but root lattice terminology is used anyway, for what that is worth. And multilinear algebra galore...
From: Carl Lumma (2007-03-24) Subject: Re: Equally dividing the pure fifth into 7 > Love the typo. Which typo is that? -Carl
From: Klaus Schmirler (2007-03-25) Subject: Re: [tuning] Re: Equally dividing the pure fifth into 7 Carl Lumma schrieb: >> Love the typo. > > Which typo is that? You hid it well - I spent some time looking for it, too. Think about what an unflat octave is. klaus
From: Carl Lumma (2007-03-25) Subject: Re: Equally dividing the pure fifth into 7 > >> Love the typo. > > > > Which typo is that? > > You hid it well - I spent some time looking for it, too. > > Think about what an unflat octave is. A pure octave? -Carl
From: Klaus Schmirler (2007-03-25) Subject: Re: [tuning] Re: Equally dividing the pure fifth into 7 Carl Lumma schrieb: >>>> Love the typo. >>> Which typo is that? >> You hid it well - I spent some time looking for it, too. >> >> Think about what an unflat octave is. > > A pure octave? Undeniably. And in quantitative terms?
From: Carl Lumma (2007-03-26) Subject: Re: Equally dividing the pure fifth into 7 > >> Think about what an unflat octave is. > > > > A pure octave? > > Undeniably. > > And in quantitative terms? I give up. -Carl
From: Klaus Schmirler (2007-03-26) Subject: Re: [tuning] Re: Equally dividing the pure fifth into 7 Carl Lumma schrieb: >>>> Think about what an unflat octave is. >>> A pure octave? >> Undeniably. >> >> And in quantitative terms? > > I give up. No you don't \ufffd= Blunt hint: the 2000 cent octave.
From: Carl Lumma (2007-03-26) Subject: Re: Equally dividing the pure fifth into 7 > >>>> Think about what an unflat octave is. > >>> A pure octave? > >> > >> Undeniably. > >> > >> And in quantitative terms? > > > > I give up. > > No you don't ¿= > > Blunt hint: the 2000 cent octave. I have no idea what you're talking about, but if anyone wants to divulge the secret my e-mail address is [email protected]. -Carl
From: Richard Eldon Barber (2007-07-09) Subject: Re: Equally dividing the pure fifth into 7 When Stanford patented FM synthesis, Allen Strange said this was akin to "patenting water". Some of us pianotech people were fortunate enough to meet Bernard Stopper in Kansas City last month. He demonstrated his tuning on a Yamaha upright and a Fazioli grand. He was touting his new pocket pc tuning software based on his 3^(1/19) pure P12 tuning system. The 19-tone P12 tuning sounded very nice- some couldn't tell the difference, others like myself reveled in the way it masked dissonance, especially in bitonal/pan-diatonic chord structures. What I found intriguing was the use of any combination of 5ths and Octaves in setting the tuning aurally. Apparently he has found the formula by which the beat found in the P4-P5-P8 sonority is cancelled by stretching the P12 just. Has anyone else worked this out, mathematically? -Richard Barber Morgan Hill, CA --- In [email protected], "Danny Wier" <dawiertx@...> wrote: > > From: "Carl Lumma" <clumma@...> > To: <[email protected]> > Sent: Friday, March 23, 2007 6:11 PM > Subject: [tuning] Re: Equally dividing the pure fifth into 7 > > > >> I was thinking someone would've come up with the idea before me. > >> But where does he get the name "OnlyPure"? There are fewer pure > >> intervals in 19-edt than in 12-edo, aren't there? > >> > >> ~D. > > > > I hate to state the obvious, but anyone who brands a tuning > > as obvious as this is an idiot. I'm almost as bad with > > the Alaska tunings, but geez. > > > > -Carl > > He also said he wanted to patent it. Can you patent a mathematical formula? > > ~D. >
From: Carl Lumma (2007-07-09) Subject: Re: Equally dividing the pure fifth into 7 > Some of us pianotech people were fortunate enough to meet Bernard > Stopper in Kansas City last month. He demonstrated his tuning on > a Yamaha upright and a Fazioli grand. He was touting his new > pocket pc tuning software based on his 3^(1/19) pure P12 tuning > system. > > The 19-tone P12 tuning sounded very nice- some couldn't tell the > difference, others like myself reveled in the way it masked > dissonance, especially in bitonal/pan-diatonic chord structures. > > What I found intriguing was the use of any combination of 5ths and > Octaves in setting the tuning aurally. Apparently he has found the > formula by which the beat found in the P4-P5-P8 sonority is > cancelled by stretching the P12 just. > > Has anyone else worked this out, mathematically? > > -Richard Barber > Morgan Hill, CA Hi Richard, There's only one equal step tuning with the P12 (what I would prefer to call 3:1) just. By P4-P5-P8 sonority, what chord or chords do you mean? And in what manner were beats cancelled? If you can explain, maybe the list will answer. -Carl
From: Richard Eldon Barber (2007-07-09) Subject: Re: Equally dividing the pure fifth into 7 By P4-P5-P8 I mean, for instance, C F G c. According to Stopper, any combination of fifths and octaves sound beatless due to a symmetry in the tuning. My guess is they are either phase-reverse cancelled, or masked by sidebands? Any thoughts? Richard --- In [email protected], "Carl Lumma" <clumma@...> wrote: > Hi Richard, > > There's only one equal step tuning with the P12 (what I would > prefer to call 3:1) just. By P4-P5-P8 sonority, what chord > or chords do you mean? And in what manner were beats cancelled? > If you can explain, maybe the list will answer. > > -Carl > > > > Some of us pianotech people were fortunate enough to meet Bernard > > Stopper in Kansas City last month. He demonstrated his tuning on > > a Yamaha upright and a Fazioli grand. He was touting his new > > pocket pc tuning software based on his 3^(1/19) pure P12 tuning > > system. > > > > The 19-tone P12 tuning sounded very nice- some couldn't tell the > > difference, others like myself reveled in the way it masked > > dissonance, especially in bitonal/pan-diatonic chord structures. > > > > What I found intriguing was the use of any combination of 5ths and > > Octaves in setting the tuning aurally. Apparently he has found the > > formula by which the beat found in the P4-P5-P8 sonority is > > cancelled by stretching the P12 just. > > > > Has anyone else worked this out, mathematically? > > > > -Richard Barber > > Morgan Hill, CA >
From: Carl Lumma (2007-07-09) Subject: Re: Equally dividing the pure fifth into 7 I doubt anything like this can be made to occur in all keys and in all registers. If beating were masked by sidebands, I should think the sidebands would themselves be quite unpleasant. And I'm not aware of any way to control the relative phases of beating intervals in a tuning. Is there a recording anywhere of this stuff, or a paper by Stopper anywhere? By the way, I'm in Los Gatos, and tangentially thinking of starting a piano tuning business in the area. If you have any interest in getting together for tea or something, feel free to write me off-list at carl at lumma dot org. -Carl --- In [email protected], "Richard Eldon Barber" <bassooner42@...> wrote: > > By P4-P5-P8 I mean, for instance, C F G c. According to > Stopper, any combination of fifths and octaves sound beatless > due to a symmetry in the tuning. My guess is they are either > phase-reverse cancelled, or masked by sidebands? Any thoughts? > Richard > > > --- In [email protected], "Carl Lumma" <clumma@> wrote: > > Hi Richard, > > > > There's only one equal step tuning with the P12 (what I would > > prefer to call 3:1) just. By P4-P5-P8 sonority, what chord > > or chords do you mean? And in what manner were beats cancelled? > > If you can explain, maybe the list will answer. > > > > -Carl > >