[1.24.15-25.11]

[1.24.15-25.11] > [Meibom:] 15> Touton houtos horismenon, pros toi 16> baryteroi ton menonton phthongon eile- 17> phtho to elachiston. pyknon d' estai to 18> ek duo dieseon enarmonion kai chro- 19> matikon elachiston. esontai duo de li- 20> chanoi eilemmenai duo genon bary- 21> tatai. he men harmonias; he de, chro- 22> matos. katholou gar barytatai 23> men hai enarmonioi lichanoi esan. 24> echomenai de, hai chromatikai. syn- 25> tonotatai de, hai diatonoi. Meta tau- 26> ta triton eilephtho pyknon pros toi 27> autoi. tetarton eilephtho pyknon to- 28> niaion. pempton de pros toi autoi, 29> to ex hemitoniou kai hemioliou diastem- 30> matos synestekos systema eilephtho. 31> hekton de, ex hemitoniou kai tonou. Hai 32> men oun ta duo ta prota lephthen- 33> ta pykna horizousai lichanoi eiren- 34> tai; he de to triton pyknon horizousa [25] 1> lichanos, chromatike men estin; kalei- 2> tai de to chroma, en hoi estin, hemiolion. 3> he de to tetarton pyknon horizousa li- 4> chanos, chromatike men estin; kaleitai 5> de to chroma, en hoi esti, toniaion. he de 6> to pempton lephthen systema horizousa 7> lichanos, ho meizon ede pyknou en. 8> epeideper isa esti ta duo toi heni, ba- 9> rytate diatonos estin. he de to hekton 10> lephthen systema horizousa lichanos, 11> syntonotate diatonos estin. > [Macran:] > Touton houtos horismenon pros toi baryteroi ton menonton > phthongon eilephtho to elachiston pyknon; touto d' estai to ek > duo dieseon autoi; touto de estai to ek duo dieseon> chromatikon elachiston. > esontai de duo li|chanoi eilemmenai duo genon barytatai, > he men harmonias he de chromatos. katholou gar barytatai men > hai enarmonioi lichanoi esan, echomenai d' hai chromatikai, > syn|tonotatai d' hai diatonoi. Meta tauta triton eilephtho > pyknon pros toi autoi; tetarton eilephtho pyknon toniaion; > pempton de pros toi autoi, to ex hemitoniou kai hemioliou > diastem|matos synestekos systema eilephtho; hekton de to ex > hemitoniou kai tonou. Hai men oun ta duo [ta] prota lephthenta > pykna horizousai lichanoi eirentai; he de to triton pyknon > horizousa || lichanos chromatike men estin, kaleitai de to > chroma en hoi estin hemiolion. He de to tetarton pyknon horizousa > lichanos chromatike men estin, kaleitai | de to chroma en hoi > esti toniaion. he de to pempton lephthen systema horizousa > lichanos, ho meizon ede pyknou en, epeideper isa esti ta duo > toi heni, barytate diatonos estin. he de to hekton lephthen | > systema horizousa lichanos syntonotate diatonos estin. > >> Given these definitions, let us take the smallest pyknon, placed >> next to the lower of the fixed notes. This will be the one >> composed of two enharmonic or two of the smallest chromatic >> diesis. The two lichanoi thus specified will be the lowest in >> each of the two genera, one in the enharmonic, the other in the >> chromatic: for we have explained that, taken overall, the >> enharmonic lichanoi are the lowest, the chromatic next, and the >> diatonic the highest. After these, consider a third pyknon >> placed next to the same note, and then a fourth one, which >> is a tone: fifthly, from the same note take the systema composed >> of a semitone and an interval one and a half times as great, and >> sixthly, that composed of a semitone and a tone. >> We have already spoken of the lichanoi bounding the first two >> pykna listed. The one that bounds the third is chromatic, and >> the chroma in which it is is called hemiolic. That bounding the >> fourth pyknon is chromatic, and the chroma in which it is is >> called tonic. The lichanos bounding the fifth systema mentioned, >> which was specified as greater than a pyknon, since the two >> intervals are equal to the one, is the lowest diatonic: that >> which bounds the sixth systema mentioned is the highest diatonic.

[1.25.11-26.7] [Meibom:] [25] 11> .................. He men oun 12> barytate chromatike lichanos tes 13> enarmoniou barytates hektoi merei to- 14> nou oxytera estin. epeideper he chro- 15> matike diesis tes enarmoniou dieseos 16> dodekatemorioi tonou meizon esti. Dei 17> gar to tou autou tritemorion tou tetartou 18> merous dodekatemorioi hyperechein. 19> hai de duo chromatikai ton duo 20> enarmonion delon hos toi diplasioi. 21> touto de estin hektemorion elatton dia- 22> stema tou elachistou ton meloidoume- 23> non. Ta de toiauta ameloideta 24> estin. ameloideton gar legomen, ho me 25> tattetai kath' heauto en systemati. He de 26> barytate diatonos tes barytates 27> chromatikos hemitonioi kai dodekate- 28> morioi tonou oxytera estin. epi men gar 29> ten tou hemioliou chromatos lichanon 30> hemitonion en ep' autes. apo de tes hemio- 31> liou epi ten enarmonion, diesis. apo 32> de tes enarmoniou epi ten baryta- 33> ten chromatiken, hektemorion. apo de 34> tes barytates chromatikes epi ten hemiolion, dodekatemorion tonou. to [26] 1> de tetartemorion ek trion dodeka- 2> temorion synkeitai. host' einai phane- 3> ron, hoti to eiremenon diastema estin 4> apo tes barytates diatonou, epi ten 5> barytaten chromatiken. he de syn- 6> tonotate diatonOS tes barytates 7> diatonou, diesei esti syntonotera. [Macran:] > He men oun barytate chromatike lichanos tes > enarmoniou barytates hektoi merei tonou oxytera estin, epeideper > he chro|matike diesis tes enarmoniou dieseos dodekatemorioi > tonou meizon esti. Dei gar to tou autou tritemorion tou tetartou > merous dodekatemorioi hyperechein, hai de duo chromatikai ton > duo | enarmonion delon hos toi diplasioi. touto de estin > hektemorion, elatton diastema tou elachistou ton meloidoumenon. > Ta de toiauta ameloideta estin, ameloideton gar legomen ho me | > tattetai kath' heauto en systemati. He de barytate diatonos > tes barytates chromatikos hemitonioi kai dodekatemorioi tonou > oxytera estin. epi men gar ten tou hemioliou chromatos lichanon | > hemitonion en ap' autes, apo de tes hemioliou epi ten > enarmonion diesis, apo de tes enarmoniou epi ten barytaten > chromatiken hektemorion, apo de tes barytates chromatikes epi > ten hemiolion dodekatemorion tonou. to || de tetartemorion > ek trion dodekatemorion synkeitai, host' einai phaneron, hoti to > eiremenon diastema estin apo tes barytates diatonou epi ten | > barytaten chromatiken. He de syntonotate diatonos tes barytates > diatonou diesei esti syntonotera. > >> Thus the lowest chromatic lichanos is higher than the lowest >> enharmonic by a sixth part of a tone, since the chromatic diesis >> is greater by a twelfth part of a tone than the enharmonic diesis. >> A third part of anything must exceed a quarter of the same thing >> by a twelfth part, and the two chromatic dieses must evidently >> exceed the two enharmonic ones by twice that amount. This is >> a sixth part, an interval smaller than the least of the melodic >> intervals. Such intervals are unmelodic, since we call 'unmelodic' >> any interval that is not placed in a systema in its own right. >> The lowest diatonic lichanos is higher than the lowest chromatic >> by a semitone and a twelfth part of a tone. We said that it is >> a semitone from the lichanos of the hemiolic chromatic, and from >> there to the enharmonic lichanos is a diesis: from the enharmonic >> to the lowest chromatic lichanos is a sixth part of a tone, and >> from the lowest chromatic to the hemiolic lichanos is a twelfth >> part of a tone. Now a quarter is composed of three twelfth parts, >> so that it is clear that from the lowest diatonic to the lowest >> chromatic lichanos is the interval stated. The highest diatonic >> lichanos is higher than the lowest diatonic by a diesis. [Monzo edit - it's a poem!] >> >> Thus the lowest chromatic lichanos >> is higher than the lowest enharmonic >> by a sixth part of a tone, >> >> since the chromatic diesis >> is greater by a twelfth part of a tone >> than the enharmonic diesis. >> >> >> A third part of anything >> must exceed a quarter of the same thing >> by a twelfth part, >> >> and the two chromatic dieses >> must evidently exceed the two enharmonic ones >> by twice that amount. >> >> This is a sixth part, >> an interval smaller >> than the least of the melodic intervals. >> >> Such intervals are unmelodic, >> since we call 'unmelodic' >> any interval that is not placed in a systema in its own right. >> >> The lowest diatonic lichanos >> is higher than the lowest chromatic >> by a semitone and a twelfth part of a tone. >> >> We said that it is a semitone >> from the lichanos of the hemiolic chromatic, >> >> and from there >> to the enharmonic lichanos >> is a diesis: >> >> from the enharmonic >> to the lowest chromatic lichanos >> is a sixth part of a tone, >> >> and from the lowest chromatic >> to the hemiolic lichanos >> is a twelfth part of a tone. >> >> Now a quarter >> is composed of three twelfth parts, >> >> so that it is clear that >> >> from the lowest diatonic >> to the lowest chromatic lichanos >> is the interval stated. >> >> The highest diatonic lichanos >> is higher than the lowest diatonic >> by a diesis.